# LO 67.5: Explain the risk-minimizing position and the risk and return-optimizing

LO 67.5: Explain the risk-minimizing position and the risk and return-optimizing position of a portfolio.
A manager can lower a portfolio VaR by lowering allocations to the positions with the highest marginal VaR. If the manager keeps the total invested capital constant, this would mean increasing allocations to positions with lower marginal VaR. Portfolio risk will be at a global minimum where all the marginal VaRs are equal for all i andy:
MVaRj = MVaR
We can use our earlier example to see how we can use marginal VaRs to make decisions to lower the risk of the entire portfolio. In the earlier example, Position A has the smaller MVaR; therefore, we will compute the marginal VaRs and total VaR for a portfolio which has $5 million invested in A and$1 million in B. The portfolio variance is:
0.062 0
0 $5′ 0.142$1

*
0.0900 + 0.0196 = 0.1096
This value is in ($millions)2. VaR is then the square root of the portfolio variance times 1.65 (95% confidence level): VaR = (1.65)($331,059) = $546,247 The VaR of$546,247 is less than the VaR of $608,490, which was produced when Portfolio A had a lower weight. We can see that the marginal VaRs are now much closer in value: cov(RA,RP) cov(RB,RP) 0.062 0 0 0.142 (cid:0)$5 \$1
0.0180 0.0196
Page 82
2018 Kaplan, Inc.
Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
The marginal VaRs of the two positions are:
MVaRA Z , x ^ A . R p ) = 1 .6 5 x
CTp
= 0.08971
j o .1096
s
MVaRB = Zc x C0V(RB>r p )
CTp
‘196 = 0.09769 VO. 1096