# The science of finance/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.