Investor A has a square root utility function (i.e., U(W) = W), whereas Investor

Question

Investor A has a square root utility function (i.e., U(W) = W), whereas Investor B has a logarithmic utility function (i.e., U(W) = ln W). Both investors have initial wealth W0 = $100 and must decide how much how much to invest in a bond and how much to invest in a stock. The current prices of the bond and stock are B0 and S0 respectively. Although neither security pays dividends or interest, both investors expect to receive income from selling these securities at their end-of-period prices, which are B1 for the bond and S1 for the stock. Since the bond is riskless, its end-of-period price is known with certainty to be B1 = B0(1+r ), where r is the riskless rate of interest. The price of the stock at t = 1 can be high or low; i.e., it will be S0(1+s) with probability .6 and it will be S0(1-s) with probability .4. Furthermore, assume that W 0= $100, r = .05, and s = .3.

A. How much (in dollar and percentage terms) of Investor A’s initial wealth should be invested in the stock and in the bond?

B. How much (in dollar and percentage terms) of Investor B’s initial wealth should be invested in the stock and in the bond?

C. Who is more risk averse - Investor A or Investor B? Explain why.

D. Recalculate your answers for parts A and B assuming initial wealth of $200 rather than $100.

Explanation / Answer

So, W0 needs to be distributed into B0 and S0. Let x amount be invested in S0 and 100 - x be invested in B0 . (100 -x as initial wealth is 100).

Finally, our investment in the bond will become (100-x)*(1.05) but there are probabilities with regards to stock moving up or down in the future.

So, expected value of utility will be

U = 0.6*sqrt( 1.05*(100-x) + x*1.3) + 0.4*sqrt( 1.05(100-x) + x*0.7) = 0.6*sqrt(105+0.25*x)+0.4*sqrt(105-0.35x)

(first part is with probability of 0.6 of stock going up and the other with probability of 0.4 of stock going down)

In order to maximise the utility, we need to differentiate this utility function and set to zero

dU/dx = 0 = 0.6*0.25*0.5/sqrt(105+0.25*x) + 0.4*(-0.35)*0.5/sqrt(105-0.35x)

We get x = 23.8 - to be invested in stock

and 100-x = 76.2 - to be invested in bond

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