Question

Consider a binomial tree with one future period (T=0,1) in which the price can go up to $30 or decrease to $15. The price of the asset at T=0 is $20, and there is a 20% probability that it goes up at T=1. The investor starts with an initial wealth of $1,000. The risk-free asset yields a return of zero percent.

What is the expected terminal wealth of the investor if she invests 70% of her wealth in the risky asset?

Select one:

1175

930

1245

None of the options provided

1088

500

Answer 1

Correct Ans is 930 = 630+300 ( Terminal nvof risky asset + terminal value of risk free asset)

Explaination : Investor invested 30% of her wealth in risk free asset (i.e. = 1000*30% = 300) which yielded a return of zero percent. Since, Risk free rate is zero percent , nil return is generated on 30% of invested amount. So, terminal Value for 30% is same i.e 300.

Now, 70% of the wealth i.e. 700(1000*70%) invested in risky asset.

Binomial Tree has been provided in the ques where it is mentioned that there are 20% chances that Prices will go up to 30 in year 1. Initial price at T=0 is 20.Thus, Prices will increase by 1.5 times ( 30/20 = 1.5)

There are 80% chances that prices will go down to 15 in year 1. Initial price at T=0 is 20. Thus, Price will decrease by 0.75 times ( 15/20 = 0.75).

Now, we have 700 to invest in risky asset

So, 700*1.5*20%Probability = 210 (20% probability of increase in price by 1.5 times)

700*0.75*80%Probability = 420 ( 80% probability of decrease in price by 0.75 times)

So, terminal value of risky asset - 210+420 = 630.