We now consider how to map and estimate VaR for specific types of position. Naturally, there is a huge variety of different financial instruments, but the task of mapping them and estimating their VaRs can be simplified tremendously by recognising that most instruments can be decomposed into a small number of more basic, primitive instruments. Instead of trying to map and estimate VaR for each specific type of instrument, all we need to do is break down each instrument into its constituent building blocks — a process known as reverse engineering — to give us an equivalent portfolio of primitive instruments. We then map this portfolio of primitive instruments.
A3.2.1 The Basic Building Blocks A3.2.1.1 Basic FX Positions
There are four types of basic building block, and the easiest of these are basic FX positions (e.g., holdings of non-interest-bearing foreign currency). FX positions are particularly simple to handle where the currencies involved (i.e., our own and the foreign currency) are included as core currencies
in our mapping system.6We would then already have the exchange rate volatilities and correlations that we require for variance–covariance analysis. If the value of our position isxin foreign currency units and the exchange rate isE, the value of the position in domestic currency units — or the mapped position, if you like — isxE. Since there is no foreign interest rate being paid,xis constant, and so the only risk attaches toE.
We can then calculate the VaR (or ETL) in the usual way. For example, if the exchange rate is normally distributed with zero mean and standard deviationσE, then the VaR is:
VaRFX= −αclσEx E (A3.1)
The VaR is minus the confidence level parameter (αcl) times the standard deviation of the exchange rate (σE) times the size of the position in domestic currency units (xE).
A3.2.1.2 Basic Equity Positions
The second type of primitive position is equity, and handling equity positions is slightly more involved. Imagine we hold an amountxA invested in the equity of firm A, but lack this particular firm’s volatility and correlation data. However, we can reasonably suppose that the firm’s return to equity,RA, is related to the equity market return,Rm, by the following sort of equation:
RA=αA+βARm+εA (A3.2)
whereαAis a firm-specific constant,βArelatesRAto the market returnRm, andεAis a firm-specific random element. The variance of the firm’s return is then:
σA2 =β2Aσm2 +σε2 (A3.3)
whereσA2is the variance of RA, and so on. The variance of the firm’s return therefore consists of a market-based componentβ2Aσm2 and a firm-specific componentσε2. Assuming zero-mean normality for the sake of argument, the VaR of the equity position is:
VaRA= −αclσAxA = −αclxA
β2Aσm2+σε2 (A3.4) Estimates of bothσm2 andβA should be publicly available, so we can easily estimateβ2Aσm2. If we also have data on the firm-specific varianceσε2, we can estimate Equation (A3.4) directly, and all is well and good.
But what do we do if we don’t have information onσε2? The answer depends on how well diversified our portfolio is: if our portfolio is well diversified, the firm-specific risks will largely net out in the aggregate portfolio and we could estimate VaR as ifσε2were zero. Our VaR would then be:
VaRA≈ −αclβAσmxA (A3.5)
In short, we map the equity risk to the market index, and use Equation (A3.5) to estimate the VaR of the mapped equity return. It is important to note that the only volatility information used is the market
6Where currencies are not included as core currencies, we need to proxy them by equivalents in terms of core currencies.
Typically, non-core currencies would be either minor currencies (e.g., the Hungarian forint) or currencies that are closely tied to some other major currency (e.g., the Dutch guilder which was tied very closely to the German mark in its later years).
Including closely related currencies as separate core instruments would lead to major collinearity problems: the variance–
covariance matrix would fail to be positive-definite, etc. The mapping of non-core currencies to baskets is much the same in principle as the mapping of individual equities to equity indices as described in the next section. The reader who wants more specific material on mapping currencies is referred to Laubsch (1996).
volatility, and the only firm-specific information we need is the firm’s market beta. This ‘market beta’
mapping allows us to approximate equity VaRs with very little firm-specific information.
The market beta approach also extends naturally to multi-asset equity portfolios, and enables us to estimate the VaR of such portfolios without needing to bother with covariance matrices. To illustrate the point, if we have a portfolio of two equities, A andB, and again assume that their expected returns are zero, the VaR of our mapped portfolio would be:
VaR= −αclβAσmxA−αclβBσmxB = −αclσm(βAxA+βBxB) (A3.6) There is no covariance matrix because both equities are mapped to the same risk factor, namely, the market risk. This approach is highly convenient — to say the least — when dealing with equity portfolios.
The only real problem with this ‘market beta’ mapping procedure is that if we hold an undiversified portfolio, then the estimate of VaR given by Equation (A3.5) will understate the true VaR because it ignores the firm-specific risk. However, if we wish to do so, we can refine the VaR estimate by using an adjustment — explained in Box A3.1 — that takes account of the extent to which the portfolio is imperfectly diversified.
Box A3.1 Adjusting Equity VaRs for Firm-specific Risk
A drawback with the market beta approach to estimating equity VaRs is that it can underestimate VaR because it ignores firm-specific risk. One way to adjust for this bias is to multiply the beta-based VaR (i.e.,−αclβAσmxA) by an adjustment factor that reflects the degree to which the portfolio isimperfectlydiversified. This adjustment factor is:
φ+(1−φ)σpu
(βAσm)
whereσpu is a hypothetical portfolio variance based on the assumption that risks are completely undiversified (i.e., perfectly correlated), andφis a diversification index given by:
φ= σpu−σp σpu−σm
whereσp is an estimate of the portfolio variance as it is. If the portfolio is perfectly diversified, thenσp =σmandφ=1. The adjustment factor is therefore also 1, and the VaR is exactly as given in Equation (A3.5). At the other extreme, if the portfolio is not diversified at all, thenσp=σpuand φ=0. The adjustment is nowσpu/(βAσm) and the VaR is−αclσpuxA, which is easily verified as the correct expression in the absence of any diversification. Finally, if the portfolio is imperfectly diversified,φtakes a value between 0 and 1 and we get a VaR somewhere between−αclβAσmxA
and−αclσpuxA. We thus have an adjustment factor that leads to correct VaRs at the extremes of perfect and zero diversification, and makes some allowance for the extent of diversification of intermediate portfolios.
It only remains to find ways of estimatingσupandσpwhen we have very little information about the volatilities and correlations of specific assets. However, if the portfolio is totally undiversified, thenσpuis just the average standard deviation of the individual assets. We can therefore estimate σpuby taking such an average. Estimatingσpis only slightly more involved. The portfolio variance can be written as:
σp2= N
i=1
wi2σi2+ N
i=1
N j=1
wiwjσi j2
wherewi is the weight of stocki in the portfolio. If we now assume for convenience that the portfolio is equally weighted in the different assets (i.e.,wi =1/N), then the portfolio variance becomes:
σ2p =(1/N) ¯σi2+[(N−1)/N] ¯σi j
which means that we can approximate the portfolio variance if we have data on the average variance and average covariance of the individual stocks in our portfolio. Provided we have these data, we can estimate bothσpuandσpand make our adjustment for the effect of imperfect portfolio diversification.7
A3.2.1.3 Zero-coupon Bonds
The third type of primitive instrument is a zero-coupon bond, known as a zero. Assuming for convenience that we are dealing with instruments that have no default risk, our task is then to map a default-free zero-coupon bond against a set of default-free reference instruments. Let us also suppose that we are using a mapping procedure of the RiskMetrics sort. The core or reference assets might be 1-month zeroes, 3-month zeroes, and so on. This means that we will have volatility and correlation data on zeroes of these particular maturities, but not on zeroes of other maturities. We will therefore lack volatility and correlation data for the particular bonds we hold, except in the fortuitous special cases where our instruments happen to be reference ones. We might hold bonds with maturities varying from 50 to 90 days, say, and yet the nearest reference bonds might be 30-day and 90-day bonds. In this case, we would have the data we want for 30-day or 90-day bonds, but not for any of the others.
So how then do we estimate the VaR of, say, an 80-day bond? The answer is that we estimate the 80-day VaR from the mapped equivalent of our 80-day bond — that is to say, from its equivalent in terms of some combination of 30-day and 90-day bonds. We can then estimate the volatility and whatever other parameters we need from the parameters for the 30-day and 90-day bonds.
Unfortunately, there is no simple way to map such bonds. We therefore need to begin by deciding on criteria that the mapping procedure should satisfy, and the best criteria seem to be those used by RiskMetrics. They suggest that the mapped position should have the same value and same variance as the old one, and should consist of cash flows of the same sign.8We can illustrate this procedure with an example adapted from pp. 117–121 of the 1996 edition of theTechnical Document. Suppose we have a cash flow coming in 6 years’ time, but the nearest reference instruments are comparable bills maturing in 5 and 7 years. The mapped 6-year instrument (I6mapped) is then a combination of the 5- and 7-year instruments,I5andI7:
I6mapped=ωI5+(1−ω)I7 (A3.7)
7There are also other ways to reduce the degree of understatement of VaR that arises from a portfolio being imperfectly diversified. One solution is to use more than one price index. Instead of using the overall stock market index, we might use more specific indices such as those for energy stocks, manufacturing stocks, and so on. These sector-specific indices would pick up more of the movement in each stock return, and thereby reduce the amount of risk left unaccounted for as firm-specific.
σε2would then be lower relative toσm2and so result in a smaller understatement of VaR.
8See RiskMetricsTechnical Document(1996, p. 118). The obvious alternative criteria are that the mapped position should have the same value and same duration as the old one. However, the duration approach only gives an exact measure of bond price volatility if the yield curve is horizontal and movements of the yield curve are strictly parallel. A mapped position that has the same duration as the original position will therefore have a different exposure to changes in the slope or shape of the yield curve.
The problem is then to find some way of choosingω. Since the mapped asset is a linear combination of the other two, we know that the variance of its return,σ62, is:
σ62=ω2σ52+(1−ω)2σ72+2ω(1−ω)ρ5,7σ5σ7 (A3.8) whereρ5,7 is the correlation coefficient and the other terms are obvious. We could solve forωif we knew all the other terms, and the only one we don’t know isσ62. Longerstaeyet al.now suggest estimating a proxy forσ62from a simple linear average forσ6, i.e.:
σ6=ωσˆ 5+(1−ω)σˆ 7, 0≤ωˆ ≤1 (A3.9) where the weight ˆω is proportional to the relative distance between the maturity of the mapped instrument and the maturities of the reference ones (i.e., in this case, 0.5). We then substitute Equation (A3.9) into Equation (A3.8) and solve the resulting equation forω. Since this equation is quadratic, there are two solutions forω, i.e.:9
ω=
−b±
b2−4ac
2a (A3.10)
where:
a=σ52+σ72−2ρ5,7σ5σ7
b=2ρ5,7σ5σ7−2σ72 (A3.11)
c=σ72−σ62
We then choose the solution that satisfies our earlier criteria (i.e., in practice, the one that gives us anωvalue between 0 and 1), and substitute this value into Equation (A3.7) to give us our mapped position.10
Once our bond is mapped, we estimate its VaR by estimating the VaR of the mapped bond (i.e., the portfolio of 5- and 7-year zeroes, with weightsωand 1−ω). This latter VaR is estimated in the same way we would estimate the VaR of any other zero-coupon bonds for which we have adequate volatility and correlation data (e.g., by using a duration approximation, etc.).
This mapping approach extends naturally to multi-instrument portfolios. The size of the covariance matrix needed will depend on how the maturities of the included bonds map against the maturities of the reference instruments, but we do at least know that it cannot exceed the size of the reference covariance matrix. So, for example, if we havendifferent zero-coupon bonds, andmfixed-income reference instruments, the size of the covariance used in our mapping process can never exceed m×m, regardless of how bign might become. After all, we only ever deal with the reference covariance matrix or some subset of it —notthe covariance matrix for ourndifferent instruments.
A3.2.1.4 Basic Forward/Futures
The fourth building block is a forward/futures position. As any derivatives textbook will ex- plain, a forward contract is an agreement to buy a particular commodity or asset at a specified
9See, e.g., RiskMetricsTechnical Document(1996, p. 120).
10This description of mapping zero-coupon bonds ignores certain complications peculiar to bonds. One of these is the tendency of bond prices to move towards par as maturity approaches (the ‘pull to par’ effect). The other is the associated tendency of bond price volatility to decline as a bond approaches maturity (the ‘roll down’ effect). These effects imply that the standard procedures to infer VaR estimates over longer horizons from VaR estimates over shorter horizons will tend to overstate the true amounts at risk, and some illustrative exercises carried out by Finger (1996, p. 7) suggest that the errors involved can be quite substantial, especially for longer holding periods. However, Finger also suggests some simple modifications to correct for these errors.
future date at a price agreed now, with the price being paid when the commodity/asset is delivered;
and a futures contract is a standardised forward contract traded on an organised exchange. There are a number of differences between these contracts, but for our purposes here these differences are seldom important. We can therefore run the two contracts together and speak of a generic forward/futures contract. To illustrate what is involved, suppose we have a forward/futures position that gives us a daily return dependent on the movement of the end-of-day forward/futures price. If we have x contracts each worth F, the value of our position isxF. IfFis normal with standard deviationσF, the VaR of our position is approximately:11
VaR≈ −αclσFxF (A3.12)
assuming again for convenience that the expected return is zero. The VaR is approximately−αcl
times the standard deviation of the forward/futures price (σF) times the value of our position (xF).
The only problem is that we may need to map the forward/futures position. How we do so would depend on the reference data available, but if we were using a RiskMetrics-style mapping system, we would have data on a number of points along a forward/futures maturity spectrum: the spot price, the 1-month forward/future price, the 3-month forward/future price, and so on. We could then map our position against these reference assets. If our position has a 4-month maturity, say, we might map it against reference contracts with maturities of 3 months and 6 months, and do so in much the same way as we previously mapped zero-coupon bonds. The mapped 4-month contract (I4mapped) would be a combination of the 3- and 6-month instruments,I3andI6:
I4mapped=ωI3+(1−ω)I6 (A3.13)
We then map this position as we did the earlier zero-coupon bonds and take the position’s VaR to be the estimated VaR of its mapped equivalent, and we can handle multiple-instrument positions in the same way as we would handle multiple bonds.
A3.2.2 More Complex Positions
Having set out our building blocks, we can now map more complex positions. We do so by reverse- engineering them — we produce synthetic equivalents for them, using our building blocks. We already know how to map these equivalent positions, and so we can take their VaRs to be approximations of the VaRs of our original positions. The main points to keep in mind are the following, which draw on established results in financial-engineering theory:
r
Coupon-paying bonds. A coupon bond can be regarded as a portfolio of zero-coupon bonds, each maturing on a different maturity date. We can therefore map coupon bonds by regarding them as portfolios of zero-coupon bonds and mapping each individual zero-coupon bond separately. The VaR of our coupon bond is then equal to the VaR of its mapped equivalent in zero-coupon bonds.r
Forward-rate agreements. An FRA is equivalent to a portfolio long in the zero-coupon bond with the longer maturity and short in the zero-coupon bond with the shorter maturity, or vice versa. We can therefore map and estimate their VaRs in the usual way.r
Floating-rate instruments. Since a floating rate note re-prices at par with every coupon payment, we can think of it as equivalent to a hypothetical zero-coupon bond whose maturity is equal to the11One reason for the approximation is that, with either contract, the investor is likely to face collateral or margin require- ments, and the cost of maintaining these margin positions will usually be interest-sensitive. With forward markets, a second source of approximation is the illiquidity of secondary forward markets. A forward VaR is based on a price in a thin market, and any estimated forward price/VaR is subject to considerable liquidity risk.
period until the next coupon payment. We can then map this hypothetical bond as we would map any other zero-coupon bond.
r
Vanilla interest-rate swaps. A vanilla interest-rate swap is equivalent to a portfolio that is long a fixed-coupon bond and short a floating-rate bond, or vice versa, and we already know how to map these instruments. All we therefore need to do is map a portfolio that is long one instrument and short the other, and the VaR follows in the standard manner.r
Structured notes. These can be produced synthetically by a combination of interest-rate swaps and conventional floating-rate notes, and we can leverage up by entering into more swaps.12r
FX forwards. A foreign-exchange forward is the equivalent of a long position in a foreign currency zero-coupon bond and a short position in a domestic currency zero-coupon bond, or vice versa.r
Commodity, equity and FX swaps. These can be broken down into some form of forward/futures contract on the one hand, and some other forward/futures contract or bond contract on the other.We could extend this list ad nauseam, but the main point is very clear: subject to a reservation to be discussed below, we can map a great many different instruments using a small number of building blocks and some elementary financial engineering theory.
And what is the reservation? Optionality. The instruments just covered all have in common the point that their returns or P/L are linear or nearly linear functions of the underlying risk factors.
Mapping is then fairly straightforward, and our results should be reasonable. Naturally, mapping involves approximation and hence error, but in the presence of linear risk factors there is no reason to be unduly concerned about such errors provided we are reasonably conscientious.
However, once we have significant optionality in any of our positions, we can easily get into serious trouble. The problem is not so much that we can’t map options positions — we can map them using any of the delta, delta–gamma and similar approaches discussed in the Appendix to Chapter 5 — but the mischief caused by the non-linearity of options’ payoffs relative to the underlying risk factors.
We will have more to say on these issues later, but it suffices for the moment to be aware that this non- linearity can seriously undermine the accuracy of any standard (i.e., linear) mapping procedure. We should therefore be very careful — if not downright wary — of using linear-based mapping systems in the presence of significant optionality. In any case, even if they were initially fairly sound, linear- based mappings of options positions can rapidly degenerate in the face of market developments: for instance, a gamma-based approximation can rapidly become useless for an expiring at-the-money vanilla option, because the gamma of such an option approaches infinity. It follows that the mapping of options positions needs to be regularly updated to reflect current market positions: mapping options is thus a dynamic process, as well as a difficult one.