The Cost of Risk (Supplementary)
We can delve deeper into this problem by trying to measure the cost of risk. Above we were trying to get some intuition as to the factors that would affect the risk-return trade off. We suggested that the concavity of the utility function and the variance of the gamble have a great affect. It is clear that the individual will never take a fair bet (i.e., 50% probability of a $1 payoff and a 50% probability of a -$1 payoff). He is better off with his initial wealth.
Now let's look at the problem from a different angle. The question is: how much would an individual pay to avoid a fair bet of size plus or minus h? Think of this as an insurance problem. The amount that you are willing to pay, p, is the insurance premium. Indifference implies:
U(W-p) = 0.5 U(W+h) + 0.5 U(W-h)
To solve the problem, we need to isolate p on the left-hand side of the equation. This is complicated by the fact that p is inside the function. The method to do this is to take a Taylor series approximation to both sides of the equation. This is a good time to review Taylor series.
The general formula for a Taylor Series expansion is:
f(x+j) = f(x) + f '(x)j + 0.5 f ''(x) j2 + ... + (1/n!) f (n) (x)jn + . . .
If this is not familiar, you can review your undergraduate calculus text. It should have a section on Taylor series. You might remember that we were faced with a similar problem earlier in the course. We wanted to get a measure of the sensitivity of the bond price to a change in the interest rate. This problem was complicated by the convexity of the bond price in the yield. Our duration measure was not very accurate for large changes in the yield. With large changes, we have to account for convexity. We can handle this with the Taylor series approximation. If f(x) to be the pricing function for the bond. The price is a function of the yield (x). Consider the change in the yield to be the j variable. A second order expansion cuts off the approximation after the second term. The first order part (f '(x)) with the duration of the bond and the second order part (f ''(x)) with the convexity of the bond. There will also be a residual part because the Taylor series is only an approximation.
Now back to our problem. Our strategy will be to take Taylor series approximations of both sides of the equation. If the size of p is small compared to the wealth W then we can ignore the second order and higher terms in the Taylor series. Note we are just plugging into the Taylor series formula above. To follow note that f (.) = U(.), x = W, and j = -p. So the left hand side is:
U(W-p) approx= U(W) - U'(W) p
The right-hand side is trickier. First, we cannot as easily ignore the higher order terms. This is because h may take on values that are considerably different than W. So on the right-hand side we will take a second order Taylor series expansion:
Note in terms of our formula for Taylor series: x = W and h = j,-j. Now equating the two expansions:
U(W) - U'(W) p = U(W) + 0.5 h2 U''(W)
We can solve for the insurance premium, p:
p = 0.5 h2 [-U''(W)/U'(W)]
In words, the insurance premium can be expressed:
Insurance premium = one half [Variance] times [Risk Aversion]
We can also think of this insurance premium as the cost of risk. The cost of risk is proportional to the variance in wealth. Note that all the terms that we expected to be in the expression are there. The ratio of the derivative controls the concavity. The variance of the gamble is also there. Note also that the cost of risk is positive because the second derivative is negative. Also note that this is only an approximation. This is sometimes referred to as coefficient of absolute risk aversion.