Suppose an investor has 3 available actions: put everything in the bank (call th

Question

Suppose an investor has 3 available actions: put everything in the bank (call this the "safe" action), split his investment between the bank and the stock market (call this the "moderate" action), or put everything in the stock market (call this the "risky" action). His utilities are as follows, depending on whether the good state (boom) or bad state (recession) occurs: Calculate his expected payoff from each action if he (wishfully) expects the bad action to happen with probability p. Show that his optimal action rule, if he expects the bad state to happen with probability p, is a_p{safe if p>1/3 moderate if 1/5

Explanation / Answer

a) Let P be the probability of occurence of bad situaion. thus the probability of occurence of good situation will be 1-P.

Now investor thinks that bad situation will occur. thus probability here will be used as P.

Expectyed pay off =Probaility of occurence of a situation * Utility of an option under such situation.

Expected payoffs are calculated below:

b) If Probaility of occurence of bad situation is > 1/3:

Lets take P = 2/3 then 1-P = 1/3 : Payy offs are:

Safe = 10*2/3 + 10*1/3= 9.999 ; Moderate = 6*2/3 + 12*1/3= 8 and Risky = -10*2/3 + 16*1/3= 2

From above the maximum outcome is from safe. Hence safe option will be choosen when P > 1/3

If Probaility of occurence of bad situation is < 1/3 and also > 1/5 :

Lets take P = 1/4 then 1-P = 3/4: Payy offs are:

Safe = 10*1/4 + 10*3/4= 10 ; Moderate = 6*1/4 + 12*3/4= 10.5 and Risky = -10*1/4 + 16*3/4= 9.5

From above the maximum outcome is from moderate. Hence moderate option will be choosen when 1/5 < P < 1/3

If Probaility of occurence of bad situation is < 1/5:

Lets take P = 1/8 then 1-P = 7/8: Payy offs are:

Safe = 10*1/8 + 10*7/8 = 10 ; Moderate = 6*1/8 + 12*7/8= 11.25 and Risky = -10*1/8 +16*7/8= 12.75

From above the maximum outcome is from risky. Hence risky option will be choosen when P < 1/5.

Options Expected Payoff Safe 10P Moderate 6P Risky -10P

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