Component versus Incremental value at risk (VaR), Level 2

[afterworkguinness]

Great turorial @David Harper CFA FRM CIPM, especially the spreadsheet.

 

[QuantFFM]

Hi, Andrew raises a good question here, with respect to GARP's sample question. The setup gives a typical two-asset portfolio with correlations and asks, "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?" I think that's a bit of a mean question because it wants the incremental VaR, but doesn't use that term. So, it's understandable to think of the component VaR.

I thought it would be useful to summarize the difference. Here is the XLS for my calcs below: http://db.tt/VexMVsNR
(I personally think this is conceptually difficult. I can't speak for you, but I almost need to review the spreadsheet to begin to come to grips with these two concepts)

Here are my assumptions for a two-asset portfolio (borrowing EUR + CAD portfolio from Jorion Chapter 7, but changing the numbers to my convenience)

• Portfolio is $100 invested equally in two currencies: CAD position = $50, EUR position = $50
• Volatility (CAD) = 30%, Volatility (EUR) = 40%, correlation(CAD, EUR) = 0.
• VaR confidence in 95%, such that normal deviate = 1.645

An FRM candidate should know how to compute 95% Individual VaRs (assume normality):

• CAD = $50 * 30% * 1.645 = $24.67
• EUR = $50 * 40% * 1.645 = $32.90

Because the correlation is imperfect (less than 1.0), we expect the diversified VaR to be less than the sum of individual VaRs:

• As portfolio volatility = 25%,
• Diversified portfolio VaR = $100 * 25% * 1.645 = $41.1

First, what are the incremental VaRs?
(again, GARP's sample question L2.2.18 is asking for the Incremental VaR: "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?")

• Incremental VaR (CAD) = the reduction in Portfolio VaR if we delete the CAD position = $41.1 - $32.9 = $8.22
• Incremental VaR (EUR) = the reduction in Portfolio VaR if we delete the EUR position = $41.1 - 24.67 = $16.45
• Or put another way, if we start with just a $50 CAD position, our one-asset portfolio VaR is $24.67. The incremental EUR VaR is the increase in our portfolio VaR as we add the EUR position: $24.67 + $16.45 = $41.1 for our new two-asset portfolio.

Second, what are the component VaRs? (we have several ways to compute this, see the XLS. We can use beta or correlation since they are directly related)

• Component VaR (CAD) = $50 * 25% portfolio volatility * 0.72 beta (CAD, Portfolio) * 1.645 deviate = $14.80
• Component VaR (EUR) = $50 * 25% portfolio volatility * 1.28 beta (EUR, Portfolio) * 1.645 deviate = $26.32
• By design, component VaRs sum to portfolio VaR: $14.80 + $26.32 = $41.1
• The only role of VaR here is to multiply volatility by the normal deviate (1.645, in this case). We could drop the deviates and just refer to the marginal risk contribution, that's what component VaR really is. In this example, the portfolio volatility is 25%. If we drop the deviates, we would find that the marginal contribution (CAD) to portfolio volatility is 9.0% and the marginal contribution (EUR) is 16%.
• Further, the marginal VaR is related to this component VaR as they both characterize the linear approximation
  • Marginal VaR is unitless partial derivative; i.e., the change in portfolio VaR with respect to a change in the position
    • For the CAD position, Marginal VaR[CAD] = Portfolio-VaR/Portfolio-Value*beta(CAD, Portfolio) = $41.1/$100.0 * 0.720 = 0.2961
      
    • For the EUR position, Marginal VaR[EUR] = Portfolio-VaR/Portfolio-Value*beta(EUR, Portfolio) = $41.1/$100.0 * 1.280 = 0.5264
  • Component VaR is the (unitless) Marginal VaR scaled by the position
  • For the CAD position, Component VaR[CAD] = marginal VaR[CAD] * CAD-Position = 0.2961 * $50.0 = $14.80
  • For the EUR position, Component VaR[EUR] = marginal VaR[EUR] * EUR-Position = 0.0.5264 * $50.0 = $26.32

What about question asked in the sample, "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?"

• Component VaR is not a terrible answer. But Component VaR (and marginal VaR) are linear approximations (first partial derivatives). Per a grand FRM theme, the larger the component, the greater the error (see Jorion Fig 7-4 to visually see the problem: but it's just like the problem with duration). Component VaR is an approximation of incremental VaR, that loses accuracy as the position size increases
• Incremental VaR is more accurate because it's a re-pricing ("full revaluation"). The subtraction $41.1 - $32.9 = $8.22 is an illustration of full revaluation: price the portfolio, re-price the portfolio, take the difference.

Hello David,

just two questions:

1. Is the difference between the incremental and the component VAR the failure due to linear approximation?
Incremental VaR (CAD) = $41.1 - $32.9 = $8.22
minus
Component VaR[CAD] = 0.2961 * $50.0 = $14.80
= Failure or inaccuracy because of linear approximation

2. Is the fololowing correct?
If the deleted position is relativ small, we can calculate the component VAR as a approxomation for the change in portfolio VAR.
If the deleted position is not small, we have to calculate the incremental VAR to get the change in portfolio VAR.

Thank you and best regards

 

[David Harper CFA FRM]

Hi @QuantFFM I hope you are doing well!

1. Yes, exactly! Component VaR is the position (in dollars) multiplied by marginal VaR, and marginal VaR is the unitless first partial derivative, ∂VaR/∂x(i). It is therefore necessarily a linear approximation: it is the instantaneous approximation scaled to the full position. In this way, given any realistically non-linear relationship between change in the position and portfolio VaR, it necessarily will mis-price the true effect, which is captured by the incremental VaR (in the way similar to how duration must mis-price the true bond/yield relationship).
2. Yes, your statements following logically. Here is Jorion Chapter 7: "Thus the component VAR indicates how the portfolio VAR would change approximately if the component was deleted from the portfolio. We should note, however, that the quality of this linear approximation improves when the VAR components are small. Hence this decomposition is more useful with large portfolios, which tend to have many small positions." --- Philippe Jorion. Value at Risk, 3rd Ed.: The New Benchmark for Managing Financial Risk (p. 172). Kindle Edition. I would just add: according to the same logic, component VaR also works as a fine approximation, if the position is large, but we are testing for a small change in the position. We don't necessarily need to simulate the exit ("deletion") of the entire position, we could just be simulating the addition/reduction (trimming) of a small amount, such that component VaR ought to be a fine approximation of an full repricing (aka, incremental VaR). I hope that's helpful!

 

[QuantFFM]

Hi @QuantFFM I hope you are doing well!

1. Yes, exactly! Component VaR is the position (in dollars) multiplied by marginal VaR, and marginal VaR is the unitless first partial derivative, ∂VaR/∂x(i). It is therefore necessarily a linear approximation: it is the instantaneous approximation scaled to the full position. In this way, given any realistically non-linear relationship between change in the position and portfolio VaR, it necessarily will mis-price the true effect, which is captured by the incremental VaR (in the way similar to how duration must mis-price the true bond/yield relationship).
2. Yes, your statements following logically. Here is Jorion Chapter 7: "Thus the component VAR indicates how the portfolio VAR would change approximately if the component was deleted from the portfolio. We should note, however, that the quality of this linear approximation improves when the VAR components are small. Hence this decomposition is more useful with large portfolios, which tend to have many small positions." --- Philippe Jorion. Value at Risk, 3rd Ed.: The New Benchmark for Managing Financial Risk (p. 172). Kindle Edition. I would just add: according to the same logic, component VaR also works as a fine approximation, if the position is large, but we are testing for a small change in the position. We don't necessarily need to simulate the exit ("deletion") of the entire position, we could just be simulating the addition/reduction (trimming) of a small amount, such that component VaR ought to be a fine approximation of an full repricing (aka, incremental VaR). I hope that's helpful!

Hi David, thank you I'm fine, although temperature is really high here. Your answer was as always really helpful.