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

[Nicole Seaman]

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
• $10.0 million position in a Materials ETF with expected excess return of 8.0%, volatility of 38.0%, and marginal VaR (ΔVaR) equal to 0.543
• The correlation between the two positions is +0.150

Each of the following is true EXCEPT for which is false?

a. The Incremental VaR of the Materials ETF position is larger than the Incremental VaR of the Consumer position
b. Each position's Component VaR is less than its Incremental VaR
c. If the correlation doubles to +0.30 then each position's Component VaR will increase
d. To either decrease the portfolio's risk (aka, portfolio VaR) or increase the portfolio's Sharpe ratio, we should reallocate from the Materials ETF to the Consumer ETF; i.e., reduce Materials position and increase the Consumer position

Answers here:

• In forum

 

[littleberries]

Hi, I'm not really getting what is the first question. Could you kindly explain please. Also there seem to be a typo in question 1 answer @David Harper CFA FRM

 

[David Harper CFA FRM]

Hi @littleberries Thank you for spotting the typo! I assume you are referring to the incorrect 0.391. So while I'm here on the typo, I just rephrased/edited the solution to 21.6.1.

Risk is minimized (i.e., the portfolio's diversified VaR is minimized) when the marginal VaRs are constant. Among the four mixes, mix #3 is clearly the choice where the marginal VaRs are nearest (i.e., 0.231 versus 0.232 is almost an exact match). Mix #1 has the greatest diversified VaR. Starting from Mix #1, risk is reduced by ADDING to the component with the lower marginal VaR (in this mix, the marginal VaR of the Tech ETF is only 0.020 while the marginal VaR of the REIT ETF is 0.461) and REDUCING the component with the higher marginal VaR. If we thusly start with Mix #1 and add 15.0% to the Tech ETF, then we "arrive" at Mix #2. However, in Mix #2 the Marginal VaR of the REIT ETF is still much higher than the Marginal VaR of the Tech ETF (i.e., 0.386 versus 0.118) so we can continue to shift toward the position with the lower Marginal VaR (which is still the Tech ETF). If we add another 15% to the Tech ETF, then we arrive at Mix #3: here the Marginal VaRs are almost equal, so we are very close to the allocation that minimizes the portfolio's diversified VaR. For the record, the diversified portfolio VaR for the four mixes are $6,131.14 (Mix #1), $5,043.03 (Mix #2), $4,621.42 (Mix #3), and $5,036.58 (Mix #4).

This is testing Jorion's 7.5.1. (From Risk Measurement to Risk Management). The concept is easier than the math: we can minimize the portfolio's VaR, which here is effectively the same thing as minimizing the portfolio's volatility (since VaR is just the volatility multiplied by the 1.645 deviate for 95% confident VaR if we assume multivariate normal!) by finding the allocation that equalizes the marginal VaRs. Equalizing the marginal VaR is effectively the same thing as equalizing each position's beta (with respect to the portfolio). If we find the allocations where marginal VaRs match, we find the minimum variance portfolio, is the idea! I hope that's helpful,

 

[littleberries]

For 21.6.2, question 2, why is it not required to take into account the position in $ to calculate the excess returns per unit of risk? if we are seeing it as a portfolio, and the goal is to find the most excess return per unit of risk, then it should be the highest combination of ER*dollar allocation/change in VaR? if that makes sense? @David Harper CFA FRM

 

[David Harper CFA FRM]

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,