Question

4. You have been given the expected return data shown in the first table on three assets—F, G, and H—over the period 2015-2018

Year	Asset F	Asset G	Asset H
2015	9	12	15
2016	8	9	16
2017	5	21	19
2018	13	6	11

1. Find the expected return, variance, std dev and coefficient of variation for each asset.
2. Now consider a portfolio that consists of 25% of F, 50% of G and 25% of H. Find the expected return, variance, std, dev and coefficient of variation for this portfolio.

Answer 1

Calculation of Expected return, Variance and Standard deviation of each of the Asset:

Year	Asset F (X) (%)	X - x̅	(X - x̅)^2	Asset G (Y) (%)	Y - ȳ	(Y - ȳ)^2	Asset H (Z) (%)	Z - Z̄	(Z - Z̄)^2
2015	9.00	0.25	0.0625	12.00	0.00	-	15	-0.25	0.06
2016	8.00	-0.75	0.5625	9.00	-3.00	9.00	16	0.75	0.56
2017	5.00	-3.75	14.0625	21.00	9.00	81.00	19	3.75	14.06
2018	13.00	4.25	18.0625	6.00	-6.00	36.00	11	-4.25	18.06
Total	35.00		32.75	48.00		126.00	61.00		32.75

Expected return of each asset:

Asset F (x̅) = Total of Return of Asset F/ no. of years

= 35/4

= 8.75%

Asset G (ȳ) = Total of Return of Asset G/ no. of years

= 48/4

= 12%

Asset H (Z̄) = Total of Return of Asset H / no. of years

= 61/4

= 15.25%

Variance of each asset:

Asset F = Total of (X - x̅)^2/ no. of years

= 32.75/4

= 8.19%

Asset G = Total of (Y - ȳ)^2/ no. of years

= 126/4

= 31.50%

Asset H = Total of (Z - Z̄)^2/ no. of years

= 32.75/4

= 8.19%

Standard deviation of each asset: (Variance)1/2

Asset F = (8.19)1/2

= 2.86%

Asset G = (31.50)1/2

= 5.61%

Asset H = (8.19)1/2

= 2.86%

Coefficient of Variance: SD/Expected return*100

Asset F = 2.86/8.75*100

= 32.69%

Asset G = 5.61/12*100

= 46.75%

Asset H = 2.86/15.25*100

= 18.75%

If the portfolio consisting 25% of Asset F, 50% of Asset G and 25% of Asset H, then

Expected return = 0.25*8.75+0.50*12+0.25*15.25

= 12%

Variance=(X2X Sd2X) + (X2Y Sd2Y) + (X2Z Sd2Z) + (2 XX XY(SdX SdY rXY)) + (2 XY XZ(SdY SdZ rYZ)) + (2 XX XZ(SdX SdZ rXZ))

= (0.252*2.862) + (0.502*5.612) + (0.252*2.862) + (2*0.25*0.50*2.86*5.61*(-0.89)) + (2*0.50*0.25*5.61*2.86*0.89) + (2*0.25*0.25*2.86*2.86*(-1))

= 0.51+7.87+0.51-3.57+3.57-1.02

= 7.87%

Standard deviation = (Variance)1/2

= 2.81%

Coefficient of Variation = SD/Expected return*100

= 2.81/12*100

= 23.42%

Calculation of Correlation between two Assets (For the purpose of calculating Variance and Standard deviation):

Year	(X - x̅)(Y - ȳ)	(Y - ȳ)(Z - Z̄)	(X - x̅)(Z - Z̄)
2015	-	-	(0.06)
2016	2.25	(2.25)	(0.56)
2017	(33.75)	33.75	(14.06)
2018	(25.50)	25.50	(18.06)
	-57.00	57.00	-32.75

Covariance of Asset F and G = Σ(X - x̅)(Y - ȳ) / no. of years

= -57/4

= -14.25

Correlation of Asset F and G = Covariance/Product of Sd of the two assets

= -14.25/(2.86*5.61)

= -0.89

Covariance of Asset G and H= Σ(Y - ȳ)(Z - Z̄)/ no. of years

= 57/4

= 14.25

Correlation of Asset G and H = 14.25/(5.61*2.86)

= 0.89

Covariance of Asset F and H = Σ(X - x̅)(Z - Z̄) / no. of years

= -32.75/4 = -8.19

Correlation of Asset F and H = -8.19/(2.86*2.86)

= -1