Workings must be shown

Question

1. John Hopkin’s portfolio had the returns on shown below during the past several years. 

Year	Return
2018	12.6%
2019	16.9%
2020	4.4%
2021	-5.8%
2022	7.2%

Using the information provided, find the arithmetic mean and the geometric mean of the John’s investment portfolio.

b) Consider a portfolio of two stocks, A and B. The expected return on Stock A next year is 22% with a standard deviation of 14%. The expected return on Stock B next year is 8% with a standard deviation of 6%. The correlation between the two stocks is 0.3.  If 70% of the portfolio is in Stock A and 30% in Stock B, what is the standard deviation of the portfolio?     

c) The Generally Accepted Company (GAC) issued a 9% bond that is to mature in 15 years. The bond had a $1,000 par value, and interest is due to be paid annually. If your required rate of return (yield to maturity) is 11%, what price would you be willing to pay for the bond?

d) Given the following information, determine the market value of a company bonds.

Par value                    $1,000

Coupon rate               13.0%

Years to maturity       8

Yield to maturity       10.0%

Interest paid               semiannually

e) Despite the pandemic, One-Nation Company paid a handsome dividend of $3.75 per share at the end of its last financial year. It is expected that the firm will continue to pay dividends for the foreseeable future at a growth rate of 3%. The required return on stocks with the similar risk is 7.5%. Given this information, calculate the current price of the stock.

                                              

Answer 1

a) To find the arithmetic mean, we need to sum up the returns and divide by the number of years:

Arithmetic mean = (12.6% + 16.9% + 4.4% - 5.8% + 7.2%) / 5 = 7.06%

To find the geometric mean, we need to multiply the returns together and take the fifth root (since there are five years):

Geometric mean = (1 + 0.126) × (1 + 0.169) × (1 + 0.044) × (1 - 0.058) × (1 + 0.072)^(1/5) - 1 = 5.08%

b) The formula for the standard deviation of a portfolio with two stocks is:

σ_p = √[w_A² × σ_A² + w_B² × σ_B² + 2w_Aw_Bσ_Aσ_Bρ_AB]

where w_A and w_B are the portfolio weights, σ_A and σ_B are the standard deviations of the individual stocks, ρ_AB is the correlation between the stocks.

Using the values given:

w_A = 0.7, w_B = 0.3 σ_A = 0.14, σ_B = 0.06 ρ_AB = 0.3

σ_p = √[(0.7)² × (0.14)² + (0.3)² × (0.06)² + 2 × (0.7) × (0.3) × (0.14) × (0.06) × (0.3)] = 0.1035 or 10.35%

c) The formula for the price of a bond is:

P = C / (1 + r)^1 + C / (1 + r)^2 + ... + C + F / (1 + r)^n

where P is the price of the bond, C is the annual coupon payment, r is the yield to maturity, F is the face value of the bond, and n is the number of years to maturity.

Using the values given:

C = 0.09 × 1000 = $90 r = 0.11 F = 1000 n = 15

P = 90 / (1 + 0.11)^1 + 90 / (1 + 0.11)^2 + ... + 90 / (1 + 0.11)^15 + 1000 / (1 + 0.11)^15 = $651.25

d) The market value of a bond can be found using the formula:

P = (C / r) × [1 - 1 / (1 + r)^n] + F / (1 + r)^n

where P is the market value of the bond, C is the semiannual coupon payment, r is the semiannual yield to maturity, F is the face value of the bond, and n is the number of semiannual periods to maturity.

Using the values given:

C = 0.13 × 1000 / 2 = $65 r = 0.1 / 2 = 0.05 F = 1000 n = 8 × 2 = 16

P = (65 / 0.05) × [1 - 1 / (1 + 0.05)^16] + 1000 / (1 + 0.05)^16 = $1,167.79