Relative Risk Aversion: Examples

Relative Risk Aversion: Examples

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Absolute risk aversion has implications for the willingness of individuals to accept risk. The higher the coefficient of absolute risk aversion, the higher the risk premium the individual is willing to pay. On the other hand, relative risk aversion is absolute risk aversion times W, indicating initial Wealth. The higher the coefficient of relative risk aversion, the higher the relative risk premium. Two examples below would explain Relative Risk Aversion based on microeconomics point of view.

1. Verify that the function u(W) = W1-α/(1-α) has a constant coefficient or relative risk aversion equal to α

Answer:

– First derivative of u(W) = W1-α/(1-α) is u’ = (1-α)W/(1-α) = W

– Second derivative of u(W) = W1-α/(1-α) or in other word First order condition (FOC) of W is u” = -αW-α-1

– Convex indifference curves imply risk aversion. A natural measure of the degree of risk aversion is the degree of convexity of indifference curve. The magnitude of second derivatives of indifference curve is proportional to -u”(W)/u'(W) or called coefficient of absolute risk aversion (Arroe-Pratt measure of absolute risk aversion). Relative risk aversion is -Wu”(W)/u'(W)

Thus, -Wu”/u’ = -W(-αW-α-1) / W = αW / W= α

2. Verify that the function u(W) = log W has a constant coefficient or relative risk aversion of 1

Answer:

– First derivative of u(W) = log W is u’ = 1/W

– Second derivative of u(W) = log W or in other word First order condition (FOC) of 1/W is u” = -1/W2

– Convex indifference curves imply risk aversion. A natural measure of the degree of risk aversion is the degree of convexity of indifference curve. The magnitude of second derivatives of indifference curve is proportional to -u”(W)/u'(W) or called coefficient of absolute risk aversion (Arroe-Pratt measure of absolute risk aversion). Relative risk aversion is -Wu”(W)/u'(W)

Thus, -Wu”/u’= -W(-1/W2) / (1/W) = (1/W) / (1/W) = 1

References

“The Structure of Economics: A Mathematical Analysis 3rd ed.” by Eugene Silberberg and Wing Suen.

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