The New Cosmos-U.S. Valuation Algorithms

by Oren Cheyette

The release this year of the Cosmos-U.S. bond portfolio management system—part of The BARRA Cosmos System™, BARRA’s suite of fixed income applications—will provide our current U.S. fixed income analytics on the Windows platform. In a subsequent release we plan to replace the existing analytical code with completely rewritten and enhanced routines. This article, based on material presented at recent BARRA seminars, describes the new algorithms that we plan to incorporate into Cosmos-U.S. We offer the material now to elicit feedback (from clients and others) that could benefit our development efforts.

The new Cosmos-U.S. analytical algorithms will include: the mean reveting Gaussian (MRG) interest rate model, used for all asset types a more efficient finite difference algorithm for bond valuation the "Quasi-Random Principal Components" (QRPC) algorithm, a new method for valuing mortgage-backed and other securities with path-dependent cash flows

The new analytics will differ from those of the B2 system (the current U.S. Fixed Income Service) in two ways: We will use a revised interest rate model, and we will no longer use the same valuation algorithm for both bonds and mortgages (though we will use the same interest rate model for both). The B2 system uses the Cox-Ingersoll-Ross (CIR) interest rate model, and the B2 valuation algorithm for both bonds and mortgages involves construction of a "tree" of interest rate scenarios, followed by analysis of the security’s cash flows along all the scenarios. In the new analytical package we will switch to the mean reverting Gaussian (MRG) interest rate model, and we will split the valuation algorithm into a finite difference method for bonds and a "Quasi-Random Principal Components" (QRPC) algorithm for MBS and other path-dependent securities.

The mean reverting Gaussian (MRG) interest rate model

In Cosmos-U.S. we will use the MRG interest rate model (also known as the "modified Vasicek" or "Hull-White" model) instead of the CIR model for two reasons. First, the empirical evidence from the last decade suggests that the MRG model better reflects the observed behavior of interest rates. Anecdotal evidence on market implied volatilities of interest rate options (such as swaptions) suggests that this is demonstrated in option pricing as well. The primary difference between the two models is that in the MRG model the absolute level of volatility (expressed as basis points per year) is independent of the short rate, while in the CIR model the absolute level of interest rate volatility is proportional to the one-half power of the short rate. Empirical estimates of the relationship based on Treasury data since 1987 give a best-fit exponent of 0.191¹—closer to the MRG value of 0 than to the CIR value of 0.5.

A second reason for the shift to the MRG model is simply that it is easier to work with than the CIR model—more efficient and accurate algorithms can be written for it. In any case, numerical evidence suggests that, provided the term structure of volatilities is suitably calibrated, it doesn’t make much difference to either valuations or measures of interest rate risk which model one uses (unless you’re an options trader running a hedged book).

The finite difference algorithm for bonds

One conventional technique for handling such cases is the "binomial lattice," in which the interest rate evolution is modeled as a branching tree, with the additional feature that the branches of the tree "recombine"—that is, an up move followed by a down move gets to the same place as down followed by up. This algorithm is the simplest of a more general class called finite difference methods. It also happens to be one of the least efficient, in terms of accuracy achieved for a given amount of computation time. What started out as a pedagogical device² has become a widely-used tool. There are significantly better finite difference methods available. The one we have applied to the bond valuation problem is the Crank-Nicolson method, and we have modified it to improve the accuracy of the results across a wide range of conditions. It is faster and more accurate than the binomial or trinomial lattice, especially for calculation of interest rate risk exposures such as effective duration, key rate duration, and effective convexity.

Figure 1: Pricing error for a 30-year bond callable at par in 1 year. The underlying bond is priced at par around the middle of the graph. The upper curve was generated using a trinomial tree algorithm with monthly time steps. The lower curve was generated using the new Cosmos-U.S. algorithm with semiannual time steps.

The lower curve in Figure 1 shows the numerical error of the Crank-Nicolson method as a function of interest rate level for a case where we have an analytical solution—namely, a bond callable only on a single date. I have chosen an example that tends to have large numerical errors: a long bond (30 years to maturity) with a short call date (1 year). The error is below 2 basis points of price across a wide range of interest rates, over which the underlying bond goes roughly from a 20-point premium to a 20-point discount. The upper curve in Figure 1 shows the error in a trinomial method calculation for the same bond. The error is much larger and, significantly, the variation in the error is large as well. This means that, as shown in Figure 2, effective durations based on the trinomial method have large errors and, moreover, are unstable. Both the size and the sign of the error change rapidly as market conditions change. (The binomial method gives similar results.) It is also worth noting that the trinomial method requires roughly three times as much computation time for this case as does our Crank-Nicolson method. This is due to the need for more frequent time steps even to achieve the accuracy shown.

Figure 2: Effective duration error based on the same calculations as for Figure 1. The new Cosmos-U.S. algorithm has errors everywhere less than 0.1 year, while the trinomial method has errors that rapidly jump between ± 0.3 years.

Our implementation of the Crank-Nicolson method uses a so-called "adaptive grid" to achieve high accuracy for problems like a short call on a long bond. The idea behind the adaptive grid is to use a coarse but wide grid for times far in the future—when the present value of any numerical errors is small, but the range of possible outcomes is large (interest rates could be anywhere in a wide band)—and to "refine" and narrow the grid for nearer events, for which the present value of errors is potentially large, but the range of possible outcomes is small. Figure 3 shows a schematic representation of the grid. The Crank-Nicolson method provides the rules for updating the values in going from one vertical line to the next earlier one. The routine determines the optimal exercise of the embedded option to find the boundary of the solid region shown in the lower portion of the figure.

Figure 3: Schematic diagram of the grid used in the new Cosmos-U.S. bond valuation algorithm for a bond with an out-of-the-money call starting at the third time step. Like the binomial lattice method, the Cosmos-U.S. method starts at the maturity date and works back to the analysis date. The bond values are tracked at the nodes. The bond value is updated at cash flow and option dates based on coupon, sinking fund and maturity payments, and the option schedule.

For path-dependent securities such as mortgage-backed securities (MBS) there is no method of comparable speed and accuracy. There is essentially no alternative to brute-force evaluation of multiple interest rate and cash flow scenarios.³ The most brutish of the brute force methods is the Monte Carlo method, which uses random numbers to generate simulations based on the interest rate model. It is quite simple to implement, is applicable to a wide range of problems, and can achieve any required degree of accuracy given sufficient computation time. The catch is in the last phrase: The algorithm converges slowly; to reduce the error by a factor of two requires four times as much calculation. In addition, in its most naïve form, the statistical error in the Monte Carlo method can be quite large, due to the use of random numbers.

The upper line and data points in figure 4 show the error in a present value calculation for a 30-year 8% FNMA pass-through using an elementary Monte Carlo implementation. (The line is a best-fit to the data points, which were obtained by repeated runs for different total numbers of paths, using different random numbers.) To achieve an accuracy of 5 basis points of price would require simulation of roughly 10,000 paths. (Perhaps coincidentally, this is the number of paths often quoted by Wall Street dealers for their MBS calculations.) On a fast computer workstation, with well-optimized code, this calculation takes about three minutes and, of course, calculation of risk exposures requires multiple runs.

Figure 4: Error and convergence rate for valuation of a 30-year 8% FNMA pass-through by the three methods described in this article

The middle line and data points show the errors obtained by applying several refinements, known generically as variance reduction, to the Monte Carlo method. One of these is to ensure that the sample paths collectively have the correct mean and variance, both of which can be calculated analytically. Another is to calculate—simultaneously with the mortgage valuation and using the same set of paths—the values of pure discount bonds that mature at each of the cash flow dates. These are of course also known exactly: They are given by the initial term structure of interest rates. The errors in the calculation of these pure discount bond prices are used to adjust the mortgage calculation. For the example shown in Figure 4, these refinements decrease the error by roughly a factor of 3.5 for a given number of paths, with very little added cost in runtime. This is equivalent to the accuracy achieved by running the naïve algorithm about 3.5 ² (@12) times longer.

The "Quasi-Random Principal Components" (QRPC) algorithm

While the variance reduction techniques improve the accuracy of the Monte Carlo method significantly, they do not improve its speed of convergence. This is still given—by the law of large numbers— to be a twofold reduction in error for a fourfold increase in runtime. The upper and middle lines in Figure 4 (shown as a log-log plot) both have slopes of approximately -0.5, reflecting this relationship. The lower data points and line are based on an entirely new approach to choosing the sample paths. This method brings together two previously known techniques for addressing the valuation of MBS.

The first technique—the Principal Components method⁴ —decomposes the sample paths into variations on different time scales. A typical path of interest rates varies on many different time scales: There are short-run daily and weekly fluctuations with no long-term trend, and there are longer-term monthly, yearly, and decade-long variations. This decomposition can be made mathematically precise, and leads to the following idea: The determinants of value for most securities are the long-run changes in rate levels; daily or weekly variations have negligible impact on value, provided that they are not part of a longer-term trend. So in constructing sample paths for valuation, we won’t go far wrong if we simply ignore the short-term fluctuations and account for only the long-term changes. The components of the variation of interest rates can be broken down into a discrete set by principal components analysis of the covariance matrix of changes in rates over time—hence the method's name. The principal component with largest weight is the one corresponding to a very long-term increasing or decreasing trend. The next one corresponds to a single "whipsaw," with rates first going one way, then coming back and ending up changing in the other direction. Subsequent ones with smaller weights correspond to two, three, and more whipsaws.

In the original version of the method, the principal components were added each with weights ±1 to obtain a sample path. While straightforward to implement, this results in a distribution of paths that doesn’t accurately reproduce the true distribution, even when many principal components are included and, consequently, the method in this form does not give very accurate valuations.

The other ingredient in our new method is to use so-called "quasi-random sequences"⁵ to determine the weights of the principal components. Quasi-random is a misnomer, derived from the term pseudo-random given to the numerical sequences used in Monte Carlo methods. (While designed to have many of the statistical properties of random numbers, they are not truly random because they are generated by deterministic and reproducible algorithms— hence "pseudo-random.") There is nothing at all random about quasi-random sequences. They are designed to overcome the statistical deficiency of Monte Carlo methods—namely, that the random number sequences they use exhibit clumps and gaps that are only gradually erased by using longer sequences (i.e., more paths).

It is these gaps and clumps that produce the poor convergence rate of the Monte Carlo method. Quasi-random sequences are contrived precisely to minimize clumps and gaps. Instead of choosing points at random, a quasi-random sequence "keeps track" of the holes in the sequence and fills them in as evenly as possible as additional points are added ⁶. The set of points that results at any stage has gaps that are as small as possible, given the number of points used, and a correspondingly minimal level of clumping.

In our new method we use a quasi-random sequence to generate a normal distribution of weights for each principal component⁷. These are summed, as before, to obtain the sample interest rate paths. This is the Quasi-Random Principal Components (QRPC) method. A typical set of sample paths of the method is shown in Figure 5.

Figure 5: 32 interest rate paths generated by the Quasi-Random Principal Components (QRPC) method. The curves show the difference over time between the simulated short rate and the initial forward rate curve.

As with the Monte Carlo method, and unlike other systematic procedures, one can keep adding additional sample paths to the calculation until a desired level of accuracy is achieved. We find that, with a desired precision of 2 basis points of option-adjusted spread (OAS), our algorithm might run 32 sample paths for a fixed rate pass-through, 128 for an adjustable-rate mortgage, and several hundred for a collateralized mortgage obligation inverse floater or an IO or PO strip. The lower line in Figure 4, fitted to the lower data points, shows the convergence rate for the new method to be given by an exponent of -0.8 for a current coupon pass-through. This is equivalent to saying that halving the error requires using roughly 2.3 times as many paths (compared to four times as many for Monte Carlo). Convergence is fastest when only the longest-term variations matter much. For ARMs and strips the convergence rate might be given by an exponent closer to -0.7 or -0.75. But even with very few paths, the absolute level of the error is consistently significantly smaller with the QRPC method than with conventional variance-reduced Monte Carlo.

The difference in rates of convergence of the Monte Carlo and QRPC methods is illustrated dramatically in Figure 6, showing the present value after different numbers of paths for the same security used in Figure 4. The QRPC method settles down to very close to the exact value after fewer than 100 paths, while the Monte Carlo method is still far from the exact value after 5,000 paths.

Figure 6: Convergence of Monte Carlo and QRPC methods for the same security shown in Figure 4. The horizontal line shows the ÒexactÓ value, computed by running the Monte Carlo with 250,000 paths.

Summary

Use of these new algorithms in Cosmos-U.S. will give improved performance for both single-asset and portfolio calculations. Accurate OAS calculations for CMOs will take no more than a few seconds on a typical PC. Calculations for bonds will take a small fraction of a second, so it will be possible to re-evaluate even a benchmark index portfolio containing thousands of securities on a lunch break.

FOOTNOTES

² Cox, John C., Stephen A. Ross, and Mark Rubenstein, "Option Pricing: A Simplified Approach," Journal of Financial Economics, 7, September 1979, pp.229-263. (return to text)

³ MBS cash flows are path-dependent because of the prepayment model, because of periodic interest rate caps (for ARMS), or because of the paydown rules (for CMOs). (return to text)

⁴ Gulrajani, Deepak, Michael Roginsky, and Ronald N. Kahn, "Advanced Techniques for the Valuation of CMOs," Chapter 7 of CMO Portfolio Management, Frank J. Fabozzi (Ed.), Frank J. Fabozzi Associates, 1994, pp.103-120. (return to text)

⁵ Niederreiter, Harald, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, 1992. (return to text)

⁶ The simplest of these sequences has a familiar analog: an English system ruler, with subdivisions of inches, half-inches, quarters, eighths, sixteenths, etc. The sequence goes: 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16 .... At each step, one takes either another of the subdivisions at the current level of refinement or, when those are exhausted, steps to a finer level, always filling in one of the largest remaining gaps, and doing so as evenly as possible. (return to text)

⁷ A closely related scheme has been described by Caflisch, Russell, and William Morokoff, "Valuation of Mortgage Backed Securities Using the Quasi-Monte Carlo Method," UCLA Dept. of Mathematics preprint, 1996. (return to text)