Question

A person with initial wealth w0 > 0 and utility function U(W) = ln(W) has two investment alternatives: A risk-free asset, which pays no interest (e.g. money), and a risky asset yielding a net return equal to r1 < 0 with probability p and equal to r2 > 0 with probability 1 (>,<,=) p in the next period. Denote the fraction of initial wealth to be invested in the risky asset by x. Find the fraction x which maximizes the expected utility of wealth in the next period.

Denote this solution by x^*. What is the condition for x^* > 0?

Answer 1

We are given utility function  U(W) = ln(W)

also we are given fraction of initial investment in risky asset as x

so out of total wealth , for investing in the risk free asset we are left with W-x

also the probability of investing in asset giving returns r1<0 is p and with r2>0 as 1-p [ in the question we are given investment in asset with returns equal to r2 > 0 with probability 1 (>,<,=) p . if there are two risky assets and the probability of investing in one is p then in the other is 1-p . here we can have the probability as only 1-p with p ,1 because p cant be more than1 or if p=1 , then investing in the other one would have no probability. ]

the following maximisation function is formed.

max {p[u(1+R0)x +(w-x)]+(1-p)[ u(1+R1)x+(W-x)]}.............................1

if we invest in x asset with R0 return , then the rest of w-x is invested in rsik free asset . this is with probablity p . we know that the probability of asset with R0 returns is p so this means that p[U(1+R0)x+(w-x)] would be the utility derived from x assets whose return is is R0 and the rest wealth in risk free .

similarly the concept goes for 1-p which is the probability for investing in R1 returns asset and the rest in risk free asset.

and we are given the utility function as U(W) = ln(w)

puttinh this value in equation 1 we get

max {p[ln(1+R0)x +(w-x)]+(1-p)[ ln(1+R1)x+(W-x)]}

now differentiating the equation with respect to x ie first order condition we get

p(R0)/( R0x+w) +(1-p)R1/(R1x+w)= 0 [as ln(x+1) would have chain rule first we ahve for differentiating the ln part we get 1/(x+1) and then differentiating (x+1) we get 1 ]

now we take LCM of both terms

and get xR1R0p + wpR0 +R1R2x+ R1w -pxR1R0 -pR1w =0 ( denominator comes to right side and gets 0)

simplifying furthur we get xR1R0= pR1w-R1w-wpR0

so x or x* = -w(R0p+ R1-pR1)/R1R0

so x* is -w(R0p +(1-p) R1)/R1R0

we know that R0<0 so denominator would be negative . also the numerator is already negative as we have -w , so we have the whole term positive (negatives cancelled out)

so x* is positive

we get dx/dw>0 ie the the investor would invest more in risky assets when he gets wealthier ie x is increasing function of w. (positively related)