Question

5. Consider two perfectly negatively correlated risky securities A and B. Your portfolio is currently weighted with 50% in A and 50% in B. A has an expected rate of return of 10% and a standard deviation of 16%. B has an expected rate of return of 8% and a standard deviation of 12%. The risk-free rate is 3%. (a) (3 points) What are the variance and return for your portfolio? (b) (4 points) What weights would give you the minimum variance portfolio for these two assets? (c) (2 points) Is your current allocation more desirable than the allocation with the weights for the minimum variance portfolio? (Hint: use the Sharpe Ratio)

Answer 1

a. The return of the portfolio is given by

weight of stock A * Return of Stock A + Weight of Stock B * Return of Stock B

= 0.5 * 0.1 + 0.5 * 0.08

= 0.05 + 0.04

= 0.09

= 9%

Variance of a porftolio is given by

(w1 * sd1)^2 + (w2*sd2)^2 + 2*w1*w2*sd1*sd2*correlation

w1 = Weight of Stock 1 = 0.5

w2 = Weight of Stock 2 = 0.5

sd1 = standard deviation of stock 1 = 0.16

sd2 = standard deviation of stock 2 = 0.12

correlation = -1

(0.5 * 0.16)^2 + (0.5 * 0.12)^2 + 2*0.5*0.5*0.16*0.12*(-1)

= 0.0064 + 0.0036 - 0.0096

= 0.01 -0.0096

= 0.0004

= 0.04%

b. I had to use excel to determine the lowest variance

A B C 1 SD1 2 SD2 0.16 0.12 3 Cr -1 4 5 W1 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 W2 0% 5% 10% 15% 20% 25% 30% 35% 40% 41% 42% 43% 44% 45% 46% 45% 46% 47% 48% 49% 50% 51% 50% 55% 60% 65% 70% 75% 80% 85% 100% 95% 90% 85% 80% 75% 70% 65% 60% 59% 58% 57% 56% 55% 54% 55% 54% 53% 52% 51% 50% 49% 50% 45% 40% 35% 30% 25% 20% 0.014400000000 0.011236000000 0.008464000000 0.006084000000 0.004096000000 0.002500000000 0.001296000000 0.000484000000 0.000064000000 0.000027040000 0.000005760000 0.000000160000 0.000010240000 0.000036000000 0.000077440000 0.000036000000 0.000077440000 0.000134560000 0.000207360000 0.000295840000 0.000400000000 0.000519840000 0.000400000000 0.001156000000 0.002304000000 0.003844000000 0.005776000000 0.008100000000 0.010816000000 0.013924000000 0.017424000000 0.021316000000 0.025600000000 28 29 30 31 32 33 34 35 15% 36 90% 95% 100% 10% 5% 37 38 0% 39 40

The lowest variance is at 43% weight of stock 1 and 57% of stock 2

c. In current allocation the return is 9% and standard deviation is 2%

In minimum variance the return is (0.43 * 0.1) + ( 0.57 * 0.08)

= 0.043 + 0.0456

= 0.0886

= 8.86%

and Standard Deviation is 0.04%

Sharpe for current allocation = (Portfolio Return - Risk Free Rate)/standard deviation

= 0.09-0.03/0.02

= 0.06/0.02

= 3

Sharpe for minimum variance = 0.0886 - 0.03/0.0004

= 0.0586/0.0004

= 146.5

The minimum variance portfolio will give better returns as the sharpe ratio is 146.5 which is exceptional