The common assumption that interest rate volatilities are constant (i.e., “affine” term structure models) and the more realistic assumption that interest rate volatility varies with the level of interest rates. Evidence from 9 countries shows that the best practice multi-factor rate dependent volatility models overwhelmingly dominate single factor models from an accuracy perspective. In this note, we use standard model validation procedures to answer this question posed by a prominent U.S. bank regulator: “How does a single factor ‘Regime Change’ term structure model compare in accuracy to the best practice HJM models for the U.S. Treasury curve?”
The Regime Change Term Structure Model
It is well known that common one factor term structure models share one of two problems in modeling risk free yield curve movements. Models that assume interest rate volatility is proportional to the level of interest rates, like those mentioned here, imply that interest rates can never be negative:
Cox, Ingersoll, and Ross[1]
Black, Derman and Toy
Black and Karasinski
Models that assume interest rate volatility is constant, the “affine” class of term structure models listed below, imply that interest rates are normally distributed.
Vasicek
Extended Vasicek/Hull and White
Ho and Lee
The January 12 analysis from Kamakura shows the assumption of constant volatility is false and the implication of normally distributed zero coupon bond yields is therefore inconsistent with the actual distribution of zero coupon bond yields.
Both classes of single factor term structure models share a common problem in modeling yield curve movements. The single factor assumption implies (in the most common implementations) that all yields will either (a) move up together, (b) move down together, or (c) remain unchanged together. The Kamakura analysis explains that this implication is false on 82.95% of the 13,487 days of yield movements in the U.S. Treasury market from 1962 through 2015.
The Regime Change model seeks to overcome immediate model validation problems that would befall the classical single factor models:
- Disqualification for the obviously false assumption that interest rates cannot be negative.
- Disqualification for the obviously false assumption that interest rate volatility does not change when the level of interest rates rise or fall.
A Regime Change model assumes that the single factor driving interest rates has a constant volatility of interest rates when the short term rate of interest is below a critical level Xlow. When the short term rate of interest is above Xlow, the Regime Change model has interest rate volatility that is proportional to the level of the short term rate of interest. As noted by Jarrow [2002] and Heath, Jarrow and Morton [1992], term structure models with proportional interest rate volatility can “blow up” if interest rate volatility is not capped at a maximum level. We impose this cap at a short rate level Xhigh. We can summarize the Regime Change interest rate volatility assumptions as follows:
If the short rate s < x_{low}, volatility = σ_{low}, a constant
If the short rate x_{low} ≤ s ≤ x_{high}, volatility = sσ, where σ is a constant and s is random.
If the short rate s ≥ x_{high}, volatility = σ_{high}, a constant
In short, the Regime Change model behaves like the extended Vasicek model when rates are very low or very high. It behaves like the Black, Derman and Toy model in between these high and low rate ranges. The model therefore implies that zero coupon bond yields are normally distributed when rates are very high or very low. This avoids the troubling implication of “no negative rates” and yet allows interest rate volatility to rise and fall over a broad spectrum of rate levels.
Using standard model validation procedures, we compare the Regime Change model with a list of challenger models to identify the best performing model. Before doing so, we clarify a common misconception of many market participants.
All Term Structure Models Fit the Starting Yield Curve Perfectly
Market participants sometimes assert, “I use the Regime Change model because it fits the current yield curve perfectly.” This statement is not false, but it is not correct either. ALL term structure models which are constructed using the no arbitrage conditions of Heath, Jarrow and Morton [1992] fit the current yield curve perfectly. Moreover, a simulation of forward looking yield movements will correctly value every single starting zero coupon bond yield and coupon-bearing bond underlying the starting yield curve in a Monte Carlo simulation. Therefore, all of the term structure models that we use as challenger models in this note have an identical ability to fit the current yield curve and to produce Monte Carlo simulations that value all initial bond prices perfectly.
Alternative Assumptions about Parameter Fitting
In the early development of term structure models, researchers assumed a known mathematical function specified how interest rate volatility varied at different points on the yield curve. The thirty year U.S. Treasury yield curve consists of one 3 month spot rate and 119 quarterly forward rates. In the Ho and Lee model, the interest rate volatility of all 120 quarter rates is assumed to be equal. In the Vasicek model, the interest rate volatility is assumed to decline exponentially as the maturity of the quarterly forward rate increases. In both cases, the parameters of the interest rate volatility function are the same for all 120 quarterly segments of the yield curve. If we have N observations on the Treasury yield curve and 120 forward rates at each observation date, the econometric analysis would be done on a data base of 120 x N observations. Each observation in this case includes the date of the observation, the forward rate, and the maturity of the forward rate in years or quarters. This procedure produces an adjusted r-squared for the entire yield curve, but it does not produce an accuracy measure for each quarterly segment of the yield curve.
An alternative modeling approach is necessary to test the accuracy of assumptions of a model like Ho and Lee or Vasicek. It is also necessary for a careful analyst who looks to history to show how interest rate varies both over time and by the maturity of the forward rate. It may well be that the sophisticated guesses of Ho and Lee and Vasicek about the change in volatility by maturity are not accurate. In order to confirm how interest rate volatility varies, both over time and by the maturity of the curve, one can derive interest rate volatility separately for each of the 119 random quarterly segments of the U.S. Treasury yield curve (note that the first segment of the yield curve, the 3 month spot rate, is known with certainty on the observation date and is therefore not random). Using this approach, we perform 119 regressions of N observations each, with a separate accuracy measure (adjusted r-squared) for each segment of the yield curve.
We use both methods below and in the Appendix to derive our conclusions.
Implementation of the Regime Change Model
We use the analysis in the appendix to set x_{low} = 0.60% and x_{high} = 8.15%. At both levels, as explained in the Kamakura Technical Guide Appendix A for the U.S. Treasury curve (January 16, 2016), the hypothesis of normality for short term interest rates cannot be rejected even at the 20% level using three popular tests for normality when rates are below 0.60% or above 8.15%. The term structure model parameters and the history of the single factor driving the full yield curve are derived jointly for all term structure models, including the Regime Change model. For mathematical convenience, term structure models are assumed to be driven by Brownian motions with a mean of zero and a standard deviation of one. Using the approaches in Appendix A, we can extract the history of the Regime Change yield curve factor such that
It is normally distributed below 0.60%
It is normally distributed above 8.15%
It has a mean of zero over the full history of the yield curve.
It has a standard deviation of 1 over the full history of the yield curve.
We then run a regression for each forward rate segment that gives us three parameters for each forward segment of the curve: σ_{low}, the σ coefficient from the volatility formula σs, and σ_{high}. The form of the regression equation used is consistent with the Heath, Jarrow and Morton [1992] no arbitrage condition as explained in the appendix.
The result of the regression for the first forward rate segment of the yield curve using quarterly data from 1962 to 2015 is given here:
The coefficient “rclowfactor” is the quarterly interest rate volatility when the short rate is less than 0.60%:
The length of the period in years Δ is 0.25, and when taken to the 3/2 power, the value is 0.125. Solving for the annualized value of σ_{low} = (0.0002069)/0.125 = 0.001655 or 0.1655%, about 16 basis points. When the short rate is 8.15% or over, the coefficient is rchighfactor = 0.0057556, consistent with an annualized volatility of 4.60448%. We allow interest rate volatility to increase linearly, not strictly proportionally, to boost explanatory power of the Regime Change model. Quarterly volatility of the first forward rate when the starting short rate is between 0.60% and 8.15% is
0.0016325 + s(0.0000426)
The variable s is the value of the short rate at the start of the observation. Note that the coefficient of the short rate, rcmidxforward, is not statistically significant but we retain the variable to be as consistent with Black, Derman and Toy as possible in this middle range of rates. The equation for the volatility of the first forward rate’s evolution has an r-squared of 0.9594, close to perfect as one would expect where the underlying single factor is derived from movements in the first forward rate itself. The equation for the second forward rate is given here:
The r-squared for the second forward rate’s movement drops to 0.9037. By the 39^{th} forward rate, the adjusted r-squared has dropped to 0.1748. Problems with the underlying theory of the Regime Change model are becoming obvious by this point on the yield curve but we postpone the discussion of those issues.
At the far end of the 30 year yield curve, the 119^{th} quarter, the r-squared has fallen to 0.0391 and none of the explanatory variables that make up the Regime Change model are statistically significant. Given that 30 year mortgages and securitizations of such loans are the largest single asset class for most banks in the United States, the weak explanatory power of the Regime Change model should be a major concern if competing models offer more power across the full length of the curve.
Initial Regime Change Model Validation Extensions
We summarize the r-squared results the Regime Change model for each of the 119 quarterly forward rates in this graph:
Explanatory power drops below 40% within a 5 year time horizon and it drops to below 20% for maturities of about 7 years and beyond. Beyond the 20 year point, explanatory power is near zero.
We now turn to the root mean squared error (per quarter) of each equation. Note that the vertical scale is the magnitude of the quarterly errors in modeling each forward rate, running from zero up to 40 basis points. Root mean squared error is 0.08% (shown above) for the first forward rate. It gets larger at the longest maturities, reaching 0.338% at the 30 year point.
We now examine the error terms of the 119 forward rate equations fitted to derive the term structure of volatilities for the Regime Change model. Note that, for forward rates maturing past the ten year point, the data set is 138 quarters instead of 215 quarters. We now ask this question: “How many of the pair-wise correlations of the errors by forward rate maturity are different from zero in a statistically significant way?” The answer is that a large majority of the 119(118)/2 = 7,021 pairwise correlations of the errors in modeling forward rates show statistically significant correlation.
This is strong initial evidence of omitted variables in the specification of the Regime Change model. We can further estimate the number of omitted variables by running a principal components analysis on the errors.
The results show that three additional variables are needed to explain at least 90% of the variation in the error terms and that 8 additional variables are needed to explain at least 99% of the variation in the error terms.
This “self-examination” of the Regime Change model gives us cause for concern. Explanatory power weakens considerably as we move longer on the yield curve, and there is strong circumstantial evidence that the Regime Change specification omits at least 8 potentially significant variables.
Challenger Models to the Regime Change Model
We now compare a series of models, one by one, to the Regime Change model. We focus on r-squareds across the yield curve, but additional statistics are available to regulatory agencies and clients of Kamakura Corporation at info@kamakuraco.com. We make a comparison with six one factor term structure models and four multi-factor term structure models:
Single Factor Models
Black, Derman and Toy
Cox, Ingersoll, and Ross
Theoretical Vasicek
Empirical Vasicek
1 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
Ho and Lee
Multi-Factor Models
2 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
3 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
6 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
9 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
Regime Change versus Black, Derman and Toy
Our first comparison is between the Regime Change model and the Black, Derman and Toy model, where interest rate volatility is proportional to the level of the short term rate of interest at the beginning of the observation.
In a reassuring bit of good news for the Regime Change model, the r-squareds for that model (in red) are consistently higher than the matched-maturity r-squareds for the Black, Derman and Toy model (in blue).
Regime Change versus Cox, Ingersoll, and Ross
We now turn to the Cox, Ingersoll, and Ross model, in which yields are driven by interest rate volatility that is proportional to the square root of the short term rate of interest.
The result is a hard-fought battle for accuracy, with the Regime Change model winning narrowly on the short and long end of the yield curve and the Cox, Ingersoll and Ross model winning at intermediate maturities.
Regime Change versus Theoretical Vasicek
Next, we compare the Regime Change model with a no arbitrage implementation of the Theoretical Vasicek model. By “Theoretical Vasicek,” we mean the extended Vasicek model with interest rate volatility that declines exponentially with the maturity of the forward rate segment being modeled.
The Theoretical Vasicek model is slightly more accurate on the very short end of the curve, lags slightly at about 7 years and 15 years, and then is essentially indistinguishable from the Regime Change model at other maturities.
Regime Change versus Empirical Vasicek
For the Empirical Vasicek implementation, the pattern of change in interest rate volatility was fitted econometrically instead of making the (arbitrary) assumption that volatility declines exponentially with the maturity of the forward rate segment being modeled.
The fit versus the Regime Change model is very similar to the fit of the Theoretical Vasicek model. The Empirical Vasicek fit is better in the very short run, lags Regime Change near the 6 year and 15 year points, and is nearly indistinguishable otherwise.
Regime Change versus 1 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
The one factor Heath, Jarrow and Morton model with rate dependent volatility does not arbitrarily split the interest rate range into regimes. Instead, allowing for a mix of both Vasicek and Black, Derman and Toy specifications via the econometric parameter fitting process increases accuracy across the board compared to the Regime Change model.
The gap between the 1 factor HJM Model with Rate Dependent Volatility and the Regime Change model is relatively large and consistent across the maturity spectrum.
Regime Change versus Ho and Lee
The Ho and Lee Model is unique among single factor models in that its focus is the entire yield curve, not the short rate and the short rate’s impact on the curve. As we note in the discussion of principal component analysis in the December 31, 2015 release of the Kamakura Heath, Jarrow and Morton model suite for U.S. Treasuries, it is not clear that the movements in the short term rate of interest are the best “first factor” in modeling yield curve movements.
The fact that the short term rate of interest is not the best “first factor” becomes obvious in testing various “yield curve shift” factors for the Ho and Lee model. We used the shift in forward rates at 9 different maturities as the first 9 factors tested and the average of those 9 shifts as the tenth factor. Not surprisingly, it was that average shift that powered the very high accuracy from 4 years on out.
While beauty is in the eye of the beholder, the Regime Change model is much less accurate than the 30 year old Ho and Lee model. This is a testimony to the intuition of Ho and Lee and to their insight that there are more factors that move the yield curve than the short rate alone.
We prove that in extending the challenger model competition to multi-factor models below.
Regime Change versus 2 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
In the two factor Heath, Jarrow and Morton model with rate dependent volatility, we add as the second factor the idiosyncratic movement in the one quarter forward rate that matures in 10 years.
The result is a very large improvement in explanatory power at both intermediate and long maturities compared to the Regime Change model. We continue to add factors at the margin as long as they have statistical significance.
Regime Change versus 3 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
The third factor added to the Heath, Jarrow and Morton framework is the idiosyncratic movement of the one quarter forward rate that matures in three years.
We get another large boost in explanatory power from the third factor. Of course, explanatory power near the 3 year maturity is the biggest improvement over the two factor model, but we again see significant improvement even on the long end of the curve. The advantage over the Regime Change model has grown even larger.
Regime Change versus 6 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
We expand the number of factors by three in the six factor Heath, Jarrow and Morton rate dependent volatility model. The factors are the idiosyncratic movement in one quarter forward rates that mature in 1, 5, and 7 years. The accuracy that results is plotted here versus the Regime Change model:
The improvement in accuracy is again very large, and the advantage versus the Regime Change model is greater still. The graph shows that the areas that can still be improved are maturities near 2 years and from maturities of 12 years and longer. We make those additions in the next model.
Regime Change versus 9 Factor Heath, Jarrow and Morton with Rate Dependent Volatility
The best practice Kamakura Heath, Jarrow and Morton model has nine factors. The additional factors are the idiosyncratic movements in the one quarter forward rates maturity in 2, 20, and 30 years. The result is an extremely accurate prediction of yield curve movements as a function of these risk factors:
The graph shows that there remains some error in forecasting yield curve shifts near the 18 month maturity point, a time horizon that falls between the 1 year and 2 year “on the run” yields reported by the U.S. Department of the Treasury. A 10^{th} factor at this time horizon can address this issue.
The 9 factor Kamakura Heath, Jarrow and Morton model has accuracy roughly 80 percentage points higher in r-squared than the Regime Change model at maturities of 6 years and over. At short maturities the advantage ranges from 3 percentage points for the first quarterly forward rate to almost 70 percentage points at 5 years.
Conclusions
The Regime Change term structure model combines the extended Vasicek model (at low rates and high rates) with the Black, Derman and Toy model (at intermediate rates) to address two problems with single factor models: the Vasicek and extended Vasicek models imply normally distributed zero coupon bond yields, but those yields are clearly not normally distributed. They are not normally distributed because interest rate volatility is not constant over time, as the Vasicek class of models assume. Interest rate volatility increases as rates rise. The Black, Derman and Toy model (like the Cox, Ingersoll and Ross model) assumes that interest rate volatility is proportional to the level of the short term rate of interest. This specification avoids the false implication of normally distributed yields, but at the cost of making negative interest rates impossible. This is a significant analytical problem in the current economic environment. The Regime Change model, by combining the two approaches in a “Regime Change” that is a function of the short rate, avoids the normality and negative rate problems.
Unfortunately, as we have seen in this model validation exercise, the Regime Change model does not address the most serious problems with one factor term structure models:
The Regime Change model (and almost all other one factor models) imply that all yields either (a) move up together, (b) move down together, or (c) remain unchanged together. As reported in the Kamakura analysis cited earlier in this paper, this implication was wrong on 82.95% of the 13,487 days on which U.S. Treasury bonds were traded from 1962 to 2015. A yield curve twist, which single factor models cannot replicate, is the norm, not the exception. Prof. Robert Jarrow summarized the issue succinctly in the context of the Regime Change model in a recent note: “[The Regime Change model] implies that, given one is in a regime, a one factor model is sufficient to determine the evolution of the term structure. This implies that all rates will be either perfectly correlated or perfectly negatively correlated within a regime. This just isn't true. This should be a grave concern to a serious analyst.”
The Regime Change model, like all other models driven by the short term rate of interest, is extremely inaccurate in predicting yield movements at all but the very shortest maturity forward rates.
The Regime Change model’s accuracy is almost indistinguishable from three term structure models: Cox, Ingersoll and Ross, the Theoretical Vasicek model, and the Empirical Vasicek model. It is clearly more accurate than one model and one model alone: the Black, Derman and Toy model on which it is partially based.
The Regime Change model fails the “effective challenge” required by best practice model validation practices and the Federal Reserve’s Dodd-Frank stress tests (DFAST) and Comprehensive Capital Analysis and Review (CCAR) procedures. It fails because it is significantly less accurate than these challenger models:
Ho and Lee Model, 1 factor
Heath, Jarrow and Morton Model with Rate Dependent Volatility
1 factor
2 factors
3 factors
6 factors
9 factors
All of the challenger models presented in this note conform to the no arbitrage constraints explained by Heath, Jarrow and Morton [1992]. That means both that the yield curves simulated both (a) start from the time 0 observable yield curve and (b) correctly price all time zero bonds in a no arbitrage Monte Carlo simulation. As such, the Regime Change model offers no advantages in this regard.
We conclude that the Regime Change model does not stand up to normal model validation standards.
Appendix A: An Introduction to Term Structure Model Construction and Accuracy Assessments
Kamakura Corporation provides regular updates of multi-factor term structure models in major bond markets around the world. The data is provided by Kamakura Corporation’s Kamakura Risk Information Services group, and the resulting parameters and supporting documentation are available by subscription. These government securities markets have been reviewed in prior notes and are available at these links:
Australia Commonwealth Government Securities
Canada Government of Canada Securities
Germany German Bunds
Japan Japanese Government Bonds
Singapore Singapore Government Securities
Spain Spanish Government Bonds
Sweden Swedish Government Securities
United Kingdom United Kingdom Government Bonds
United States U.S. Treasury Bonds
In all nine of these studies, one factor models failed basic model validation tests and were judged unacceptable from an accuracy point of view.
Term Structure Model Parameter Derivation
The procedures used to derive the parameters of the single factor models discussed above and the multi-factor Heath, Jarrow and Morton model are described in detail in these documents:
Jarrow, Robert A. and Donald R. van Deventer, “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Version 2.0, Kamakura Corporation, June 30, 2015.
Jarrow, Robert A. and Donald R. van Deventer, Appendix A, Version 1.0: “U.S. Treasury Yields,” to “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Kamakura Corporation, January 2016.
Jarrow, Robert A. and Donald R. van Deventer, “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model,” Technical Guide, Version 4.0, Kamakura Corporation, June 16, 2015.
We followed these steps to estimate the parameters of each term structure model discussed above:
- We extract the zero coupon yields and zero coupon bond prices for all quarterly maturities out to 30 years for all daily observations for which the 30 year zero coupon yield is available. For other observations, we extended the analysis to the longest maturity available, 10 years. This is done using Kamakura Risk Manager, version 8.1, using the maximum smoothness forward rate approach to fill the quarterly maturity gaps in the zero coupon bond data.
- We drop the daily observations that are not the last observation of the quarter, to avoid overlapping quarterly observations and the resulting autocorrelated errors that would stem from that.
- We calculate the continuously compounded changes in forward returns as described in the parameter technical guide.
- We then begin the process of creating the orthogonalized risk factors that drive interest rates. These factors are assumed to be uncorrelated independent random variables that have a normal distribution with mean zero and standard deviation of 1.
- In the estimation process, we added factors to the model as long as each new factor provided incremental explanatory power.
We use the resulting parameters and accuracy tests to conduct a model validation exercise for each model.
Functional Form Used for Term Structure Estimation
It is well known that zero coupon bond yields, zero coupon bond prices, continuous forward rates, discrete forward rates, and forward returns (one plus the uncompounded discrete forward rate) are linked mathematically. For discrete modeling, the formulas are summarized in the Kamakura Technical Guide “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model,” Appendix 3.
As explained by Jarrow [2002], it is convenient both from a mathematical and expository point of view to work with forward returns. The “forward return” at time t for a one period uncompounded forward rate f(t,t+jΔ) maturing in j periods of length Δ is simply F(t,t+jΔ)=1+f(t,t+jΔ). We emphasize again that the forward rates are uncompounded and unannualized and they are expressed as a decimal, not as a percent. In Jarrow’s equation 15.4, which is reproduced on page 8 of “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model,” the link between the one period forward return of a given maturity in the next period is a function of the current one period forward with the same maturity date:
The current time is time t. The length of the discrete periods, in years, is Δ. The forward returns are related to zero coupon bond prices as follows:
The full explanation of the notation is given in Appendix 2 in “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model.” The term μ is the risk neutral drift for a one period holding period for a one period forward return maturing at time t+jΔ. The risk neutral drift μ is determined by the interest rate volatilities σ which come out of the estimation process. The formula for μ is given beginning on page 8 of “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model.”
The regression formulation for parameter estimation is a modification of the equation above. We divide the expression above by F(t,t+jΔ) and then take the natural logs of both sides. We define the left hand side variable, the continuously compounded change in the one period forward return of the given maturity, as “cReturn”:
If we have both the functional form of the interest rate volatilities σ and a time series of the random shocks dW, we can estimate the parameters of the interest rate volatilities. For the simplest case, the interest rate volatilities for an n factor term structure model are n scalar constants σ_{i} . In that special case we can run the ordinary least squares regression, one regression for each of the forward return segments underlying the current yield curve. This gives 119 regressions
where e is the random error term. The β coefficient values allow us to solve for the n factor sigma values (which change by maturity of the forward return) using the following formula:
The parameter α is not the risk neutral drift μ, because we are using the empirical forward returns in our estimation, not the risk neutral forward returns. Instead, α is an estimate of the empirical drift in the one period forward return maturing in j periods of length Δ from current time t. The use of empirical drifts to construct the term premium above and beyond the expected future level interest rates that is reflected in the current yield curve is discussed in a separate series of papers by Kamakura Corporation.
Example: Fitting the Theoretical and Empirical Vasicek Models
Both the Theoretical and Empirical Vasicek models assume the following:
One factor drives the entire yield curve
That factor is the random variation in the “short rate,” which (to be precise) is the change in the (random) first quarterly forward rate (initially maturing six months from now) and the spot 3 month rate that will prevail in 3 months (since the forward rate ultimately moves forward in time to become the spot 3 month rate).
The change in this forward rate as it becomes the 3 month spot rate must be consistent with the no arbitrage formulation above.
There are different interest rate volatilities associated with each maturity of forward rate, but these volatilities do not change over time.
We perform the following steps.
Step 1: We assemble the time series of continuously compounded changes in forward rates for the first forward rate. We label this variable cReturn[1].
Step 2: We run a regression with a constant term and no explanatory variables to isolate the random component of cReturn[1], the error terms e in this regression:
Step 3: We define our single risk factor dW in this equation, which normalizes the error terms to have mean zero and standard deviation of 1:
Step 4: For the Empirical Vasicek model, we let the data tell us how interest rate volatility changes as the maturity of the forward rate changes. Accordingly, we run this regression for each forward rate with maturity i, which runs from 1 through 119 quarters. We denote the error term as e* to distinguish from the errors e above:
Step 5: We solve for the annualized volatility σ_{i} for each forward rate i=1,119 using the beta from the regression:
Step 6: We report the r-squared from this regression for model validation (shown in the graphs above) and store the sigma values for simulation. We calculate the risk neutral drift μ from the sigma values per Heath, Jarrow and Morton [1992].
Step 7: For the theoretical Vasicek model, we do not use the volatilities we have just derived. Instead, we are told that the interest rate volatilities decline exponentially according to the following formula where i is the number of periods of length Δ to maturity of the one period forward rate:
In Step 4, we ran 119 regressions with N observations each. Using the formulation in Step 7, we need to run just 1 regression with N x 119 one period forward rates with 119 different maturities.
Step 8: We extract the coefficients b and c from this non-linear regression, which can be estimated in any number of ways. In the model validation above, we used non-linear least squares:
Step 9: We solve for the 119 sigma values using this formula and our new-found knowledge of the parameters for b and c.
Step 10: We report the r-squared from our non-linear equation fitting for model validation. If we need the accuracy by each forward rate segment (as we did above), we predict the values of cReturn[i], calling them pcReturn[i], and get the r-squared from 119 linear regressions of this form:
Example: Fitting the Cox, Ingersoll and Ross Model
We follow a similar process for the Cox, Ingersoll and Ross (“CIR”) model. The model assumes
One factor drives the entire yield curve
That factor is the random variation in the “short rate” s
There are different interest rate volatilities associated with each maturity of forward rate. These volatilities change over time, because they are assumed to be proportional to the square root of the short rate s.
Step 1: Generate a time series equal to the square root of the short rate:
Step 2: Create the single time series consisting of the ratio of cReturn[1] (the first forward rate’s change in return) and the square root of s, which we label Z:
Step 3: Run this regression with no constant term and no explanatory variables to extract the random component of the adjusted cReturn[1], Z[1]:
Step 4: Define the yield curve factor dW from the error term, normalizing as above so that the mean of the factor is zero and the standard deviation is 1:
Step 5: Run this regression, where the explanatory variable is the product of the square root of the short rate and the yield curve factor dW. There will be one regression (and therefore different volatilities), which may or may not include a constant term, for each of the 119 quarterly forward rate maturities:
Step 6: The interest rate volatility for each quarterly segment i of the yield curve is
The parameter σ_{i} for each yield curve segment is defined by solving this equation from the β parameter values:
Deriving Other Term Structure Model Parameters
The process for deriving the parameters of other term structure models is similar. For additional information, please contact Kamakura Corporation at info@kamakuraco.com.
Appendix B: Further Reading for the Technically Inclined Reader
References for random interest rate modeling are given here:
Black, Fischer, E. Derman, W. Toy, "A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options," Financial Analysts Journal , 1990, pp. 33-39.
Black, Fischer and Piotr Karasinski, "Bond and Option Pricing when Short Rates are Lognormal, Capital Standards: Interest Rate Risk, " Financial Analysts Journal,1991 pp. 52-59.
Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, "A Theory of the Term Structure of Interest Rates," Econometrica, 1985, pp. 385-407.
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.
Ho, Thomas S. Y., and Sang-Bin Lee, "Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance 41, December 1986, pp. 1011-1029.
Hull, John and Alan White, "Pricing Interest-Rate Derivative Securities," Review of Financial Studies, 3 1990b, pp. 573-592.
Hull, John and Alan White, "One Factor Interest Rate Models and the Valuation of Interest Rate Derivative Securities,” Journal of Financial and Quantitative Analysis, 1993, pp. 235-254.
Hull, John and Alan White, "Numerical Procedures for Implementing Term Structure Models I: Single-Factor Models," Journal of Derivatives, 1994a, pp. 7-16.
Hull, John and Alan White, "Numerical Procedures for Implementing Term Structure Models II: Two-Factor Models," Journal of Derivatives, 1994b, pp. 37-48.
Jarrow, Robert, Modelling Fixed Income Securities and Interest Rate Options, second edition, Stanford University Press, Stanford, California, 2002.
Longstaff, Francis A. and Eduardo S. Schwartz, "A Two Factor Interest Rate and Contingent Claims Valuation," Journal of Fixed Income, Vol. 2, No. 3, Dec. 1992, pp. 16-23.
Shimko, David C., Naohiko Tejima, and Donald R. van Deventer, “The Pricing of Risky Debt when Interest Rates are Stochastic,” Journal of Fixed Income, September, 1993, pp. 58 – 66.
Vasicek, Oldrich A, "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, 1977, pp. 177-188.
References for term structure parameter estimation are given here:
Jarrow, Robert A. and Donald R. van Deventer, “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Version 2.0, Kamakura Corporation, June 30, 2015.
Jarrow, Robert A. and Donald R. van Deventer, Appendix A, Version 1.0: “U.S. Treasury Yields,” to “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Kamakura Corporation, January 2016.
Jarrow, Robert A. and Donald R. van Deventer, “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model,” Technical Guide, Version 4.0, Kamakura Corporation, June 16, 2015.
van Deventer, Donald R. “An Updated Multi-Factor Heath Jarrow and Morton Model For U.S. Treasuries, 1962-2015,” Kamakura Corporation memorandum, January 12, 2016.
References for non-parametric methods of model testing are given here:
Bharath, Sreedhar and Tyler Shumway, "Forecasting Default with the Merton Distance to Default Model," Review of Financial Studies, May 2008, pp. 1339-1369.
Jarrow, Robert, Donald R. van Deventer and Xiaoming Wang, “A Robust Test of Merton’s Structural Model for Credit Risk,” Journal of Risk, fall 2003, pp. 39-58.
References for modeling traded securities (like bank stocks) in a random interest rate framework are given here:
Amin, Kaushik and Robert A. Jarrow, "Pricing American Options on Risky Assets in a Stochastic Interest Rate Economy," Mathematical Finance, October 1992, pp. 217-237.
Jarrow, Robert A. “Amin and Jarrow with Defaults,” Kamakura Corporation and Cornell University Working Paper, March 18, 2013.
[1] Professor Jarrow notes, “CIR models are nonnegative if a restriction on the parameters is imposed. The usual model has this restriction imposed.”