Query on CHAPTER 2

[Adithyan]

There's a question on page 23 of chapter 2 on utilitlity theory where they want us to say if A or B is more risk averse.

I didn't understand as to how B is considered more risk averse. because with change in wealth his absolute risk aversion reduces. Kindly help as to how is this to be understood.

 

[Adithyan]

The question is as below, I don't get the solution though:

Suppose Investor A has a power utility function with r (gamma) = 1 , whilst Investor B has a power utility
function with r (gamma) = 0.5 .
(i) Which investor is more risk-averse (assuming that w > 0 )?
(ii) Suppose that Investor B has an initial wealth of 100 and is offered the opportunity to buy
Investment X for 100, which offers an equal chance of a payout of 110 or 92. Will the
Investor B choose to buy Investment X?

(i) Which investor is more risk-averse?
Investor B is more risk-averse because they have a lower risk aversion coefficient r . We can
show this by deriving the absolute risk aversion and relative risk aversion measures for each
investor.
For Investor A:
A(w) = R(w) = 0
ie Investor A is risk-neutral.
For Investor B:
A(w) = 1/ 2w, R(w) = 1/2 > 0

Hence, Investor B is strictly risk-averse for all w > 0.

 

[Anna Bishop]

Hello Adithyan

I wonder if an easier way of looking at it is to consider the utility function itself.

U(w) = (w^γ - 1) / γ

For investor A, with γ = 1, we have:

U(w) = w - 1

For investor B, with γ = 0.5, we have:

U(w) = 2 (√w - 1)

Can you visualise a plot of these two functions?

Pages 10 to 12 in Chapter 2 show some illustrations of the shapes of the U(w) graphs for investors that are risk-averse, risk-neutral and risk-seeking.

For a risk-averse investor, we are looking for diminishing marginal utility of wealth. For each additional $1 of wealth, the extra utility derived reduces. The graph of U(w) against w will be concave. The second derivative, U''(w) < 0.

For a risk-neutral investor, we are looking for constant marginal utility of wealth. For each additional $1 of wealth, the extra utility is the same. The graph of U(w) against w will be a straight line. The second derivative, U''(w) = 0.

For an investor to be risk-seeking, we are looking for increasing marginal utility of wealth. For each additional $1 of wealth, the extra utility derived increases. The graph of U(w) against w will be convex. The second derivative, U''(w) > 0.

Investor A's utility curve is a straight line. U(w) = w - 1. Hence, Investor A is risk-neutral. We can check that U'(w) = 1 and U''(w) = 0.

Investor B's utility curve is a graph of the square root of w, which has a concave shape. Hence, Investor B is risk-averse. We can check that U'(w) = w^(-0.5) and U''(w) = - 0.5w^(-1.5) <0 if w > 0.

I'm not convinced that we need to consider A(w) and R(w) to answer this question. However, I agree with your comment that Investor B exhibits decreasing absolute risk aversion. This means, as Investor B becomes wealthier, he or she becomes less risk averse. However, Investor B is still risk averse (just to a lesser degree as wealth increases) whereas Investor A is not risk averse at all!

I'll make a note that we look to revise this solution going forwards - it's not wrong - but could be simplified.

Does this help?
Anna

 

[Adithyan]

Hello Adithyan

I wonder if an easier way of looking at it is to consider the utility function itself.

U(w) = (w^γ - 1) / γ

For investor A, with γ = 1, we have:

U(w) = w - 1

For investor B, with γ = 0.5, we have:

U(w) = 2 (√w - 1)

Can you visualise a plot of these two functions?

Pages 10 to 12 in Chapter 2 show some illustrations of the shapes of the U(w) graphs for investors that are risk-averse, risk-neutral and risk-seeking.

For a risk-averse investor, we are looking for diminishing marginal utility of wealth. For each additional $1 of wealth, the extra utility derived reduces. The graph of U(w) against w will be concave. The second derivative, U''(w) < 0.

For a risk-neutral investor, we are looking for constant marginal utility of wealth. For each additional $1 of wealth, the extra utility is the same. The graph of U(w) against w will be a straight line. The second derivative, U''(w) = 0.

For an investor to be risk-seeking, we are looking for increasing marginal utility of wealth. For each additional $1 of wealth, the extra utility derived increases. The graph of U(w) against w will be convex. The second derivative, U''(w) > 0.

Investor A's utility curve is a straight line. U(w) = w - 1. Hence, Investor A is risk-neutral. We can check that U'(w) = 1 and U''(w) = 0.

Investor B's utility curve is a graph of the square root of w, which has a concave shape. Hence, Investor B is risk-averse. We can check that U'(w) = w^(-0.5) and U''(w) = - 0.5w^(-1.5) <0 if w > 0.

I'm not convinced that we need to consider A(w) and R(w) to answer this question. However, I agree with your comment that Investor B exhibits decreasing absolute risk aversion. This means, as Investor B becomes wealthier, he or she becomes less risk averse. However, Investor B is still risk averse (just to a lesser degree as wealth increases) whereas Investor A is not risk averse at all!

I'll make a note that we look to revise this solution going forwards - it's not wrong - but could be simplified.

Does this help?
Anna

Thanks a ton for your response. It is extremely helpful indeed!