Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 4.0%. The probability distributions of the risky funds are:

	Expected Return	Standard Deviation
Stock fund (S)	10%	32%
Bond fund (B)	7%	24%

The correlation between the fund returns is 0.1250.

What is the Sharpe ratio of the best feasible CAL?

Answer 1

Expected return stock = E(R₁ )

Expected return bond = E(R₂ )

Standard deviation sock= σ1

Standard deviation bond= σ₂

weight of stock = w₁

weight of bond = w₂

The expected return of the risky portfolio= w₁E(R₁) + w₂E(R₂)

the covariance of risky portfolio= w²₁σ²₁ + w²₂σ²₂ + 2ρ(R₁, R₂) w₁w₂σ₁σ₂, using ρ(R₁, R₂), the correlation of R₁ and R₂

hence, we get the following risk weights and retuns for the risky portfolio as shown in Table 1

Now we compute the complete portfolio which

The expected return of a complete portfolio is given as:

E(R_c) = w_pE(R_p) + (1 − w_p)R_f (see table 2 for computation of risk free returns based on weights )

And the variance and standard deviation of the complete portfolio return is given as:

Var(R_c) = w²_pVar(R_p), σ(R_c) = w_pσ(R_p) (see table 2 for computation of risk free returns based on weights )

The line E(R_c) = R_f + S_pσ(R_c) is the capital allocation line (CAL). The slope of the line, S_p, is called the Sharpe ratio

hence, by replacing the values in the formula we get

6.25%=4%+Sp*2.24%

i.e Sp= 1.00446429

Table 1

weight of stocks	Weight of bonds	Riski portfolio standard deviation	Expected return
0.1	0.9	4.94%	7.30%
0.2	0.8	4.40%	7.60%
0.3	0.7	4.15%	7.90%
0.4	0.6	4.17%	8.20%
0.5	0.5	4.48%	8.50%
0.6	0.4	5.07%	8.80%
0.7	0.3	5.94%	9.10%
0.8	0.2	7.09%	9.40%
0.9	0.1	8.52%	9.70%
1	0	10.24%	10.00%

Table 2

riski portfolio weights	Risk free assets weights	0.049408	0.044032	0.041472	0.041728	0.0448	0.050688	0.059392	0.070912	0.085248	0.1024
0.1	0.9	4.33%	4.33%	4.33%	4.33%	4.33%	4.33%	4.33%	4.33%	4.33%	4.33%
0.2	0.8	4.72%	4.72%	4.72%	4.72%	4.72%	4.72%	4.72%	4.72%	4.72%	4.72%
0.3	0.7	5.17%	5.17%	5.17%	5.17%	5.17%	5.17%	5.17%	5.17%	5.17%	5.17%
0.4	0.6	5.68%	5.68%	5.68%	5.68%	5.68%	5.68%	5.68%	5.68%	5.68%	5.68%
0.5	0.5	6.25%	6.25%	6.25%	6.25%	6.25%	6.25%	6.25%	6.25%	6.25%	6.25%
0.6	0.4	6.88%	6.88%	6.88%	6.88%	6.88%	6.88%	6.88%	6.88%	6.88%	6.88%
0.7	0.3	7.57%	7.57%	7.57%	7.57%	7.57%	7.57%	7.57%	7.57%	7.57%	7.57%
0.8	0.2	8.32%	8.32%	8.32%	8.32%	8.32%	8.32%	8.32%	8.32%	8.32%	8.32%
0.9	0.1	9.13%	9.13%	9.13%	9.13%	9.13%	9.13%	9.13%	9.13%	9.13%	9.13%
1	0	10.00%	10.00%	10.00%	10.00%	10.00%	10.00%	10.00%	10.00%	10.00%	10.00%

Table 3

riski portfolio weights	Risk free assets weights	Riski portfolio standard deviation	Risk of risk free portfolio
0.1	0.9	4.94%	0.49%
0.2	0.8	4.40%	0.88%
0.3	0.7	4.15%	1.24%
0.4	0.6	4.17%	1.67%
0.5	0.5	4.48%	2.24%
0.6	0.4	5.07%	3.04%
0.7	0.3	5.94%	4.16%
0.8	0.2	0.070912	5.67%
0.9	0.1	0.085248	7.67%
1	0	0.1024	10.24%

Table 1