Question

You are interested in using short selling to increase the possible returns from your portfolio. You have short sold $200 of ABC and invested $1,200 in DEF. The following data are available on ABC and DEF:

	ABC	DEF
Expected return	3%	15%
Standard deviation	7%	35%

The correlation between ABC and DEF is 0.4. Calculate the expected return and standard deviation of the portfolio.

(Hint: The total invested is $1,000, and while individual weights can be greater than one or less than zero, the sum of the weights must still be one.)

Answer 1

ABC expected return (ABC return) = 0.03; DEF expected return (DEF return) = 0.15; ABC standard deviation (ABC sd) = 0.07; DEF standard deviation (DEF sd) = 0.35; Correlation(r) = 0.4; ABC Weight = -short value/Total investment = -200/1000 =-0.2; DEF Weight = invested value/Total investment = 1200/1000 = 1.2

Expected return of the portfolio = (ABC return*ABC weight) + (DEF return*DEF weight) = (0.03*-0.2)+(0.15*1.2) = -0.006+0.18 = 0.174 = 17.4%

Standard deviation of the portfolio = Square root of {[(ABC weight^2)*(ABC sd^2)]+[(DEF weight^2)*(DEF sd^2)]+[2*ABC weight*DEF weight*ABC sd*DEF sd*r]}

= Square root of {[(-0.2^2)*(0.07^2)]+[(1.2^2)*(0.35^2)]+[2*-0.2*1.2*0.07*0.35]}

= Square root of {(0.04*0.0049)+(1.44*0.1225)-0.01176}

= Square root of{0.000196+0.1764-0.01176}

= Square root of {0.164836} = 0.406 = 40.6%

Portfolio return = 17.4%

Portfolio standard deviation = 40.6%