Extensions of VaR: CVaR, IVaR, and MVaR
If you look closely, under the exterior of most popular automobiles lies a frame that is the exact same as other cars made by the same manufacturer, including those that are produced for the luxury car market. It's no surprise that a car manufacturer would use a common frame to build multiple types of vehicles—that's getting good bang for your buck.
The same thought process is illustrated in changing VaR to fit various specific needs.
For example, say you're concerned about the maximum loss possible.
Would VaR be a good fit?
Not quite.
VaR's going to give you the minimum loss.
Of course!
VaR isn't the best choice for the maximum loss, but it can be modified to give you the loss that would occur when VaR is exceeded. That's called the __conditional VaR (CVaR)__ or __expected tail loss__ or __expected shortfall__, which is the average loss conditional on exceeding the VaR cutoff. CVaR can be estimated through observing the entire return distribution, even if the distribution isn't standard.
Which VaR method wouldn't be a great choice for CVaR?
Bingo!
The parametric method isn't a great choice for CVaR because there could be several different types of distributions, and you need to capture all the returns beyond the VaR calculation. The Monte Carlo and historical simulations are capable of capturing all these returns.
No, actually.
A historical simulation will allow you to analyze the entire return distribution, so it's actually a great choice for CVaR.
No.
Any distribution that's not standard will typically be analyzed with the Monte Carlo simulation method, so it will be used for CVaR.
But that's not all that VaR is good for.
Say that you're going to significantly increase a single equity position in your portfolio relative to your other holdings. In that case, you'd want to know how that positional increase would impact the VaR. It's called __incremental VaR (IVaR)__ .
Calculating IVaR is pretty straightforward: it's the difference between the new VaR and the old VaR.
And for more detail when it comes to VaR and positional changes, you can also analyze marginal VaR. __Marginal VaR (MVaR)__ uses calculus formulas to estimate the impact of a very small change in the portfolio. It can be used to analyze a diversified portfolio of holdings by calculating each position's VaR and weighting each VaR to the portfolio's VaR.
So as you can imagine, MVaR and IVaR can provide substantial benefits to asset managers. What's one major benefit?
No.
That's a benefit of CVaR.
That's it!
MVaR and IVaR can help analyze trades prior to executing them so that the impact on VaR can be analyzed. That helps managers understand the risk that the portfolio is taking with a possible position.
Another VaR tool to help with portfolio management is ___ex ante_ tracking error__ or __relative VaR__, which calculates the difference that the performance of a given holding might differ from its benchmark.
To measure relative VaR, you'd take the VaR of the difference between the portfolio and benchmark. You would need to enter in the benchmark's holdings as a short position into the VaR model, so the _ex ante_ tracking error will be at or near zero when the benchmark and the portfolio perfectly match. That also means that as the _ex ante_ tracking error gets larger, the portfolio and the benchmark will have a greater difference.
To sum it up:
[[summary]]
No
Yes
Parametric method
Historical simulation
Monte Carlo simulation
Analyzing a trade prior to execution
Analyzing the expected shortfall of the maximum loss
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