Using quarterly U.S. Treasury yield curves from 1962, we illustrated the best practice approach for establishing the most accurate number of risk factors. In general, if there are N inputs to a yield curve at time 0 and N inputs to a yield curve one period later, it will take as many as N (and in fact, usually N) inputs to explain the yield curve shift fully. Using a quarterly history of the U.S. Treasury curve since 1962, we showed that at least 9 interest rate risk factors were necessary to model the curve with high degree of accuracy. In this note, we provide a worked example of stress testing and “reverse stress testing” with a 9 factor Heath, Jarrow and Morton model of the U.S. Treasury yield curve.
Conclusion: Stress-testing, reverse stress-testing, and single factor stress-testing are straightforward in a realistic 9 factor Heath, Jarrow and Morton no-arbitrage model. The mathematical principles used are more than 300 years old, and the key Heath, Jarrow and Morton insights have been extensively studied and validated over the last quarter of a century. The resulting models are highly accurate, and the relevant interest rate coefficients for multi-factor models are readily available.
A Summary of Multi-Factor Interest Rate Model Set-up
In this section, we briefly summarize the highlights of how a diligent interest rate risk analyst sets up a multi-factor interest rate model. We then use such a model to analyze one yield curve shift and stress test each of the 9 factors in sequence to show their marginal contribution to the full movement of the yield curve. We note here that random interest rates are at the core of all of the typical branches of risk management analytics, so risk managers in credit risk, market risk, liquidity risk, stress testing, capital adequacy, and regulatory reporting all need the same interest rate risk capabilities.
Step 1: Assemble as much relevant data as possible
The Board of Governors of the Federal Reserve makes daily U.S. Treasury yield data available on its website beginning with data from 1962. “Relevant data” may well include data from other countries with more experience in particular rate environment than one’s own country. For example, an interest rate analyst in the United States who has not analyzed the low-rate experience in Japan is operating on a data set that is much less complete than it could be.
Step 2: Smooth the yield curve data with a high quality “no arbitrage” yield curve smoothing technology like the maximum smoothness forward rate approach.
See van Deventer, Imai and Mesler (2013), chapters 5 and 17, for the details of this approach first published 20 years ago. Jarrow (2014) notes that the Svensson (1995) smoothing function and its special case, the Nelson Siegel (1987) approach, are inconsistent with a no-arbitrage market and should be avoided.
Step 3: Extract zero coupon bond prices with the same maturity interval as the periodicity to be done.
If the modeling is to be done on a forward-looking basis by quarter, then the relevant zero coupon bond prices should have quarterly maturities. If the modeling will use monthly time intervals, then the zero coupon bond prices should have monthly maturities. From now on, we will discuss quarterly modeling without loss of generality.
Step 4: Create the “forward returns” from the zero coupon bond prices.
A forward “return” is one plus the uncompounded, unannualized forward rate. At time 0, the quarterly 3 month return for the quarterly period maturing at the end of month 6 is the ratio P(1)/P(2). P(1) is the zero coupon bond price maturing in 1 quarter (the end of month 3) and P(2) is the zero coupon bond price maturing in 2 quarters (the end of month 6).
Step 5: Using a procedure like Gram-Schmidt, derive the quarterly history of the risk factors that one believes are the drivers of the yield curve.
Later steps in the procedure will validate how many factors are important and the degree to which they are (a) statistically significant and (b) accurate. If there are N inputs to the yield curve, then one should assume as many as N factors will be necessary. The Gram-Schmidt procedure, following the approach of Cauchy (born in 1789), allows the analyst to derive the quarterly history of N independent, uncorrelated and identifiable risk factors. In the March 5 paper, we discussed these 9 risk factors and we will use them again in this note:
Factor101, the change in forward returns for forward maturing in month 6
Factor102, residuals of change in forward returns for forward maturing in year 10
Factor103, residuals of change in forward returns for forward maturing in year 3
Factor104, residuals of change in forward returns for forward maturing in year 7
Factor105, residuals of change in forward returns for forward maturing in year 5
Factor106, residuals of change in forward returns for forward maturing in year 1
Factor107, residuals of change in forward returns for forward maturing in year 2
Factor108, residuals of change in forward returns for forward maturing in year 30
Factor109, residuals of change in forward returns for forward maturing in year 20
Note that the analyst should drop the first observation after the Federal Reserve has made a change in the maturities reported on the H15 statistical release. We do this because the availability or lack of availability of a maturity point will change the shape of the smoothed yield curve. The changes in yield curve data are obvious in the data history that can loaded from the website of the Board of Governors of the Federal Reserve.
Step 6: Impose the no-arbitrage restrictions of Heath, Jarrow and Morton (references below in the appendix) on the form of the econometric relationship which links the 3 month spot rate and the 119 3 month forward returns in the 30 year Treasury yield curve to the 9 risk factors driving interest rates.
One of the many key insights of the Heath, Jarrow and Morton approach is that the drift in the term structure of interest rates is not independent of the risk factors and their volatilities. For background on Heath, Jarrow and Morton, please our March 5 note and the appendix below.
Step 7: Derive the coefficients linking the 9 risk factors and the changes in the 119 forward returns using 119 econometric relationships, one for each of the quarterly forward returns.
This takes less that 60 seconds using a modern statistical package. Alternatively, a firm like Kamakura Corporation supplies the coefficients to clients.
Step 8: Perform the desired yield curve analysis.
There are three extremely commonly uses of such a multi-factor model.
- Forward-looking simulation, in which the 9 risk factors are simulated again and again in sequence in order to derive M different scenarios for yield curve movements that are consistent with a no arbitrage economy.
- Stress-testing the entire yield curve, much like the Federal Reserve has done with the 3 scenarios for 13 quarters in its 2014 Comprehensive Capital Analysis and Review.
- Reverse stress-testing, deriving the risk factor shifts that must come about to produce a given yield curve and/or risk management calculation result (a VAR level, a net income level, or a given gain or loss).
- Stress-testing each risk factor individually to better understand how the factors drive interest rate risk
- Hedging analysis, which we cover in the next installment of this analysis.
We cover the first four points in this note.
Forward-Looking Analysis and Reverse Stress-Testing
The graph below shows the par coupon yield curve that prevailed in the U.S. Treasury market on March 7, 2014, in blue. The red curve is the par coupon yield curve that will prevail in one quarter, on June 7, 2014.
The red curve could be given to us by someone else, like the Federal Reserve. In that case, it is helpful to understand what changes in risk factors would produce the curve shift from the blue curve to the red curve. Alternatively, we could ask the opposite question: if our 9 risk factors are shocked, what shape of yield curve will that produce? Heath, Jarrow and Morton assume that the interest rate risk factors are independent of each other and that the changes in the factors are normally distributed (over instantaneous time intervals), although the changes in interest rates that result from these risk factor changes need not be normally distributed. Each “shock,” then, is a number of standard deviations by which the risk factors shift from zero, their assumed mean. The assumed standard deviation of each risk factor is 1. The coefficients from the econometric relationship control the absolute size (and shape) of interest rate volatility contributed by each factor. Let’s call these shift factors for the 9 risk factors xi for i = 1, 2,…9. There are 9 observable points on our quarterly yield curve (there are 11 maturities on the H15 statistical release but we only use 9 of them for a quarterly simulation). We have 9 equations in 9 unknowns. For each of the 9 observable points on the yield curve, the quarterly change in the yield at maturity i from time 0 to quarterly period 1 ci (1)= yi(1)-yi(0) is a function of the risk factor shocks:
The form of the function f is determined by the Heath, Jarrow, and Morton no-arbitrage restrictions. If we are given x1 through x9, then we know the yield curve shift; if we are given the yield curve shift, we can derive x1 through x9 because we have 9 equations f1, f2,…f9 in 9 unknowns. Forward-looking simulation and reverse stress testing are linked as the same way as bond prices and yield to maturity are linked.
What shift in risk factors produced the yield curve shift from the blue curve on March 7 to the red curve? Using constant coefficients from 1962 to the present, the answer is given in this chart. The values in red are the number of standard deviations by which each risk factor must shift to produce the red yield curve from the starting point of the blue curve:
We now stress test these factors in sequence and “build up” the yield curve shift from the blue curve to the red curve.
Stress-Testing Individual Risk Factors
We can now shift the factors one at a time, adding one each time, until all 9 factors have shifted and we arrive at the simulated red yield curve for June 7. Note, however, that even if there are no shocks to the yield curve from the risk factors, the yield curve will still shift due to “drift,” just like the vega component in the Black-Scholes options model. Moving upwards from the bottom of this table, we will look at the yield curve after each incremental change to understand the contribution of time and risk factors 1 through 9 to the total movement of the curve.
We start with this chart, which shows the impact of the passage of time alone on the base yield curve. The yield curve on June 7, 2014 takes on the light blue shape if all risk factors are zero:
Next, we add the impact of the shift in risk factor one, which was a negative shift of 0.25298 standard deviations. The risk factors are selected in order of presumed importance, but the absolute shift (shown in green) is small and the result is a slight upward movement in the yield curve from the light blue curve that would have prevailed if all risk factors were zero:
Now we drop the light blue line, the drift curve, from the graph and add another curve that shows the impact of drift and risk factors 1 and 2. The incremental impact of a downward shift of 0.34632 in risk factor 2 moves the curve from the green curve to the yellow curve.
Now we drop the 1 factor curve (in green) and add a curve that reflects the cumulative impact of drift and factors 1, 2, and 3. The shift in risk factor three is another downward shift, this time 1.19472 standard deviations, and moves the curve from yellow to orange. This is one of the biggest contributors to the total movement in the curve.
Continuing with the same pattern, we drop the curve reflecting the impact of 2 factors (in yellow) and add a curve that contains the total impact of drift and factors 1, 2, 3, and 4. The change in the fourth risk factor (from the table above) is another downward shift of 1.14632 standard deviations. The result is another significant move in the curve, this time to the curve in light blue. We are gradually converging to the final shape of the curve, as we should, since the incremental impact of the risk factors on average declines as we move in sequence from factor 1 through factor 9.
We proceed to the next step. We drop the orange line, the 3 factor yield curve, and add the incremental curve that reflects the impact of drift and factors 1, 2, 3, 4, and 5. The change in risk factor 5 is a negative shift of 1.39725 standard deviations. The result is the curve in light green, below, which is very close to the final curve in red.
We now drop the light blue line, the 4 factor model, and add the six factor model. The contribution of the sixth factor is a positive shift of 1.12135 standard deviations, shown in light blue.
We drop the 5 factor model in light green and add the 7 factor model. The shift in the seventh factor is a very small positive shift of 0.05207. The seven factor model is shown in light green and is barely distinguishable from the final curve.
Lastly, we drop the 6 factor model in light blue and add the 8 factor model. The eighth factor has a positive shift of 1.19101 standard deviations, but the coefficients on this risk factor are small and the resulting impact on the shape of the curve, shown in light blue, are again barely discernable in this particular scenario.
After a downward shift in the 9th factor of 0.77044 standard deviations, we end up with the fully shifted curve in red:
Summing Up Insights on Stress Testing Multi-Factor Rate Models
This largely pictorial presentation was intended to show that stress testing, reverse stress-testing, and single factor stress testing for multi-factor no-arbitrage models of the yield curve is both very accurate and straightforward:
- The risk factors can be extracted using the same principles as Cauchy derived more than 300 years ago.
- The resulting risk factors are identifiable points on the yield curve, unlike the unnamed “components” that we saw in the March 5 analysis from principal components analysis.
- The risk factors represent the idiosyncratic contribution that these yield curve points make to the over-all movement of the curve, so there is zero correlation among the risk factors.
- The mathematical formulae linking the risk factors to forward returns has been in the public domain since Heath, Jarrow and Morton first published their model in the late 1980s.
- The coefficients are readily derivable or available from a firm like Kamakura Corporation.
- The coefficients can readily be loaded into and scenarios generated on best practice risk management software.
The result is risk management that is highly accurate, very realistic, reflective of actual experience, and a solid framework for the incremental analysis of default risk, liquidity risk, prepayment risk, mortality risk, and so on.
In the next two installments of this series, we address two important questions:
- Assuming the model used so far is true, how does one hedge interest rate risk in a multi-factor model? How does that hedging procedure compare with the results from a single factor model like “duration,” PVBP, or an extended-Vasicek Hull-White model?
- Especially in a low rate environment, what modifications above and beyond a normally distributed shock to the risk factors might improve accuracy still further?
We close with an appendix on the Heath, Jarrow and Morton fundamentals. Thanks for reading.
Author’s note: The author wishes to thank Prof. Robert Jarrow for his careful comments on this note.
Donald R. van Deventer
March 19, 2014
The Heath, Jarrow, and Morton Approach
In a series of papers originally written in the late 1980s, Heath, Jarrow, and Morton describe a much different analytical process for interest rate modeling that is designed to achieve both of the necessary conditions for interest rate modeling accuracy that we noted above:
- A consistent and accurate reproduction of yield curve history
- The accurate pricing of all relevant observable interest-rate related securities
The Heath, Jarrow and Morton approach is perfectly suited to this task because the authors take the current shape of the yield curve as a given, instead of deriving what the shape of the yield curve must be given a set of mathematical assumptions. The Heath, Jarrow and Morton process for interest rate modeling is as follows:
- Take the current yield curve as given
- Assume a volatility structure for interest rates, including the number of factors and the nature of those factors’ movements
- Constrain the drift in yield curve so that no arbitrage is possible
- Given those constraints, derive the movements in the yield curve
For an excellent introduction to the Heath, Jarrow, and Morton approach, see Jarrow and Chatterjea (2013) and the worked 1, 2, and 3 factor examples in chapters 6 through 9 of van Deventer, Imai and Mesler. For the mathematically confident, the original papers will someday earn the authors the Nobel Prize in Economic Science:
Heath, David, Robert A. Jarrow and Andrew Morton, "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approach," Journal of Financial and Quantitative Analysis,1990, pp. 419-440.
Heath, David, Robert A. Jarrow and Andrew Morton, "Contingent Claims Valuation with a Random Evolution of Interest Rates," The Review of Futures Markets, 9 (1), 1990, pp.54 -76.
Heath, David, Robert A. Jarrow and Andrew Morton, ”Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation,” Econometrica, 60(1),1992, pp. 77-105.
Heath, David, Robert A. Jarrow and Andrew Morton, "Easier Done than Said", RISK Magazine, October, 1992.
Kamakura Corporation provides videos of 50 years of daily U.S. Treasury yield movements that may be of interest to analysts who need to be convinced that multiple points on the yield curve are in fact random:
Par coupon bond yields
Other sources are listed in Chapter 3 of Advanced Financial Risk Management, 2nd edition.