−1.60− (−1.08) −0.52, and 1.10− (−0.35) 0.75, and the monthly volatility is 0.0029, or 0.29% per month. Assuming that the expected relative return is zero, the benchmark-relative VaR is
or 0.48% per year.
USING THE PRINCIPAL COMPONENTS WITH MONTE CARLO
means that some risks are not measured. For example, restricting attention to the first three principal components implies that the value-at-risk captures only the risks of changes in the level, slope, and curvature of the yield curve.
More complicated yield curve reshapings are assumed to be impossible. While in many situations this is reasonable because such complicated yield curve reshapings account for only a small fraction of the portfolio risk, restricting attention to the first few principal components treats these events as impossible, when in fact they can happen.
The error in measuring risk resulting from the reduction of dimensional- ity is likely not a problem for most investment portfolios, because the bulk of the risk of such portfolios typically stems from their exposures to the first three principal components: changes in the level, slope, and curvature of the yield curve (in addition to spread risks). However, the error resulting from the reduction in dimensionality can be a problem for well-hedged portfolios.
Once changes in the level and slope of the term structure have been hedged, the remaining risk in the portfolio will be due to more complicated yield curve reshapings. These are precisely the risks that are not captured by the first few principal components. In other forms, this problem pervades risk measurement systems. All risk measurement systems involve some simplifica- tion and reduction in dimensionality and thus omit some risks; if these omit- ted risks are important for the portfolio, the value-at-risk will be understated.
The interaction of the reduction in dimensionality and hedging is particu- larly a problem for grid Monte Carlo approaches based on principal compo- nents. In such approaches, one strategy for reducing the computational burden of portfolio revaluations is to use a cruder grid for the second and third princi- pal components. The justification is that these components explain a much smaller proportion of the variability in interest rates; thus, there is little cost to using a crude approximation to measure their impact on the portfolio value. But if the effect of the first one or two principal components is hedged, then the risk of the portfolio is due primarily to the second principal component, or the com- bination of the second and third. In this case, measuring the impact of the sec- ond and third principal components on the value of the portfolio using a crude approximation can have a significant impact on the value-at-risk.
NOTES
Numerous books on multivariate statistics discuss the principal components decomposition, for example Anderson (1984), Joliffe (1986), and Mardia, Kent, and Bibby (1979), and it is implemented in most statistical software packages.
This chapter approached the principal components decomposition as a particular kind of factor model, and equation (8.9) was interpreted as saying that the principal components p1 and p2 explain the random vector x through the factor loadings However, the principal components decomposition is equivalent to a decomposition of the covariance matrix into its eigenvectors and eigenvalues, and these need not be given a statistical interpretation. In mathematical terms, the vectors 1 (1121)′ and 2 (21 22)′ make up an orthonormal basis for ℜ2 (or more generally, ℜK), which can be seen from equation (8.9) expressing the vector in terms of the two
orthogonal vectors and and the coeffi-
cients p1 and p2. Eigenvectors and eigenvalues are developed in this way in many books on linear algebra, which are too numerous to mention here.
Litterman and Scheinkman (1991) was one of the first papers carrying out the principal components decomposition of changes in the term struc- ture of interest rates. Since then, the use of principal components in model- ing changes in interest rates has become standard, appearing for example in Golub and Tilman (2000), Hull (2000), James and Webber (2000), Jarrow (1996), Rebanoto (1998), Wilson (1994), and many others. Frye (1997), Golub, and Tilman (1997) and Singh (1997) were among the first to describe the use of principal components in computing value-at-risk. Princi- pal components grid Monte Carlo is described by Frye (1998).
Multiple term structures in different currencies can be handled either by finding a set of principal components that explains the common variation across multiple term structures, as in Niffikeer, Hewins, and Flavell (2000), or by carrying out a principal components decomposition of each term struc- ture. As above, the K principal components that describe each term structure will be uncorrelated with each other; however, the principal components of the term structures for different currencies will be correlated. In general, all of the principal components will be correlated with changes in exchange rates. Reflecting these nonzero correlations, the covariance matrix used in the value-at-risk calculation will not be diagonal. While this does add some complexity, the advantage of the separate decompositions of each market vis-à-vis a single “simultaneous” principal components decomposition of all world fixed-income markets is that the market factors (the principal compo- nents) resulting from the separate decompositions will produce the market factors most useful for the separate risk analyses of each market. In particu- lar, the separate decompositions will yield for each market principal compo- nents that have natural interpretations as level, slope, and curvature.
if.
= =
x = (x1 x2)′
1 = (11 21)′ 2 = (21 22)′
9
135
Stress Testing
After considering all of the issues, your organization has chosen a method and computed the value-at-risk. Using a critical probability of ␣ or confi- dence level of 1–␣, the value-at-risk over the chosen holding period is con- sistent with your organization’s risk appetite. You have also decomposed the risk and confirmed that you are within budgeted limits. You are almost ready to relax. But before you manage to sneak out the door, your boss finds you with a series of questions: “What happens in extreme market conditions?” “When the value-at-risk is exceeded, just how large can the losses be?” And finally, “What risks have been left out of the VaR?”
Stress testing provides partial answers to these questions. The phrase stress testing is a general rubric for performing a set of analyses to investi- gate the effects of extreme market conditions. Stress testing involves three basic steps. (i) The process usually begins with a set of extreme, or stressed, market scenarios. These might be created from actual past events, such as the Russian government default during August 1998; possible future mar- ket crises, such as a collapse of a major financial institution that leads to a
“flight to quality”; or stylized scenarios, such as assumed five or 10 stan- dard deviation moves in market rates or prices. (ii) For each scenario, one then determines the changes in the prices of the financial instruments in the portfolio and sums them to determine the change in portfolio value.
(iii) Finally, one typically prepares a summary of the results showing the estimated level of mark-to-market gain or loss for each stress scenario and the portfolios or market sectors in which the loss would be concentrated.
It seems clear that such analyses can provide useful information beyond the value-at-risk estimate. Most obviously, value-at-risk does not provide information on the magnitude of the losses when the value-at-risk is exceeded. Due to possible nonlinearities in the portfolio value, one can- not reliably estimate such losses by extrapolating beyond the value-at-risk estimate. Further, to say that the confidence value-at-risk over a horizon of one month is only a small percentage z of the portfolio value does not reveal what will happen in the event of a stock market crash. It
1–␣
could mean that the portfolio has no exposure to a stock market crash, or it could mean that it has a significant exposure but that the probability of a crash (and all other events that result in a loss greater than z) is less than ␣. Second, the value-at-risk estimate provides no information about the direction of the risk exposure. For example, if the value-at-risk is z, the risk manager does not know whether a loss of this magnitude is realized in a stock market decline or in a sudden rise in prices.
Value-at-risk also says nothing about the risk due to factors that are omitted from the value-at-risk model, either for reasons of simplicity or due to lack of data. For example, the value-at-risk model might include only a single yield curve for a particular currency, thereby implicitly assuming that the changes in the prices of government and corporate bonds of the same maturity move together. Even if it includes multiple yield curves, it may not include sufficient detail on credit and maturity spreads to capture the risks in the portfolio.
Stress tests address these shortcomings by directly simulating portfolio performance conditional on particular changes in market rates and prices.
Because the scenarios used in stress testing can involve changes in market rates and prices of any size, stress tests can capture the effect of large mar- ket moves whose frequency or likelihood cannot reliably be estimated.
Combined with appropriate valuation models, they also capture the effect of options and other instruments whose values are nonlinear functions of the market factors. Because they examine specific selected scenarios, stress tests are easy for consumers of risk estimates to understand and enable meaningful participation in discussion of the risks in the portfolio. Pro- vided that the results are presented with sufficiently fine granularity, stress tests allow portfolio and risk managers to identify and structure hedges of the unacceptable exposures. Finally, a consistent set of stress tests run on multiple portfolios can identify unacceptable concentrations of risk in extreme scenarios.
Despite recent advances in approaches to stress testing, there is no stan- dard way to stress test a portfolio, no standard set of scenarios to consider, and even no standard approach for generating scenarios. This chapter illus- trates a number of available approaches for constructing stress scenarios using the example equity portfolio discussed previously in Chapters 3, 5, and 6. As described there, this portfolio includes a cash position in large capitalization U.S. equities, a short position in S&P 500 index futures con- tracts, written S&P 500 index call options, a long position in FT-SE 100 index futures contracts, and written FT-SE 100 index options. As a result, it has exposures to the S&P 500 index, the FT-SE 100 index, and the dollar/
pound exchange rate.