Question

he following information indicates percentage returns for stocks L and M over a 6-year period:

Year	Stock L Returns	Stock M Returns
1	14.79%	20.57%
2	14.3%	18.19%
3	16.47%	16.2%
4	17.86%	14.43%
5	17.6%	12.53%
6	19.39%	10.98%

In combining [L−M] in a single portfolio, stock M would receive 60% of capital funds.

Furthermore, the information below reflects percentage returns for assets F, G, and H over a 4-year period, with asset F being the base instrument:

Year	Asset F Returns	Asset G Returns	Asset H Returns
1	16.23%	17.04%	14.1%
2	17.48%	16.33%	15.32%
3	18.11%	15.41%	16.2%
4	19.25%	14.1%	17.22%

Using these assets, you have a choice of either combining [F−G] or [F−H] in a single portfolio, on an equally-weighted basis.

Required: Calculate the absolute percentage difference in the coefficient of variation (CV) between the stock portfolio [L−M] and the portfolio which outlines the optimal combination of assets.

Answer% Do not round intermediate calculations. Input your answer as a percent rounded to 2 decimal places (for example: 28.31%).

Answer 1

In order to calculate the Coefficient of Variation we need to calculate the mean and standard deviation of the portfolios.

Coefficient of Variation = Standard Deviation / Mean

Where Mean = Sum of the returns / No of returns

Standard Deviation = rootover (x - mean of x) ^2 / n considering population

Portfolio return = sum of (Weight of stocks in portfolio * Expected return of portfolio )

Standard Deviation of the Portfolio = ( w1^2*mean of portfolio1 ^2 + w2^2mean of portfolio2 ^2 + 2 * + 2* w1*w2*Covariance between the stocks)

In order to calculate the mean and standard deviation of the portfoilio [ L- M]

We need to calculate the mean and standard deviation of the individual stock L & M

Mean for Stock L = 99.43 / 6 = 16.57

Mean for Stock M = 93.22/ 6 = 15.54

Standard deviation for stock L = (17.91 / 6 )^0.5 = 1.73

Standard deviation for stock M= (76.27 / 6)^0.5 = 3.57

Covariance between Stock L & Stock M = -35.51/ 6 = 5.92

Mean for Portfolio [ L-M] = 0.4 * 16.57 + 0.6 * 15.54 = 15.95%

Standard Deviation of Portfolio[ L-M] = ( 0.4^2*1.73^2 + 0.6^2*3.57^2 + 2*0.4*0.6*-5.92) ^0.5= 3.59%

Coefficient of Variation for Portfolio[ L-M] = 3.59/15.95 = 0.14

Mean for Stock F = 70.65 / 4 = 17.66

Standard deviation for stock F = ( 11.16 / 4 )^0.5 = 1.67

Mean for Stock G = 63.12/4 = 15.78

Standard deviation for stock G = ( 5.34 / 4) ^0.5 = 1.16

Mean for Stock H = 63.07/4 = 15.77

Standard deviation for stock H = (5.42/4)^0.5 = 1.16

Covariance between Stock F & Stock G = -4.60/4 = 1.15

Covariance between Stock F & Stock H = 6.75/4 = 1.69

Mean for Portfolio [ F-G] = 0.5 * 17.66 + 0. 5* 15.78 = 16.72%

Standard Deviation of Portfolio[F-G] = ( 0.5^2*1.67^2 + 0.5^2*1.16^2 + 2*0.5*0.5*-1.15) ^0.5= 0.46%

Coefficient of Variation for Portfolio[F-G] = 0.46/16.72 = 0.03

Mean for Portfolio [ F-H] = 0.5 * 17.66 + 0.5 * 15.77 = 16.53%

Standard Deviation of Portfolio[ F-H] = ( 0.5^2*1.67^2 + 0.5^2*1.16^2 + 2*0.5*0.5*1.69) ^0.5= 1.74%

Coefficient of Variation for Portfolio[ F-H] = 1.74 / 16.53 = 0.11

portfolio which outlines the optimal combination of assets is the portfolio having lower Coefficient of Variation

Portfolio [ F – G ] has lower Coefficient of Variation hence it is optimal Portfolio

SO we need to calculate the difference between Coefficient of Variation of portfolio [ L-M] & Portfolio [ F-G]

Difference in Coefficient of Variation = 0.14 – 0.03 = 0.11

% Difference in Coefficient of Variation = 0.11 / 0.14 * 100 = 78.57%

	Years	Return on Stock L ( X)	A=(X - MEAN OF X)	B =(X - MEAN OF X) ^2	Return on Stock M ( Y)	C=(Y - MEAN OF Y)	D = (Y - MEAN OF Y) ^2	A*C
	1	14.85	-1.721666667	2.964136111	20.79	5.253333333	27.59751111	-9.044488889
	2	14.34	-2.231666667	4.980336111	18.7	3.163333333	10.00667778	-7.059505556
	3	16.02	-0.551666667	0.304336111	16.62	1.083333333	1.173611111	-0.597638889
	4	17.1	0.528333333	0.279136111	14.21	-1.326666667	1.760044444	-0.700922222
	5	17.7	1.128333333	1.273136111	12.53	-3.006666667	9.040044444	-3.392522222
	6	19.42	2.848333333	8.113002778	10.37	-5.166666667	26.69444444	-14.71638889
	Total	99.43		17.91408	93.22		76.27233	-35.51146667

	Years	Return on Stock F ( X)	A=(X - MEAN OF X)	B =(X - MEAN OF X) ^2	Return on Stock G ( Y)	C=(Y - MEAN OF Y)	D = (Y - MEAN OF Y) ^2	A*C
	1	16.01	-0.561666667	0.315469444	17.42	1.883333333	3.546944444	-1.057805556
	2	17.06	0.488333333	0.238469444	16.1	0.563333333	0.317344444	0.275094444
	3	18.17	1.598333333	2.554669444	15.24	-0.296666667	0.088011111	-0.474172222
	4	19.41	2.838333333	8.056136111	14.36	-1.176666667	1.384544444	-3.339772222
	Total	70.65		11.16474	63.12		5.33684	-4.596655556

	Years	Return on Stock F ( X)	A=(X - MEAN OF X)	B =(X - MEAN OF X) ^2	Return on Stock H ( Y)	C=(Y - MEAN OF Y)	D = (Y - MEAN OF Y) ^2	A*C
	1	16.01	-0.561666667	0.315469444	14.39	-1.146666667	1.314844444	0.644044444
	2	17.06	0.488333333	0.238469444	15.04	-0.496666667	0.246677778	-0.242538889
	3	18.17	1.598333333	2.554669444	16.29	0.753333333	0.567511111	1.204077778
	4	19.41	2.838333333	8.056136111	17.35	1.813333333	3.288177778	5.146844444
	Total	70.65		11.16474	63.07		5.41721	6.752427778