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by Romain Berry
This article is the seventh in a series of articles exploring risk management for institutional investors. We spent the last six articles describing at a high level the various methodologies to compute the Value-at-Risk (VaR) for a portfolio of financial instruments as well as the most standard stress testing and back testing techniques that generally accompany any risk reporting. From VaR, there are few other risk measures based on the same methodology that are widely used in risk management. Amongst them is Incremental VaR, which is used to identify the risk contributions of each position compared to the overall risk of the portfolio. In this article, we define Incremental VaR, illustrate its usefulness with an example, and discuss its pros and cons. Definition Incremental VaR (IVaR) is sometimes confused with Marginal VaR. We use in this article the most widely accepted definition of IVaR as the analysis of a portfolio with an incremental position. Therefore, we state that Incremental VaR of a given position = VaR of the portfolio with the given position – VaR of the portfolio without the given position IVaR can be positive or negative depending on whether the given position contributes to the risk of the portfolio or whether it diversifies (reduces) the risk of the portfolio, respectively. IVaR helps to understand the dynamic between the various positions within a portfolio. The main drivers of VaR are the volatility of each position and the correlations between these positions. As we remove one position, the Variance-Covariance matrix will change accordingly and so will the VaR of the portfolio. Correlation is a rather loose concept in risk management and does not provide a satisfactory explanation for the influence of one position on another one. Indeed, when we say that two positions are (linearly) correlated, we only infer that when one position moves in one direction, it will trigger a move in the other position with the direction (up or down) of the sign of the correlation and with the amplitude of the value of the coefficient expressed between 0 and 100% (on an absolute scale). But the coefficient of correlation does not provide any insight into the reason for this correlation, on whether that correlation will remain stable as time goes by, or on whether these two positions may be correlated through a third position that mutually influences the two positions separately. Calculating IVaR can help grasp the dynamic of the correlations amongst all positions that compose a portfolio. As we remove a position and calculate its IVaR, we will be able to assess the significance of the interaction of that position with the other assets in the portfolio. The aforementioned definition can be understood and is practically used in two ways: Ex-Post or Ex-Ante. Ex-Post means that we could analyze the portfolio if we had sold the position. In that case, we state that IVaR (a) = VaR (P) – VaR (P - a) where P represents the portfolio value and a stands for the market value of a given position. Ex-Post IVaR may answer the questions of whether the portfolio is still well diversified, of the ongoing effectiveness of a hedge, or of spotting a change in the interaction of that position with the other positions (meaning that the correlation of this position with the other assets present in the portfolio may be changing). The limitation of IVaR is that it cannot isolate the specific assets in the portfolio that are mainly affected by the removal of that position. Ex-Ante IVaR can be expressed as IVaR (a) = VaR (P + a) – VaR (P) In that case, our interest lies in measuring the impact an incremental position will have on the portfolio. If IVaR (a) is positive, then this incremental position will add on more risk to the portfolio. If IVaR (a) is negative, this incremental position will act as a hedge and be a factor of risk diversification for the portfolio. Ex-Ante IVaR could thus be included in a pre-trade analysis or a stock selection. According to the objective (mitigate or take on more risk), a portfolio manager can calculate the IVaR of the potential stock candidates and immediately be able to gauge which one would meet his objective the better. Example Let us illustrate IVaR through a simple example. In order to keep the example straightforward, we will only compute Analytical VaR (using the methodology explained in our second article on VaR1). Exhibit 1 shows a portfolio worth $100m equally comprised of three assets. Asset 1 has an expected mean of 0.2% return with a volatility (standard deviation) of 3%, Asset 2 has an expected mean of 1% return with a volatility of 7%, and Asset 3 has an expected mean of 0% return with a volatility of 1%. Asset 3 then seems to follow a standard normal distribution. Assets 1 and 2 are positively correlated with a coefficient of 50%, while Asset 3 is negatively correlated with the two other assets (-10% and -25%, respectively). Asset 2 appears to be the riskier asset, while Asset 3 does not seem to generate any return and is the least volatile of the three assets. From our definition of IVaR, we could guess that Asset 2 may contribute the most to the risk of the portfolio (having the highest volatility) and that Asset 3 may act as a (poor) hedge as it is negatively correlated with the two other assets. Calculating IVaR will put a precise number on our intuition and will facilitate the risk management of the portfolio through a better understanding of the interaction between these three assets. Exhibit 1 – Calculating Incremental VaR
We calculated the VaR of the portfolio to be $4.376m or 4.38% of the value of the portfolio. The mean and volatility of the portfolio are 0.4% and 2.9%, respectively. We could thus feel that the assets with a higher volatility will add more risk to the portfolio, while the assets with a lower volatility of the portfolio may potentially diversify the risk of the portfolio. In order to calculate IVaR for each individual asset, we have considered what would have been the VaR for the (new) portfolio if we had sold each asset in turn (Ex-Post IVaR). For instance, in order to calculate IVaR1, the Incremental VaR for Asset 1, we calculated the VaR (noted VaR2,3) of a new portfolio without Asset 1 (worth $66.66m equally weighted between Asset 2 and Asset 3, with a mean of 0.5% and a volatility of 3.41%). VaR2,3 amounts to $3.405m. Therefore, IVaR1 equals $0.971m ($4.376m - $3.405m). Similarly, we calculated IVaR2 and IVaR3 to be $2.762m and -$0.097m, respectively. Note that Asset 2 does contribute the most to the risk profile of the portfolio and that Asset 3 does reduce the overall risk of the portfolio (even if rather marginally). Exhibit 2 shows the various IVaRs relative to the VaR of the portfolio. Note also that IVaR is not additive as its two components (VaR with and without a given asset) are sub-additive. Exhibit 2 – Incremental VaRs for the three assets
Conclusion As one might imagine, calculating IVaR for every single instrument in a large portfolio could be rather time-consuming and would require a powerful computer. Nowadays, risk packages handle that task rather effectively (meaning, a good ratio of accuracy vs. time), and therefore time should not be an issue. Along with VaR, IVaR has become a standard tool to make informed decisions about adding a new instrument to a current portfolio, and is widely used in Risk Management as well as in Investment/Portfolio Management. IVaR is mainly used to assess the (incremental) instruments that will provide the most optimal hedge to a portfolio. To view the next article, From Alpha to Omega: The Omega Measure , click here.
1Value-at-Risk: An Overview of Analytical VaR, J.P. Morgan Investment Analytics & Consulting Quarterly Newsletter, September 2008 (available on IAC web site or on request). |
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