# The science of finance/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°.
O1 is the optimal unit risk project. std(O1) = 1. Its average surplus profit E(O1) = k, where k is the risk price constant.
Theorem: the irreducible risk of a project with surplus profit X is equal to its covariance with the optimal unit risk project O1. E(X°)/k = cov(X,O1), where k is the risk price constant.
Proof:
* If E(X°) > 0, 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). But var(Y) = (E(Y°)/k)² = (E(X°)/k)² and cov(X,Y) = E(X°)cov(X,O1)/k. Hence the theorem.
* 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°.
Proof of the theorem: E (−X°) > 0 so E(−X°)/k = cov(−X°,O1) = −cov(X°,O1). Hence E(X°)/k = cov(X,O1).
* If E(X°) = 0, E(X°+O1)/k = cov(X°+O1,O1) = cov(X°,O1) + cov(O1,O1) = E(O1)/k = 1, so cov(X°,O1) = cov(X,O1) = 0.

## 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 two 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 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.