Question

Only securities A.B.C with the following characteristics exist in the market.

1.Securities A: E (R) = 10% / Standard deviation = 12.0%
2.Securities B: E (R) = 6% / Standard deviation = 6.0%
3.Securities C: E (R) = 4% / Standard deviation = 0.0%

Suppose that the correlation coefficient between the returns of Securities A and B is 0.5.

(a) When constructing a portfolio with securities A and B, change the composition ratio to (1.0, 0), (0.7, 0.3), (0.5, 0.5), (0.3, 0.7), (0.1, 0) and expect How do returns and standard deviations change?

(b) Use only Securities A and B. We want to achieve an expected return of 8%. What is the minimum risk faced at this time?

(c) When constructing a portfolio with securities A, B, and C, what is the composition ratio of a portfolio with an expected return of 8% and a minimum risk? At this time, what is the composition ratio between risky assets A and B?

(d) What is the compositional ratio of the portfolio with the expected return of 6% and the minimum risk when constructing the portfolio with securities A, B, and C? At this time, does the composition ratio between risky assets A and B differ from that in (c)?

Answer 1

(a)

We calculate the expected return and the standard deviation of the portfolios using excel

Expected return on the portfolio = w(x)*E(x) + w(y)*E(y)

Standard deviation of portfolio =

where x and y are the securities

Weight of security A	Weight of secuity B	E(p)	Standard deviation of Portfolio
1	0	10.00%	12.00%
0.7	0.3	8.80%	8.59%
0.5	0.5	8.00%	6.71%
0.3	0.7	7.20%	5.53%
0	1	6.00%	6.00%

(b)

We use an excel solver for this.

First, we enter the formulas and any random weights (say 0.5,0.5 in this case)

We put the following constraints such that weight should be greater than or equal to zero and expected return should be 8%

Solving we get, the weight of portfolio A= 0.5

Weight of portfolio B= 0.5

The minimum risk faced = 7.91%

(c)

Here, again we solve the question using an excel solver.

We input random numbers in weights

C is the risk-free asset since the standard deviation is 0

We input the following conditions in excel solver, such that the standard deviation of the portfolio is minimized and the expected return is equal to 8%

Solving, we get

	Weights	E[r]	Std. dev	Correlation
A	0.571443	0.1	0.12	0.5
B	0.285721	0.06	0.06
C	0.142836	0.04	0
Portfolio	1	8%	0.07856

Hence the weights of the securities A.B.C are in the table above

The composition between A and B

Here we consider a risky portfolio consisting of only A and B. The proportion of weights, however, remains same

Composition of A in the risky-portfolio = 0.571443 / (0.571443+0.285721) = 0.67

Composition of A in the risky-portfolio = 1- 0.67 = 0.33

(d)

Similar to part c).This time we set the expected return constraint to 6%

Solving, we get

	Weights	E[r]	Std. dev	Correlation
A	0.285714	0.1	0.12	0.5
B	0.142857	0.06	0.06
Risk-free	0.571429	0.04	0
Portfolio	1	6%	0.039279

Even here,  

Composition of A in the risky-portfolio = 0.285714/ (0.285714+0.142857) = 0.67

Composition of B in the risky-portfolio = 1- 0.67 = 0.33

Hence we observe that the composition of A and B in the risky asset portfolio has not changed from part c)

This is because these weights are the optimum risky portfolio and any change in expected return will change the weights of A, B, and C, but the weight of A and B in proportion to each other haven't changed since they lie on the same risky-asset portfolio.