Question

9. Compute the average annual return and standard deviation of an evenly weighted portfolio of stocks and 10-year government bonds over the past five years ended December 31, 2018?

Answer 1

We are assuming that there are 4 securities in a portfolio i.e. stock 1, stock 2, bond 1 and bond 2.

Below mentioned are the returns of portfolio of 5 years:

Securities Name	Weight (w)	2014 (r)	2015 (r)	2016 (r)	2017 (r)	2018 (r)
Stock 1 (S1)	0.25	20%	25%	10%	15%	20%
Stock 2 (S2)	0.25	15%	20%	25%	15%	20%
Bond 1 (B1)	0.25	10%	15%	10%	5%	10%
Bond 2 (B2)	0.25	5%	10%	10%	15%	10%

Yearly portfolio return is calculated as SUM(S1w * S1r + S2w * S2r + B1w * B1r + B2w * B2r)

Securities Name	Weight (w)	2014 (r)	Returns * Weight
Stock 1 (S1)	0.25	20%	5.00%
Stock 2 (S2)	0.25	15%	3.75%
Bond 1 (B1)	0.25	10%	2.50%
Bond 2 (B2)	0.25	5%	1.25%
Portfolio Return			12.50%

Years	2014	2015	2016	2017	2018
Portfolio Returns	12.50%	17.50%	13.75%	12.50%	15.00%

Average returns of a portfolio is calculated as sum of last 5 year returns divided by no. of years.

Portfolio Avg. return of last 5 years= (12.5%+17.5%+13.75%+12.5%+15%)/5 = 14.25%

Below mentioned is calculation of standard deviation of portfolio:

Portfolio Std. Dev formula = sq. root of (S₁w^{^2} * S₁sd^{^2} + S₂w^{^2} * S₂sd^{^2} + B₁w^{^2} * B₁sd^{^2} + B₂w^{^2} * B₂sd^{^2} + 2 * S₁w * S₂w * B₁w * B₂w * S₁sd * S₂sd * B₁sd * B₂sd * Correlation(S_1,S_2,B_1,B₂))

Securities Name	Weight	2014 Std. Dev	2015 Std. Dev	2016 Std. Dev	2017 Std. Dev	2018 Std. Dev
Stock 1 (S1)	0.25	20%	25%	10%	15%	20%
Stock 2 (S2)	0.25	15%	20%	25%	15%	20%
Bond 1 (B1)	0.25	0%	0%	0%	0%	0%
Bond 2 (B2)	0.25	0%	0%	0%	0%	0%

Assuming Correlation(S_1,S_2,B_1,B₂) = 0.5

Securities Name	Weight	2014 Std. Dev	W^2 * Std^2	W * Std
Stock 1 (S1)	0.25	20%	0.25%	5.00%
Stock 2 (S2)	0.25	15%	0.14%	3.75%
Bond 1 (B1)	0.25	0%	0%	0%
Bond 2 (B2)	0.25	0%	0%	0%

2014 Portfolio Std. Dev= Sq. root(0.25% + 0.14% + 0% + 0% + 2 * 0.5 * 5% * 3.75% * 0% * 0%)

= Sq. root(0.39%)

=6.25%

Years	2014	2015	2016	2017	2018
Portfolio Std Dev	6.25%	8.00%	6.73%	5.30%	7.07%

5 year Avg Portfolio Std Dev= sum of last 5 year std dev divided by no. of years, which is 6.67%.