# The science of finance/Printable version

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# Introduction

« I strengthened you so that you might go and bear fruit, fruit that will last » (John 15,16)
What's the point of working? If it is true that our industries are destroying the Earth and everything that lives on it, wouldn't it be better to stop everything? Natural resources are fruits of the Earth. Work that destroys them is worse than work that bears no fruit. But if we no longer get up to work, how will we eat? Be treated ? How will we get what we need to live well?
Heaven on Earth: love one another. To love others is to live for their good: to perceive, to be moved, to imagine, to think, to want, to speak and to act for their good, therefore always to be of service to them. If we could not provide services, we could not truly love, effectively. Love one another, truly, means: provide services to one another.
Economic goods are wealth because they provide us with services. We can serve others by producing wealth that serves them.
We produce wealth with wealth. In a production project, revenues are always preceded by costs, because wealth must be brought forward to produce new wealth. This is why production projects always need to be financed. To finance is to advance wealth to carry out projects. In general, money must be advanced to purchase services, supplies and production goods.
Services require work and resources. To obtain all the services that we need, or that make life better, without exhausting workers, and without wasting natural resources, which are in limited quantity, we must be well trained and well equipped, because then we can fully develop our potential, be productive and provide services to everyone, without wasting our lives earning it, and without destroying the planet.
To train and equip workers, we just need to finance them. Our financing capacities are limited only by the resources available and the intelligence that gives us the means to use them. Finance give us the means to make the Earth a paradise, because resources are gigantic. So what are we waiting for? Why hasn't finance already made heaven on Earth?
Thanks to productivity gains, which are sometimes very large, an economy where everyone is at least modestly rich is a goal that seems attainable. We just need to invest so that everyone can be productive. We want good investments.
This short treatise on finance provides some answers, or rather beginnings of answers, to the following questions: What is a good investment?
Who should decide on investments and their financing methods?
Why are we not always encouraged to choose the good investments? Why are we sometimes encouraged to choose the bad ones?
To invest, we generally have to advance money. But where does the money come from? How do we have enough money to finance all the good projects?
To choose an investment carefully, we must anticipate the value of the projects. How to count the value of the projects in which we invest?
The science of finance is often astonishing. Here are some examples which will be explained in this treatise:
To be good producers of wealth, we have to be like Mozart.
Wealth equivalent to several billion dollars can disappear without anything tangible having disappeared and without anyone having been robbed.
When we drink a bottle of Champagne, we can save its value.
Wealth is sometimes created without work, without anyone having been robbed, and even without any tangible goods having been produced.
Some goods are almost eternal. They are like gooses that lay golden eggs, because they produce profit without being consumed.
Financial logic can encourage us to choose the most thieving projects, or the most destructive to the environment.
The most profitable project is not necessarily the most thieving project, it can be only the most intelligent project.
The circulation of money is like the multiplication of loaves.
Money creation can increase the production of wealth.
Central banks refused to pay the debts in gold that they had agreed to pay.
Central banks can create all the money they want, to lend it, to buy any asset, to pay any expense, to pay dividends to governments or to give it. They destroy the money they have created when their loans are repaid, when they sell assets, and when they receive interest on their loans.
Commercial banks create money when they lend, when they buy an asset, when they pay any expense, and when they pay dividends to their shareholders. They destroy money when their loans are repaid, when they sell assets and when they receive income.
We are no longer held back by the bridle of gold to finance all the good investments, because we can create all the money we need.
When agents restrict their spending, they collectively become poorer, rather than richer, and do not achieve the desired goal of increasing their monetary reserves.
Financial logic leads us to devalue wealth preserved for future generations.
A risk-free project that earns a regular profit has optimal value if and only if its net present value is zero.
Finance provides the means to make profits with a profit rate as high as one wants, even an infinite profit rate.
To be good financiers, we have to think like communists.
The surplus profits of optimal projects are all positive multiples of the same random magnitude.
Very risky projects or financial assets sometimes need to be valued as risk-free assets.
There is a risk price constant k which makes it possible to measure the cost of risk. kR is the cost of a risk R, measured by the standard deviation of the profit. 0 < k < 1. k seems to be approximately 1/2.
A one in two chance of winning 100 costs 50(1-k) = 25 if we play against an irreducible financial risk, and if k = 1/2.
It is possible to play against fate, without any other counterpart, at (1+k)/(1-k) against 1, with equal probabilities. (1+k)/(1-k) is strictly greater than 1 and could be quite large, 2 or 3, or more.
There are negative risks that have a positive value. If we increase a negative risk in absolute value without decreasing the average profit, we increase the value of the company.
The space of all projects with zero net present value is Euclidean. Its metric is the covariance between the surplus profits of the projects.
A general mathematical solution can be found to all problems of financial risk calculation.
Humans are buying and selling trillions of dollars of assets that have no value: cryptoassets.
Cryptoassets sellers are thieves and arsonists.
An optimal agent can predict her future decisions or their probabilities, because she knows that she will make optimal decisions. This method of anticipation leads to a general method of calculating the optimal value of all risky projects, and therefore the value of all wealth.

# What is wealth?

Economic science often invites us to wish for growth, in order to fight against unemployment, because it is good for public finances and because it is supposed to improve the quality of our lives. Pollution, the depletion or destruction of natural resources, the psychological distress that often accompanies a consumerist lifestyle, the wasting of wealth to satisfy vanity, and many other effects, are enough to show that there is something insane about the idolatry of growth. But on the other hand, the defense of degrowth does not seem very sensible either, because we must produce wealth to live and live well. Choosing between growth and degrowth is foolish. We want both because we want the growth of the good and the degrowth of the bad, quite simply. But what is good? And what is bad? What is really wealth?

## Goods and services

Work has value as soon as it bears fruit. But what is real fruit?
To be in good health, to perceive well, to be moved well, to imagine well, to think well, to want well, to act well, and all that constitutes a good life of a mind, are all fundamental goods for all minds. Similarly, to be in bad health, to perceive badly, to be moved badly, to imagine badly, to think badly, to want badly, to act badly, and all that constitutes a bad life of a mind, are all fundamental evils for all minds.
Derivative goods are means to achieve fundamental goods. Derivative evils are causes of fundamental evils. Fundamental goods and evils are mutually exclusive but derivative goods and evils are not, because a means to achieve a fundamental good can at the same time be a cause of a fundamental evil.
Goods may be essential or only desirable. Essential goods can be more or less essential, the same goes for desirable goods. Evils can be intolerable or bearable. Intolerable evils can be more or less intolerable, the same goes for bearable evils.
Preventing intolerable evil is an essential good. Being deprived of an essential good is an intolerable evil. Preventing a bearable evil is generally a desirable good. Being deprived of a desirable good is generally not an evil, because desirable goods are far too numerous for one to be able to have them all.
We sometimes distinguish between goods and services. But services are also goods, and even more fundamental goods than others, because a good which is not a service is a good because it provides a service. For example, food is a good because it provides the service of nourishing. Some products are not goods because they provide no service. They generally have a negative value, because getting rid of them has a cost.
Services are consumed at the time they are produced. Goods that are not services are consumed after a certain period of time, short (fresh products) or more or less long (durable goods, including stocks of non-perishable foods). Some durable goods are almost eternal (quality housing, jewelry, works of art, etc.). Others are consumed by use over their lifespan. Even near-eternal goods generally require work to maintain.
A durable good is like a service put in a bottle, can or container. Those who produce the durable good provide the service. Those who use and consume the durable good receive the service. A good is a good only if it provides a service. Good is always to be of service. The economy as a whole is a system of exchange or gift of services.
Workers are sometimes competed with and replaced by durable goods, because these are also service providers.
The wealth accumulated and retained is not only the sum of all the tangible and durable goods that we keep for the services they will render to us, because projects in progress and companies are also durable goods. As with all durable goods they are expected to provide services. A project or company is profitable if the value of the goods and services provided, the revenue, is greater than the value of the costs, the goods and services consumed.
Wealth is always a service or a means of providing services. A service is wealth because it improves the quality of life, or because it is a means of producing other wealth. Workers, tangible and durable goods, projects and companies are wealth because they provide services.
Final consumption is the consumption of goods and services which directly improve the quality of life (in principle, because they can also deteriorate it): food, clothing, housing, health, education, transport, sport and entertainment, communication to long distance... Intermediate consumption is the consumption of goods and services which are used in the production chain of final goods and services. Certain goods, such as means of transport, computers and smartphones, can serve as both intermediate goods and final goods. The line between intermediate goods and final goods is often blurred, because final goods are also generally intermediate goods which are used to produce new goods.
The quality of life does not depend only on final consumption: having a good job and benefiting from good working conditions, feeling secure in the present, for one's future, that of one's children, one's country and all of humanity, respect and be respected, love and be loved, know how to meditate and relax, be at peace with oneself and with others, not despair, breathe good air, benefit from a good climate and a welcoming nature...
The fruits of the Earth are all the riches given to us by Nature, plus all those that we can produce. We too are fruits of the Earth.

## How is wealth created?

Nature constantly creates wealth. Sunlight, rain and wind, Earth, seas and rivers, fauna and flora are perpetually renewed riches.
Workers and durable goods create wealth by providing services.
It is generally necessary to consume wealth to produce new wealth. Supplies must be consumed. Workers must consume wealth to reproduce their labor power. Production goods are consumed by their use, unless they are quasi-eternal.
Wealth combined can produce more wealth than wealth separated. A team of workers can do what separate workers cannot do. Well-equipped workers can produce wealth that cannot be produced without such equipment. They also need supplies. Production goods generally require workers to be used. Projects and companies create wealth with workers, supplies, and productive goods, which could not produce such wealth if they were not thus brought together. Projects joined together can produce more wealth than separate projects, because the completion of one project can increase the value of another project, when there are synergies. The art of producing wealth is always an art of composing wealth already present, just as a symphony is a composition of all the talents of the musicians of an orchestra. Bringing together wealth and carrying out several projects at the same time to find synergies is like finding harmony between several voices. Finding the right progression and rhythm for the projects put together is like finding a beautiful melody and its rhythm. To be good producers of wealth we have to be like Mozart.

## Real wealth and market wealth

Real wealth (capital) at a given time is the set of all durable goods that exist at that time.
Real wealth must include the intelligence, skills and health of human beings (human capital) and natural resources (seas, oceans, rivers and lakes, landscapes, wildlife and natural flora...). Market wealth is the market value of real wealth. It is evaluated based on market prices. When goods are not sold, their market value is assessed based on the market prices of equivalent goods. As human beings are not sold as slaves, their market value cannot be assessed, except by indirect means (discounted lifetime income or price of risk) which are highly questionable. The valuation of natural resources poses similar problems.
Market wealth depends on long-term expectations. Durable goods have a market value because it is anticipated that they will be used, and that they can be sold. Companies have a market value because we anticipate that they will make profits. But lifestyles, and anticipations of future lifestyles, can vary. Such variations are difficult to predict. If, for example, humans give up on air tourism, all the infrastructure and equipment intended to produce and consume airplanes, including the airplanes themselves, automatically lose their value. If fiancés lose the habit of offering diamonds, the market value of diamond stocks will be greatly diminished. Expectations are very fluctuating. They vary with the occurrence of unforeseen events (disasters...) and are often irrational (animal spirits) because no one can predict with certainty what the future holds. This is why the market value of shares can vary suddenly. Billions of dollars can disappear in a day without a note having been burned, simply because human beings have changed their minds.

## The ubiquity of options

The freedom to choose is the most fundamental good. If we remove freedom, we remove most fundamental goods. If everything is prescribed in advance, minds are nothing more than servants or slaves. And the freedom to choose is also generally a condition of efficiency and intelligence. A program that wants to prescribe everything in advance is most often too rigid, prevents us from adapting to novelty and condemns us to failure.
Having the freedom to choose means having options. We have an option when we have the possibility or the right to do something but not the obligation. For example, a lottery ticket is an option on its eventual gain. We have the right but not the obligation to collect the gain if there is one.
When exercising an option is simply to cash in an immediate gain, the agent's freedom to exercise the option is more or less fictitious. In general, agents do not refuse to cash in their gains. But the same is not true if the exercise of an option exposes to the risk of loss.
Usually we reason about options that offer only two possible choices: either we exercise the option, or we don't. But we can also reason about options that have many possible choices. To exercise the option is then to choose one possibility among the many offered. For example, if A and B are two two-choice options, to be exercised on the same date, the two together can be considered a single four-choice option: exercise A and B, exercise A without B, exercise B without A, exercise neither A nor B.
Not to exercise an option is to exercise the option not to exercise it. When we have a two-choice option, we always have at the same time the opposite option of not exercising it. We exercise one when we do not exercise the other.
An option is European style (more commonly we say European) when the date of its exercise is fixed in advance. It is American style when the date of the exercise can be chosen. A durable and consumable good is an option on its consumption. One acquires the option by acquiring the good, one exercises the option when one consumes it. It is an American option whose maturity is the consumption limit date.
A capital good is an option on its use. If not worn out by use, there is an unlimited succession of European options, one option for each day, or period, of use. But if worn by use, it is similar to a batch of American options. Each time we use it, we consume part of its potential use, which amounts to exercising an American option.
A natural wealth is an option on its use. If it is renewable, like land that is not degraded by its use, it is an unlimited succession of European options, one for each day, or each period, of use. If consumed by its use, such as a natural reserve of oil, it is similar to a batch of American options.
A skill is an option on its exercise. It is a succession of European options for every day, or all periods, of work.
Designing a project means acquiring the option to carry it out. If the project is dated, it is a European option. If the project is not dated, if we can choose the moment of its realization, it is an American option.
A buy or sell decision is usually the exercise of an option and the acquisition of a new option at the same time.
When we have 1000 dollars, we acquired the option to spend them, to buy everything that is sold within the limit of 1000 dollars. The option is exercised by spending the 1000 dollars. It is an American option with perpetual duration.
As soon as there is uncertainty about the value of the expected services, a purchase is similar to the purchase of an option. It's like buying a lottery ticket. As soon as one is free to choose the dates of the expected services, a purchase is similar to the purchase of an American option. The only purchasing decisions that do not look like option purchases are those for which there is no uncertainty either about the dates or about the value of the services expected.
When we acquire a durable good, we acquire the option of reselling it. It's an American option. The exercise of the option, the sale, is at the same time the acquisition of a new option, the sum of money transferred by the buyer.
A loan, if there is a risk of default, is like an option on its repayment. It is a European option, or a succession of European options, if the repayment dates are fixed in advance. To exercise the option is to be reimbursed, if possible. Ownership of a business is like an option on its profits. It is a succession of European options, for all the dates of payment of the dividends. To exercise the option is to receive the dividends, if any.
To hire an employee is to acquire an option on the services they can render.
To freely have the means to provide services is always to have a portfolio of options, because freely providing a service to others or to oneself is the exercise of an option. Wealth is always a wealth of options. The means to provide service and the freedom to make good use of them are the foundations of wealth.

## Savings and investment

Everything that is produced is consumed or saved. It is the law of the excluded middle: everything that is produced is consumed or is not consumed. To be saved is to be preserved, is not to be consumed. A stock of durable goods is a savings, like a squirrel's stock of nuts.
Accumulated wealth is the sum of all the durable goods we have preserved and all the projects in progress. A durable good can be considered as a project, the plan to use it. Conversely, a project can be considered a durable good. Its purchase price is the money that must be paid to realize it. Like all other durable goods, it is purchased to provide services. Its revenues are the services it produces.
When we advance the money to carry out a project, we create a durable asset, the project, and we buy it at the same time, we become the owner of the project. The project we created is a durable asset that we keep until it is finished. The money that we have advanced, that we have invested in a project, is therefore saved.
One way of investing is to buy production goods to carry out the project of using them, but we can also invest without purchasing any production goods, because we can carry out projects by renting all the goods which we need. What matters for there to be investment, and therefore savings, is not the purchase of production goods, but the money advanced to carry out projects that we hope will be profitable.
A company can be considered a project. To own shares is to be co-owner of a project. The money saved by the shareholder was invested in a project.
Increasing a stock of unsold goods is usually unwanted savings. It can also be counted as an unwanted investment, because we plan to sell unsold items. If we always count stocks of goods as investments, savings equals investment. This is an accounting equality. With the law of the excluded middle, we then obtain: everything that is produced is consumed or invested.
A company party can be counted as an investment, because it can increase the value of the company, by encouraging team spirit for example. All the money spent on the party is invested and therefore saved, since an investment is always a savings. By drinking bottles of Champagne, we save their value. The adage is therefore confirmed: a bottle drunk is a bottle won, not a bottle lost. Lost bottles are those that are never drunk.
Net investment equals net saving. It is the change in overall wealth over a given period. It is equal to the change in value of the sum of all durable goods that are preserved, if we count the projects in progress as durable goods retained, and to the change in value of all projects in progress, if counts all durable goods kept as projects in progress, plans to use them, or sell them.
The purchase of a durable good intended for consumption, a pair of shoes for example, is a saving as long as the good is not consumed, but it is generally counted as consumption, not as an investment, because we anticipate the consumption for which it is intended.
Buying a bond, or any other way of lending your money, is saving because the bond is kept. When we lend our money, the lender's savings are offset by the borrower's dissavings, and overall savings are zero, because we have not retained more wealth. No wealth has been created.
Keeping money in a bank account is a way of saving by lending money to the bank. As with bonds, overall savings are zero. No wealth is created. The customer's savings are offset by the bank's dissavings.
Buying a lottery ticket is a savings because it is kept until the day of the draw. Games of chance are generally zero-sum games. Whatever is gained by some is lost by others, and vice versa. The sale of lottery tickets is a savings for the buyers and a dissaving for the seller, who will have to pay the winnings. Overall savings are zero. As with bonds, no wealth is created from the sale of lottery tickets.
Financial products, except stocks and bonds, often resemble lottery tickets in zero-sum games. In such cases, the savings of the buyer of a financial product is offset by the dissavings of the seller, and the overall savings is zero. No wealth is created.
Acquiring an option is a savings, because the option is retained. It is consumed on the day of exercise. If the exercise of an option is the acquisition of a new option, it is a saving which replaces the previous one.
Cryptocurrencies are like lottery tickets. Buying them is therefore a savings, but it is not a good investment.
Even final consumption can be accompanied by a form of savings, because it can produce good memories that are kept. A good memory is wealth that has been saved.

## What is a liability?

An asset is a durable wealth, or a right to receive wealth. A liability is a debt, or a duty to return or provide wealth.
Both assets and liabilities can be risky. A lottery ticket is a risky asset for the buyer and a risky liability for the seller.
A liability is not necessarily the duty to pay money. It can be the duty to provide a good or service.
Short selling is the sale of an asset that has been borrowed. It must be returned when the loan is due and therefore repurchased that day, if it has not been repurchased before. When we have sold short, the asset that we must return is a liability. Short selling is the financial technique for playing on falling prices, because we make a profit if the price of the asset that we have sold short decreases.
If a project is risky, it happens that we do not know if it will bring in revenue or on the contrary cause a cost that we will have to pay. Such a project is neither an asset nor a liability, but a random asset-liability. This is what happens when a company has unlimited liability. Limited liability companies do not expose their owners to the risk of paying debts and are therefore always assets.
A wealth or a portfolio is composed of assets and liabilities, risky or not, and random asset-liabilities.

## What's the point of saving?

We can save to carry out projects and thus produce new wealth.
We can also save to preserve wealth. If these riches will be useful to us, this amounts to carrying out the project of using them and thus producing new wealth. But if these preserved riches remain useless, like gold buried in a garden, nothing is produced. Now wealth is given to us to produce wealth. This is why Jesus condemns hoarding as a crime: “throw that worthless servant outside, into the darkness, where there will be weeping and gnashing of teeth.” (Matthew 25:30). Buying gold means locking it in a safe or a vault. It's like burying it in a garden.
Gold in a vault produces nothing, and its conservation consumes a little wealth: guarding costs. Cryptocurrencies produce nothing, and their conservation consumes a lot of wealth, particularly the electrical energy consumed. This is why cryptocurrencies are not a good investment, but only a money pit.

# Income
## Profit

A project is profitable when the value of the wealth it produces is greater than the value of the wealth it consumes.
The costs of a project are the wealth it consumes, its revenues are the wealth it produces. Supplies, labor, and wear and tear on production goods are costs. The goods and services produced are revenues. Profit is the difference in value between revenues and costs.
When a durable good is a final consumption good, its revenues are the services provided by its consumption, its cost is its acquisition price. If the value of the service consumed is counted by the value of the good consumed at the same time, the value of the revenue is equal to the value of the cost. Generally, just stocking our pantry isn't enough to make a profit. A final consumer good produces a service, but it generally does not produce a profit, because the value of the service it provides is not increased by the plan to retain it.
A durable good is quasi-eternal when it produces services without being consumed. In general, there are also usage costs, heating a home for example, or the electricity bill for a computer. If the services produced have a value greater than user costs, a quasi-eternal good resembles a goose that lays golden eggs, because it produces a profit without being consumed.
When goods or services are consumed in a production project, the value of the wealth they produce depends on the project. The more intelligent a project is, the greater the wealth it produces for a fixed cost. Profit is possible because wealth can be used to produce more wealth. The intelligent use of wealth to produce new wealth brings profit.
A business is generally an open-ended project. For a finite duration project, we count the profit made on the day the project closes. To count the profit of a company, we can consider it as a succession of projects with a finite duration. The company is like a project that is renewed each period. The value of the company at the beginning of the period is counted as an initial cost of the project for that period, the value of the company at the end of the period is counted as final revenue. To count the profit of a company, we must count its value, its capital, and its appreciation, or depreciation.
A worker is a profitable enterprise for himself. His income is the remuneration for his work. His costs are those he must pay to work. For a worker, an increase in his labor income is an increase in his profit. For the company that employs him, this is an increase in its costs, which is removed from its profit.
The calculation of profits depends on prices. If prices vary, a profitable project can become ruinous, and vice versa.

## Is profit a theft?

Profit from a business or project is property income, not labor income. If there is any work involved in managing the property or business, it should be counted as a cost of the project, which is subtracted from the profit. The profits of a business or project are income paid to owners who have not worked, whereas a worker must earn his profit, the income from his work, by the sweat of his brow. Should we conclude that the owners' profits are wealth stolen from the workers?
An unfair price is a theft in disguise. If it is too high, the buyer is harmed, as if robbed by the seller. If it's too low, it's the opposite. But if the price is right, there is no theft. When labor is underpaid, it is stolen by those who purchase its services. The unfair price of labor means that in this case, profit is theft.
We often think about projects as if all prices were fixed and imposed, because there is a tyranny of prices. We often don't choose the cost prices and we can't set a price for the revenue that strays too far from the prices that already exist, if we want to hope to sell. If the price system is fair, there is no theft. A profitable project is not necessarily a thieving project. Profit is wealth created by the completion of the project. For a profitable project, revenues are greater than costs, because the project is intelligent, because the means have been well chosen to produce goods or services. The wealth created by a project is a fruit of intelligence. If prices are right, the most profitable project is not the most thieving project, only the most intelligent project.
Owners take profits from businesses and leave nothing for others, who may feel they have been robbed, if property is very unfairly distributed. But it is the injustice in the distribution of profits that causes theft, not the profit itself.
Even if the property is public, we must still count the profits of a project, if we want to know if we have made good use of our wealth, because profit is a creation of wealth. Even in a purely socialist economy, we want profitable projects, because we want to create wealth.
To repair the injustice of the distribution of property, one might believe that workers should always be owners of their means of production: the land to those who work it.. In this way, the wealth produced returns entirely to those who produce it and the owners cannot earn anything without working. The redistribution of wealth to workers is sometimes a measure of justice, but it can also lead to senseless consequences if it is generalized excessively. The responsibilities of an owner are not the same as those of a worker. We may want to work without being an owner. Requiring a flight attendant to be a shareholder in the airline in which she is recruited would be insane, and would pose recruitment difficulties. Additionally, capital per worker (the value of the firm divided by the number of workers) varies a lot depending on the firm. Some workers should therefore be much richer than others. The redistribution of the means of production to workers could therefore worsen wealth inequalities instead of reducing them.
The owners of a project are those who have provided the money or wealth to carry it out. If they cannot receive the profits from a project and only suffer its losses, they have no incentive to advance their wealth. Abolishing profits from private property means removing incentives to make good use of property.
If the workers do not want to put up the money for a project, or cannot, because they are not rich enough, if the private owners do not want to either, all that remains is the State- Providence to finance the project. Private property is not necessary for freedom of enterprise because the state can play the role of a universal bank that finances all projects as soon as they merit financing.
With a magnificent state, always intelligent, competent, honest and dedicated to serving its citizens, all big property could be public without infringing on private liberties. But the state isn't always beautiful. We can fear, not without reason, that the State is sometimes tyrannical.
Private property often provides good incentives: take care of wealth and make good use of it to carry out profitable projects. But it can also provide bad incentives. If prices are unfair, the most profitable projects can also be the most thieving. In this case, the search for profit pushes us to steal. Private interests can also be contrary to the public interest by ignoring costs paid by the public. Owners do not always pay all the costs of their projects. If the unpaid costs are costs of environmental degradation, the search for private profits can lead to ecological crime.
Large projects are generally those that concern the greatest number of citizens. If wealth is very concentrated, the very rich can behave like tyrants and make decisions based on their private interests that concern everyone, and which sometimes harm everyone. When projects concern many citizens, it may seem desirable that ownership be public, because a good state always makes its decisions in the general interest. Public power can therefore be a way of limiting the tyranny of the very rich. But if the State is unjust, we have the choice between the plague and cholera, the tyranny of the very rich or the tyranny of the State.
Social democracy is about taking the best of both worlds, private wealth and public wealth. It is the most commonly chosen option, on the left, the center and the right.

## Creating wealth without work

Nature constantly produces wealth without our assistance, the light that the Sun constantly sheds on the Earth for example.
A fully automated or robotic production unit can operate with almost no labor, provided that supervision and maintenance costs can be neglected.
Good quality housing is an almost eternal good which provides a great service, being housed, almost without work. We still have to clean up.
If we consider reading as a pleasure and not as work, a book is an almost eternal good which produces wealth, the pleasure of reading, without work and without being consumed.
If new, very useful uses are found for a raw material, the value of the stocks already accumulated is automatically increased. This is a real increase in wealth, because stocks are like reservoirs of the services they can provide. The owners of the stocks can make a profit by reselling their stocks for more than the purchase price. This is profit earned without any work having been done and without anyone having been robbed. Knowledge is a wealth that can create wealth without requiring any work, other than that of acquiring the knowledge.

## What is income?

An agent's income is the sum of their labor and property income. It is consumed or saved.
Property income is profit, which includes capital appreciation, net of depreciation.
Depreciation of capital must be distinguished from consumption. When a durable good is consumed by its use, a car or a pair of shoes, for example, it is a consumption of capital, not a depreciation. Depreciation of capital is the decrease in its value when it is not consumed, diminished or partly sold.
We increase our capital by saving a part of our income, therefore by investing our money: by buying durable goods or company shares, or by paying the initial costs of profitable projects. Appreciation of capital is the increase in its value when we have not bought anything.
When a company reinvests its profits, it increases its capital by advancing money, as if shareholders had received dividends and used them to buy new shares of the company.
We can consume by spending part of our income, by consuming or selling part of our capital, or by borrowing. Borrowing to consume is dissaving, because it increases liabilities without increasing assets. The value of a property is the difference between assets and liabilities.
Savings can always be counted as an investment. They are the difference between the value of the property held at the end of the period and the value of the property held at the beginning of the period.

# Where does the money come from?
## Money and the multiplication of goods and services

Money is a multiplier of goods and services.
Everyone is encouraged to provide goods services to earn money, and to request goods and services, as soon as we can afford them. In this way, money incentivizes everyone to provide and demand services. This incentive is permanent. Money must be destroyed, or prevented from circulating, to cancel this incentive, because as soon as money is available, people are encouraged to spend it, and therefore to circulate it.
One person's expenses make another person's income, because the goods and services purchased are always sold. So the more people spend money, the more they earn. Money stimulates economic activity by encouraging spending. The income generated by the offer of goods and services leads to demand, and therefore to offer, new goods and services, as if the goods and services already offered could be multiplied, like the multiplication of loaves.
We can measure the multiplication of goods and services by money with its speed of circulation. This speed is the number of times during a given period that a unit of currency was used to purchase a service or a new good.
When we increase the money supply, the money put into circulation encourages people to spend more. This may lead to an increase in activity, prices, or both. If there is available production capacity, producers can increase quantities without increasing prices. In this case the increase in the money supply immediately leads to an increase in activity, because the money created encourages agents to spend more. This revival by demand has a permanent effect. The increase in demand recurs in each period, as long as the money created is not destroyed, or withdrawn from circulation, and prices do not increase, because the money created always provides an incentive to spend more. The increase in prices can cancel out this revival by demand, because the money created is then used to pay more for the same quantities than before.

## Cash

Cash is notes and coins put into circulation by a central bank.
Even when their notes were convertible into gold, central banks created money by printing them, because they printed more notes than their gold reserves.
Central banks put money into circulation when they lend it and when they purchase assets.
When the notes were convertible into gold, they were like debts of the central bank, which it had to repay in gold on demand, as if the notes corresponded to gold deposits. This is why banknotes are counted as liabilities of a central bank.
As long as the notes were convertible into gold, printing more notes than the gold reserves was risky, because central banks could be exposed to massive demands for gold withdrawals that they could not honor. Gold reserves were therefore a constraint which limited monetary creation. The increase in the money supply was held back by the bridle of gold.
Banknotes are no longer convertible into gold. Everything happened as if the central banks had defaulted, as if they had definitively refused to repay their debts. The notes are still counted as their liabilities, but it is a debt that they no longer have to repay.
The abandonment of gold convertibility could have led to the abandonment of bank notes, if the agents had decided that this paper currency no longer had any value and if they had chosen another form of money. But this is not what happened, because there was no other currency capable of replacing the one proposed by the central bank.
When a central bank sells assets, or when loans are repaid to it, it withdraws money from circulation and reduces its assets and liabilities at the same time . This is why it makes sense to count notes as liabilities of the central bank, even if they are not really repayable debts, because a central bank can choose to repay its liabilities.
Proponents of cryptocurrencies claim that they could take the place of centralized currencies. According to them, we could do without central banks and pay for all our transactions in cryptos. But it's a lie. The energy cost of crypto payments is very high. All the energy resources on the entire planet would not be enough to replace currencies with cryptos. Until now we have not found anything better than the central currency and the banking system to produce all the money we need.

## The miracle of bank money

Bank accounts are money lent to banks. Since current accounts are not paid, banks borrow money without paying interest, while individuals must pay interest when they borrow. Banks make money by the interest of lending money that has been lent to them without interest.
One might believe that banks do not create money because they only lend what has been lent to them before: deposits make credits. Banks cannot lend more money than they have been given. But when they grant a new loan, they increase the borrower's current account without reducing the other current accounts: the credits make the deposits. Now the sum of all current accounts is part of the money supply. So this increases with each new bank loan. Every time a bank makes a new loan, an equal amount of money is created.
An individual can only lend money he already has. A bank can lend money it doesn't have, and receive interest for that loan, because it creates the money by lending it.
The money created by a bank loan has a counterpart: the borrower's obligation to repay. When the bank loan is repaid, the currency initially created is ultimately destroyed. We therefore do not have to fear being drowned under a disproportionate flood of new money. If there are more new bank loans than repaid loans, the money supply increases. If, on the other hand, there are fewer new loans than repaid loans, the money supply decreases.
Money creation by banks looks like dishonest privilege, because they create money every time they lend it. But we must rather see this freedom of monetary creation as a blessing. To carry out projects, we generally have to advance money. In the absence of monetary creation, we are limited by the available money supply. Money creation makes it possible to advance money to carry out projects without being limited by the money available at the start. A good banker is on the lookout for companies and good projects that deserve to be financed. Creating money to finance companies and their projects is part of the daily work of banks. This is reality, not a utopia.
If money creation leads to an increase in demand without a parallel increase in supply, it leads to inflation, it increases prices without increasing activity. But if money creation is devoted to good investments, it leads to an increase in production capacities, and producers can then increase quantities without increasing prices. It is therefore possible to create money without causing inflation, provided that the money created is used for truly productive investments.

## The origin of money creation by banks

Money in circulation can exist in several forms: cash C held by individuals or companies other than commercial banks, bank money B and reserves R of commercial banks. Public administrations, except the central bank, are considered companies.
B is the sum of all current accounts of individuals and companies in commercial banks.
R is the sum of bank cash reserves and current accounts of commercial banks at the central bank.
The central bank has no cash, no monetary reserves, because it does not need them. Why keep money in reserve when you can create it whenever you need it?
Cash reserves are similar to a bank account at the central bank, but it is the customers who counts their money, not the central bank.
M1 = C + B is the sum of monetary reserves of individuals and companies, other than commercial banks.
M0 = C + R is central money, the liability of the central bank.
M = C + B + R is the sum of all circulating monetary reserves held by individuals and companies, including commercial banks.
When an individual deposits 100 in cash in the bank, C decreases by 100 while R and B both increase by 100. M is therefore increased by 100. We create money by depositing cash in the bank, because this deposit is counted twice as a reserve, once as the bank's reserve, and a second time as the customer's reserve.
If an individual withdraws 100 in cash from his bank, C increases by 100 while R and B both decrease by 100. M is therefore reduced by 100. We destroy money when we withdraw money in cash, because reserves once present twice are now present only once.
If a bank lends 100 in cash to an individual, C increases by 100 and R decreases by 100. M is therefore not changed. Cash lending leads to monetary creation only if it is deposited in a bank. If a bank loan to an individual of 100 is repaid in cash, R increases by 100 and C decreases by 100. M is therefore not changed.
If a bank lends 100 in bank money to an individual, B increases by 100, and R is unchanged, because the decrease in the lending bank's reserves is offset by the increase in the borrower's bank's reserves. A bank loan to an individual is a creation of money if it is granted in bank money. The lending bank does not create the money it lends, because it is withdrawn from its reserves. The money created is the additional reserve of the borrower's bank.
If a bank loan to an individual of 100 is repaid in bank money, B decreases by 100 and R is not changed, because the increase in the lending bank's reserves is offset by the decrease in the bank's reserves of the borrower. Repaying a bank loan is a destruction of money if it is repaid in bank money. The lending bank does not destroy the repaid money, because it puts it in its reserves. The money destroyed is the decrease in the reserves of the borrower's bank.
If a bank buys in cash an asset from an individual at a price of 100, R decreases by 100 and C increases by 100, so M is not changed.
If a bank buys in bank money an asset from an individual at a price of 100, B increases by 100 and R is not changed, because the decrease in the purchasing bank's reserves is offset by the increase in the bank's reserves from the seller. So money is created every time a bank buys an asset and pays in bank money. The purchasing bank does not create the money with which it buys the asset, because it is withdrawn from its reserves. The money created is the seller's bank's additional reserve.
Banks can increase the money supply by purchasing assets. Why then don't they buy all the assets, since they can collectively create all the money they want to buy them?
Banks have capital requirements. When they create money by purchasing assets, they increase their assets and liabilities at the same time. A company's equity is the value of the company, the difference between its assets and its liabilities. Banks therefore cannot create money by purchasing assets unlimitedly if they meet their capital requirements.
When a bank sells in cash an asset to an individual at a price of 100, R increases by 100 and C decreases by 100, so M is not changed.
When a bank sells in bank money an asset to an individual at a price of 100, B decreases by 100 and R is not changed, because the increase in the selling bank's reserves is offset by the decrease in the bank's reserves of the buyer. So money is destroyed every time a bank sells an asset and is paid in bank money. The bank does not destroy the money it receives because it puts it in its reserves. The money destroyed is the decrease in the reserves of the buyer's bank.
Even the current expenses and revenues of commercial banks are accompanied by monetary creation or destruction, when paid in bank money. Money is created every time a bank pays its expenses in bank money. Money is destroyed every time a bank receives revenue in bank money.
Banks create money every time they spend. If all they have to do is create money to afford everything they want, why don't they spend more? Like all businesses, they have a budgetary obligation. Costs must be offset by revenues. Since the money they create by paying their costs is destroyed when they receive revenues, they cannot increase the money supply unlimitedly by increasing their spending.
If a bank pays a profit of 100 to its shareholders in bank money, B increases by 100 and R is not changed because the decrease in the reserves of the bank paying profits is offset by the increase in the reserves of the banks of the shareholders. M is therefore increased by 100. The bank does not create the money it pays to its shareholders because it is withdrawn from its reserves. The money created is the additional reserve of the shareholders' banks.

## How is money put into circulation?

The central bank puts money into circulation by lending it, buying assets, paying its current expenses and paying its profits to the State or States for the euro zone. It withdraws money from circulation when the loans it has made are repaid, when it sells assets and when it receives interest on its loans.
If the central bank lends 100 to a commercial bank, it credits it to its current account at the central bank. R is increased by 100 and therefore M too. The central bank creates the money it lends by lending it.
If the central bank buys an asset from an individual at a price of 100 in bank money, B and R are both increased by 100, because the reserves of the seller's bank increase. M is therefore increased by 200. When the central bank buys an asset it creates twice as much money as the price of the asset.
The central bank can always create as much money as it wants to lend, to buy assets, to give, or for any other expenditure.
Bank money is created when banks lend, purchase assets, pay expenses, and pay profits in bank money. It is destroyed when bank loans are repaid, when banks sell assets and when they receive income, always in bank money.
Cash is put into circulation at the request of individuals and businesses as soon as they withdraw cash from their bank. The central bank prints all the notes that are requested. The quantity of banknotes in circulation depends on demand from individuals and businesses. In particular, the central bank prints all the notes with which criminals fill suitcases, for their payments and savings.

## The momentum of money

How to inject a billion into an economy? There are two ways to do this:
* If the billion already exists and is immobilized, it is injected by putting it into circulation.
* If the billion does not already exist, it is created by putting it into circulation.
In the first case, the money supply has not changed. Only the velocity of circulation of money changes, because the billion previously immobilized is put into circulation. In the second case, the money supply is increased and the velocity of circulation has not changed. In both cases the effect is the same, because what matters for an economy is not the money supply M or the velocity V of circulation of money taken separately, but their product MV, the momentum of money.
GDP measures the production of wealth within a country during a given period. Nominal GDP is GDP valued at current prices. The velocity V of money circulation is by definition the nominal GDP divided by the money supply M:
V = GDP/M
The momentum of money, MV, is equal to the nominal GDP.

## The paradox of spending restriction

Suppose that on average agents decide to spend less money, because they have less confidence in the future and want more money in their bank account or in cash, for security. The loss of confidence in the future can also lead agents to forgo investment spending, because one must hope to invest in projects that one believes to be profitable. Since the income of some depends on the spending of others, the income of all decreases on average. Spending restriction leads to a reduction in productive activity. Moreover, agents cannot all have more money in reserve, if the money supply is constant, because the latter is the sum of all their reserves. By restricting their spending, they reduce their income and cannot achieve their objective of increasing their reserves. They obtain an effect opposite to the desired effect, an impoverishment instead of an enrichment. This is the paradox of spending restriction. It is generally called the paradox of thrift, but this could be a misleading expression, because investment expenditure can be a kind of careful spending, a good saving, and its increase does not lead to a contraction of productive activity.
When agents restrict their spending, they keep their money longer and therefore cause V to decrease. If M is constant, GDP = MV decreases by the same amount.
The loss of confidence in the future can lead agents to increase their reserves of real wealth, like a squirrel saving for the winter. This increase in savings is at the same time an increase in investment and does not lead to a contraction of activity. But fear of the future can also lead agents to restrict their spending, because they hope in vain to increase their monetary reserves, and it thus causes a contraction of activity.
The paradox of spending restriction shows that the loss of confidence in the future is enough to cause an increase in unemployment. Therefore, an economy can enter a recession for purely psychological reasons.
When money does not circulate, it has no direct effect on purchases, sales, and prices, as if it did not exist, as if it were no longer part of the money supply. But this immobile money still has an economic effect, because it influences the decisions of its owner. Agents generally want to have a minimum of monetary reserves. If their reserves decrease, they want to replenish them and are discouraged from spending. This is why even money that does not circulate can stimulate activity.
If a recession is caused by spending restrictions, in principle it is sufficient to create money to solve the problem. The money created allows agents to increase their reserves as they wish.

# The value of a risk-free project

We calculate the value of a project by anticipating its profit, therefore its costs and revenues.

## Calculating costs and benefits

Calculating costs and benefits is a general method of evaluating decisions. Market rules require such calculations. A company that does not correctly count its expenses and revenues generally goes bankrupt. But the importance of calculating costs and benefits does not stop with business accounting. For most projects, even non-profit, even with only philanthropic intentions, there is an interest of evaluating the costs and benefits, in order to make the best choices, or at least reasonable ones, choices that are likely to be satisfactory. The calculations do not need to be very precise. Rough assessments can be enough to make good decisions.
When it comes to irreplaceable natural resources, the calculation of costs and benefits is rapid: the cost of their disappearance is infinite, so no benefit justifies their sacrifice.
In general, companies do not pay, or not much, for their environmental damage. If they were made to pay this cost by evaluating it by the replacement cost of lost wealth, they would have to take it into account in their selling prices. But since market prices largely ignore environmental costs, they encourage us to make bad decisions, to choose products that cost us much more than their purchase price. If we want to correctly evaluate the costs and benefits, we must also take into account the hidden costs or benefits, ignored in the accounts of companies or individuals.

## The devaluation of the future

If we know the costs and revenues, calculating profit seems easy: profit is the sum of all revenues minus the sum of all costs. If the project is short-term, this is an accurate calculation, but if the project is long-term, costs and revenues are poorly estimated if deadlines are ignored. A revenue of 100 tomorrow does not have the same value as a revenue of 100 in a year. The same goes for costs. An intertemporal exchange rate is needed to convert the value of future payments into present payments. This is called the discount rate. It is estimated using interest rates on risk-free loans.
With an interest rate of 2% per year, one receives 102 next year if one invested 100 today. 102 one year from now is therefore worth as much as 100 today. 100x(1.02)^20 = 148 twenty years from now is as much as 100 today. 100x(1.02)^100 = 724 one hundred years from now is as much as 100 today. 100 one hundred years from now is therefore as much as 100/7.24 = 13.8 today. 13.8 is the present value of 100 a hundred years from now. Financial logic leads to the systematic devaluation of future goods. In financial calculations. The interests of future generations are therefore badly taken into account by financial logic.
The fundamental financial error, the capital sin from the point of view of finance, is to let wealth lie dormant, not to use it to produce more, to bury one's gold in one's garden, for example, instead of finance a productive enterprise. Financial logic therefore invites us to make the most of all available wealth. But if we apply this logic to non-renewable natural resources, we come to an absurd conclusion: it would be wrong to conserve them, because they are unused wealth. Why leave them to future generations when we can use them right away to earn a lot of money? In our financial accounts, the wealth kept for future generations is worth nothing or almost nothing, it would be much better to exploit it right away.
Financial logic underestimates the value of long-lived goods, because it does not take into account their value for those who are not yet born. The demand for goods makes their value, but the absent are always wrong. When we ignore the interests of future generations, it's their fault, because they don't ask for anything, because they aren't born.
The present economic system is destroying our future. Every day the planet is more degraded than the day before. Natural wealth is disappearing at breakneck speed. We work to impoverish ourselves. If economic development is left to laissez-faire, to the law of the market, where goods are valued by those who can pay for them, it leads us straight to the precipice, because the market devalues the long-term future.

## The profit of a risk-free project

Profit is the difference between the final revenue and the initial cost.
The initial cost is evaluated on the day the project is launched, and the final revenue on the day the project is closed.
The simplest risk-free projects have a single cost and a single revenue, paying 100 today to receive 102 in a year for example. They are equivalent, from an accounting point of view, to a zero-coupon bond. A bond is a debt. The issuer of the bond borrows money. The buyer of the bond lends his money. The issuer of the bond must pay the interest and repay the principal. A bond's coupons represent the interests that must be paid before the principal is repaid. A zero-coupon bond is repaid in one go, principal and interest. For example, we can buy a bond for 100 today which commits the issuer to repay 102 in a year.
During a production project, costs precede revenue. Initial costs are costs that are not paid for by past revenues. The duration of a project can always be divided into two periods, one where money must be advanced to cover the costs, because they are not paid from previous revenues, and the next period where it is no longer necessary to advance such money, because the revenues are sufficient to cover the costs. Initial costs are the net costs of the first period. Final revenues are the net revenues of the second period. Initial costs are the money that must be paid upfront to complete the project. Final revenue is the money left in the treasury after initial costs have been paid and the project is completed.
Money that is not used is money that is dormant, which does not earn any interest. This is why a company has no interest in keeping a large amount of cash. Rather than leaving the money in the fund, it is better to invest it and earn interest without risk. We can thus manage our cash flow as closely as possible by buying and selling bonds without risk. Treasury costs are the costs we pay if we do not manage our cash flow as accurately as possible, if we let money sleep in the cash register. To ignore them, we can assume that the treasury is always invested with a risk-free interest rate, as if it were always managed as closely as possible. For a small treasury or a short-term project, the treasury costs are very low, and can be ignored, but they can be very significant for large treasuries over a long period.
If the treasury costs have been reduced to zero, the final revenue is the sum of the final revenues updated on the closing day of the project.
If for example the discount rate is 2% annually, a revenue of 100 in one year is equivalent to a revenue of 102 in two years.
With the same discount rate, a cost of 102 in a year is equivalent to a cost of 100 today, because by paying 100 today and placing it at the risk-free rate, we can pay 102 in a year. If we have reduced the treasury costs to zero, the initial cost of a risk-free project is the sum of the initial costs discounted on the day the project is launched.
The profit rate is the profit divided by the initial cost. The profit rate must be counted per unit of time. A 21% profit rate for a two-year project is a 10% annual profit rate.

## The net present value of a risk-free project

The value of a project on the day it closes is its final revenue.
The value of a risk-free project on the day it is launched is the discounted value that day of its final revenue.
If for example the final revenue is 102 in one year and if the discount rate is 2% annually, then the value of the project today is 100.
The net present value of a risk-free project is the difference between the value of the project and its initial cost, therefore the value of the project net of its initial cost. The net present value on the day the project is launched is the difference between the present value of the anticipated final revenue on that day and the initial cost. The net present value is not the profit, because the final revenue must be discounted to the day the project is launched.
If its net present value is strictly greater than zero, a risk-free project is a windfall. Its value is greater than its initial cost, its price. If its net present value is zero, it is an optimal project, which pays as much as regular optimal risk-free projects, and its value is equal to its price. If its net present value is strictly less than zero, it is a suboptimal project, earning less than regular optimal risk-free projects, or losing money, and its value is less than its price. This is why one of the rules of finance is to refuse a project if its net present value is negative.
When a firm is doing well, it is expected to make the best use of its available resources and to have a net present value of zero, ignoring windfalls, because it is making an optimal surplus profit for its initial cost. Zero net present value therefore means that a firm is worth its initial cost because it is being managed optimally. If the net present value is strictly greater than zero, it is a sum of windfalls. If a company is poorly managed, its net present value falls below zero and is like a sum of all the costs of management errors.
The surplus profit of a project is the excess profit compared to the profit of a project which pays at the risk-free interest rate and which has the same initial cost.
Theorem: the net present value of a risk-free project is the value on the day the project is launched of the anticipated surplus profit.
Proof: the value on the closing day of the project of the initial cost is equal to the initial cost plus the profit that this initial cost would have yielded if it had been invested at the risk-free rate. The surplus profit is therefore the difference between the final revenue and the value of the initial cost on the day the project is closed. The value of the anticipated surplus profit on the day the project is launched is therefore the value on that day of the anticipated final revenue and the initial cost, therefore the net present value.
Theorem: the net present value of a project is the sum of all revenues minus the sum of all costs, all discounted on the day the project is launched.
Proof: let r be the annual discount rate. This means that a value x on date t1 is worth x(1+r)^(t2-t1) on date t2. a^b is a exponent b. r = 5% means r = 5/100 = 0.05. If r = 5%, 1+r = 1.05. Dates are measured in years. Let 0 be the project launch date, t1, the date of the first day when all initial costs are paid, and t2 the date of the project closing day. The revenues and costs are R(t) and C(t). The initial cost C is the sum over t from 0 to t1, t1 excluded, of (C(t)-R(t))(1+r)^(-t). The final revenue R is the sum over t from t1 to t2 of (R(t)-C(t))^(t2-t). The sum over t from 0 to t2 of (R(t)-C(t))^(-t) is therefore equal to -C + R(1+r)^(-t2) = (R - C(1 +r)^t2)(1+r)^(-t2). This is the desired result because the net present value is the present value of the anticipated surplus profit R - C(1+r)^t2.
Theorem: a risk-free project that earns a regular profit has optimal value if and only if its net present value is zero.
Proof: if a risk-free project has a surplus profit strictly greater than zero, it is necessarily a windfall that cannot be repeated regularly. If we could, it would be enough to borrow at the risk-free rate to make unlimited profit. But the laws of finance do not allow unlimited profits. A risk-free project which brings in a regular profit therefore necessarily has a surplus profit less than or equal to zero. Therefore a risk-free project which brings in a regular profit is optimal if and only if its surplus profit is zero, hence the theorem.
Theorem: the net present value of the sum of risk-free projects is the sum of their net present values.
Proof: Let C1 and C2 be the initial costs of two risk-free projects evaluated on the same day. R1 and R2 are the values ​​on that same day of their final revenues. The net present value of project 1 is NPV1 = R1 - C1, that of project 2 is NPV2 = R2 - C2, that of project 1+2 is NPV(1+2) = R1+R2-(C1+C2) = NPV1 + NPV2. The net present value of the sum of two risk-free projects is therefore the sum of the net present values ​​of the two component projects. We can conclude by reasoning by recurrence that the net present value of a sum of n projects is the sum of the n net present values ​​of the components.
The composition of projects can create value because the initial cost and final revenue of one project may depend on the existence of another project. To calculate the net present value of a sum of projects, one must count the initial costs and final revenues after taking this composition effect into account. When calculated in this way, the net present value of a sum of projects is always the sum of the net present values ​​of the component projects. The net present value of the sum of two risk-free projects is therefore the sum of the net present values ​​of the two component projects. We can conclude by reasoning by recurrence that the net present value of a sum of n projects is the sum of the n net present values ​​of the components.

## Leverage

We can benefit from leverage if a project has a higher rate of profit than the rate at which money can be borrowed. Leverage increases the rate of profit to infinity by borrowing all or part of the funds needed for the project. If we can borrow all the funds, there is no money to advance and the rate of profit is infinite. If we only borrow a portion of the funds, we increase the rate of profit, because we gain on the difference between the rate of profit of the project and the rate at which we borrow.
An example: if we invest 100 in a company with a profit rate of 20% a year, we make a profit of 20 after one year. If we borrowed 50 at the rate of 2%, we have to pay back 51 after one year, the profit is only 19, but we have advanced only 50. The profit rate is therefore 19/50 = 38%. By borrowing, the rate of profit has been increased by leverage from 20% to 38%.
A borrower can always reduce the initial cost of a project by borrowing some of the funds advanced. This reduction in the initial cost is accompanied by a reduction in the final revenue, because interest must be paid on the borrowed money. The value of a project is the value of its final revenue and is therefore reduced by leverage. But if the money is borrowed at the risk-free rate, the reduction in the value of the project is exactly offset by the reduction in the initial cost.
Theorem: if a project is financed by borrowing at the risk-free rate, its surplus profit is not modified.
Proof: Let C be the initial cost of the project, E the amount borrowed at the risk-free rate r and R the final revenue. r is an annual rate. For simplicity, we assume that R is obtained after one year. If the project is not financed by borrowing, the profit is R - C and the surplus profit is R - C - rC = R - C(1+r). If the project is financed by borrowing, the profit is R - (C-E) - E(1+r) = R - C - rE. rE is the portion of the profit that was given up to repay the loan. The surplus profit is the profit less interest on the initial cost: R - C -rE - r(C-E) = R - C - rC. The surplus profit is therefore not modified by the method of financing.
The initial cost of a production project can be varied without varying its surplus profit. The initial cost is therefore not a relevant quantity for assessing the capacity to produce a surplus profit. The same production project creates the same surplus profit regardless of its method of financing, therefore regardless of its initial cost, even if it is zero. If the initial cost is zero, the profit is equal to the surplus profit.
Theorem: the net present value of a risk-free project is not modified by its financing method.
Proof: this is an immediate corollary of the previous theorem, because the net present value of a risk-free project is the present value of its surplus profit. We will show later that the previous theorem can be generalized to risky projects.
Theorem: if we can borrow at the risk-free rate, we can always multiply the surplus profit rate of a project by leverage.
Proof: let r be the risk-free rate, p the profit rate of the project. s=p-r is the surplus profit rate. If we finance the project by borrowing a fraction L of the funds advanced, the surplus profit is not modified, but the initial cost is multiplied by 1-L, the surplus profit rate is therefore s/(1-L).
Leverage therefore makes it possible to obtain a profit rate as large as desired. If we borrow the entire initial cost of the project, there is no money to advance and the profit rate is infinite.
Leverage, when one can benefit from it, looks like a magnificent windfall, since it allows to increase the rate of profit as much as we want. If the project is not risky, there is no reason to deprive oneself of such a windfall. But projects are usually risky. If the realized rate of profit is lower than the rate at which one has borrowed, one must support a loss, which is all the more important that one borrowed more. Leverage increases the risk of a project and can lead to bankruptcy. This is why companies are generally required to have sufficient capital, not be solely financed by loans. These funds are like a sort of cushion, which allows the company to bear possible losses (Admati & Hellwig 2013). If a company is abusing leverage, having low capital compared to what it borrows, it runs the risk of bankruptcy and puts lenders at risk of default. Leverage is therefore a way to increase the expected rate of profit while increasing risks, and by offloading some of these risks on lenders.
It is desirable, if only for reasons of social justice, so that even the less fortunate can undertake, that some projects be financed solely by borrowing, without requiring initial capital, so that they benefit from infinite leverage. But in this case the lenders must know that they take on the project risks.
Banks are the primary beneficiaries of leverage, because they can borrow at a very low rate, possibly zero, when bank accounts are unpaid.

## The optimal risk-free rate

The optimal risk-free rate is the smallest rate at which one can borrow under market conditions, if one is a risk-free borrower. It is also the largest rate at which one can lend to a risk-free borrower, under market conditions.
Market conditions are regular conditions, which exclude windfalls, and which can be repeated in principle as much as one wants. Borrowing at a rate lower than the optimal risk-free rate is a windfall, because one can lend what one has borrowed at a higher rate and thus benefit from infinite leverage. Similarly, lending at a rate higher than the optimal risk-free rate is a windfall, if the borrower is risk-free, and if one is oneself a risk-free borrower, because one can borrow what one lends at a lower rate and also benefit from infinite leverage.
A windfall cannot be repeated as often as one wants, otherwise one could earn an unlimited profit. An infinite rate of profit is not impossible. For a risk-free project, it is a windfall. But an infinite profit, or a profit as large as one wants, is not possible.
Since bank accounts are not remunerated, one might think that banks permanently benefit from an infinite rate of profit, since they can borrow at zero interest. But they do not really borrow at zero interest. The distribution of banknotes, checks, and other services are provided free of charge, or almost free of charge, by banks to their customers. For banks, these are costs of borrowing money from their customers, as if they had to pay interest.
The optimal risk-free rate is the discount rate that should be chosen to value all costs and revenues of all projects, because it is an intertemporal exchange rate for risk-free borrowers, who can invest money in risky or not risky projects.

# Risk calculation
## Probabilities in Economics

The mathematical theory of risk is the theory of probability. It was first designed for games of chance. It enables us to calculate their average gains, and their risks, provided that there is no cheating. It can also be applied to physical systems that contain a large number of molecules. Their random motion is brownian and cannot cheat. Probabilities are in Nature.
An economy is not a casino. Economic agents are not in brownian motion. What meaning can we then give to probabilities in economics?
We can measure a probability when a random experiment is reproducible. The precision of the measurement increases with the number n of repetitions of the experiment. For physical systems, n can be as large as the number of molecules, therefore billions of billions of billions, because the molecules are identical. This is why probabilistic physical measurements can be very precise.
The precision of an experiment is measured by the relative margin of error, the margin of error divided by the measured quantity. It can be shown that the relative margin of error is equal to the inverse of the square root of n, when n is large, for a random experiment repeated n times.
Economic agents are all different from each other. One is never the reproduction of the other. The conditions in which they are placed are also not reproducible, because times change, because we never go back. It therefore seems that in economics the maximum number of repetitions of an experiment is equal to one. This is not enough to measure a probability.
Economic agents are not identical but they are sometimes very similar. The same goes for the conditions in which they are placed. Economic probabilities are therefore sometimes measurable, with a precision that depends on the number of repetitions of the measurement and the greater or lesser resemblance between the various measurements. Often we must be content with very imprecise estimates.
Mathematical theory is useful first because it teaches us to reason about risks. When probabilities are measurable, mathematical models can also be good representatives of reality. How to measure risk?
The risk of a project is measured by the dispersion of its anticipated final revenue.
A random quantity X is defined with probabilities. If there are n possible outcomes X(i) where i varies from 1 to n, we assign to each of them a probability p(X = X(i)) between 0 and 1, both included. A probability equal to 1 means that the outcome is certain, or almost. It is infinitely unlikely that the outcome will not occur. If the probability is zero, the outcome is infinitely unlikely, almost impossible.
The sum over all i of the p(X=X(i)) is equal to 1, because it is certain that the outcome is one of the X(i).
$\sum _{i}p(X=X(i))=1$
The mean E(X), also called the average or the expected value of X, is the sum over all i of p(X = X(i)) X(i).
$E(X)=\sum _{i}p(X=X(i))X(i)$
The mean of the absolute value of the deviations from the mean is a measure of dispersion, but the standard deviation, the square root of the mean of the squares of the deviations from the mean, is generally preferred because it is often easier to calculate.
The variance var(X) is the mean of the squared deviations from the mean.
$var(X)=\sum _{i}p(X=X(i))(X(i)-E(X))^{2}=E((X-E(X))^{2})$
The standard deviation std(X) is the positive square root of the variance.
$std(X)={\sqrt {var(X)}}$
The standard deviation measures the dispersion of a random quantity but it is not the only indicator of risk, because deviations from the mean can be dispersed in very different ways for the same standard deviation. The distribution of deviations from the mean, not just their standard deviation, can influence the assessment of risk. But in most cases, the standard deviation is considered a sufficient measure of risk.
The risk of a project is the standard deviation of its anticipated final revenue.
Profit is the difference between the final revenue and the initial cost. If the initial cost is fixed, the standard deviation of the profit is equal to the standard deviation of the final revenue and is therefore a measure of the same risk. The surplus profit is the difference between the profit of the project and the profit it would have earned if its initial cost had been invested at the risk-free rate. If the initial cost is fixed, the standard deviation on the surplus profit is therefore equal to the standard deviation on the profit and is also a measure of the same risk. We have therefore proven:
Theorem: if its initial cost is fixed, the risk of a project is the standard deviation of its anticipated surplus profit.

## The compensation of risks

The risks of one project may be offset by the risks of one or more other projects. Risk reduction by compensation is an example of value creation by composition of projects or options, because risk must be counted as a cost.
Consider a coin toss shooter. One can bet on tails by risking 1 with a 1 in 2 chance of winning 2. Betting on tails means acquiring an option to win 2. The price of this option is 1. The expectation of winning is also 1=0.5x2 . According to financial theory, the value of a project is not equal to its expected gain, the risk must be taken into account. For the same expected gain, a project has less value the more risky it is. We should therefore conclude that the price 1 to bet on tails and hope to win 2 is overvalued, since the project is risky, but this conclusion is false. We can compose the projects. The expected gain of several projects is the sum of the expected gain of each of them. If we bet heads and tails at the same time, we get a risk-free project to win 2. If the options to bet heads and tails cost less than 1, we could compound them and get a risk-free project to win 2 by paying less than 2. In this way, one could obtain without risk an unlimited profit from any initial bet, which is impossible. So the options to bet on heads or tails are correctly evaluated by their expected value. One can ignore their risk because it can be offset. The risk of betting heads can be offset by the risk of betting tails to get a risk-free project.
We can compose a risk-free portfolio with very risky options. The return on the risk-free portfolio thus composed is the weighted sum of the returns on the assets that make it up. If these assets had a higher return than the return of the risk-free assets, the risk-free portfolio thus composed would have a higher return than that of the other risk-free portfolios, and one could make an unlimited profit, without risk, simply by selling risk-free portfolios and buying a risk-free portfolio with a higher return. But the financial markets do not allow us to make unlimited profit without risk. So risky assets should be valued as if they were risk-free, as soon as they can be part of a risk-free portfolio. To evaluate a risky asset, one must take into account the risk, but not the risk inherent in the asset, only the minimal risk of a portfolio of which the asset is a component, because one can reduce the risks by composing portfolios, because one risk can be offset by another risk. A risk has a cost only if it cannot be offset. When valuing a financial asset, irreducible risk must be taken into account. It is the risk that cannot be further reduced by building a portfolio. Financial options and other assets should be valued as risk-free assets as soon as they can be part of a risk-free portfolio, because their risk can be reduced to zero.
A project, or an option, should not be evaluated as if it were isolated, separated from other projects, because then the cost of the risk could be overestimated. To evaluate a project, we must evaluate the irreducible risk, we must therefore evaluate the contribution of the project to the value of an optimal project, made up of several projects whose risks compensate each other partially or totally, in an optimal way. The same project can contribute to different projects, which have different risks, but if they are optimal projects, the value of its contribution is always the same. We reduce the risks by diversifying them, provided that they are independent, or not very dependent. When a project can be repeated several times, its risk can be reduced if its success each time is independent or little dependent on its success on previous and subsequent occasions.
Reducing risks may take time. Present risks can be offset by risks taken at later times. Present risks can be offset by risks taken at later times. Bad years can be offset by good years. The job of an insurance company is to reduce risks by offsetting them. If it does not reduce risks, or does not do so well enough, it is itself a risky business. Being insured by a company that is at risk of failure is about the same as not being insured at all.

## Independence, covariance and correlation

To calculate risk compensation, we must reason on the independence and covariance between random profits.
Two events A and B are independent if and only if the probability of their conjunction is the product of their respective probabilities, p(A and B) = p(A) p(B).
Two random quantities X and Y are independent if and only if all events X = X(i) are independent of all events Y = Y(j), p(X=X(i) and Y=Y(j)) = p(X=X(i)) p(Y=Y(j)), for all i and all j.
The covariance between two random quantities measures the correlation between the variations of one and the variations of the other. If the variations of one have on average the same sign as the variations of the other, the covariance is positive. If the variations of one have on average an opposite sign, the covariance is negative. Positive covariance means that the quantities vary more often in the same direction than in the opposite direction. Negative covariance means that they vary more often in the opposite direction than in the same direction. Zero covariance means that they vary as often in the same direction as in the opposite direction.
The covariance cov(X,Y) of two random quantities is the average of the products of their deviations from the average E( (X-E(X))(Y-E(Y)) )
cov(X,Y) = sum over all i and all j of p(X=X(i) and Y=Y(j))(X(i)-E(X))(Y(j)-E(Y)).
Theorems: for all random quantities X, Y, Z and any real number a,
* cov(X,Y) = cov(Y,X)
* cov(X,X) = var(X)
* cov(X,a) = 0
* cov(X,Y+Z) = cov(X,Y) + cov(X,Z)
* cov(X,Y+a) = cov(X,Y)
* cov(X,aY) = a cov(X,Y)
Proofs: they follow immediately from the definition of covariance.
* var(X+Y) = var(X) + 2cov(X,Y) + var(Y)
Proof: var(X+Y) = cov(X+Y,X+Y) = cov(X,X) + 2cov(X,Y) + cov(Y,Y)
Theorem: if the random quantities X and Y are independent then their covariance is zero.
Proof: $cov(X,Y)=\sum _{i,j}p(X=X(i))p(Y=Y(j))(X(i)-E(X))(Y(j)-E(Y))=\sum _{i}p(X=X(i))(X(i)-E(X))\sum _{j}p(Y=Y(j))(Y(j)-E(Y))=0$, because $\sum _{i}p(Y=Y(j))Y((j)=E(Y)$ and $\sum _{j}p(Y=Y(j))=1$.
The correlation coefficient cor(X,Y) of two random quantities X and Y is their covariance divided by the product of their standard deviations, cov(X,Y)/(std(X) std(Y)).
Theorem: if the correlation coefficient between two random quantities X and Y is strictly smaller than 1 then the risk of their sum X+Y is strictly smaller than the sum of their risks.
Proof: $(std(X+Y))^{2}=var(X+Y)=var(X)+2cov(X,Y)+var(Y)<var(X)+2std(X)std(Y)+var(Y)$ if $cor(X,Y)<1$. $(std(X)+std(Y))^{2}=var(X)+2std(X)std(Y)+var(Y)$, so $(std(X+Y))^{2}<(std(X)+std(Y))^{2}$ and $std(X+Y)<std(X)+std(Y)$.
In particular, if X and Y are risky and independent, the risk of their sum is strictly smaller than the sum of their risks.
Theorem: if the correlation coefficient cor(X,Y) between two random quantities X and Y is equal to 1 then there exist two real numbers a and b, a > 0, such that Y = aX +b almost always.
A statement is true almost always, or almost everywhere, when its probability is equal to 1.
Lemma: if var(X) = 0 then X = E(X) almost always.
Proof of the lemma: if the probability that X is different from E(X) is not zero, then the probability that (X - E(X))² > 0 also, and var(X) > 0. Proof of the theorem: let a = std(Y)/std(X). var(Y - aX) = var(Y) - 2a cov(X,Y) + a²var(X) = 0 because cov(X,Y) = std(X)std(Y). So Y - aX = E(Y) - a E(X) almost always. Hence the theorem. When Y = aX +b for two constants a and b, we say that Y is an affine function of X.
In the following, we will not distinguish a statement that is almost always true from a statement that is simply true.

## Optimal projects

If an economy could be divided into many independent projects, such that the success or failure of one project did not depend on the success or failure of the others, then it would be possible to offset all the risks, and to obtain for the economy as a whole a risk almost equal to zero. But projects in the same economy are not generally independent. The prosperity of some depends on the prosperity of others. The ruin of one can lead to the ruin of others. This is why there are risks that cannot be offset. Risks are sometimes irreducible because the agents of the same economic system are interdependent. Irreducible risks are systemic.
A project is optimal if and only if it has the smallest risk among all projects that have the same average profit and the same initial cost. The risk of an optimal project is irreducible, in the sense that it cannot be reduced without reducing the average profit.
The previous definition of an optimal project is equivalent to the following: a project is optimal if and only if it has the largest average profit among all projects that have the same risk and the same initial cost.
Optimal profits should be evaluated with market prices, average prices or ordinary prices. They represent the investment opportunities available to the economy as a whole. If there are bargains, very favorable prices compared to ordinary prices, they should not be counted when evaluating optimal profits, because they are only special conditions of a lucky agent , and they do not represent the economy as a whole.

## Leverage on an optimal project

Leverage varies the initial cost of a project by varying its final revenue. One might hope that it could transform a suboptimal project into an optimal project, but this hope is vain:
Theorem: if a project is optimal, it remains optimal if it is partially or totally financed by a loan at the risk-free rate, therefore taking advantage of the leverage effect.
Proof: borrowing reduces the average profit, because the interest must be repaid, but it does not change the dispersion of profits, because the interest is fixed in advance. Therefore, the risk of the project is not changed by borrowing. The surplus profit is not changed by the financing method, and it is optimal for the risk of the project. The project is therefore optimal regardless of its financing method.
A risky project is represented by a series of random costs and revenues, all dated, from which we can calculate an initial cost, a final revenue, a profit and a surplus profit, all random. Let X be the random quantity that represents the surplus profit of a risky project.
Theorem: if X is the random surplus profit of an optimal project whose initial cost C is not random, then X is also the random surplus profit of an optimal project whose initial cost is D, whatever D.
Proof: if D < C, it is sufficient to borrow C - D at the risk-free rate to bring the initial cost back to D without varying the surplus profit. If D > C, it is sufficient to lend D - C at the risk-free rate to increase the initial cost from C to D without varying the surplus profit.
If the initial cost is random, it can be set at an arbitrary value, possibly zero, by deciding to borrow all the costs that are not covered either by this initial sum fixed in advance or by revenues. An optimal project is therefore characterized only by its random surplus profit, not by its initial cost:
Theorem: a project is optimal if and only if it has the smallest risk among all projects that have the same average surplus profit.
Proof: This is an immediate consequence of the previous theorem.

## The composition of optimal projects

Theorem: if X is the random surplus profit of an optimal project, then aX is also the surplus profit of an optimal project, if a > 0.
Proof: if a < 1, it is enough to buy a share a of project X to obtain an optimal surplus profit aX. If a > 1, it is enough to increase the size of project X by a factor a.
A project is optimal for market conditions, which are assumed to be unlimitedly reproducible. This is why it is assumed that the size of an optimal project can always be increased. This is a theoretical simplification. In reality, there are always limits to the increase in the size of projects.
Theorem: if X and Y are the random surplus profits of two optimal projects, then X + Y is also the random surplus profit of an optimal project.
Proof: if we buy X and Y, we obtain a project whose random surplus profit is X + Y. If the risk of X + Y were smaller than the sum of the risks of X and Y, the risks of X and Y could be reduced by pooling them and sharing their common risk and X and Y would not be optimal surplus profits. So the risk of X + Y cannot be smaller than the sum of the risks of X and Y and therefore cannot be reduced.

## The correlation between all optimal projects

Theorem: two risky optimal projects cannot be independent. Proof: if they were independent, the risks could be reduced by combining them. However, optimal projects have a risk that cannot be reduced. Therefore, they are not independent.
In particular, repeating the same risky optimal project does not reduce its risk because successive projects are not independent of each other.
The dependence between optimal projects is very strong. All optimal projects are very closely correlated:
Theorem: the surplus profits of optimal risky projects are all strictly positive multiples of the same random quantity.
Proof: Let X and Y be the surplus profits of two risky optimal projects. The risk std(X+Y) of their sum is equal to the sum std(X) + std(Y) of their risks, otherwise combining them would reduce the risk and they would not be optimal. So cor(X,Y) = 1. So Y = aX + b, where a and b are constants, and a > 0. Y and aX are both optimal surplus profits, so b = 0. The surplus profits of the risky optimal projects are all multiples of each other, so all multiples of only one of them. Hence the theorem.
This theorem is very surprising, almost incredible, and one can even be afraid that it could lead to absurdities. Optimal projects can be carried out in different places and at different times. However, it is enough to know the final revenue of a single optimal project to know the final revenue of all optimal projects. For example, the final revenue of an optimal project that ends here and now should be enough to know the final revenues of present or future optimal projects everywhere in the world. Carrying out a single optimal project should therefore be like a crystal ball that would enable one to predict the results of all present and future optimal projects. But then these projects would not be risky any more since their final revenues would be known in advance. Carrying out a single optimal project and observing its result should therefore be enough to reduce all risks to zero, and we would no longer need risk theory and insurance companies.
We cannot find this crystal ball because we can never know if a risky project is optimal. We cannot know it before carrying it out, because the probabilities of the final revenues cannot be known precisely. We cannot know it after carrying it out either, for the same reason.
When we estimate the risk to identify an optimal project, we cannot conclude that it is really optimal, because our estimates are never precise enough, we can only conclude that it is perhaps not very different from an optimal project.
The existence of a single random quantity representative of all risky optimal projects is a consequence of the mathematical model. It assumes that all probabilities of all events are exactly defined in advance, as if all probabilities were written in advance with an infinite number of decimal places. Such exactness of probabilities cannot exist in reality, because nothing is ever exactly reproducible. That is why a single risky optimal project that represents all the others cannot exist in reality. It has only a mathematical existence.
Even if it exists only in a mathematical way, the unique random quantity representing all optimal risky projects has a realistic meaning. It means that agents who carry out optimal risky projects are all in the same boat. They all win together or they all lose together, but the losses of some cannot be compensated by the gains of others, otherwise the risk would be reducible.
When we bet against irreducible risk, we bet on the success of all those who also bet against irreducible risk, so we are all united, we do not play against each other. We are encouraged to bet if we believe that we will all succeed together. We are discouraged from betting if we believe that we will all lose together. The incentive to carry out risky optimal projects is based on the solidarity between all those who take risks and their hopes.
When we bet against an irreducible risk, we acquire a right to a share of the profits of the hoped-for collective success, but we commit at the same time to suffer a share of the losses, if it is a collective failure.
Risk takers are those who have the means to advance money, therefore the capitalists. To optimize their investments, they have an interest in all being united, therefore in thinking like socialists or communists. So we have proven:
Theorem: to be good financiers, we have to think like communists.

## Risk and time

Time can have several effects on risk:
* Time can increase risk, because it takes time to make profits. The more time passes, the more profits can increase, the more their dispersion also increases, so the risk increases.
* Present risks can be offset by future risks. The passage of time therefore reduces risk by intertemporal compensation: good years compensate for bad years.
* The further away a project is in time, the more difficult it is to anticipate its final revenue. Therefore, the revenue of a project is riskier if it is further away in time. This distance from the final revenue is reduced as time passes. Therefore, the passage of time reduces risk by reducing uncertainties.
To assess risks, we must estimate probabilities by taking into account all available information.
A project is relatively optimal when it is optimal for given probabilities.
A portfolio is managed dynamically when its composition is modified over time.
A portfolio is static if its composition is constant.
New information arrives at all times and can lead us to improve our probability estimates and reduce uncertainties. Relatively optimal projects can therefore change over time. The more time passes, the better our assessment of optimal projects and the more we are able to reduce risks. If we do not manage a project dynamically, we neglect this possibility of reducing risk and therefore risk losing more money. A portfolio or project must therefore be managed dynamically to remain relatively optimal.

# The cost and the benefit of risk
## The cost of risk

When a project is risky, investors demand compensation for taking risk, in the form of surplus profit.The surplus profit of a project is the excess of its profit compared to the profit that one would have obtained if one had invested money at the risk-free interest rate.
The cost of risk is the optimal average surplus profit that can be obtained for a given risk.
The variation in the average surplus profit of an optimal project as a function of risk makes it possible to measure the cost of risk: the cost of risk is the average surplus profit required to compensate for the risk.
Theorem: the cost of risk is proportional to the risk.
Proof: Suppose that ownership of an optimal project is shared among several shareholders who share the profits. The standard deviation of surplus profit is shared among all shareholders in the same way as surplus profit. The compensation received by each shareholder is therefore proportional to the risk they have taken on themselves, because risk can be measured by the standard deviation of surplus profit. The cost of risk divided by the risk is therefore a constant k. We have therefore proven:
Theorem: there exists a risk price constant k such that kR is the cost of a risk R.
This risk price constant is dimensionless, because the standard deviation on profit has the same dimension as surplus profit. Risk and the cost of risk are measured in dollars, if the monetary unit is the dollar. We will show later that the risk price constant k is necessarily less than 1. Is it really constant and universal? No, because the attitude towards risk and the compensation required for the same risk can vary over time. Is it the same for all companies and all projects? Not necessarily, because the standard deviation is not the only condition that characterizes a risk. Different projects can have very different distributions of profits and losses while having the same standard deviation of profit. These distributional differences can influence the perception of risk and the requirement for compensation. But the standard deviation of profit can be considered a good measure of risk for most projects. This is why the risk price constant k can be considered the same for all projects and companies.
It is enough to know the average profits of an optimal risk-free project and an optimal risky project to calculate the risk price constant k and from there the costs of all risks and therefore the value of all projects. An optimal risk-free project and an optimal risky project are like measuring standards against which we can measure the value of all projects, whether optimal or not.
How much is k? The discount rate is the optimal risk-free profit rate. 2 or 3% per year are realistic values, perhaps more, up to 4 or 5%, if the owners are very advantaged, perhaps less, in a recession. An average profit rate of 10% per year with a standard deviation of 15% is representative of a well-managed company that takes risks while remaining prudent. With a discount rate of 2 or 3% per year, this makes a surplus profit rate of 7 or 8%, for a risk of 15%. If we assume that these values ​​represent an irreducible risk for an optimal project, k is about 1/2.

## Discounting the cost of risk

The cost of risk is kR if the standard deviation of the final revenue is R. This cost is evaluated on the project closing day. To calculate the anticipated value of the project, this cost must be discounted on the launch day.
Theorem: the cost of risk must be discounted with the same discount rate as the other costs and revenues.
Proof: if we place final revenue at the risk-free rate, we obtain with a delay new final revenues that have simply been multiplied by the same discount factor. The standard deviation on the final revenues is therefore also multiplied by this same discount factor. Since no new risk has been taken, the anticipated cost of risk must not be modified. Hence the theorem.

## Two common mistakes about the cost of risk

Sometimes the cost of risk is assessed by changing the discount rate used to calculate the value of the project. This way of calculating seems to makes sense to those who use it, because the true discount rate is assessed from risk-free zero-coupon bonds. They conclude that another discount rate should be used for risky projects. But this reasoning is nonsense. The same discount rate is used to value costs and revenues. There is no sense in devaluing losses because they are risky. Risky losses do not cost less but more than risk-free losses equal on average, because they increase the risk of a project. The discount rate depends on the conditions of the whole economy at a given date, not on the projects it is used to assess. All costs and revenues of all projects, whether risky or not, should be assessed with the same discount rate.
Risk and its cost are sometimes estimated using the standard deviation of the annual profit rate, because this standard deviation seems like a good measure of risk. But such a calculation of the cost of risk is not exact. For example, consider a two-year project that has a two-year surplus profit rate of 60% or -20% with equal probabilities. The average surplus profit rate is 20% over two years. The standard deviation is 40%, so this project is optimal if k = 1/2. Let r = 2% be the annual discount rate. The two-year profit rate is therefore 64.04% or -15.96%. 64% biannually is 1.64^(1/2) - 1 = 28.1% annually. -16% biannually is (0.84)^(1/2) - 1 = -8.3%.The annual surplus profit rate is therefore 26.1% or -10.3% with equal probabilities The average annual surplus profit rate is (26.1-10.3)/2 = 7.9% and the standard deviation on the annual surplus profit rate is (26.1+10.3)/2 = 18.2%.The profit and surplus profit rates differ only by a constant, so they have the same standard deviation. If we were to evaluate the risk using the standard deviation of the annual profit rate, we would conclude that this project is suboptimal, when it is optimal. The standard deviation of the annual profit rate is therefore not a good measure of risk for a project that lasts for several years.

## The expected value of random gains and losses

The price of risk makes it possible to calculate the value of random gains or losses whose risk is irreducible. The value of a random gain is the average gain minus the cost of risk. The value of a random loss is the average loss increased by the cost of risk:
Consider a gain of 100 with probability 1/2, therefore an average gain of 50. The standard deviation of the gain, therefore the risk, is equal to 50. If the risk price constant is k, the cost of this risk is 50k, since the risk of 50 is assumed to be irreducible. The value of this random gain is therefore equal to 50(1-k) if its risk is irreducible. Consider a loss of 100 with probability 1/2, therefore an average loss of 50. The standard deviation of the loss, therefore the risk, is equal to 50. The cost of this risk is 50k. The value of this random loss is therefore equal to 50(1+k) if its risk is irreducible.
If k = 1/2, a one in two chance of winning 100 costs 25 for someone who plays against an irreducible financial risk. In a game of heads or tails, this chance costs 50. In the national lottery, it costs 100. Those who like to take risks therefore have an interest in playing against irreducible financial risks.
Theorem: the risk price constant k is always strictly less than 1.
Proof: if k were equal to 1, a non-zero average gain without risk of loss would have a zero value, as if a lottery ticket could be free. We could therefore benefit from unlimited profit without taking the risk of losing a single penny. Such profit is not permitted by the laws of finance. If k is strictly greater than 1, a non-zero average gain without risk of loss would have a negative value. This means that we could be paid to accept it, which is impossible.

## The benefit of risk

Risk is counted as a cost when comparing projects that have the same average profit. But it can also be counted as a profit if we compare optimal projects that have the same initial cost, because then the higher the risk, the higher the average profit. Risk is also a benefit if we consider an optimal project whose initial cost is zero. Such a project is possible when we can borrow at the risk-free rate to fully finance a risky project. We then benefit from infinite leverage.
Suppose the discount rate is 2% annually, and the risk price constant k is 0.5. This means that a standard deviation of 1 in profit must be offset by an increase of 0.5 in average profit. Consider a project that costs 100 today and whose only revenue is 126 or 94 in a year, each with the same probability 1/2. The average profit is 10. The standard deviation of the profit is 16. The average surplus profit is 8. This risky project is optimal, because a risk equal to 16 has been compensated by an increase of 16k = 8 in the average profit. Such compensation justifies taking risks.
Suppose we can borrow 100 at the risk-free rate to finance the previous risky project. We have to repay 102 in a year. So we have a one in two chance of winning 24 and a one in two chance of losing 8. It's like playing a coin toss 3 against 1.
The odds of 3 to 1 depend on the risk price constant k=0.5, but it is always greater than 1 to 1 for an optimal risky project, as soon as the cost of risk is greater than zero.
Consider a project that costs 100 today and whose only revenue is 118 + 16k or 86 + 16k in a year, each with the same probability 1/2. The average profit is 2 + 16k. The standard deviation of profit is 16. The average surplus profit is 16k. This project is optimal because we compensated a risk equal to 16 by an increase of 16k in average profit. If we borrow 100 at the risk-free rate to finance the previous project, we have a one in two chance of winning 16 + 16k and a one in two chance of losing 16 - 16k. We therefore play at 1+k against 1-k, therefore at (1+k)/(1-k) against 1, with equal probabilities. These odds only depend on the risk price constant k, not on the discount rate. We therefore proved:
Theorem: if the risk price constant is k, we can play (1+k)/(1-k) against 1 with equal probabilities.
Only irreducible risk yields such a profit. If the risk can be reduced to zero, as in an ordinary game of heads or tails, the odds of heads or tails must be 1 to 1, otherwise one of the players is harmed. An irreducible risk is taken against fate. There is no other counterpart.
When we can play (1+k)/(1-k) against 1 with equal probabilities, we cannot repeat the game several times to increase the profits while decreasing the risks, because then we would reduce the risk. However, it was assumed that the risk was irreducible.

# The value of a risky project
## Only irreducible risk has a cost

To assess the cost of risk of a project that is not optimal, its intrinsic risk must not be taken into account, because it can be reduced without cost, if it is compensated by other risks. If a risky project were sold while ignoring this possibility of reduction, the buyer would make a gain to the detriment of the seller, simply by compensating for the risk.
The cost of risk of a project is the cost of its irreducible risk. If the risk is fully compensable, it can be cancelled and then has no cost. Only irreducible risk has a cost.
The cost of risk of a project is the average surplus profit of an optimal project that has the same irreducible risk and the same initial cost.
The value of a project on the day of its launch is the present value on that day of its average final revenue reduced by the cost of its irreducible risk.

## How to measure irreducible risks?

If a project is optimal, its irreducible risk is its risk. But if a project is not optimal, its risk is not irreducible. How then to measure its irreducible risk?
We reduce the risk by compensating it with other risks, thus by integrating a risky project into a larger project. The projects thus brought together are like the components of a portfolio. We reduce the risks as much as possible by incorporating a project into an optimal portfolio.
If a project can be a part of an optimal portfolio that has the same average surplus profit rate, its irreducible risk is its share of the irreducible risk of the optimal portfolio, thus the risk of the optimal portfolio multiplied by the initial cost of the project divided by the initial cost of the portfolio.
But a project can also have an average surplus profit rate different from the optimal portfolio of which it is a component. How then to attribute its share of the portfolio risk to it?
For the value of the optimal portfolio to be the sum of the values ​​of its components, its risk must be the sum of the irreducible risks attributed to each of its components.
Let us consider two projects whose surplus profits are X and Y. We assume that X+Y is the surplus profit of an optimal portfolio and that its risk is R.
* If E(X+Y) > 0, the irreducible risk of X in X+Y is R E(X)/E(X+Y) and that of Y is R E(Y)/E(X+Y).
* If E(X+Y) = 0 then the irreducible risks of X and Y are equal and opposite, because an optimal project with zero average surplus profit is risk-free.
Since risk is a standard deviation, it is always a positive number. But if we distribute the risk of an optimal project over its various components, they receive a negative share if their average surplus profit is negative. This is why the irreducible risk of a project can be negative. Reducing a negative risk is increasing its absolute value. The cost of a negative risk is negative. This means that it is not a cost but a benefit.
Theorem: the irreducible risk of X does not depend on the optimal portfolio X + Y in which it is measured.
Proof: Let X + Y and X + Z be two optimal portfolios. X + Z = a(X + Y) where a >= 0. If E(X+Y) > 0, Rx = E(X)/E(X+Y) std(X+Y). If a > 0, Rx = a E(X)/E(X+Z) std(X+Z)/a = E(X)/E(X+Z) std(X+Z). If a = 0, Z = -X and the irreducible risk Rx of X is equal and opposite to that Rz of Z. Rx = - Rz. Let W be such that Z + W is optimal and E(Z + W) > 0. There exists b > 0 such that Z + W = b(X + Y). Rz = E(Z)/E(Z + W) std(Z + W) = -b E(X/E(X+Y) std(X + Y)/b = -E(X)/E(X+Y) std(X + Y), so equal to -Rx when Rx is measured in X + Y, as it should be. If E(X + Y) = 0 and a > 0, the roles of Y and Z are reversed, but the proof is the same. If E(X + Y) = 0 and a = 0, then Y = Z = -X, and Rx = - Ry = -Rz. Hence the theorem.
The existence of negative risks poses a difficulty for the definition of optimal projects. Reducing a negative risk can mean reducing its absolute value or, on the contrary, increasing its absolute value. A project whose risk is negative is never optimal in the first sense, because its average surplus profit can be increased by reducing its risk in absolute value, but it can be optimal in the second sense, because its risk cannot be increased in absolute value without reducing its average surplus profit. We can therefore reason on projects with optimal negative risk.
A project is with optimal negative risk when its irreducible risk is negative and cannot be increased in absolute value without decreasing the average surplus profit of the project.
Projects with optimal negative risk are very paradoxical, very different from optimal projects with positive risk, and they are not optimal if we understand risk reduction in its ordinary sense, where the risk is always positive, because it is a standard deviation.
A project cannot be incorporated into an optimal portfolio if its initial cost is too high, because any portfolio that would contain it would be suboptimal. If its initial cost is too low, it is a windfall, and cannot be incorporated into an optimal portfolio, because these exclude windfalls.
The risk of a project does not depend on its initial cost, if it is fixed in advance. The irreducible risk does not depend on it either. We can vary the initial cost of a project without varying its risk and thus find an initial cost such that the project can be incorporated into an optimal portfolio.
The irreducible risk of a project is the irreducible risk of the project that has the same final revenue and whose initial cost has been adjusted to be part of an optimal portfolio.
All projects can be divided into three categories, depending on whether their irreducible risk is positive, zero, or negative. Let X be the surplus profit of a project and X° the surplus profit of the same project when its initial cost has been adjusted to be part of an optimal portfolio. The irreducible risk of X is positive if E(X°) > 0, zero if E(X°) = 0, and negative if E(X°) < 0.
Theorem: the absolute value of the irreducible risk of a project whose surplus profit is X is equal to the risk of an optimal project whose average surplus profit is equal to |E(X°)|.
Proof: let Y be the surplus profit of a project such that X°+Y is the surplus profit of an optimal project. Let R be the risk of X°+Y. R = ect(X°+Y). Let Rx and Ry be the irreducible risks of X and Y respectively.
* If E(X°) > 0 or < 0, Rx = R E(X°)/E(X°+Y). |E(X°)|/E(X°+Y) (X°+Y) is an optimal project whose average surplus profit is |E(X°)| and its risk is R |E(X°)|/E(X°+Y), therefore equal to |Rx|.
* If E(X°) = 0, Rx = 0. The risk of X°+Y is zero. X°+Y is the surplus profit of an optimal project without risk, so E(X°+Y) = 0 = E(X°).
Theorem: if we increase an irreducible negative risk in absolute value without decreasing the average profit, we increase the value of a project.
Proof: the value of a project is the value of its average surplus profit minus the cost of the irreducible risk. If the irreducible risk is negative, the cost of the risk is negative and therefore increases the value of the project.

## The net present value of a risky project

The net present value of a risky project is its value net of its initial cost.
As with risk-free projects, if the net present value of a risky project is less than zero, it seems that the project should be rejected because it is not worth its initial cost. This rule must be applied flexibly, because risks and their costs are often difficult to measure. In such cases, rough estimates must be made. If the net present value of a risky project is zero, the project is correctly valued by its initial cost. If the net present value of a risky project is greater than zero, the project is a windfall, because its value is greater than its initial cost.
The net present value of a risky project is not the average surplus profit, because the cost of risk must be taken into account:
Theorem: the net present value of a risky project is the average of its surplus profit minus the cost of its irreducible risk. If X is the surplus profit, Rx the irreducible risk of X and k the risk price constant, NPV(X) = E(X) - k Rx.
Proof: The net present value of a risky project is the present value of its average final revenue minus the cost of its irreducible risk minus the initial cost. The average surplus profit is the difference between the average final revenue and the value, on the day the project closes, of the initial cost. The present value, on the day the project starts, of the average surplus profit is therefore the difference between the present value of the average final revenue and the initial cost. Hence the theorem.
Theorem: the net present value of a project is not modified by its financing method.
Proof: when we use leverage, we do not modify the surplus profit of a project, we therefore do not modify either its average surplus profit or its irreducible risk.
Theorem: the net present value of a sum of projects is the sum of the net present values ​​of the component projects.
Proof: This theorem has already been proven for risk-free projects. Let X and Y be the surplus profits of two projects, risky or not. The irreducible risk of X+Y is the sum of the irreducible risks of X and Y. The average of the net present value of X+Y is the sum of the averages of the net present values ​​of X and Y. Therefore the net present value of X+Y is the sum of the net present values ​​of X and Y. By reasoning by recurrence, we establish this theorem for any number of component projects.
To calculate the net present value of a sum of projects, we must first take into account the effect of value creation by composition, because the initial costs and final revenues of the various projects may depend on the existence of the other projects.
Theorem: the net present value of an optimal project is zero.
Proof: the risk of an optimal project is its irreducible risk and it is exactly compensated by the average surplus profit.
The converse is not true for a risky project. A risky project can have a net present value of zero without being optimal, if its risk is not irreducible.
Lemma: if a project can be part of an optimal portfolio then its net present value is zero.
Proof: Let X and Y be the surplus profits of two risky projects such that X+Y is an optimal project. If the net present value NPV(X) > 0, the average surplus profit E(X) > k Rx, where Rx is the irreducible risk of X, and X would be a windfall. If NPV(X) < 0, E(X) < k Rx. R = Rx + Ry. E(X+Y) = k R = k Rx + k Ry = E(X) + E(Y) so E(Y) > k Ry, and Y would be a windfall. Now an optimal portfolio must not contain any windfall. Therefore the net present value of each of its shares is zero.
In particular, if X° is the surplus profit of a project X whose initial cost has been adjusted so that it can be part of an optimal portfolio, the net present value of project X° is zero.
Theorem: if X is the surplus profit of a project and X° = X + C the surplus profit of the same project when its initial cost has been adjusted so that it can be part of an optimal portfolio, then the net present value of X is the constant -C.
Proof: the net present value of X is that of X° minus C, therefore equal to -C, because the net present value of X° is zero.
Theorem: the net present value of a project is zero if and only if it can be part of an optimal project.
Proof: If the net present value of a project with surplus profit X is zero, then X = X° and can therefore be part of an optimal project. The converse has already been proven.

## Short selling

Let a project P be defined by a fixed initial cost C and a random final revenue R. Selling P short is selling it after having borrowed it with the obligation to return it. R is the final value of P, therefore also the amount that must be paid to return it. C is the price that must be paid to acquire P, therefore also the amount that is received if it is sold short. Selling P short is therefore the project that has a fixed initial revenue C and a random final cost R. An initial revenue can be considered as a negative initial cost, and a final cost as a negative final revenue. Selling P short is therefore the project -P whose initial cost is -C and final revenue -R.
Theorem: if X is the surplus profit of a project P, -X is the surplus profit of the project -P of selling P short.
Proof: X = R - C(1+r)^t where r is the discount rate and t is the duration of the project. -X = -R - (-C)(1+r)^t is therefore the surplus profit of the project whose initial cost is -C and final revenue -R.
Theorem: the risk of short selling a project P is equal to the risk of P.
Proof: R = std(X) = std(-X) is both the risk of P and the risk of short selling P.
Theorem: the net present value of short selling a project P is equal and opposite to the net present value of project P.
Proof: let X be the surplus profit of P. NPV(X-X) = NPV(0) = 0 = NPV(X) + NPV(-X). Therefore NPV(X) = -NPV(-X).
If we buy P by paying its initial cost at the same time as we sell it short, we realize a risk-free project that has a zero initial cost and a zero surplus profit, so it has a zero net present value.
Theorem: the irreducible risk of short selling a project P is equal and opposite to the irreducible risk of project P. Proof: NPV(X) = E(X) - k Rx. NPV(-X) = E(-X) - k Rx-, where Rx- is the irreducible risk of -X. Rx- = ( E(-X) - VAN(-X) )/k = ( -E(X) + VAN(X) )/k = -Rx.
Theorem: a project is with optimal negative risk if and only if it is equivalent to short selling an optimal project. Proof:
* if a project is equivalent to short selling an optimal project then its surplus profit X is such that -X is the surplus profit of an optimal project and its initial cost is -C where C is the initial cost of this optimal project. If Y is the surplus profit of a negative-risk project P that has the same cost -C, short selling P has a cost C and a surplus profit -Y. Since X is optimal, -E(X) > -E(Y), so E(X) < E(Y). E(X) = k Rx and E(Y) = k Ry, so Rx < Ry and |Rx| > |Ry|. Therefore all negative-risk projects that have the same initial cost as X have an irreducible risk smaller than that of X in absolute value. Therefore X is with optimal negative risk.
* if X is the surplus profit of an optimal negative-risk project whose initial cost is C and irreducible risk Rx, then -X is the surplus profit of short selling this project and its irreducible risk is -Rx. The initial cost of this short sale is -C. Let Y be the surplus profit of a project P that has the same cost -C. -Y is the surplus profit of short selling P and has the same cost C as X. Since X is at optimal negative risk, the irreducible risk Ry- of -Y is smaller in absolute value than that of X: Rx < Ry-. Therefore Ry = -Ry- > -Rx. Therefore -X is an optimal project. Therefore X is equivalent to short selling an optimal project.

## The vector space of projects

Theorem: all projects form a vector space. Proof: a project is identified with the random variable of its surplus profit. The sum of two projects X and Y is the project whose surplus profit is X+Y, so the union of projects X and Y. Project aX is the acquisition of a shares of project X if a is positive or the short sale of |a| shares of project X if a is negative. The space of all projects is therefore a vector space.
The surplus profits of all projects, regardless of their dates and durations, must all be evaluated, that is to say discounted, on the same day, so that they can be compared and added together.
Theorem: in the vector space of projects, the null vector represents a risk-free project whose profit is that obtained if we had placed the initial cost of the project at the optimal risk-free rate.
Proof: std(0) = 0 so a project with surplus profit X = 0 is risk-free. X = R - C(1+r)^t where R is the final revenue, C the initial cost, t the duration of the project and r the optimal risk-free rate. Therefore R = C(1+r)^t. Therefore the profit is R - C = C(1+r)^t - C.
Theorem: NPV(aX) = aNPV(X) Proof: NPV(aX) = E(aX) minus the irreducible risk of aX. Whether a is positive or negative, the irreducible risk of aX is a times the irreducible risk of X. Therefore NPV(aX) = aE(X) minus a times the irreducible risk of X = a NPV(X).
Theorem: in the vector space of all projects, projects with zero net present value form a vector subspace.
Proof: if NPV(X) = 0 and NPV(Y) = 0 then NPV(X+Y) = NPV(X) + NPV(Y) = 0 and NPV(aX) = a NPV(X) = 0.
Theorem: a project is optimal if and only if it is optimal in the vector space of projects with zero net present value.
Proof: all components of an optimal project have zero net present value, because an optimal project must not contain a windfall, hence no project whose net present value is strictly greater than zero, and because it must not contain the short sale of a windfall, because this would be an error that would reduce the value of the project. Theorem: the space of projects with zero net present value is Euclidean.
Proof: it is a vector space with a positive symmetric bilinear form, the covariance between two random variables. We assume that it is of finite dimension, because we reason on the projects that can be carried out with today's means. It remains to show that the covariance is positive definite in the space of projects with zero net present value. If cov(X,X) = var(X) = 0 then std(X) = 0 and X = 0 because NPV(X) = 0. Hence the theorem.

## Irreducible risk and covariance with an optimal project

X° is the surplus profit of a project with surplus profit X whose initial cost has been adjusted so that it can be part of an optimal portfolio. If the net present value of the project with surplus profit X is zero, X = X°.
Theorem: if the irreducible risk of X is strictly positive, so if E(X°) > 0, then this irreducible risk is the positive square root of the covariance of X with an optimal project whose average surplus profit is E( X°). Moreover, the covariance of X with all optimal risky projects is strictly positive.
Proof: let Y be the surplus profit of an optimal project that has the same average surplus profit as X°. Let Z = aX° + (1-a)Y be the surplus profit of a portfolio that contains a share a of project X° and a share (1-a) of project Y. The average surplus profit of Z is the same as that of X° and Y. If a is negative, Z contains (1+|a|)Y as an asset and |a|X° as a liability. This means that to constitute Z, we sold |a|X° short. Since Y is an optimal project, d/da var(Z) = 0 at a = 0. var(Z) = a²var(X°) + 2a(1-a)cov(X°,Y) +(1-a)²var(Y). Therefore d/da var(Z) = 2a var(X°) + (2-4a)cov(X°,Y) + (2a - 2)var(Y). At a=0, d/da var(Z) = 2cov(X°,Y) - 2var(Y) = 0. Therefore var(Y) = cov(X°, Y) = cov(X,Y). Hence the first part of the theorem, because var(Y) is the square of the irreducible risk of project X. cov(X,Y) > 0 because var(Y) > 0. All optimal projects are strictly positive multiples of the same random quantity, so their covariance with X is always strictly positive, since cov(X,aY) = a cov(X,Y).
If E(X°) < 0, there is no optimal project that has the same average surplus profit as X°, because they all have a profit at least equal to the risk-free profit, so a positive or zero surplus profit.
Lemma: if X is the surplus profit of a project, (-X)° = -X°.
Proof: NPV((-X)°) = 0 = E((-X)°) - k Rx-, where Rx- is the irreducible risk of -X. NPV(-X°) = 0 = E(-X°) - k Rx-. Therefore E((-X)°) = E(-X°). Now (-X)° = -X° + C where C is a constant. Therefore C = 0 and (-X)° = -X°.
Theorem: if the irreducible risk of X is strictly negative, so if E(X°) < 0, then this irreducible risk is the negative square root of the opposite of the covariance of X with an optimal project whose average surplus profit is -E(X°), and the covariance of X with all optimal projects is strictly negative.
Proof: if the irreducible risk of X is strictly negative then the irreducible risk Rx- of -X is strictly positive. Rx- is the positive square root of the covariance of -X with an optimal project whose average surplus profit is E((-X)°) = E(-X°) = - E(X°). Now cov(-X,Y) = -cov(X,Y) for all Y. Hence the theorem.
Theorem: the irreducible risk of X is zero if and only if the covariance of X with all optimal projects is zero.
Proof:
* According to the previous theorems, if the irreducible risk of X is not zero, then the covariance of X with all optimal projects is not zero. So if the covariance of X with all optimal projects is zero then the irreducible risk of X is zero.
* Let X be the surplus profit of a project whose irreducible risk is zero. E(X°) = 0. Let Y be the surplus profit of a risky optimal project. The irreducible risk of X°+Y is the same as that of Y, since the irreducible risk of X° is zero. Therefore the irreducible risk of X°+Y is std(Y). According to the previous theorem, the irreducible risk of X°+Y is the square root of the covariance of X°+Y with an optimal project Z that has the same average surplus profit as X°+Y. Since Y and Z are optimal and have the same average surplus profit, Y = Z. cov(X°+Y, Z) = cov(X°,Y) + var(Y). Therefore var(Y) = cov(X°,Y) + var(Y). Therefore cov(X°, Y) = 0 = cov(X,Y). Since the surplus profits of all optimal projects are all multiples of each other, cov(X,W) = 0 for optimal surplus profits W.
We have therefore proven:
Theorem: the irreducible risk of a project always has the same sign as its covariance with all optimal risky projects.
In other words:
* The irreducible risk of X is strictly positive if and only if the covariance of X with all optimal risky projects is strictly positive.
* The irreducible risk of X is zero if and only if the covariance of X with all optimal projects is zero.
* The irreducible risk of X is strictly negative if and only if the covariance of X with all optimal risky projects is strictly negative.

## How to construct a vector space of projects?

* Choose a finite number of random quantities, all of zero expected value, and a risk price constant strictly between 0 and 1.
* Choose among these random quantities and their multiples a random quantity Op whose variance is equal to 1. All optimal projects are then represented by a(Op + k), for any positive number a, zero included.
* If X is one of the random quantities initially chosen, X° = X + k cov(X,Op) represents a project whose net present value is zero. Si Z = aX + bY, Z° = aX° + bY° = Z + k cov(Z,Op). In particular Op° = Op + k.
* The vector space of projects with zero net present value is the vector space generated by the X°, for all the random quantities X initially chosen.
* The vector space of all projects is the space of all random quantities Z = Y + a, for any number a and any random quantity Y which represents a project with zero net present value. a is the net present value of Z. Z is the random surplus profit of the project.
With such a vector space, one can prove, due to its construction, all theorems on net present value and irreducible risk. Here are three examples:
* The average surplus profit E(X) of an optimal project X is equal to k std(X).
Proof: E(a(Op + k)) = a E(Op) + k a = k a. std(a(Op + k)) = a std(Op) = a, because a > or = 0. Therefore E(a(Op + k)) = k std(a(Op + k)).
* If E(X) > 0 is the average surplus profit of a project with zero net present value, then E(X) is equal to k cov(X, Y)^(1/2) where Y is an optimal project that has the same average surplus profit as X.
Proof: E(X)/k Op + E(X) is such an optimal project. cov(X, E(X)/k Op + E(X)) = E(X)/k cov(X,Op). Or E(X) = k cov(X,Op). So cov(X, E(X)/k Op + E(X)) = cov(X,Op)² and E(X) = k cov(X, E(X)/k Op + E(X))^(1/2).
* If a project is optimal then it is optimal in the space of zero net present value projects.
Proof: Let X be an optimal project and Y a zero net present value project that has the same expected value as X. X = a(Op + k), so std(X) = a and E(X) = ka. E(Y) = k cov(Y,Op) = E(X) = ka, so cov(Y,Op) = a = std(X). By the Cauchy Schwarz inequality, cov(Y,Op)² < or = var(Y) var(Op) = var(Y). So std(Y) > or = std(X). X has the smallest risk among all zero net present value projects that have the same average surplus profit and is therefore optimal in the space of zero net present value projects.
Such a vector space is the general solution to all problems of financial risk calculation, because one can always reduce the mathematical problem to the study of such a vector space.

## The Modigliani-Miller theorem

We must distinguish between the initial price and the value of a share of ownership of a project. If P is a project of value V whose initial cost is C, then the initial price of a share x (a number between 0 and 1) of P is xC and its value is xV. The initial price and the value can be different, except if the net present value of the project is zero, because then it is worth its initial cost: V = C.
For a project whose value is V and initial cost C, the value of an initial stake of 1 is V/C.
Modigliani-Miller theorem: if the net present value of a project is zero, then leverage does not change the value of an initial stake to finance the project.
We can give several proofs of this theorem:
* Leverage does not change the net present value of a project. Therefore V = C with or without leverage, for a project whose net present value is zero. So the value of an initial stake of 1 is always 1, regardless of the leverage chosen.
* If the leverage multiplies the surplus profit X by a factor a, it multiplies at the same time the irreducible risk Rx, by this same factor. Now V = C + E(X) - k Rx. For a project with zero net present value, E(X) = k Rx and E(aX) = a(EX) = k a Rx. If we finance a project by leverage, we vary the average surplus profit E(X) and the cost of risk k Rx by the same amount. Since one exactly compensates for the other, the value of an initial stake is not modified.
For the same initial stake, the leverage increases the risk, because we invest in a larger project, financed both by the initial stake and by borrowing. The increase in the average surplus profit by leverage is the compensation for an increase in risk.
If the irreducible risk of a company is negative, it must be counted as revenue. Leverage increases the risk in absolute value and therefore increases this revenue, but at the same time it decreases the average surplus profit, because this is negative. The decrease in the average surplus profit is compensated by the increase in absolute value of the negative risk. This is why the value of an initial stake is not changed.
The efficient markets hypothesis is that all firms are quoted at their fair value, so their net present value is always zero. This is why Modigliani and Miller used this hypothesis to prove their theorem.

## The zero value of cryptoassets

Theorem: the value of cryptoassets is always zero.
We can give several proofs of this theorem:
* The value of a good is the value of the services it can provide. But cryptoassets do not provide any services. Therefore their value is zero.
* The value of a project is the value of its final revenue minus the cost of the irreducible risk. The earnings of cryptoassets sellers depend on the existence of buyers. If there are no more buyers, cryptoassets can no longer be sold and the final revenue of their owners will be zero. In all likelihood, human beings will understand that cryptoassets are a scam and they will stop buying them. Therefore the final revenue will be zero. The risk is the standard deviation of the final revenue and is therefore also zero. The irreducible risk is also zero. Hence the theorem.
When you buy a cryptoasset, you buy an asset whose value is certain, because it is zero. It is therefore a sure way to lose all the money you have advanced. If you want to make money, or not lose too much, after buying cryptoassets, You have to find gullible people who are willing to buy assets that are worthless. Cryptoassets are like lottery tickets, where you bet on the existence of people gullible enough to buy them when their value is zero.
For cryptoassets to be a currency, you would have to agree to pay hundreds of dollars or more in transaction fees every time you buy a sandwich. So the idea that cryptoassets could be used as a currency is a lie.
How can cryptoassets producers make a lot of money when they produce no wealth?
They steal from savers by selling at high prices assets that have no value. Cryptoasset producers and promoters are therefore crooks and thieves. They take advantage of the gullibility of savers. Selling cryptoassets is theft, because it is selling assets that are worthless at a high price. Cryptoasset buyers are robbed when they buy and robbers when they resell. Cryptoasset producers are the first thieves in this chain of thieves. “Rob your neighbor as you were robbed” could be the motto of cryptoasset sellers.
Cryptoasset buyers become cryptoasset sellers. By encouraging savers to buy cryptoassets, cryptoasset sellers encourage buyers to become thieves, crooks and arsonists. Cryptoasset sellers are therefore criminals who push savers into crime.
Being a buyer of cryptoassets is already being a thief, since one buys with the intention of selling, therefore of stealing, and one is a thief when one intends to steal.
Cryptoassets do not produce any wealth but they consume a lot of it, enough to provide electricity to an entire country. The dragon Crypto is a glutton. It devours riches that could support millions of people. Even if savers ask those who ruined them to reimburse their wiped out savings, they will not get all their money back, because it is used to pay the gigantic production costs of cryptoassets. The dragon Crypto has already swallowed approximately two trillions of dollars.
Finance has always been the open door to all kinds of scams, because those who finance buy wealth that does not yet exist. Scammers sell wealth that does not exist and will never exist. Honest entrepreneurs sell wealth that will really exist. By its scale and its duration, the sale of cryptoassets is the biggest scam in the history of finance. Never before have savers lost so much money due to financial dishonesty.
Overconsumption of energy is turning the planet into a desert, due to global warming. We received from our ancestors a temperate planet, where life is good, and we are delivering to our children a burning, desert planet, where life has become almost impossible. Cryptoasset sellers and buyers want to get rich by burning the planet, without producing any wealth that could be useful to our children. They think: "after me the Flood!" They do not care about the future and they have made their greed their God. They are already ruined, because they bought assets that are worth nothing. Cryptoassets sellers and buyers are thieves and arsonists.
The trillions of dollars sunk in cryptoassets could have been invested to prepare our future and that of future generations. We would have a better future and savers would not be ruined. But cryptoasset sellers and buyers do not care about future generations. They prefer to ruin savers and burn the planet.
If all the savers in the world learn the truth about cryptoassets, if they finally understand that their true value is zero, they will stop buying them because they will know that they will not be able to resell them, or only resell them at a loss. Then the sellers will no longer be able to sell, because there will be no more buyers. The cryptoasset industry will disappear, as it must, because its existence is the perpetuation of crime.
Cryptoasset sellers and buyers believe that it is impossible for this industry to disappear. But to know what is possible or not, you need to know the laws. It is impossible for the cryptoasset industry not to disappear. It is a necessary consequence of the laws of finance.
Can the price of cryptoassets increase further? It depends on the intelligence of savers. For it to increase, savers must accept losing more money. For example, if the price of bitcoin goes from $60,000 to $200,000, savers will have collectively lost about 20,000,000 x 140,000 = 2.8 trillion dollars more, which they will never be able to recover. The maximum price of bitcoin is a measure of the maximum stupidity of savers. The gullibility of savers is like a deposit that criminals want to exploit. Is this deposit exhausted? If so, the price of cryptoassets will not increase any more. But if there is still stupidity to exploit, the price of cryptoassets can still increase.

## Who pays the cost of risk?

When a risky project is sold, the seller pays the cost of the irreducible risk, because this risk reduces the value of the project, and therefore the price at which the project can be sold. The buyer is paid to take the risk.
When a risky project is realized, the value of the project, net of its initial cost, is the surplus profit realized. The average surplus profit realized is the average net present value and it ignores the cost of risk. When a risky project is realized, the cost of risk is therefore not paid on average, as if ultimately no one paid it.
Risk takers pay for the risk when they bear losses, but the surplus profit they hope to realize does not take into account the cost of risk.

## The value of a decision

The theory of the value of durable goods, projects, companies, assets or portfolios is always a theory of the value of a decision: what is the value of the decision to buy it? If this value is higher than the proposed price, then the purchase is a windfall, if it is lower, it is better to give up. Since the purchase price is an initial cost, the theory of the net present value of projects is a general theory of the value of decisions.
The gains or losses resulting from a decision depend on subsequent decisions. To know future gains and losses, an agent must anticipate her upcoming decisions and their value. To know the value of a decision, an agent must know the value of the decisions that will follow. How is it ? Isn't there an infinite regress? To know the value of a decision to be made today, I must know the value of the decisions that will have to be made tomorrow, but to know the value of the decisions of tomorrow I must know the value of the decisions the day after tomorrow, and and so on. How then can we know the value of decisions?
An optimal agent always chooses the maximum value when making a decision. What is the most valuable possibility of all that one can choose? Which is better, to exercise an option or not to exercise it? Which is the better option, the option to exercise an option or the option not to exercise it?
To know her future decisions, an optimal agent must reason about the decisions of an optimal agent. An optimal agent can predict her future decisions or their probabilities, because she knows that she will make optimal decisions. (Bellman).
An optimal agent can reason from the end. She must anticipate gains and losses for all possible purposes of the project, at time t. Then she anticipates the gains and losses at the previous stage, at time t-1. Since she knows that she will choose the best decision, she can anticipate her decision at time t-1. Then she can anticipate the gains and losses of a decision at time t-2, and so on. The behavior of an agent can be modeled with a decision tree.
If the environment is predictable, a node represents a moment in a possible destiny where the agent makes a decision. The branches that start from the same node represent the possible choices. Each node can be associated with a gain or a loss. These are the gains and losses that immediately result from the decision made at the earlier node. We start by assuming that these gains or losses are predictable and risk-free. We can therefore ignore the costs of risk.
A decision tree represents all possible sequences of decisions made by an agent and allows the calculation of the associated gains or losses. Only decisions that are relevant to the value of the project are included, those decisions that may have an effect on the value of the decision to purchase the project.
To find a destiny chosen by an optimal agent, we can reason starting from the end, to calculate a function V which assigns a value to each node of the project. Let t be the last instant of the project and z a terminal node at this instant. V(z) is the immediate gain or loss associated with z. Let x be a node at time t-1. V(x) is the sum of the immediate gain or loss associated with x and the present value at time t-1 of the maximum Vmax of V(y) for all nodes y at time t that follow the node x. In this way we can calculate V for all nodes at time t-1, if we already know V for all nodes at time t. One can repeat the process until the initial moment and thus obtain V for all the nodes. We find at the same time the destiny chosen by an optimal agent (or the destinies that she can choose if there are several). An optimal agent always makes a decision that maximizes V at the next node.
If an agent's environment is random, we can model her behavior with a two-player decision tree, as if she were playing with her environment. Decisions are made by the agent at even times, and randomly at odd times by the environment. Each even node is associated with an immediate gain or loss and its probability of being reached by the odd node that precedes it. We can define a function V for all the nodes of this tree as before. For an odd node, V is the probability-weighted average of the V(y) for all subsequent even nodes y. An optimal agent must take risk into account when evaluating possible choices. For an even node, it is therefore necessary to seek not the maximum of V for the odd nodes which follow, but the maximum of V less the cost of the risk which follows a decision. Vmax is not the maximum of V but the value of V which maximizes V less the cost of the risk. For an even node, V is the sum of the immediate gain or loss and the present value (at the time of the node) of Vmax associated with that node. The value of a decision is the value of V at the odd nodes, minus the cost of the risk that follows this decision. V is the expectation of the sum of the present values, at the instant the decision is made, of all the gains and losses that follow this decision for an optimal agent. V is an expectation or anticipation of wealth. An optimal agent must take into account the risk to make the best choice, she always chooses the highest value of the expected wealth minus the cost of the risk when she makes a decision.
The cost of risk must be counted at the time the decision is made, to evaluate the decision, but it is not counted in the expected wealth, because it is a cost that is ultimately not paid on average. To assess risk, an optimal agent must calculate V by discounting all final revenues or losses to the day she makes the decision, and calculate the standard deviation of V. A well-designed project is optimal. The intrinsic risks of all decisions are always irreducible, because the project was designed to compensate for all the risks that could be. If a project is not so well designed, it is suboptimal, because its risks are not irreducible. When evaluating a suboptimal project, one must take into account the cost of the risk that has not been reduced, one must evaluate all decisions as if their risks were irreducible, to take into account the loss of value caused by these risks that could have been reduced.
The cost of risks in the decision tree of a suboptimal project must be counted as if the intrinsic risks of the decisions were irreducible, even if they are not.
Formally, an obligation can be considered as an option with only one possible choice, because an option always obliges us to choose one of the possibilities proposed. An obligation to pay has a negative value for the obligor. One asks to be paid to acquire an obligation to pay. Similarly an option can have a negative value if all possible choices are losses. Such an option is a random liability. When an optimal agent has to exercise a negative option, it chooses the minimum loss. A seller of a positive-valued option is paid to acquire a negative-valued option, because he agrees to pay any gains to the buyer of the positive-valued option. The present theory of the value of decisions and options is fully general. It includes all assets and liabilities, whether risky or not, all options with positive or negative value, and all random assets-liabilities. It can be used to reason about all economic decisions, all buying and selling, consumption, saving and investment decisions.
This theory of the value of decisions can be generalized to several players to model competition and cooperation between economic agents.

# Thierry against the dragon Crypto

Winter is coming and it will be hard. People are stocking up. They are not consuming all their wealth. They are saving it, because they do not want to starve to death when winter comes.
Crypto is a very gluttonous and very thieving dragon. He devours almost all the wealth saved for the winter. So people are afraid.
When Thierry learns this, he says to himself that something must be done : "We cannot let this dragon starve us." But Thierry is small. He is not muscular and he has no weapons. So he goes to see the Goddess, the Truth, and he says to her "Madam Truth, I would like a sword." The Goddess answers him in a stern tone: "But why do you want a sword? - It is because of the dragon Crypto. He devours all our wealth and we are afraid of dying this winter."
Then the goddess smiles at him and gives him the most beautiful sword of all, very strong, very sharp, and very light, because she sees that Thierry is not muscular. Thierry thanks her and gets ready to go in search of Crypto. But the Goddess stops him: "Wait, you also need a shield. Crypto spits flames and he could burn you. But with this shield you will be protected. He always returns to the sender the projectiles and flames that bounce off it"
She gives Thierry the most beautiful, the lightest and most powerful shield.
When Thierry arrives in front of Crypto, he tells him: "Dragon Crypto, if you don't stop devouring our wealth, I will kill you."
Crypto laughs: "But Thierry, did you see yourself? You are very small, so small that I don't even want to eat you. Your sword and shield are ridiculous. You can't do anything against me. Go away quickly before I get angry, because if I get angry you will be burned by my flames."
Thierry stays straight in front of the dragon. He is not afraid because he knows that the Truth protects him. He insists: "Dragon Crypto, if you do not promise to stop devouring our wealth, I will kill you now."
Crypto gets angry at this affront: "Little insolent, you will receive the punishment you deserve." And he spits his flames. Thierry brandishes his shield, the flames bounce and Crypto has his eyes burned by his own flames. Then it is easy for Thierry to thrust his sword into the throat of Crypto, who has become blind. And the dragon dies.

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