Question

The 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

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 1

Variance = Sum of Square deviations/ number of values = (Σ(Xi-Xm)^2) / n

Covariance between X & Y = Σ((Xi-Xm) x (Yi-Ym)) / n

Standard Deviation = Variance ^ (1/2)

Portfolio Mean = Wa x Ra + Wb x Rb

where Wa & Wb are weights of the assets A & B in the portfolio

& Ra & Rb are rate of returns on A & B

The Standard deviation of two asset portfolio is given by

σp = (wa^2 x σa^2 + wb^2 x σb^2 + 2 x wa x wb x Cova,b)^(1/2)

where,

wa & wb are weights of the assets A & B in the portfolio

σa & σb are standard deviation of A & B respectively

Covab is the covariance between A & B

Coefficient of Variation = Standard Deviation / Mean x 100%

The table below shows the calculation, mean variance & Covariance for L&M

Year	Stock L returns	Stock M returns	Deviation from the mean for L (Li-Lm)	Deviation from the mean for M (Mi-Mm)	Square of deviation for L (Li-Lm)^2	Square of deviation for M (Mi-Mm)^2	(L-Lm) x (M- Mm)
1	14.79%	20.57%	-1.95%	5.09%	0.00038	0.00259	(0.0010)
2	14.30%	18.19%	-2.44%	2.71%	0.00059	0.00073	(0.0007)
3	16.47%	16.20%	-0.26%	0.72%	0.00001	0.00005	(0.0000)
4	17.86%	14.43%	1.13%	-1.05%	0.00013	0.00011	(0.0001)
5	17.60%	12.53%	0.86%	-2.95%	0.00007	0.00087	(0.0003)
6	19.39%	10.98%	2.66%	-4.50%	0.00070	0.00203	(0.0012)
Total	100.41%	92.90%	0.00%	0.00%	0.001885	0.006383	(0.003237)
Mean	16.74%	15.48%
Variance/ Covariance				0.00031	0.00106	(0.00054)
Standard Deviation				1.77%	3.26%

Mean for L = 16.74%, SD for L= 1.77%, Mean for M = 15.48%, SD for M = 3.26%, CovLM = -0.00054

Mean of Portfolio with Weight of L = 40% & Weight of M = 60%

= 40% x 16.74% + 60% x 15.48%

= 15.98%

The Standard deviation of two asset portfolio is given by

σp = (wa^2 x σa^2 + wb^2 x σb^2 + 2 x wa x wb x Cova,b)^(1/2)

= (40%^2 x 1.77%^2 + 60%^2 x 3.26%^2 + 2 x 40% x 60% X (-0.00054)) ^ (0.5)

= 0.00005 + 0.00038 - 0.00026

= 0.00017

= 1.32%

Coefficient of Variation of this portfolio of L&M

= 1.32% / 15.98%

= 8.26%

For Portfolio of F&G

Year	Stock F returns	Stock G returns	Deviation from the mean for F (Fi-Fm)	Deviation from the mean for G (Gi-Gm)	Square of deviation for F (Fi-Fm)^2	Square of deviation for G (Gi-Gm)^2	(F-Fm) x (G- Gm)
1	16.23%	17.04%	-1.54%	1.32%	0.00024	0.00017	(0.0002)
2	17.48%	16.33%	-0.29%	0.61%	0.00001	0.00004	(0.0000)
3	18.11%	15.41%	0.34%	-0.31%	0.00001	0.00001	(0.0000)
4	19.25%	14.10%	1.48%	-1.62%	0.00022	0.00026	(0.0002)
Total	71.07%	62.88%	0.00%	0.00%	0.000476	0.000484	(0.000471)
Mean	17.77%	15.72%
Variance/ Covariance				0.00012	0.00012	(0.00012)
Standard Deviation				1.09%	1.10%

Mean for F = 17.77%, SD for F= 1.09%, Mean for G = 15.72%, SD for G = 1.10%, CovFG = -0.00012

Mean of Portfolio with Weight of F = 50% & Weight of G = 50%

= 50% x 17.77% + 50% x 15.72%

= 16.74%

The Standard deviation of two asset portfolio is given by

σp = (wa^2 x σa^2 + wb^2 x σb^2 + 2 x wa x wb x Cova,b)^(1/2)

= (50%^2 x 1.09%^2 + 50%^2 x 1.10%^2 + 2 x 50% x 50% X (-0.00012)) ^ (0.5)

= 0.00003 + 0.00003 - 0.000059

= 0.00001

= 0.103%

Coefficient of Variation of this portfolio of F&G

= 0.103% / 16.74%

= 0.0062

= 0.62%

For Portfolio of F&H with 50% weights each

Year	Stock F returns	Stock H returns	Deviation from the mean for F (Fi-Fm)	Deviation from the mean for H (Hi-Hm)	Square of deviation for F (Fi-Fm)^2	Square of deviation for H (Hi-Hm)^2	(F-Fm) x (H- Hm)
1	16.23%	14.10%	-1.54%	-1.61%	0.00024	0.00026	0.0002
2	17.48%	15.32%	-0.29%	-0.39%	0.00001	0.00002	0.0000
3	18.11%	16.20%	0.34%	0.49%	0.00001	0.00002	0.0000
4	19.25%	17.22%	1.48%	1.51%	0.00022	0.00023	0.0002
Total	71.07%	62.84%	0.00%	0.00%	0.000476	0.000526	0.000499
Mean	17.77%	15.71%
Variance/ Covariance				0.00012	0.00013	0.00012
Standard Deviation				1.09%	1.15%

Mean for F = 17.77%, SD for F= 1.09%, Mean for H = 15.71%, SD for H = 1.15%, CovFH = 0.00012

Mean of Portfolio with Weight of F = 50% & Weight of G = 50%

= 50% x 17.77% + 50% x 15.71%

= 16.74%

The Standard deviation of two asset portfolio is given by

σp = (wa^2 x σa^2 + wb^2 x σb^2 + 2 x wa x wb x Cova,b)^(1/2)

= (50%^2 x 1.09%^2 + 50%^2 x 1.10%^2 + 2 x 50% x 50% X 0.00012) ^ (0.5)

= 0.00003 + 0.000032 + 0.000062

= 0.000125

= 1.18%

Coefficient of Variation of this portfolio of F&H

= 1.18% / 16.74%

= 0.0668

= 6.68%

F&G is a better portfolio with same returns as F&H but lesser SD

% Difference in Coefficient of Variation of L&M &  F&G

= (8.26% - 0.62%)

= 7.64%