A5.4 RECOMMENDED READING
6.3 FIXED-INCOME V A R AND ETL
6.3.1 General Considerations
Our third topic is the estimation of the risks of positions in interest-sensitive instruments, usually known as fixed-income instruments: these are bonds, floating-rate notes, structured notes, interest-rate derivatives such as interest rate swaps and futures, and swaptions. However, fixed-income problems can be difficult to handle using analytical or algorithmic methods because they usually involve a range of spot rates across the term structure, not just one or two spot interest rates, and fixed-income problems can be particularly difficult to handle where they involve interest-rate options, because of the extra complications of optionality (e.g., the need to take account of volatilities, etc.). But, fortunately, fixed-income problems are often also very amenable to simulation methods.8
When estimating the fixed-income VaR or ETL, we generally have to take account of two distinctive features of fixed-income positions: the types of stochastic processes governing interest rates, and the term structure.
6.3.1.1 Stochastic Processes for Interest Rates
Interest-rate processes differ from stock-price processes in that interest rates are usually taken to be mean-reverting — if interest rates are high, they have a tendency to fall, and if interest rates are low, they have a tendency to rise. This means that interest rates are expected to fall if they are relatively high, and are expected to rise if they are relatively low. In this respect interest rates differ from stock prices which show no mean reversion: for instance, under the archetypal assumption of a random walk or martingale, the best forecast of a future stock price is today’s stock price, regardless of how high (or low) that price might be. There is a considerable literature on interest-rate processes (see, e.g., Hull (2000, ch. 20–21), James and Webber (2000, ch. 3 and 7), etc.), but one of the most popular of these processes is the Cox–Ingersoll–Ross (CIR) process:
dr =k(µ−r)dt+σ√ dt√
r d z (6.8)
whereµis the long-run mean interest rate, or the reversion level to which interest rates tend to revert, σ is the annualised volatility of interest rates,kindicates the speed with which interest rates tend
7However, we should also be careful with scenario simulation: the results of Abken (2000, p. 27) suggest that the performance of scenario simulation can sometimes be erratic and results can be significantly biased compared to those of standard Monte Carlo simulation and principal components simulation approaches. Convergence can also be slow for some types of portfolio, and Abken recommends that users of scenario simulation should periodically check its results against those of standard methods.
8There is a very diverse specialist literature on fixed-income VaR, and I haven’t space in this book to cover it properly.
However, a good starting point is D’Vari and Sosa (2000), Niffikeeret al.(2000) and Vlaar (2000). For those who are interested in the VaR of mortgage-backed securities, see Jakobsen (1996).
to revert to their mean values, andd zis a standard normal random variable (Cox et al.(1985)).
This process is popular because it captures the three major stylised facts of empirical interest-rate processes9— namely, that interest rates are stochastic, positive and mean-reverting.10
6.3.1.2 The Term Structure of Interest Rates
The other distinctive feature of fixed-income positions is more important, and harder to deal with.
Most fixed-income positions involve payments that are due on multiple future dates, so the valuation of fixed-income instruments requires us to take account of a number of different points along the spot rate term structure. For example, if we have a coupon-paying bond that pays a coupon every 6 months, we can only price this bond if we have information about the spot rates at 6-monthly intervals along the term structure.11 There are exceptions — most notably those relating to zero- coupon bonds, which can be priced with only one spot rate — but the main point is inescapable: in general, we need information about the spot rate term structure, and not just about one individual spot rate, to be able to price interest-sensitive instruments.
From the perspective of VaR (or ETL) estimation, this means that we need information about the prospectiveterm structureat the end of the holding period. The VaR depends on the P/L, and the P/L depends on how the prices of our instruments change over the holding period. We must already know the current prices of our fixed-income instruments, but their prices at the end of the holding period will depend on the spot rate term structure that prevails at that time, and possibly on other future variables as well. So we need a prospective future term structure to price our instruments at the end of the holding period, so that we can then estimate prospective P/L and hence estimate VaR and ETL.
6.3.2 A General Approach to Fixed-income VaR and ETL
So how do we estimate fixed-income VaR and ETL? At first sight, this is quite a daunting prospect, because fixed-income positions are very diverse and some fixed-income problems are extremely difficult. However, most of these difficulties actually relate to pricing, and most pricing problems have now been resolved. Indeed, the state of the art has advanced to the point where the vast majority of fixed-income positions can now be priced both accurately and quickly. Since pricing is the key to VaR (and ETL) estimation, it follows that, if we can price positions, then we can also estimate their VaRs; consequently, there are — in theory — few real obstacles to VaR estimation. If we encounter any problems, they would be the usual practical issues of accuracy and calculation time.
To estimate the VaR (or ETL) of a fixed-income position, we therefore need to simulate the distribution of possible values of a fixed-income portfolio at the end of the holding period. If we are dealing with the simpler fixed-income instruments — such as bonds, floating-rate notes or swaps — then a terminal term structure provides us with enough information to price the instruments and, hence, value the portfolio, at the end of the holding period. The term structure information is sufficient
9However, none of the simpler models of interest-rate processes can fully capture the dynamics of interest rates. We tend to find that after a model has been fitted to the data, the goodness of the fit then tends to deteriorate over time, and this makes the use of such models for pricing purposes somewhat problematic unless they are frequently recalibrated. For more on these issues and how to deal with them, see, e.g., James and Webber (2000).
10That said, it is still an open question how much mean reversion really matters when it comes to VaR estimation — but when in doubt, it is usually best to play safe and put a reasonable mean-reversion term into our interest-rate process.
11I realise that we can also price such a bond if we have the yield to maturity, but the yield to maturity is only a (bad) surrogate for the term structure anyway, and we can’t calculate the yield in the first place without the term structure information I am referring to. So one way or the other, we still need the term structure to price the bond.
because we can value these instruments using standard pricing methods based on present-value calculations of the remaining payments due.
However, if we are dealing with interest-rate options — such as interest-rate calls, puts, floors, ceilings, etc. — then informationonlyabout the term structure of spot rates willnotbe enough to value our instruments. To value positions involving interest-rate options, we also generally need information about the terminal volatility term structure (i.e., we want the volatilities associated with each of a number of spot rates). Moreover, in these circumstances, we can no longer price our instruments using simple present-value methods, and we need to resort to an appropriate option pricing model as well.
There is a very large literature on such models, and the models themselves vary a great deal in their sophistication and flexibility.12Fortunately, given the availability these days of good software, it is no longer difficult to implement some of the better models, including the Heath–Jarrow–Morton (HJM) model.13 The only problem then remaining is how to obtain the terminal volatilities, and we can forecast these volatilities either by assuming that the terminal volatilities are equal to the currently prevailing volatilities (which is permissible if we can assume that volatilities are random walks or mar- tingales, which are best predicted by their current values) or by using GARCH or similar approaches.
Since accuracy and calculation time are often significant issues in fixed-income problems, it also makes sense to look for ways of improving accuracy and/or reducing computation time. We should therefore explore ways to use variance-reduction methods and/or principal components methods.
A good approach for these problems is scenario simulation, which is, as we have seen, a form of speeded-up principal components simulation. If we follow Jamshidian and Zhu (1997) and have three principal components, with seven possible states for the first principal component, five for the second, and three for the third, then we would have 105 distinct principal component scenarios. This would also give us 105 spot rate (or term structure) scenarios, and the only simulation required would be to pick a random scenario from the multinomial scenario-selection process. The random number generation should therefore be very quick.
We now have to come up with plausible principal component scenarios, and these are best obtained by combining information on principal component volatilities with the state probabilities specified in Equation (6.5). For example, if a principal component has five possible states, then Equation (6.5) implies that the probabilities of these states are 1/16, 1/4, 3/8, 1/4 and 1/16, and we might then say that the middle scenario reflected no change in the principal component, the two adjacent states reflected changes of minus or plus, say, one standard deviation, and the other two states reflected changes of, say, two standard deviations.
Box 6.1 Estimating the VaR and ETL of Coupon Bonds
Suppose we wish to estimate the VaR and ETL at the 95% confidence level of a $1 position in a coupon-paying bond with 10 years to maturity, with a coupon rate of 5%, over a holding period of 1 year. These coupon payments should be accounted for in our P/L, and one simple way of
12These models are covered in textbooks such as Hull (2000, ch. 21–22), James and Webber (2000, part 2) or Rebonato (1998, part 4).
13The HJM model is highly flexible, and can price the vast majority of fixed-income instruments, including options, based on information about term-structure spot rates and term-structure volatilities. HJM can also be implemented in MATLAB using the HJM functions in the Financial Derivatives Toolbox.
doing so is to assume that coupon payments are reinvested at the going spot rate until the end of the holding period. We assume for convenience that the term structure is flat at 5%, so all spot rates are 5%. We also assume that the spot interest-rate process is a CIR process like that given in Equation (6.8), withµ=0.05,k=0.01 andσ =0.05, and discretise this process intoN =10 steps, sodt =1/10 measured in years, and assume a number of trials,M, equal to 1,000. Given these parameters, our estimated VaR and ETL — estimated with the function ‘bondvaretl’ — turn out to be 0.011 and 0.014 respectively.