A universe of securities includes a risky stock (S), a bond fund (B), and T-bill

Question

A universe of securities includes a risky stock (S), a bond fund (B), and T-bills. The data are:

Security

Expected return

Standard deviation

S

13%

33%

B

6%

12%

T-bills

2%

0%

The correlation coefficient between S and B is -0.04.

Calculate the weights on S and B that create the optimal risky portfolio (call it “O”).

Calculate the expected return of O.

Calculate the standard deviation of O.

Calculate the Sharpe ratio of O.

Calculate the Sharpe ratio of S.

Calculate the Sharpe ratio of B.

Why is Sharpe ratio of O higher than that of S or B?

Suppose an investor places 1/2 of the complete portfolio in the risky portfolio O and the remainder in T-bills:

Calculate the complete portfolio’s expected return.

Calculate the complete portfolio’s standard deviation.

Calculate the complete portfolio’s Sharpe ratio.

Security

Expected return

Standard deviation

S

13%

33%

B

6%

12%

T-bills

2%

0%

Explanation / Answer

Ws = ( (Rs - Rf)*Sb2 - (Rb - Rf)*Cov ) / ( (Rs - Rf)*Sb2 + (Rb - Rf)*Ss2 - [Rs - Rf + Rb - Rf]*Cov )

Cov = Cor*Sb*Ss = -0.04*0.33*.012 = -0.0001584

Rs - Rf = 13% - 2% = 0.11

Rb - Rf = 6% - 2% = 0.04

Ss2 = 0.332 = 0.1089

Sb2 = 0.122 = 0.0144

Ws = (0.11*0.0144 - 0.04*-0.0001584) / ( 0.11*0.0144 + 0.04*0.1089 - (0.11+0.04)*-0.0001584)

Ws = 0.00159 / 0.00596376 = 0.266667 = 26.67%

Wb = 1 - Ws = 1 - 26.67% = 73.33%

expected return of O = 26.67%*13% + 73.33%*6% = 7.87%

standard deviation of O = Sqrt (26.67%2 * 0.1089 + 73.33%2 * 0.0144 + 2*26.67%*73.33%*-0.0001584) = 12.42%

Sharpe ratio of O = 7.87% - 2% / 12.42% = 0.4726 = 47.26%

Sharpe ratio of S = 13% - 2% / 33% = 0.3333 = 33.33%

Sharpe ratio of B = 6% - 2% / 12% = 0.3333 = 33.33%

It is higher because, it has better returns for lower risk

complete portfolio’s expected return = 0.5*7.87% + 0.5*2% = 4.94%

complete portfolio’s standard deviation = Sqrt(0.5*(7.87%-4.94%)2 + 0.5*(2%-4.94%)2 ) = 2.94%

complete portfolio’s Sharpe ratio = 4.94% - 2% / 2.94% = 1 = 100%

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