Question

The realized returns for stock A and stock B from 2004-2009 are provided in the table below Year 2004 2005 2006 2007 2008 2009 Stock A -9% 21% 6% -4% 3% 10% Stock B 23% 9% 32% -1% -6% 27% (a) Calculate the expected returns (as percents) over the next year for the stocks assuming the average annual realized returns and past volatility from 2004-2009 are unbiased estimators of expected returns and future volatility. stock A 4.5 stock B 14 Calculate the volatilities (as percents) for returns over the next year for the stocks. (Round your answers to two decimal places.) stock A 10.60 stock B 15.65 (b) Calculate the expected return and volatility (as percents) of an equally-weighted portfolio. The correlation between the returns of the two stocks is 6.27%. (Round your answer for volatility to two decimal places.) expected return volatility (c) Explain why the portfolio has a lower volatility than the average volatility of the two stocks. on the The correlation of 6.27% is low, so most of factors that affect the returns of one stock have ---Select-- returns of the other asset. Consequently, the risk is ---Select-- when they are combined in a portfolio.

Answer 1

1]

a]

expected return = average annual realized return

average annual realized return is calculated using AVERAGE function in Excel

volatility is measured by standard deviation

standard deviation is calculated using STDEV.S function in Excel

1 Year 23% Stock A - Stock B - Returns Returns 2004 -9% 2005 21% 9% 2006 6% 32% 2007 -4% 2008 3% -6% 7 2009 10% 27% 8 Average 4.50% 14.00% 9 Std. Dev. 10.60% 15.65% -1%

-0.09 1 Year Stock A - Returns 2 2004 3 2005 0.21 4 2006 0.06 5 2007 -0.04 6 2008 0.03 7 2009 0.1 8 Average =AVERAGE(B2:B7) 9 Std. Dev. STDEV.S(B2:B7) Stock B - Returns 0.23 0.09 0.32 -0.01 -0.06 0.27 =AVERAGE(C2:07) -STDEV.S(C2:07)

b]

In an equally weighted portfolio, the weight of each stock is 0.50.

Expected return of two-asset portfolio R_p = w₁R₁ + w₂R_2,

where R_p = expected return

w₁ = weight of Asset 1

R₁ = expected return of Asset 1

w₂ = weight of Asset 2

R₂ = expected return of Asset 2

Expected return = (0.50 * 4.50%) + (0.50 * 14.00%)

Expected return = 9.25%

volatility is measured by standard deviation

standard deviation for a two-asset portfolio σ_p = (w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov_1,2)^1/2

where σ_p = standard deviation of the portfolio

w₁ = weight of Asset 1

w₂ = weight of Asset 2

σ₁ = standard deviation of Asset 1

σ₂ = standard deviation of Asset 2

Cov_1,2 = covariance of returns between Asset 1 and Asset 2

Cov_1,2 = ρ_1,2 * σ₁ * σ_2, where ρ_1,2 = correlation of returns between Asset 1 and Asset 2

σ_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov_1,2

σ_p = ((0.50² * 0.1060²) + (0.50² * 0.1565²) + (2 * 0.50 * 0.50 * 0.0627 * 0.1060 * 0.1565))^1/2

σ_p = 9.72%

volatility = 9.72%

c]

The correlation of 6.27% is low, so most of the factors that affect the returns of one stock have no impact on the returns of the other asset. Consequently, the risk is lower when they are combined in a portfolio.