Question

Please could you show full steps and no excel please

Miss M had R100,000 to invest in the financial market. She chose to invest R10,000 in Asset A, which returned 12.0%; R40,000 in Asset B which returned 15.0%; R20,000 in Asset C which returned –5.0%, and the remaining in Asset D which returned 5%. The correlation between the assets are ρAB = 0.75, ρAC = 0.35, ρAD = 1, ρBC = –0.5, ρBD = 0.25, and ρCD = -1. The standard deviations are 0.75, 0.45, 0.30, and 0.50 for A, B, C, and D respectively. To find the variance of an N -stock portfolio, you must add the entries in a NxN matrix. The diagonal cells contain variance terms (X2σ2 ) and the off-diagonal cells contain covariance terms (XAXBσAB), where XA = proportion invested in stock A and σAB = covariance between stocks A and B.

A. What is the average real return for each if inflation rate is 3%?

B. Find the proportion of investment in each stock.

C. Calculate the expected returns of the portfolio.

D. What is the variance of the portfolio returns?

E. Find the standard deviation.

Answer 1

a. The formula for calculating real rate of return is:

where r is the real rate of return, n is nominal rate of return, i is the inflation rate.

So, r_a = (0.12 - 0.03)/(1 +0.3) = 0.8738 which is 8.738%

r_b = (0.15 - 0.03)/(1.03)
=0.1165 which is 11.65%

r_c =(-0.05 - 0.03)/1.03
=-0.078 which is -7.8%

r_d  =(0.05-0.03)/1.03
=0.019 which is 1.9%

b. The proportion of each stock is the relative share of each stock in the overall investment.

The following table states the proportion of each stock:

Stocks	Amount Invested(R)	Proportion in Total Investment (%)
A	10,000	10%
B	40,000	40%
C	20,000	20%
D	30,000(1,00,000- Rest)	30%
Total	1,00,000	100%

c. The expected return of the portfolio could be calculated with the following formula:

where w_A is the weight of stock A and so on while r_A is the return on the stock A and so on.

Over here, it is important to note that the proportion of investment has been considered as the weights.

Therefore, E(r_p) = 0.1*12% + 0.4* 15% + 0.2 * -5% + 0.3 * 5%

= 1.2% + 6% - 1% + 1.5% = 7.7%

So, the expected returns of the portfolio = 7.7%

d. The following information has been provided to us:

Particulars	A	B	C	D
Mean	12	15	-5	5
Standard Deviation	0.75	0.45	0.3	0.5

The formula for variance is:

So, var(p) = 0.75^2*0.12^2 + 0.45^2*0.15^2+ 0.3^2* -0.5^2 + 0.5^2*0.5^2 + 2(0.75*0.12*0.45*0.15*0.75) + 2(0.75*0.12*0.3*-0.5*0.35) + 2(0.75*0.12*0.5*0.5*1) + 2(0.3*-0.5*0.45*0.15*-0.5) + 2(0.5*0.5*0.45*0.15*0.25) + 2(0.3*-0.5*0.5*0.5*-1)

So var(p) = 0.240

d. Standard deviation of a portfolio is sq.root of variance of a portfolio i.e. sq.root(0.240) i.e. 0.490 approx.