Question

  

Suppose you want to invest in a portfolio comprising an index fund that reproduces the market portfolio (such as the S&P/TSX composite index) and also T-bills. Your investment budget is $10,000. The market index has an expected return of 16% and standard deviation of 12%, and the return on T-bills is 4%. Your goal is to get an expected return of 14% on the combined portfolio (Index fund and T-bills).

1. How much do you need to invest in the index fund and how much in T-Bills?
2. What is the standard deviation of this combined portfolio?
3. What is the reward to variability ratio of the index fund?
4. What is the reward to variability ratio of the combined portfolio (index fund and T-bills)?

           

Answer 1

Return of Portfolio = Weight of Index fund*Return of Index fund+Weight of T bill*Return of T bill

14 = 16*Weight of Index fund+4*(1-weight of Index fund)

Weight of Index fund = 0.8333

Weight of T bill =1-weight of Index fund=1-0.8333=0.1667

Amount to invest in Index fund = index weight *investment amount = 0.833333*10000= 8333.33

Amount to invest in T bill = t bill weight *investment amount = 0.166666*10000= 1666.67

Std dev of risk free asset as well as correlation between risky and risk free asset is always 0 (hence not given in the question)

therefore:

Variance	=( w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB))
Variance	=0.83333^2*0.12^2+0.16667^2*0^2+2*0.83333*0.16667*0.12*0*0
Variance	0.01
Standard deviation=	(variance)^0.5
Standard deviation=	10.00%

Sharpe ratio(reward to variability) portfolio

=(Return-risk free rate)/std dev

=(14-4)/10

=1

Sharpe ratio(reward to variability) Index fund

=(Return-risk free rate)/std dev

=(16-4)/12

=1