Question

Assume that your cousin holds just one stock, Eastman Chemical Bonding (ECB), which he thinks has very little risk. You agree that the stock is relatively safe, but you want to demonstrate that his risk would be even lower if he were more diversified. You obtain the following returns data for Wilder's Creations and Buildings (WCB). Both companies have had less variability than most other stocks over the past 5 years. Using the standard deviation of returns, by how much would your cousin's risk have been reduced if he had held a portfolio consisting of 70% in ECB and the remainder in WCB?

	Year	ECB	WCB
	2007	40.00%	40.00%
	2008	-10.00%	15.00%
	2009	35.00%	-5.00%
	2010	-5.00%	-10.00%
	2011	15.00%	35.00%

*Show your computations and/or excel functions you used in your solution.

Answer 1

I'll be using Geometric mean to calculate the mean returns

Expected Return for ECB =( (1+ Return_i))^{1 / N}

Expected Return for ECB = ((1+40%) * (1+(-10%)) * (1+35%)* (1+(-5%))*(1+15%))^1/5

Expected Return for ECB = 13.19%

Expected Return for WCB = ( (1+ Return_i))^{1 / N}

Expected Return for WCB = ((1+40%) * (1+15%) * (1+(-5%)) * (1+(-10%)) * (1+35%))^1/5

Expected Return for WCB = 13.19%

Variance of ECB = ( (Return_i - Expected Return for ECB)²) / N

Variance of ECB = ((40% - 13.19%)² +(-10% - 13.19%)² +(35% - 13.19%)² +(-5% - 13.19%)² +(15% - 13.19%)²) / 5

Variance of ECB = 4.13%

Standard Deviation of ECB = Variance of ECB

Standard Deviation of ECB = 20.33%

Variance of WCB = ( (Return_i - Expected Return for WCB)²) / N

Variance of WCB = ((40% - 13.19%)² +(15% - 13.19%)² +(-5% - 13.19%)² +(-10% - 13.19%)² +(35% - 13.19%)²) / 5

Variance of WCB = 4.13%

Standard Deviation of WCB = Variance of WCB

Standard Deviation of WCB = 20.33%

Covariance between ECB & WCB = ( (Return_i - Expected Return for ECB) * (Return_i - Expected Return for WCB)) / N

Covariance between ECB & WCB = ((40% - 13.19%) * (40% - 13.19%) + (-10% - 13.19%) * (15% - 13.19%) + (35% - 13.19%) * (-5% - 13.19%) + (-5% - 13.19%) * (-10% - 13.19%) + (15% - 13.19%) * (35% - 13.19%)) / 5  

Covariance between ECB & WCB = 1.48%

Standard Deviation of Portfolio = (Weight OF ECB * Standard Deviation of ECB)² + (Weight OF WCB * Standard Deviation of WCB)² + 2 * Weight OF ECB * Weight OF ECB * Covariance between ECB & WCB / No of observations

Standard Deviation of Portfolio = (70% * 20.33%)² + (30% * 20.33%)² + 2 * 70% * 30% * 1.48%

Standard Deviation of Portfolio = 0.030195

Standard Deviation of Portfolio = 17.38%

The resulting portfolio has lower risk than the single stock ECB than my cousin is holding since Standard deviation of the portfolio < Standard devaition of ECB i.e. (17.38% < 20.33%)