Question

Suppose that you have built a simple portfolio based on allocating 70% of your funds in a mix of S&P 500 index shares and the remaining 30% of your funds in a mix of utility sector shares. If we denote the monthly return for S&P 500 index by X and the monthly return for utility index by Y , then the portfolio monthly return variable will be R = 0.7X + 0.3Y.

(a) Knowing that ??= 0.298%, ??= 0.675%, ??= 4.453%, ??= 4.403% and the correlation between X and Y is corr(X; Y ) = ??? = 0.495, compute the expected monthly return on your portfolio and the variability in your portfolio's monthly return as measured by its variance and standard deviation.

(b) Recalculate the variance and standard deviation of the return on your portfolio knowing that the two variables are uncorrelated.

(c) Compute the variance and standard deviation of the return on your portfolio knowing that the two variables are perfectly positively correlated.

(d) Compute the variance and standard deviation of the return on your portfolio knowing that the two variables are perfectly negatively correlated.

(e) Suppose you do not know the exact mean returns for your variables X and Y and based on a year's worth of previous data, somebody estimated for you a 95% confidence interval for the mean return of your portfolio. The reported interval is [0.377%; 0.453%]. Based on this information is it true that P(?? ∈ [0.377%; 0.453%]) = 0.95. Explain.

Answer 1

Ans: A) as given w1 = 70% & w2 = 30%
  ??= 0.298%, ??= 0.675%,  ??= 4.453%, ??= 4.403% and corr(X; Y ) = ??? = 0.495
Hence sdX = 21.10% & sdY = 20.98%
E[XY] = w1*?? + w2* ?? = 70% * 0.298% + 30% * 0.675%
= 0.41%
sd = √(w1^2 * ?? + w2^2 * ?? + 2 * w1 * w2 *corXY* sdX * sdY)
= √(70%^2 * 4.453% + 30%^2 * 4.403% + 2 * 70% * 30% * 0.495 * 21.10% * 20.98%
= 18.70%

B) When two variable are uncorrelated ??? = 0
Hence sd = √(w1^2 * ?? + w2^2 * ?? + 2 * w1 * w2 *corXY* sdX * sdY)
= √(70%^2 * 4.453% + 30%^2 * 4.403% + 2 * 70% * 30% * 0 * 21.10% * 20.98%
= 16.06%
Var = sd ^2 = 16.06% ^ 2 = 2.57%

C) When two variable are positively correlated ??? = 1
Hence sd = √(w1^2 * ?? + w2^2 * ?? + 2 * w1 * w2 *corXY* sdX * sdY)
= √(70%^2 * 4.453% + 30%^2 * 4.403% + 2 * 70% * 30% * 1 * 21.10% * 20.98%
= 21.07%
Var = sd ^2 = 21.07% ^ 2 = 4.43%

D) When two variable are negatively correlated ??? = -1
Hence sd = √(w1^2 * ?? + w2^2 * ?? + 2 * w1 * w2 *corXY* sdX * sdY)
= √(70%^2 * 4.453% + 30%^2 * 4.403% + 2 * 70% * 30% * -1 * 21.10% * 20.98%
= 8.48%
Var = sd ^2 = 8.48% ^ 2 = 0.719%

E) n = 12, sd = 18.70%, mean = 0.41%, t-score = 2.2
Confidence interval =
μ = M ± t(s_M)
where s_M = standard error = √(s²/n)
μ = M ± t(s_M)
μ = 0.0041 ± 2.2*0.05
μ = 0.0041 ± 0.118814

Hence the true confidence interval falls between -0.114714 and 0.122914