Question

Miss Maple is considering 2 securities, A and B, and the relevant information is given below:

State of Economy Prob. Return on A Return on B

Bear 0.4 0.03 0.065

Bull 0.6 0.15 0.065

(a). Calculate the expected returns and standard deviations of the two securities.

b) Suppose Miss Maple invested $2,500 in security A and $3,500 in security B. Calculate the expected return and standard deviation of her portfolio.

(c). Suppose Miss Maple borrowed from her friend 40 shares of security B which is currently sold at $50, and sold all of the shares. She promised her friend to pay back in a year with the same number of shares of security B. Then she bought security A with the proceeds obtained in the sales of B shares and the cash of $6,000 she owned. Calculate the expected return and standard deviation of the portfolio.

Answer 1

a. Calculate the expected returns and standard deviations of the two securities.

Solution: a

Given:

State of Economy	Probability	Return on Security A (%)	Return on Security B (%)
Bear	0.6	3.0%	6.5%
Bull	0.4	15.0%	6.5 %

To Calculate:

Expected Return and Standard Deviation of both securities A and B:

1. Expected Return of Security A:

Formula:

Expected Return on Security A = (Probability in Bearish Market× Return on A in Bearish Market) + (Probability in Bullish Market × Return on A in Bullish Market)

On putting the values in the formula, we get,

E(RA) = 0.6 (0.03) + 0.4 (0.15) = 0.078 = 7.80%

Expected Return on Security A, E(RA) = 7.80 %

2. Expected Return of Security B:

Formula:

Expected Return on Security B = (Probability in Bearish Market × Return on B in Bearish Market) + (Probability in Bullish Market × Return on B in Bullish Market)

On putting the values in the formula, we get,

E(RB) = 0.6 (0.065) +0.4 (0.065) = 6.5%

Expected Return on Security B, E(RB) = 6.5 %

3. Standard Deviation of Security A:

Formula:

Standard Deviation of A = (Probability in Bearish Market × (Return on A in Bearish Market - Expected Return on A)) + (Probability in Bullish Market × (Return on A in Bullish Market - Expected Return on A))

On putting the values in the formula, we get,

σ ^2 A = (0.6 × (0.03 - 0.078) ^ 2) + (0.4 × (0.15 - 0.078) ^ 2) = 0.003456

σ A = (0.003456) ^ 1/2 = 0.05878 = 5.878 %

Standard Deviation of A, σ A = 5.878 %

4. Standard Deviation of Security B:

Formula:

Standard Deviation of B = (Probability in Bearish Market × (Return on B in Bearish Market - Expected Return on B)) + (Probability in Bullish Market × (Return on B in Bullish Market - Expected Return on B))

On putting the values in the formula, we get,

σ ^2 B = (0.6 × (0.065 - 0.065) ^ 2) + (0.4 × (0.065 - 0.065) ^ 2)

= (0.6 × 0) + (0.4 × 0) = 0

σ ^2 B = 0

So, σ B = 0

Standard Deviation of B, σ B = 0

Ans: a) Expected Return on Security A, E(RA) = 7.80 % & Expected Return on Security B, E(RB) = 6.5 %

Standard Deviation of A, σ A = 5.878 % & Standard Deviation of B, σ B = 0

b.) Suppose Miss Maple invested $2,500 in Security A and $3,500 in security B. Calculate the expected return and standard deviation of her portfolio.

Solution:

Given:

Investment in Security A = $ 2,500

Investment in Security B = $ 3,500

To Calculate:

Expected Return and Standard Deviation of Portfolio:

1. Expected Return of Portfolio:

Firstly, we will find Weight of A and B,

Weight of A, WA = Investment in A / Investment in B

WA = $2,500 / $6,000 = 0.417

Weight of B, WB = 1 – Weight of A

WB = 1- 0.417 = 0.583

WA = 0.417 and WB = 0.583

Now, we calculate Expected Return on Portfolio:

Formula:

Expected Return on Portfolio, E(RP) = (Weight of A × Expected Return on Portfolio A) + (Weight of B × Expected Return on Portfolio B

E(RP) = 0.417(0.078) + 0.583(0.065) = 0.0704 = 7.04%

Expected Return on Portfolio E (RP) = 7.04 %

2. Standard Deviation of Portfolio:

Formula:

Standard Deviation of Portfolio = (Weight of A × Standard Deviation of A) + (Weight of B × Standard Deviation of B)

σ P = WA × σ A + WB × σ B

= (0.417 × 0.05878) + (0.583 × 0)

= 0.0006

σ P = (0.0006) 1/2 = 0.0245 = 2.45%

Standard Deviation of Portfolio σ P = 2.45 %

Ans: b) Expected Return on Portfolio E (RP) = 7.04 % and

Standard Deviation of Portfolio σ P = 2.45 %

(c). Suppose Miss Maple borrowed from her friend 40 shares of security B which is currently sold at $50 and sold all of the shares. She promised her friend to pay back in a year with the same number of shares of security B. Then she bought security A with the proceeds obtained in the sales of B shares and the cash of $6,000 she owned. Calculate the expected return and standard deviation of the portfolio.

Solution:

Given:

Number of shares borrowed of B = 40

Price per share = $ 50

Amount borrowed = Number of shares × Share Price = 40 × $ 50= $ 2000

Cash Owned = $ 6000

Investment in A = Sale of 40 shares of B + Cash Owned

Investment in A = $ 2,000 + $ 6,000 = $ 8,000

Now, we will find Weight of A and B,

Weight of A, WA = Investment in A / Investment in B

WA = $ 8,000 / $ 6,000 = 4/3

Weight of B, WB = 1 – Weight of A

WB = 1- 4/3 = -1/3

WA = 4/3 and WB = 1/3

Now, we calculate

1. Expected Return on Portfolio:

Formula:

Expected Return on Portfolio, E(RP) = (Weight of A × Expected Return on Portfolio A) + (Weight of B × Expected Return on Portfolio B

E(RP) = (4/3) (0.078) + (-1/3) (0.065) = 0.0823 = 8.23%

Expected Return on Portfolio E (RP) = 8.23 %

2. Standard Deviation of Portfolio:

Formula:

Standard Deviation of Portfolio = (Weight of A × Standard Deviation of A) + (Weight of B × Standard Deviation of B)

σ P = WA × σ A + WB × σ B

= (4/3 × 0.05878) + (-1/3 × 0)

= 0.07837 ≈ 7.837

σ P = 7.837 %

Standard Deviation of Portfolio σ P = 7.837 %

Ans: c) Expected Return on Portfolio E (RP) = 8.23 % and

Standard Deviation of Portfolio σ P = 7.837 %