COMPONENT V A R

A5.4 RECOMMENDED READING

7.2 COMPONENT V A R

r

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:

r

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.

r

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.

homogeneous function of the positions in the instruments (or asset classes, etc.) concerned.⁵This linear homogeneity allows us to apply Euler’s theorem, which tells us that:

VaR= n

i=1

wi

∂VaR

∂wi

= ∇VaR(p)w (7.6)

If we now define the component VaR for instrumenti,CVaRi, as:

CVaRi =wi∂VaR

∂wi

(7.7) we can substitute Equation (7.6) into Equation (7.7) to get:

VaR= n

i=1

CVaRi (7.8)

which gives us a breakdown of the VaR into component VaR constituents that satisfies both incremen- tality and additivity properties.⁶The key to CVaR is thus Equation (7.7), which specifies theCVaRi

in terms of the position sizes (i.e., thewi) and the marginal VaRs or mathematical first derivatives of the VaR with respect to thewi.

It is sometimes more convenient to express CVaRs in percentage terms, and we can do so by dividing Equation (7.8) throughout by the VaR itself:

1= 1 VaR

n i=1

CVaRi = n

i=1

%CVaRi (7.9)

The percentage CVaRs, the %CVaRi terms, give us the component VaRs expressed as percentages of total VaR.

Component VaRs give us a good idea of the distribution of risks within our portfolio and, as with incremental VaR, we can distinguish between three main cases:

r

High contributions to risk. High CVaRs represent high pockets of risk, which contribute strongly to overall portfolio VaR.

r

Moderate contributions to risk. Moderate positive CVaRs represent moderate pockets of risk.

r

Negative contributions to risk. Negative CVaRs represent natural hedges that offset some of the risk of the rest of the portfolio. Natural hedges are very useful, because they indicate where and how we can reduce overall risks.

It is important to note that these CVaRs reflect marginal contributions to total risk, taking account of all relevant factors, including correlations and volatilities as well as position sizes. As a result, we cannot really predict CVaRs using only position-size information or volatility information alone:

r

A position might be relatively large in size and have a small or negative CVaR, and another position might be relatively small and have a large CVaR, because of volatility and correlation effects.

r

A position in a high-volatility instrument might have a low or negative CVaR and a position in a low-volatility instrument might have a high CVaR, because of correlation and position-size effects.

5A functiony=(x1, . . . ,xn) is linearly homogeneous if multiplying the inputs by some positive constantλleads to the output multiplying by the same proportion (i.e.,λy=(λx1, . . . , λxn)).

6The latter is obvious; the other is a useful exercise.

The impact of correlation factors can also be appreciated by considering an important special case.

If P/L or arithmetic returns are normal, we can show thatCVaRiis approximately:

CVaRi ≈ωiβiVaR(p) (7.10)

whereωi is the relative share of instrumenti in the portfolio, and is assumed to be ‘small’,βi is the beta coefficient of instrumentiin the portfolio, orσi,p/σp², whereσi,pis the covariance between the returns toiand p(see, e.g., Dowd (1998b, p. 32) or Hallerbach (1999, pp. 8–9)). As we might expect in a normal world, the CVaR for instrument i reflects that instrument’s beta: other things being equal, a high beta implies a high CVaR, a low beta implies a low CVaR, and a negative beta implies a negative CVaR. Correlations therefore have important effects on CVaRs — and because of the complicated ways in which correlations interact with each other, these effects are seldom obvious at first sight.⁷

However, we should also keep in mind that the CVaR risk decomposition outlined in Equations (7.6)–(7.9) has an important limitation: it is a linear marginal analysis. The component risks add up to total VaR because of linear homogeneity working through Euler’s theorem, but the price we pay for this additivity property is that we have to take each component VaR to be sim- ply the position size multiplied by the marginal VaR. This is restrictive, because it implies that the component VaR is proportional to the position size: if we change the size of the position byk%, the component VaR will also change by k%. Strictly speaking, this linear-proportionality is only guaranteed if each position is very small relative to the total portfolio; and where the position size is significant relative to the total portfolio, the component VaR estimated in this way is likely, at best, to give only an approximate idea of the impact of the position on the portfolio VaR. If we want a

‘true’ estimate of the latter, we would have to resort to the IVaR, and take the difference between the VaRs of the portfolio with and without the position concerned. The IVaR then gives us an exact estimate of the impact of the portfolio. Unfortunately, this exactness also has its price: we lose the additivity property, and the component VaRs no longer add up to the total VaR, which makes it difficult to interpret these IVaR-CVaRs (or whatever else we call them) as decompositions of the total risk. In short, when positions are significant in size relative to the total portfolio, we can only hope for our CVaRs to give approximate estimates of the effects of the positions concerned on the portfolio VaR.⁸

7.2.2 Uses of Component VaR 7.2.2.1 ‘Drill-down’ Capability

The additivity of component VaRs is, as we have seen, very useful for ‘explaining’ how the VaR can be broken down into constituent components. Yet it also enables us to break down our risks at multiple levels, and at each stage the component risks will correctly add up to the total risk of the unit at the next level up. We can break down the firm-wide risk into component risks associated with large

7The IVaR also depends on the relative position size as well (as reflected here in the position size orωiterms), but does not depend particularly on any individual instrument or asset volatility.

8This can cause problems for capital allocation purposes in particular. If we want to use component VaRs to allocate capital, we want the component VaRs to be accurate and to satisfy additivity, but we can’t in general satisfy both conditions.

This leaves us with an awkward choice: we can satisfy additivity and base our capital requirements on potentially inaccurate component risk measures; or we can make our IVaR-CVaR estimates accurate, and then they don’t add up properly. In the latter case, we could find that the component VaRs add up to more than the total VaR, in which case a bottom-up approach to capital requirements would leave us excessively capitalised at the firm-wide level; or, alternatively, we might find that the components add up to less than the total VaR, in which case we would have a capital shortfall at the aggregate level. Either way, we get into messy overhead allocation problems, and any solution would inevitably be ad hoc — and therefore probably inaccurate anyway. Additive accurate CVaRs would certainly make life much easier.

Drill-down to Firm-wide VaR

Level of large business unit

CVaR of large business unit

CVaR of large business unit

Level of small business unit

CVaR of small business unit

CVaR of small business unit

CVaR of small business unit

CVaR of small business unit Figure 7.4 Multiple-level risk decomposition and drill-down capability.

business units (e.g., by country or region); we can break these down in turn to obtain the component risks associated with smaller units (e.g., individual branches); and so forth, right down to the level of individual desks or traders. This breakdown is illustrated in Figure 7.4. The key point is that the component risks correctly add up, and this implies that we can break down our risks to obtain the component VaRs at any level we choose: we can break down our firm-wide VaR into component VaRs at the level of large business units, at the level of smaller units, or at any other level, including the level of the individual desk, the individual trader, or the individual instrument. The additivity of component VaRs therefore gives rise to a ‘drill-down’ capability — an ability to decompose a risk figure, or identify its components, down to any level we choose. So, for example, an institution might use drill-down to establish how each and every unit, at each and every level — each trader, instrument, asset class, desk, branch, region, or whatever — contributes to overall risk. Drill-down capability is, needless to say, of immense practical usefulness — for determining the positions or units that need attention, identifying hidden sources of risk, setting limits, making investment or trading decisions, determining capital requirements, establishing remuneration schedules, and so on.

7.2.2.2 Reporting Component VaRs

Given especially that many component risks are less than obvious, it is very important to report component risks meaningfully, and in ways that interested parties (e.g., senior managers, etc.) can understand without too much difficulty. This suggests that:

r

We should ‘slice and dice’ component VaRs, and report them accordingly, in ways geared to each particular audience, business unit, etc.

r

Reports should be as short and straightforward as possible, and avoid unnecessary information that can distract from the key points to be communicated.

r

Reports should identify key assumptions and spell out possible consequences if those assumptions are mistaken.

It follows, then, that there are many possible ways of reporting CVaR information. These might include, among many others, reports of CVaR by asset class (e.g., equities, commodities, etc.), market risk factors, individual trades or positions, types of counterparty (e.g., government counterparties, swap counterparties, etc.), individual counterparties, and so on, and each of these is good for its own particular purpose.

CVaR and IVaR information can also be presented in the form of ‘hot spots’, ‘best hedges’, ‘best replicating portfolios’ and ‘implied views’ reports:

r

Hot spots reports give the CVaRs ranked in terms of their size — the top-ranking CVaRs are the

‘hot spots’, or the biggest sources of portfolio risk — and these give a very immediate indication of where the portfolio risks are coming from.

r

Best hedges reports give the best hedges — for each instrument or asset, the trade (long or short) that would minimise portfolio VaR. For a position with a negative IVaR, the best hedge would involve a further purchase or investment; for a positive IVaR, the best hedge would involve a sale or short position. Best hedges are very useful benchmarks for portfolio management (see, e.g., Litterman (1997b, p. 40)).

r

Best replicating portfolios (BRPs) are those portfolios, made up of small numbers of positions, that best replicate the risks of our ‘real’ portfolio: we select a small number of assetsn, estimate the BRP using regression analysis (see Litterman (1997b, pp. 40–41)), and report the BRPs of a range ofn-values. Best replicating portfolios are very useful for identifying macro portfolio hedges — hedges against the portfolio as a whole. They also help us to understand the risks we face: if we have a very large portfolio, it can be difficult to understand what is going on, but if we can replicate the portfolio with one that has only a small number of different assets, we can get a much better picture of the risks involved. BRP reports are therefore particularly useful when dealing with very large or very complex portfolios.

r

Implied views are the views about future returns that make the current portfolio an optimal one.

Comparing implied views about returns with actual views is a useful tool in helping to understand how portfolios can be improved. They are also useful in helping to macro manage a portfolio whose composition is subject to delegated decision-making. A good example, suggested by Litterman (1997b, p. 41), is in big financial institutions whose portfolios are affected by large numbers of traders operating in different markets: at the end of each day, the implied views of the portfolio can be estimated and compared to the actual views of, say, in-house forecasters. Any differences between actual and implied views can then be reconciled by taking positions to bring the implied views into line.