Question

WeVest Financial Advisors suggests an investment in two stocks (40% in Stock A and 60% in Stock B). They claim the investment will reduce risk through diversification, but they need proof. This is the historical returns for the two stocks.

Year	Returns (%)
	Stock A				Stock B
2012		15.47	%			13.36	%
2013		16.50				15.20
2014		12.09				7.31
2015		10.45				10.01
2016		10.80				5.71

a. Using a 40/60 split, what is the weighted average standard deviation of the two stocks? (Enter your answer as a percent rounded to two decimal places.)

b. Recalculate the standard deviation of a portfolio of the two stocks. (Enter your answer as a percent rounded to two decimal places.)

c. What is the reduction in standard deviation that results from the creation of a portfolio of the two stocks? (Enter your answer as a percent rounded to two decimal places.)

Answer 1

First, with the given historical returns, we need to calculate the standard deviation of returns of individual stocks. We also need to calculate the correlation of returns of two stocks (which we would need for our future purpose.

Standard deviation of a dataset is calculated as:

n = number of years (or frequency of dataset)

Average of Return for Stock A, Mean (X) = (15.47+16.50+12.09+10.45+10.80)%/5 = 13.06%

Average of Return for Stock B, Mean (Y) = (13.36+15.20+7.31+10.01+5.71)%/5 = 10.32%

	Stock A		Stock B
Year	Xi	[Xi - Mean (X)]^2	Yi	[Yi - Mean (Y)]^2
2012	15.47%	0.000579846	13.36%	0.000925376
2013	16.50%	0.001181984	15.20%	0.002383392
2014	12.09%	0.000094478	7.31%	0.000904806
2015	10.45%	0.000682254	10.01%	0.000009486
2016	10.80%	0.000511664	5.71%	0.002123366

Standard deviation of Return for Stock A = ((0.000579846 + 0.001181984 + 0.000094478 + 0.000682254 + 0.000511664)/(5-1))^(1/2) = 2.8%

Standard deviation of Return for Stock B = ((0.000925376 + 0.002383392 + 0.000904806 + 0.000009486 + 0.002123366)/(5-1))^(1/2) = 4.0%.

Calculating correlation coeffient requires using the formula:

	Stock A	P		Stock B	Q
Year	Xi	Xi - Mean (X)	[Xi - Mean (X)]^2	Yi	Yi - Mean (Y)	[Yi - Mean (Y)]^2	P * Q
2012	15.47%	2.41%	0.000579846	13.36%	3.04%	0.000925376	0.073%
2013	16.50%	3.44%	0.001181984	15.20%	4.88%	0.002383392	0.168%
2014	12.09%	-0.97%	0.000094478	7.31%	-3.01%	0.000904806	0.029%
2015	10.45%	-2.61%	0.000682254	10.01%	-0.31%	0.000009486	0.008%
2016	10.80%	-2.26%	0.000511664	5.71%	-4.61%	0.002123366	0.104%

Substituting the sum of 4th, 7th and 8th column at respective places like in formula,

Correlation coefficient r = (0.00383/(0.00305 * 0.00635)) = 0.87

A)

Weighted average of std deviation with 40% in A, 60% in B

Wtd avg = (40% * 2.8%) + (60% * 4%) = 3.495%

B)

Portfolio standard deviation is calculated by the formula given below:

Plugging values in formula,

Std dev of portfolio =

= 3.42%

C)

Decline in std deviation in portfolio over individual stocks = 3.495% - 3.419% = 0.075%

=> Portfolio standard deviation is lower than standard deviation of weighted standard deviation of stocks