# The science of finance/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.