5. Consider two assets A and B for which return distributions can be summarized
ID: 2717301 • Letter: 5
Question
5. Consider two assets A and B for which return distributions can be summarized as follows
E[Ra]=3% E[Rb]= 7%
"variance"= 1%^2 "variance"=4%^2
"standard deviation"= 1% "standard deviation"= 2%
rAB = 0
What is the risk of the minimum risk portfolio composed of these two Stocks? (Hint: Use the calculus to minimize sp2). Is the risk of the minimum risk portfolio below that of every constituent asset? What is the expected ROR on the minimum risk portfolio?
Consider two other assets A’ and B’, which are identical (in statistical summary), respectively, to A and B above except that rAB = 1. Write down the answers to the same question as in 5.
Explanation / Answer
Risk of the minimum risk portfolio composed of these two Stocks =
minimum risk of portfolio in asset A =
[Standard deviation of stock B*2 - correlation (p)* SD of A * SD of B] / [SD*2 of A + SD*2 of B - 2 * correlation * SD of A * SD of B]
= [0.02*2 + 0* 0.02* 0.01] / [0.02*2 + 0.01*2 - 2*0*0.02*0.01]
= (0.0004-0) / (0.0004 + 0.0001 - 0 )
= 0.8
minimum risk of portfolio in asset B = 0.2
Standard deviation average = 0.8* .01 + 0.20*.02 = .012
Standard deviation of portfolio = [(0.8*2)*(.01*2) + (.2*2)*(.01*2) + 2*0.8*0.2*0*.01*.02]^2
=
Diversification benefit = 1.2 -.82 = 0.38
Expected return = 0.8*.03 + .2 * .07 = 3.8%
Risk of the minimum risk portfolio composed of these two Stocks if rAB = 1
Risk of the minimum risk portfolio composed of these two Stocks =
minimum risk of portfolio in asset A =
[Standard deviation of stock B*2 - correlation (p)* SD of A * SD of B] / [SD*2 of A + SD*2 of B - 2 * correlation * SD of A * SD of B]
= [0.02*2 + 1* 0.02* 0.01] / [0.02*2 + 0.01*2 - 2*1*0.02*0.01]
= (0.0004-0.0002) / (0.0004 + 0.0001 - 0.0004 )
= 2
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