Hi,
I have been doing some research and couldn't find any solution to how calculate the coefficient of absolute risk aversion (CARA) and coefficient of relative risk aversion (CRRA).
A lot of articles and lectures in universities are always showing the formulas, (CARA=U''(x)/U'(x)) but I couldn't find a concrete example of how these were calculated.
So I was struggling to calculate my exercise:
a. Suppose a risk-averse agent is given an asset that will pay off 1 with probability .5 and will pay off 2 with probability .5. He is just indifferent between keeping this asset and exchanging it for an asset that will pay off 1.25 for sure. What can you say about his coefficient of absolute risk aversion?
Expected Value asset A E(x) = (0.5 * 1 = 0.5) + (0.5 * 2 = 1.0) = 1.5
Utility of Expected Value is = ln (1.5) = 0.41
Lower Utility for risk-averse agent = 0.5*ln(1) + 0.5*ln(2) = 0.35
Certainty Equivalent = ln(X) = 0.35 = e^0.35 = 1.42
Risk Premium = Expected Value – Certainty Equivalent = 1.5 – 1.42 = 0.08
Expected Value asset B = E(x) = 1 * 1,25 = 1,25
Arrow-Pratt measure of absolute risk aversion A = U’’(x)/U’(x) = ?
b. Now suppose instead that the same agent as in (a) is given an asset that pays off either 2 or 4 with equal probability. If his coefficient of absolute risk aversion is the same as in (a) what is the certain payoff that would leave him equally well-off as holding the risky gamble?
c. Now suppose as in (b) that the agent is given an asset that pays off either 2 or 4 with equal probability. You learn that he is indifferent between keeping this risky gamble and exchanging it for an asset that pays 2.5 for sure. What can you say about his coefficient of absolute risk aversion in this case?
Could somebody help a bit please?
In your exercise you assume log utility, so U(W)=ln(W) where W is uncertain wealth. The coefficient of absolute risk aversion (ARA) then is -U''(W)/U'(W) = 1/W. The coefficient of relative risk aversion is 1 (log utility has the specific feature of constant - that is, wealth-independent- relative risk aversion).
As you see, ARA is wealth-dependent. Consequently, your initial example needs to be modified. You say the investor is indifferent between the lottery and a sure payment of 1.25. However, that will only be true for a specific level of initial wealth. Let's call that level X. Then, indifference in your example requires:
ln(X+1.25)=0.5ln(X+1)+0.5ln(X+2)
This only holds for a negative X (X=-0.875, to be precise).
For the solution in b you do not need ARA. Just solve
0.5ln(X+2)+0.5(X+4)=0.5ln(X+Y)
For any GIVEN level of wealth, X, you can now calculate the amount Ysuch that the investor is indifferent between Y and the lottery.
a)To measure Arrow’s Pratt risk aversion:
rA = -U(w)''/U(w)' , where U(w) = Ln(W) (assumed by the exercise).
U(w)' = 1/W and U(w)'' = -1/W2, and rA = 1/W>0 (positive = risk averse)
Since W is the denominator, as wealth increases, rA decreases and as W→∞,
rA→0 (becomes neutral)
b) In the case of a risk - neutral person, the following equation needs to be considered:
U(P X1 + (1-P) X2)= P U(x1) + (1-P) U (x2)
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