Stock GS BRK.A
E(R)
sigma 10 5
Correlation 0
10 9
Weight in GS Weight in BRK.A Exp. Ret. on ptf. Variance of ptf sigma of ptf exp.ret.of ptf -1 2 0 424 20.5912603 0 0 1 5 81 9 5 0.25 0.75 6.25 51.8125 7.19809002 6.25 0.5 0.5 7.5 45.25 6.72681202 7.5 0.75 0.25 8.75 61.3125 7.83022988 8.75 1 0 10 100 10 10 2 -1 15 481 21.9317122 15
Holding a straightedge up to the graph, I see that the line from 2% on the Y-axis touches the curve at approximate I then ask, what combination of GS and BRK.A has E(R ) of 8.5%? To find this, I must solvethe equation w*10+(1-w)*5=8.5 where w is the weight of the portfolio in GS, and 1-w is the weight in BRK.A w=3.5/5= 0.7
Check that this portfolio has the correct sigma: Weight in GS Weight in BRK.A Exp. Ret. on ptf. Variance of ptf sigma of ptf exp.ret. of ptf 0.7 0.3 8.5 56.29 7.50266619 8.5
Okay, so our "tangency portfolio of these two risky assets" is 0.7 in GS 0.3 in BRK.A Now we consider investing in this portfolio and the riskfree asset. The risky-asset portfolio has a sigma of 7.5%. W The standard deviation of a portfolio made up of w in the risky and 1-w in the riskfree is given by abs(w)*sigma where sigma is the sigma of the risky asset we want this to equal 12%, so we set abs(w)*7.5=12, or abs(w)= 1.6 So we can choose w=1.6 or w=-1.6 (both give the same abs(w)) But a portfolio with w=-1.6 has an expected return of -1.6*8.5+(1-(-1.6))*2= -8.4 And a portfolio with w=+1.6 has an expected return of +1.6*8.5+(1-1.6)*2= 12.4 So we pick the one with w=+1.6.
Since we have $300 to invest, we invest 1.6*300=480 in the "tangency portfolio of the two risky assets" and (1-1. Our investment in BRK.A is thus 0.3*480= 144 And our investment in GS is 0.7*480= 336
exp.ret.of ptf
16
(
0
5
10
15
20
25
ely E(R)=8.5 sigma=7.5.
/e pick the weight of this portfolio such that the standard deviation of our "total" portfolio is 12%
6)*300=-180 in the riskfree, i.e., we borrow $180 at 2%