INCREMENTAL V A R

A5.4 RECOMMENDED READING

7.1 INCREMENTAL V A R

7.1.1 Interpreting Incremental VaR

If VaR gives us an indication of portfolio risks, IVaR gives us an indication of how those risks change when we change the portfolio itself. In practice, we are often concerned with how the portfolio risk changes when we take on a new position, in which case the IVaR is the change in portfolio VaR associated with adding the new position to our portfolio.

The relationship of the IVaR to the new position is very informative. This relationship is illustrated in Figure 7.1, which plots the IVaR against the size of the new position relative to the size of the existing portfolio. There are three main cases to consider:

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High IVaR. A high positive IVaR means that the new position adds substantially to portfolio risk.

Typically, the IVaR not only rises with relative position size, but also rises at an increasing rate.

The reason for this is that as the relative position size continues to rise, the new position has an

1The literature on incremental and component risks focuses on the VaR as the baseline risk measure; however, it should be obvious by now that we can translate the analysis of incremental and component VaRs so that it applies to the ETL as well.

We can estimate the incremental ETL using the ‘before and after’ approach discussed in the text below or using estimates of marginal ETL comparable to the marginal VaRs that are dealt with in the text. The latter will then also suffice to give us estimates of the component ETLs.

‘Best hedge’

Negative IVaR

High IVaR

Moderate IVaR

Relative position size IVaR

Figure 7.1 Incremental VaR and relative position size.

ever-growing influence on the new portfolio VaR, and hence the IVaR, and increasingly drowns out diversification effects.

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Moderate IVaR. A moderate positive IVaR means that the new position adds moderately to portfolio risk, and once again, the IVaR typically rises at an increasing rate with relative position size.

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Negative IVaR. A negative IVaR means that the new position reduces overall portfolio risk VaR, and indicates that the new position is a natural hedge against the existing portfolio. However, as its relative size continues to rise, the IVaR must eventually rise because the IVaR will increasingly reflect the VaR of the new position rather than the old portfolio. This implies that the IVaR must have a shape similar to that shown in the figure — it initially falls, but bottoms out, and then rises at an increasing rate. So any position is only a hedge over a limited range of relative position sizes, and ceases to be a hedge when the position size gets too large. The point (or relative position) at which the hedge effect is largest is known as the ‘best hedge’, and is a useful benchmark for portfolio risk management.²

7.1.2 Estimating IVaR by Brute Force: The ‘Before and After’ Approach

The most obvious way to estimate IVaR is a brute force, or ‘before and after’, approach. This approach is illustrated in Figure 7.2. We start with our existing portfolio p, map the portfolio, and then input

2Although IVaRs are good for telling us the impact of trades on VaR — a positive IVaR means that the trade increases VaR, a negative IVaR tells us that it decreases VaR, etc. — they give us a rather incomplete basis for comparing alternative trades. The point is that we can always alter the IVaR by altering the size of the trade: if positionAhas a lower IVaR than positionB, we can usually reverse the ranking simply by changing the sizes of the trades. We therefore need to standardise or normalise the basis of comparison so we can make more meaningful comparisons between alternative trades. The key to this is to nail down what we mean by a trade’s size, and we can define a trade’s size in various ways: in terms of the amount invested, the notional principal, its standalone VaR, its expected returns, and so on (see, e.g., Garman (1996b, pp. 62–63;

1996c, pp. 4–5)). Each of these alternatives has its own strengths and weaknesses, and we should take care to use one that is appropriate to our situation.

Portfoliop Portfoliop+a

Mapping

Market data Mapped p Mappedp+a

VaR(p) VaR(p+a)

IVaR(a)

Figure 7.2 The ‘before and after’ approach to IVaR estimation.

Note: Adapted with permission from Garman (1996c, figure 1).

market data to obtain our portfolio VaR,VaR(p). We then consider the candidate tradea, construct the hypothetical new portfolio that we would have if we went ahead with the trade, and do the same for that portfolio. This gives us the new portfolio VaR,VaR(p+a), say. The IVaR associated with trade/positiona,IVaR(a), is then estimated as the difference between the two:

IVaR=VaR(p+a)−VaR(p) (7.1)

However, this ‘before and after’ approach has a fairly obvious drawback. If we have a large number of different positions — and particularly if we have a lot of optionality or other forms of non-linearity — then estimating each VaR will take time. Many financial institutions often have tens of thousands of positions, and re-evaluating the whole portfolio VaR can be a time-consuming process. Because of the time they take to obtain, IVaR estimates based on the ‘before and after’ approach are often of limited practical use in trading and real-time decision-making.

7.1.3 Estimating IVaR Using Marginal VaRs 7.1.3.1 Garman’s ‘delVaR’ Approach

An elegant way to reduce the computational burden is suggested by Garman (1996a–c). His sugges- tion is that we estimate IVaR using a Taylor-series approximation based on marginal VaRs (or, if we like, the mathematical derivatives of our portfolio VaR). Again, suppose we have a portfoliopand wish to estimate the IVaR associated with adding a positionato our existing portfolio. We begin by mapping pandato a set ofninstruments. The portfolio pthen has a vector of (mapped) position sizes in these instruments of [w1, . . . , wn] (sow1is the size of our mapped position in instrument 1, etc.) and the new portfolio has a corresponding position-size vector of [w1+w1, . . . , wn+wn].

Ifais ‘small’ relative top, we can approximate the VaR of our new portfolio (i.e.,VaR(p+a)) by

taking a first-order Taylor-series approximation aroundVaR(p), i.e.:

VaR(p+a)≈VaR(p)+ n

i=1

∂VaR

∂wi

dwi (7.2)

wheredwi ≈wi (see Garman (1996b, p. 61)). The IVaR associated with positiona,IVaR(a), is then:

IVaR(a)=VaR(p+a)−VaR(p)≈ n

i=1

∂VaR

∂wi

dwi (7.3)

where the partial derivatives, ∂VaR/∂wi, give us the marginal changes in VaR associated with marginal changes in the relevant cash-flow elements. If we wish, we can rewrite Equation (7.3) in matrix notation as:

IVaR(a)≈ ∇VaR(p)dw (7.4)

wheredwis the transpose of the l×nvector [dw1, . . . ,dwn] and∇VaR(p), known as ‘delVaR’ is the 1×nvector of partial derivatives ofVaR(p) with respect to thewi.³Equation (7.4) gives us an approximation to the IVaR associated with positiona given information on the∇VaR(p) anddw vectors: the latter is readily obtained from mapping the position, and the former (which depends only on the existing portfolio p) can be estimated at the same time thatVaR(p) is estimated. This means that we can approximate the IVaR associated with positiona using only one set of initial estimates — those of VaR(p) and ∇VaR(p) — relating only to the original portfolio, and the only information we need about the position itself is its (readily available) mapped position-size vector [dw1, . . . ,dwn]. This, in turn, means that we can estimate as many different IVaRs as we like, given only one set of estimates ofVaR(p) and∇VaR(p). This ‘delVaR’ approach is very useful because it enables us to estimate and so use IVaRs in real time — for instance, when assessing investment risks and specifying position limits.

The process of estimating IVaR using the delVaR approach is illustrated in Figure 7.3. We begin by mapping our portfolio and using market data to estimate the portfolio VaR and delVaRs. Observe, too, that these depend on the portfolio we already have, and not on any candidate trades. Once we have the portfolio VaR and delVaRs, we can then take any candidate tradea, map the trade, and use the mapped trade and delVaRs to estimate the IVaR associated with that candidate trade.

The only question that remains is how to estimate∇VaR(p), and we can always estimate the terms in this vector by suitable approximations — we can estimate∂VaR/∂wi by estimating the VaR for position sizeswiandwi+wi, and taking∂VaR/∂wi ≈(VaR(p|wi+wi)−VaR(p|wi))/wi, whereVaR(p|wi) is the VaR ofpwith position sizei equal towi, etc.

In some cases, we can also solve∇VaR(p) algebraically. For example, where P/L is normally distributed with mean vector␮and variance–covariance matrix,∇VaR(p) is:

∇VaR(p)= −␮+ wαcl

[w^Tw]^1/2 (7.5)

3My exposition differs from Garman’s in that I prefer to deal in terms of position sizes whilst he couches his discussion in terms of cash-flow vertices. The two are much the same, but I believe positions sizes are easier for most people to follow.

The ‘delVaR’ approach has since been patented by Financial Engineering Associates under the tradename ‘VaRDelta’, and further details can be found on their website, www.fea.com.

Portfoliop Candidate trade a

Mapping

Market data Mapped p Mappeda

IVaR(a)

VaR(p) DelVaR(p)

Figure 7.3 The delVaR approach to IVaR estimation.

Note: Adapted with permission from Garman (1996c, figure 3).

(see, e.g., Gourierouxet al.(2000, p. 228),⁴and Garman (1996b)). Equation (7.5) allows us to estimate

∇VaR(p) making use of information about the position size vectors for the existing portfolio (w) and the new position (dw), the mean vector␮and the variance–covariance matrixΣ— all of which are readily available or already known.

The delVaR approach could be implemented on a daily cycle. At the start of each trading day, we would estimate both the VaR and delVaR of our existing portfolio. As we require our various IVaR estimates throughout the day, we would obtain them using Equation (7.4) and Equation (7.5), as appropriate. These estimates could be based on our initial daily estimates ofVaR(p) and∇VaR(p), and could be done extremely quickly without the arduous process of re-estimating portfolio VaRs throughout the day, as we would have to do using a ‘before and after’ approach. Experience suggests that this approximation is pretty good for most institutions most of the time.

Box 7.1 Estimating IVaR

We can estimate IVaR (and CVaR) by any of the standard methods: parametric estimation methods, non-parametric (e.g., HS) methods, or simulation methods:

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Parametric methods are appropriate when we can solve for the delVaRs (e.g., as we can for normal VaR).

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We can apply HS methods by using a ‘before and after’ approach using HS to estimate the

‘before’ and ‘after’ portfolio VaRs.

4Gourierouxet al.(2000) also provide a more general approach to delVaR applicable for other parametric assumptions, and Scaillet (2000b) does the same for non-parametric ETL.

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We can also apply simulation methods using a ‘before and after’ approach, but this can be inaccurate if the user is not careful. If we run two separate ‘before’ and ‘after’ paths, the variance of the IVaR (or CVaR) estimator will behave much like the variance of an option-delta estimator in such circumstances: the variance will get very large as the ‘increment’ gets small (see Boyleet al.(1997, p. 1304)). The solution is to run one set of price paths, and infer the

‘before’ and ‘after’ portfolio VaRs from that. This estimator is of order 1, and will therefore get small as the increment gets small. We can also apply simulation methods to estimate the original portfolio VaR and the delVaR terms, and can then plug these estimates into Equation (7.4) to obtain our IVaRs.

7.1.3.2 Potential Drawbacks of the delVaR Approach

Nonetheless, the delVaR approach only approximates IVaR, and is therefore only as good as the approximation itself. When the position or trade considered is ‘small’ relative to the size of the original portfolio, the approximation should be a good one and we could expect the delVaR approach to be reliable. However, there are two circumstances in which this procedure might not be reliable:

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If we are dealing with very large trades, the first-order Taylor series might not give us a good approximation for the VaR of the new portfolio, and in this case the resulting IVaR approximation might be poor.

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If we have a large number of small trades accumulating during the day, the sum of daily trades will cause the intra-day portfolio to drift away from the start-of-day portfolio, and the VaR and delVaRs of the latter will be increasingly poor proxies for the VaR and delVaRs of the former.

Inaccurate VaR and delVaR estimates can then lead to inaccurate IVaR estimates due to drift in the portfolio composition, even if individual trades are all ‘small’.

Whether these problems are significant will depend on our circumstances, but if we wish to make our IVaR estimates more accurate, we can do so by re-estimating the portfolio VaR and delVaR more frequently: for instance, we can re-estimate VaR and delVaR after a particularly big trade, or after a specified number of trades have taken place, or every so often (e.g., every few minutes) during the trading day.