# P2.T9.21.6. Marginal value at risk (VaR) in portfolio management

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Apply the concept of marginal VaR to guide decisions about portfolio VaR. Explain the risk-minimizing position and the risk and return-optimizing position of a portfolio. Explain the difference between risk management and portfolio management and describe how to use marginal VaR in portfolio management.

Questions:

21.6.1. Emily manages a $20.0 million portfolio allocated into two positions (aka, two components): a Technology ETF, and a Real Estate Investment Trust (REIT) ETF. She compares four different allocations, where the weight assigned to the Tech ETF component is either 35.0%, 50.0%, 65%, or 80.0%. If her goal is to minimize the portfolio's diversified value at risk (VaR), which of the above allocations is best? a. Mix #1 with 35.0% allocated to the Tech ETF b. Mix #2 with 50.0% allocated to the Tech ETF c. Mix #3 with 65.0% allocated to the Tech ETF d. Mix #4 with 80.0% allocated to the Tech ETF 21.6.2. Peter manages a$10.0 million portfolio allocated into two positions (aka, two components): an Energy ETF, and a Healthcare ETF. He compares four different allocations, where the weight assigned to the Energy ETF component is either 45.0%, 60.0%, 75%, or 90.0%. The Energy ETF offers an expected excess (i.e., in excess of the riskfree rate) of 6.0% while the Healthcare ETF offers an excess return of 12.0%.

Peter's goal is not risk minimization, but rather to maximize the ratio of expected excess return to risk. Put another way, he seeks the most efficient portfolio. Given his goal, which of the above allocations is best?

a. Mix #1 with 45.0% allocated to the Energy ETF
b. Mix #2 with 60.0% allocated to the Energy ETF
c. Mix #3 with 75.0% allocated to the Energy ETF
d. Mix #4 with 90.0% allocated to the Energy ETF

21.6.3. Consider a $30.0 million portfolio with two positions: •$20.0 million position in a Consumer ETF with expected excess return of 4.0%, volatility of 12.0%, and marginal VaR (ΔVaR) equal to 0.122

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @littleberries Per Jorion 7.5.2. (and in particular formula 7.38 and its surrounding discussion), we can identify the optimal portfolio (aka, portfolio with the highest Sharpe ratio) by finding the mix (aka, allocation between components, in this case the allocation between Energy and Healthcare) where the the ratios, Excess_Return/Marginal_VaR and/or Excess_Return/Beta, are equalized. You make an excellent point about position size. However, it is implicit in the Marginal VaR and Beta; e.g., the beta is a beta of the position with respect to the portfolio (that includes the position, in a self-referencing way). As we shift the allocation, the marginal VaR and betas adjust based on the allocation; they include size because they are a function of size relative to the portfolio. Okay, but what about the dollar size of the overall portfolio? This does not matter: Sharpe is return/volatility so, just like volatility, if we optimize (minimize) in percentage terms, we are doing it for the portfolio regardless of its incidental dollar size. I hope that's helpful,