Can You Derive Market Volatility Forecasts from
the Observed Yield Curve Convexity Bias?
Wesley Phoa
Vice President, Research
Capital Management Sciences
Suite 300, 11766 Wilshire Boulevard
Los Angeles, CA 90025
phone (310) 479 9715
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ABSTRACT
In theory, long bond yields incorporate a convexity adjustment reflecting anticipated volatility in the bond market, making it possible to deduce the market's expectations about longterm volatility from the shape of the yield curve. This has been carried out using a four factor yield curve model derived from an expectations theory of the term structure, which can accommodate a bond market risk premium. Analysis of 19531996 US Treasury bond data shows that expected volatility is sometimes efficiently priced into bond yields, and sometimes barely priced in at all: there are two distinct regimes which seem to alternate.
INTRODUCTION
The yield on a 30year Treasury bond is often, but not always, lower than the yield on a 20year bond. A common and plausible explanation is that since the 30year bond has more convexity than the 20year bond, it will tend to outperform when bond yields fluctuate. Thus investors pay for convexity by accepting a lower yield.
The convexity bias in observed yields has been the subject of much attention in the swap market: see Burghardt and Hoskins [1995] for a popular exposition. In this case, it is observed that the 10year swap rate is significantly lower than the yield on an equivalent 10year strip of Eurodollar futures. This is because the swap has convexity while the futures position does not; a dealer who is receiving swap and short Eurodollar futures benefits from being long convexity.
There are various rules of thumb for computing an appropriate yield adjustment to take this convexity effect into account; for example, a simple but robust formula is reproduced in section I of the present paper. Exhibit 2 graphs the convexity adjustment versus bond maturity, based on this rule of thumb and assuming an absolute yield volatility of 80 bp per annum. One could also take a more rigorous approach, and derive the theoretical convexity bias using a formal term structure model.
Since convexity only delivers benefits if yields are volatile, the size of the convexity adjustment depends on how volatile yields are expected to be. There are thus two approaches one can take. First, one can estimate volatility separately, and then use this to calculate what the appropriate convexity adjustment should be.
Alternatively, one can observe actual swap or bond yields, and attempt to deduce what level of volatility is implied by those yields, i.e. market participants' expectations about future (medium to long term) volatility, as reflected in observed yields. In this paper, we refer to such a volatility estimate as the implicit volatility.
Implicit volatility is clearly different from historical volatility. The market's expectations about future volatility are only partly determined by historical experience. However, since changes in longterm volatility  as opposed to shortterm volatility spikes  tend to be triggered by structural shifts in the economy, a large discrepancy between historical volatility and implicit volatility should be regarded as surprising, and worthy of further investigation.
Implicit volatility may also be different from the implied volatilities observed in the interest rate derivatives markets. The value of the convexity in a long bond is only fully realized over a long period of time  10 years or more. (Note that it is irrelevant that the bond may change hands over this period, provided the potential future value of convexity is priced into each transaction.) Short dated implied volatilities of bond futures options are clearly irrelevant; cap/floor volatilities are not directly relevant, because they are the implied volatilities of money market rates and not bond yields. A comparable implied volatility would be that on (say) a 10year swaption on a 10year swap, but prices on such longdated contracts are not routinely quoted.
Furthermore, because the importance of the convexity bias has only been appreciated relatively recently, there may well be a discrepancy between implicit volatility and OTC option implied volatilities even where they are directly comparable, such as in the swap market. In theory, this gives rise to an arbitrage opportunity; in practice, it may be difficult to exploit a convexity arbitrage efficiently.
This paper focuses on bond market data and attempts to estimate implicit volatility from long bond yields. For reasons explained in section I  briefly, the need to incorporate the effect of overall yield curve slope, to discount the effect of yield anomalies on specific bonds, and to take into account the sometimes aberrant behavior of the short end of the yield curve  the method adopted may appear somewhat roundabout. Section I describes a method of parametrizing yield curve data based on a formal, economically justified theory of the structure of investor expectations about future bond market returns; the parameters are short and longrun expected returns, a rate of adjustment coefficient, and the implicit volatility which we wish to estimate. Section II describes the estimation procedure, a constrained nonlinear optimization problem which turns out to require some care. Section III presents the results. Section IV relates the results to historically observed bond yield volatilities and presents some possible interpretations, while the conclusion outlines some possible implications for long bond trading strategies with a medium term horizon, and some future areas of research. There is also a brief discussion of alternative theories of the term structure.
An attempt has been made to use methods which are as simple and direct as possible. While there is some subtlety in the formulation of concepts, the technical details are mostly straightforward. In particular, the exposition should be readily accessible to investment managers who are unfamiliar with the mathematics of term structure models. However, readers should bear in mind that although section I gives the economic motivation for the method used, the true justification lies not in an economic or mathematical derivation but in the empirical results in sections II and III.
This study makes use of constant maturity Treasury yields provided by the US Treasury (see Exhibit 1). This made it possible to study a relatively long history, going back to 1953, and thus extending over several radically different periods in the development of the US fixedincome markets. Note that although some yield histories are available back to 1921, long bond yields for specific maturities are not provided, making it impossible to extend the study beyond 1953 using this data.
The author wishes to thank Charles Belvin, Richard Mason and Michael Shearer for fruitful discussions. In particular, an important initial motivation for this paper was an attempt to determine what was missing from a formal expectations theory advanced by Richard Mason. In the process, unfortunately, some of the more elegant aspects of Mason's theory have had to be abandoned.
I. FORMALIZATION OF THE EXPECTATIONS THEORY
Although the convexity bias in long bond yields is most apparent in the 2030 year yield spread, there are a number of reasons why it is undesirable to adopt a method of analysis which focuses solely on this spread. The most obvious reason is that this would restrict us to studying periods where both 20year and 30year historical yields were available, severely constraining the historical scope of the analysis.
The second reason is that we need to take the overall slope of the yield curve into account. If the yield curve as a whole is nearly flat, the convexity bias will lead to an appreciable downward slope from the 20year to the 30year point; if it is steeply positive, and the same convexity bias is present, the curve may be flat or even upward sloping from 20 years to 30 years. Thus short bond yields are relevant to the analysis.
The third reason is that any analysis which relies too heavily on the yields of specific bonds is vulnerable to yield anomalies which affect those particular bonds. These may arise from temporary supply/demand or liquidity conditions, or may be more persistent (as in the case of the ontherun 10year note: Carayannopoulos [1996]). Thus it is desirable to somehow take all bond yields into account in the analysis.
An important caveat is that very short dated bond yields must be used with caution. These often show low correlations with other bond yields, and principal component analysis suggests that there are factors which specifically affect the short end of the yield curve, such as the formation of humps. This is consistent with market experience: the short end of the curve is often "distorted" when current monetary policy settings are perceived as aberrant. Furthermore, market segmentation, the relatively low liquidity of short maturity bonds and the interaction with the Eurodollar futures market all make the behavior of this part of the curve more complex. These phenomena, while interesting, are not relevant to the measurement of the convexity bias at the long end of the curve and may well interfere with the analysis by introducing instability into the measurement of yield curve slope.
The approach adopted here is to use observed bond yields to construct a smooth "theoretical" yield curve, which eliminates yield anomalies and shortend aberrations as far as possible. Note that naïve spline methods are ruled out, since the results would then be sensitive to the (arbitrary) choice of knot points; a further problem with splines is that they are inappropriate if too few bond yields are available. Instead, the smoothing process is based on an analysis of the theoretical structure of investor expectations about future money market yields.
In order to derive a formal expectations theory for the shape of the yield curve, we will: (a) define the "long rate" and the "short rate", and regard the forward rate curve as linking these two rates; (b) observe that the structure of investor expectations determines the functional form of the forward rate curve; (c) show how a bond market risk premium may be incorporated, and (d) show how a convexity adjustment may be incorporated. Although the technical details are very simple, it is worthwhile explaining the conceptual foundations in some detail, since this critically affects the interpretation of the results.
The long rate is defined to be the expected longterm return on bonds. That is, it is the "expected return on a bond with infinite duration", i.e. the limit of the expected return on a zero coupon Treasury bond as the maturity approaches infinity. The long rate is not equal to the limiting zero coupon yield, since yields incorporate a convexity adjustment. The long rate is determined by the market's expectations of long run inflation and growth in the economy.
The short rate is defined to be the expected shortterm return on bonds; this assumes that since the bond market is liquid and homogeneous, any difference in expected shortterm returns between bonds will be arbitraged away. The short rate need not be equal to any specific bond yield, since the expected shortterm return on a bond incorporates not just an income effect but an expected price effect as reflected in observed forward bond prices. The short rate is determined by the market's expectations of imminent monetary policy, as influenced by near term inflation and growth prospects; clearly, it is related to the short term money market yield.
Note that the long rate l is not the consol yield: it is "longer" than a consol, because a consol has shortdated cashflows. Also note that the short rate s is not a money market yield, but must be interpreted as a bond market return which does not take account of a misalignment between bond market expectations and current monetary policy. For example, if the Fed Funds rate is currently 5% but the market expects that a 1% tightening is imminent, the short rate is likely to be closer to 6% than 5%. Duffie and Kan [1993] remark that the short rate should be regarded as a limiting, rather than an actual, yield; a key insight due to Mason is that in constructing a model of bond yields, the short rate should be regarded as a limiting bond yield, factoring out the idiosyncrasies of the short term money market.
Ignoring convexity adjustments for the moment, the short rate and the long rate can be thought of as being joined by an instantaneous forward rate curve g(t), so that _{ } and g(t) can be integrated appropriately to obtain any specific spot or forward bond yield. (Note that g(t) may be viewed as the forecast short rate at time t.) The hypothesis made in this paper is that the shape of the forward rate curve is determined by the market's expectations of future bond market returns, adjusted for risk. A number of steps are required to spell this out.
First assume that expected bond market returns are equal to expected money market yields; g(t) may then be interpreted as the expected money market yield at time t. Make the further assumption that the long run money market yield _{ } is equal to the sum of the long run expected inflation rate and the long run expected real interest rate, which must be true in equilibrium. Frankel [1995] (the original paper appeared in 1982) shows that a wide variety of macroeconomic models  those including a moneydemand equation and a priceadjustment equation  imply that the rationally expected path of money market yields g(t) must have the functional form:
_{}
That is, assuming that investors form their expectations in accordance with a reasonable macroeconomic model  which may occur on a conscious or unconscious level  then their expectations about future money market yields must take this form, at least beyond the near term. In the near term, as pointed out above, more detailed knowledge about likely monetary policy can result in a more complex structure of shortterm rate expectations.
Note that the result even holds for macroeconomic models which allow future random disturbances to the level and trend of the money supply, provided these have expectation zero. Also note that we have not assumed that the economy is correctly described by a macroeconomic model of this kind  we have only assumed that investors (explicitly or implicitly) believe that it can be so described, at least when forming their expectations. Given the intuitive foundations of models of this kind, this appears to be an initially plausible assumption.
The coefficient k is a rate of adjustment parameter, or a mean reversion parameter, which is derived from certain elasticity parameters in the macroeconomic model (which say, for example, how quickly prices respond to excess demand). Note that the value of k is determined by the structure of the economy itself, and is unrelated to the fundamental determinants of interest rates such as the economic cycle. In forming expectations, then, investors would not expect k itself to vary over time.
As explained above, s = g(0) should not be identified with the current money market yield; rather, it is derived from the bond market's prediction of the "imminent" money market yield, which factors out any perceived shortterm aberrations in monetary policy. Put differently: when the bond market perceives current money market yields as reflecting a disequilibrium state, s will reflect its judgment of where the short term equilibrium money market yield should be, based on its assessment of current economic conditions.
In practice, there is arguably a risk premium for holding Treasury bonds instead of money market securities (quite distinct from a convexity adjustment, which applies only to long bonds and has the opposite sign). For example, Ilmanen [1996] finds that although the historical risk premium is extremely difficult to estimate, it does appear to exist: moreover, it is approximately constant across bonds with maturities of 3 to 20 years. Our framework can easily accommodate an expected bond market risk premium: one can add a risk premium to s and l. Note that the expected risk premium need not be constant by maturity, but it is constrained to vary smoothly with maturity, with the same rate of adjustment parameter k.
The final step is to determine how to incorporate a convexity adjustment to obtain an assumed forward rate curve f(t); to do this, we will derive a rule of thumb which is commonly employed in the swap market. Consider an arbitrary bond without embedded options. If the yield of the bond at some fixed future date t is y, write the bond price as B(y). For example, the current tforward price of the bond is:
_{}
where _{ } is the current tforward yield.
Now, the expected bond price at time t is just _{ }; but if the bond yield is volatile, it is not the case that _{ }; in fact, the difference between the forward yield and the expected future yield is precisely the convexity adjustment.
If the actual yield of the bond at time t is y rather than _{ }, then we can write:
_{}
where D is the duration of the forward bond position and C is its convexity. Taking expected values, this gives us:
_{}
Now _{ } where _{ } is the expected basis point volatility of long bond yields. Therefore, by cancellation,
_{}
In particular, the convexity adjustment for a FRA is approximately ½s^{2}t^{2}. This gives the relationship between the expected future short rate and the forward rate at a future time t. To summarize, then, our expectations theory states that the forward rate curve has the functional form:
_{}
where s is the expected absolute volatility of long bond yields. The four parameters need to be estimated from a given set of Treasury bond yields, i.e. a given yield curve. Note that the presence of the mean reversion parameter means that this model does not fit into the framework of Duffie and Kan [1993].
This formulation has the undesirable feature that very long forward rates become negative, causing very long forward bond prices to explode. However, this is not a significant problem in practice. Taking l = 6% (i.e. a historical low for the post1970 period) and s = 0.75% (corresponding to 12.5% proportional volatility, a high estimate for longterm volatility), forward rates only become negative at a forward date of 46 years, and the 30year forward rate is approximately 3.5%. Thus, it may be hoped that the model gives reasonable answers when applied to Treasury bonds, although more sophisticated methods would have to be devised to analyze 100year bonds.
More reasonable asymptotic behavior could be obtained by assuming that long bond yields were mean reverting. Unfortunately, this introduces another parameter which is extremely difficult to estimate  much more so than k, which is itself hard to estimate, as pointed out in the next section. Moreover, the mean reversion time scale would necessarily be long (for example, long bond yields broadly trended upwards from 1953 to 1982, cf. Exhibit 5). Thus, the practical impact on the analysis of Treasury bond yields would be minimal.
The point is that even though the above formulation has implausible asymptotic properties, it may still do a good job of modeling the investor expectations which determine Treasury bond yields. In fact, as a model of the structure of investor expectations, it is probably more plausible than one based on a mathematically sophisticated term structure theory with a larger number of unobservable parameters, even if the latter is mathematically more consistent. The test, of course, is to fit the model to historical Treasury yields and see whether the results are meaningful.
Note that the model can be fit to a single yield curve, i.e. to crosssectional data. In particular, we do not need to make any assumptions about the time series properties of Treasury yields in order to estimate implicit volatility (or the mean reversion coefficient) using the model. Besides making the methodology much more robust, this also means that much more information can be extracted from the data, in the form of historical time series of parameter estimates.
II. ESTIMATION PROCEDURE
The forward rate curve in the previous section can be integrated to obtain the following formulas for the discount function and par yield curve:
_{}
_{}
In principle, then, the estimation problem is simple: given a set of CMT yields, find _{ } which give the best fit to those yields. The first question is: what is the meaning of "best fit"? This paper adopts the normal procedure of initially using _{ } estimation, and then reviewing the distribution of errors to determine whether this was in fact a reasonable thing to do.
It is helpful to make some observations before proceeding. Firstly, provided the model of investor expectations is realistic, one would not expect serious outliers. Deviations from the model may be thought of as indicating anomalous liquidity or supply/demand conditions, and under this interpretation a discrepancy no larger than 515 bp (and generally less than this for longer bonds) would be expected. Thus it should not be necessary to use more robust statistical estimators.
However, as noted above, the model need not be realistic for shortdated bonds. Thus, if 1year CMT yields are included in the data set, much larger discrepancies would be expected under certain circumstances  for example, at times where there was a sharp divergence between Fed policy and the market's views. It should also be noted that, in economic terms, the same basis point error is less significant for a shorter than for a longer bond. It might thus be possible that somewhat larger discrepancies could arise for shorter bonds.
The solution adopted was to weight basis point errors by bond duration, i.e. to minimize price errors rather than yield errors. This turns out to give empirically good results for 230 year bond yields. It gives the 1year CMT minimal weight without discarding it from the data set altogether.
The present study implemented _{ } estimation using the routines provided in the Optimization Toolbox of MATLAB^{â}: see Grace [1994] (the initial reference for the algorithms referred to below). These proved to be satisfactorily robust and efficient, and have the advantage of providing useful diagnostics; for example, they report condition numbers of any severely illconditioned matrices which arise.
It might seem most natural to regard this as a constrained nonlinear optimization problem, where natural constraints are _{ }. Negative longterm rates are economically unreasonable; negative shortterm rates have only ever existed for brief periods, under very special circumstances; and negative volatilities or mean reversion coefficients makes no sense. Efficient methods for solving constrained problems have been developed relatively recently.
Unfortunately, the presence of k as a fitted parameter in the model specification considerably complicates the estimation process. Bond yields are affine functions of l, s and s^{2} but not k. For very small and very large k, derivatives involving k are badly behaved, leading to severe illconditioning in methods which make use of numerically computed gradients. The k term also makes the behavior of any such algorithm overly sensitive to the choice of initial guess. For this reason, it was decided to solve the unconstrained problem first, using the robust NelderMead simplex algorithm: see Press et al. [1992].
As expected, this algorithm converged in all cases but often gave meaningless results, including negative volatilities and/or absurdly large values presumably resulting from an unlucky choice of initial guess. However, where convergence to a meaningful set of values was achieved, k generally lay between 0.1 and 2 and s was generally less than 2%. Thus, this exercise did establish that it was valid to impose the constraints _{ }, _{ } and _{ }. This has the effect of restricting the search to economically reasonable values, and of excluding regions where the k derivatives are badly behaved.
The next step was to solve the new constrained problem using a sequential quadratic programming algorithm, which employs an active set strategy at each iteration. Because of the constraints on k, this algorithm converged rapidly in all cases, and yielded intuitively reasonable long and short rates (as illustrated in Exhibit 5). Furthermore the time series of fitted long and short rates do not exhibit more volatility than the time series of observed long and short bond yields, implying that the estimation process is stable.
Exhibit 3 shows histograms of basis point errors for the 5year and 10year bond; note that except in the case of the 1year CMT yield, errors were generally small (less than 10 bp 90% of the time); also, errors seem to be approximately normally distributed, which suggests that it was reasonable to employ _{ } estimation. There was no bias except for the 10year CMT yield, which was generally overestimated by the model; this however is consistent with the findings of Carayannopoulos [1996] regarding the systematic overpricing of the 10year Treasury note. Exhibit 4 shows the time series behavior of the error for the 10year and 30year CMT yield; the error appears to be a stationary process which is uncorrelated with the level or slope of the yield curve, or with any other obvious economic variable. It may be concluded that the yield curve model we have adopted is a realistic one.
Interestingly, the distribution of estimates of k is approximately bimodal: values cluster roughly around 0.1 (the lower constraint) and 0.5 (using an initial guess of 0.4), with the latter peak being significantly more important. Examination of the time series shows that, apart from shortterm spikes, k only moves significantly away from 0.5 in the periods 19591969, when it was generally close to 0.1, and in the period 19911996, when it fluctuated between 0.1 and 0.75: see Exhibits 6 and 8.
It is possible to view k as a curvature parameter: the predicted degree of curvature of the yield curve will be roughly proportional to k(l  s). This means that estimates of k are primarily influenced by 3year to 7year bond yields. By contrast, estimates of s are mainly influenced by 10year to 30year bond yields. Thus, if we are interested in estimating s, then variations in k should not make too much difference  although in principle all four quantities are coupled.
In the light of this observation, the constrained problem with k = 0.5 was solved using a standard LevenbergMarquardt algorithm. This converged rapidly in all cases, and always gave meaningful estimates for the long and short rates. In most cases it also gave estimates for implicit volatility consistent with the previous estimates, though there was some divergence when the yield curve was very steep. However, in the period 199396, the estimates for s (setting k = 0.5) were about 50% of the estimates for s (allowing k to be a fitted parameter). That is, when the "actual" value of k diverges significantly from 0.5, as it did during most of this period, a procedure which assumes that k = 0.5 can generate consistently inaccurate estimates for s.
Thus, as one might expect the rate of mean reversion should not generally have a significant impact on the pricing of long bonds, provided that a "ballpark" value is assumed (but 0.5 is not always a suitable "ballpark" value). That is, the problem of estimating implicit volatility is more robust than it might have appeared at first. However, estimating the mean reversion coefficient as well as the other parameters using crosssectional data is still likely to give more reliable results.
It is perhaps fortunate that accurate estimates for k are not required, since estimating this quantity is an inherently unstable problem  partly because, when the yield curve is flat or nearly flat, almost any value of k will give rise to the same theoretical yields. Note that the estimation procedure adopted here, which weights errors by duration and hence gives a high weight to long bond yields, probably does not generate the most accurate possible estimates of k.
III. RESULTS, 19531996 US TREASURY DATA
Exhibit 7 shows the monthly estimates of implicit volatility for the period 19531996. Exhibit 9 shows the daily estimates for the period February 1977September 1996. All results are expressed in percent per annum. As one would expect, the results are quite consistent.
The major observation is that implicit volatility is not constant, nor does it seem to be a continuously varying random quantity. Instead, it seems to be either switched "on" or "off": there appear to be two distinct regimes (see Exhibit 10).
Exhibit 11 compares implicit volatility with historical 10year bond yield volatility, both expressed in percent per annum. (Implicit volatility is smoothed by taking an annual median, to make the graph easier to read.) Historical volatility is exponentially weighted with a decay factor of 0.9, and is thus a reasonably longterm measure, while still responding quickly to volatility shocks.
Implicit volatility can be switched "off" even while bond yields are still quite volatile. However, a volatility shock tends to switch implicit volatility "on". The most recent examples of this occurred in 197980 and 199394.
The other interesting observation is that in basis point terms, historical volatility tends to be higher when yields are higher, while the level of implicit volatility (when it is "on") seems to be independent of absolute yield levels. Remembering that implicit volatility is meant to encapsulate the market's volatility expectations over long time frames, during which outright yields will vary greatly, it seems a priori reasonable that the fact that yields are currently high or low should not unduly bias implicit volatility, and this does in fact appear to be the case.
If one believes that the volatility of yields is proportional to the outright level of yields, then an implicit volatility of 7080 bp is consistent with the hypothesis that the market expects longterm proportional volatility to be around 11%, and that long bond yields will fluctuate around 7.5%, i.e. are mean reverting (cf. the comment at the end of section 1). But the evidence for this interpretation is not very strong.
Although complete data was not available for this study, it appears that implicit volatility is not closely related to longdated OTC implied volatilities. This is not surprising, for the reasons mentioned in the introduction.
Finally, the mean reversion coefficient itself seems to exhibit a kind of regimeswitching behavior. During the periods 195357, 197292 and 1994 it appears to fluctuate around 0.5, while during 195871 and 199396 (excepting 1994) it appears to fluctuate around 0.10.2. A very tentative interpretation would be that a low value of k appears to coincide with longterm expectations of strong growth coupled with low inflation and a belief that the business cycle no longer applies  these are clearly secular, rather than cyclical, shifts in attitudes.
However, it would be unwise to attempt to draw any definitive conclusions since, as mentioned in the previous section, the estimation procedure was not optimized to generate the most accurate possible estimates of k. A more refined procedure would probably employ different weights, and should make use of the yields of traded bonds, rather than interpolated constant maturity Treasury yields.
CONCLUSIONS
A series of studies beginning with Litterman and Scheinkman [1991] have attempted to isolate the major factors which drive shifts in the yield curve. The tool of choice has been principal component analysis. Consistently, the two dominant factors turn out to be a parallel shift in the yield curve and a shift in yield curve slope, with a curvature shift generally emerging as the third factor. These may be interpreted, in terms of our model, as changes in l, (l  s) and k.
This has some implications for risk management. In the discussion following Brown and Schaefer [1995], it is noted that two or three factors appear to suffice when attempting to capture yield curve risk. The present study suggests the existence of an additional source of yield curve risk  fluctuations in implicit volatility s  which should be taken into account in the risk monitoring process.
This source of risk may not be apparent from a principal component analysis since, as we have seen, s does not exhibit small random fluctuations like a "typical" random quantity, but instead seems to be switched "on" or "off". Furthermore, studies tend to use data sets whose observations are much more closely clustered at short maturities, thus reducing the statistical weight given to observed fluctuations in long bond yields. Thus a shortend "hump" factor is much more likely to be detected, and will appear relatively more important than a "convexity bias" factor driven by changes in implicit volatility.
Volatility risk is, of course, very familiar from the interest rate derivatives markets. In fact, in those markets, where volatility shows a distinct term structure, it is necessary to subdivide volatility risk by, for example, separating exposure to shortdated and longdated volatility. In the present study, by contrast, it does not make practical sense to introduce more than one volatility parameter. Although theoretically attractive, this would make the estimation process unstable, and would probably lead to meaningless results. (Note that mean reversion implies that physical bond yield volatilities have some term structure anyway; i.e. the model is already realistic in that it does not predict flat volatilities.)
It would be interesting to determine how implicit volatility risk could be hedged. This question might be studied in the framework of a model of the stochastic evolution of l, (l  s), k and s. For example, the regimeswitching behavior of k and s could be modeled in a continuous framework using noisy Van der Pol oscillators. As various technical obstacles arise, this line of investigation must be postponed.
On a more practical note, the present line of investigation has potential applications to fixed income portfolio management. For example, it could be argued that when implicit volatility is low, as it was in 19871993 except for brief periods, 30year bonds are undervalued relative to shorter bonds. This has implications for an active portfolio strategy with a relatively long time horizon: in effect, in periods such as these, longdated long bond volatility can be bought very cheaply.
There are a number of reasons why this strategy is not entirely obvious, which may explain why such "anomalies" can persist for long periods of time. In particular, note that one cannot exploit the opportunity simply by executing a durationmatched switch from, say, 20year bonds into 30year bonds, as this would change the exposure of the portfolio to a shift in yield curve slope. Instead, one must adopt a neutral strategy which leaves slope risk as well as duration unaffected (see, e.g., Willner [1996]): a 20year position must be rebalanced into 10year and 30year holdings, with a correspondingly smaller convexity pickup. Thus the trade is only worth executing in volume. Incidentally, this strategy ignores curvature risk; kimmunization is possible via "condor" rather than "butterfly" trades, but is unwieldy and subject to additional execution risk.
Note that the fact that these "anomalies" may persist for a long time, combined with the costs of funding short bond positions, means that it probably does not make sense to execute this kind of convexity arbitrage on a leveraged basis. Thus one would not necessarily expect hedge fund activity or proprietary trading to drive such anomalies away.
Alternatively, rather than regarding low implicit volatility as an anomaly, one might choose to interpret it as suggesting a risk premium for 30year bonds over and above the overall bond market risk premium (versus money market returns). Thus during 19871993 this implied risk premium would have averaged 4050 bp. But it is difficult to see why this risk premium should vanish at points when long bond yields become very volatile, which is what it seems to do. One would also be forced to conclude that long bonds had a negative risk premium in the early 1960s. Although this is possible  in recent experience they had certainly been much less volatile than short rates  it is somewhat counterintuitive.
A final practical observation is that the shape of the long end of the curve  which is mainly determined by s  is, to a large degree, independent of the shape of the mid part of the curve  which is mainly determined by k, although the formulation of Willner [1996] may perhaps be more useful for practical applications. That is, investment managers may be justified in assuming that long bond "value" and midrange bond "value" are not too closely coupled, and adopting different methods to analyze different parts of the curve.
The present study potentially sheds some light on alternative theories of the term structure, in particular the preferred habitat theory. It may be possible to relate regime switching to structural changes in the fixedincome market triggered by volatility. However, this would require a detailed institutional analysis, and is beyond the scope of this paper.
We conclude by noting the usefulness of examining a long historical time period when carrying out any analysis of bond market behavior. In the present case, the data exhibited noticeable patterns over time frames longer than an economic cycle, presumably related to secular changes in the structure of the economy or the US bond market. Furthermore, an analysis of (say) 19901996 data alone would have been potentially misleading. This observation should be borne in mind when extending the analysis to bond markets other than the US.
REFERENCES
Brown, R. and Schaefer, S., "Interest rate volatility and the shape of the term structure", in Howison, S., Kelly, F. and Wilmott, P. (eds.), Mathematical Models in Finance, Chapman and Hall, 1995.
Burghardt, G. and Hoskins, B., "A question of bias", Risk, March 1995.
Carayannopoulos, P, "A seasoning process in the US Treasury bond market: the curious case of newly issued tenyear notes", Financial Analysts Journal, January/February 1996.
Duffie, D. and Kan, R., "A yieldfactor model of interest rates", working paper, 1993.
Dybvig, P. and Marshall, W., "Pricing long bonds: pitfalls and opportunities", Financial Analysts Journal, January/February 1996.
Frankel, J., Financial Markets and Monetary Policy, MIT Press, 1995.
Grace, A., Optimization Toolbox For Use with MATLAB^{â}, The MathWorks Inc., 1994.
Ilmanen, A., "Market rate expectations and forward rates" / "Does duration extension enhance longterm expected returns?", Journal of Fixed Income, September 1996.
Litterman, R. and Scheinkman, J., "Common factors affecting bond returns", Journal of Fixed Income, June 1991.
Press, W., Teukolsky, S., Vetterling, W., Flannery, B., Numerical Recipes in C: The Art of Scientific Computing (Second Edition), 1992.
Willner, R., "A new tool for portfolio managers: level, slope and curvature durations", Journal of Fixed Income, June 1996.
EXHIBITS
EXHIBIT 1 n US Treasury yield data used in this study

Monthly series 
Daily series 


from 
to 
from 
to 
1 year 
Apr53 
Sep96 
2/15/1977 
9/30/96 
2 year 
Jun76 
Sep96 
2/15/1977 
9/30/96 
3 year 
Apr53 
Sep96 
2/15/1977 
9/30/96 
5 year 
Apr53 
Sep96 
2/15/1977 
9/30/96 
7 year 
Jul69 
Sep96 
2/15/1977 
9/30/96 
10 year 
Apr53 
Sep96 
2/15/1977 
9/30/96 
20 year (old) 
Apr53 
Dec86 
 
 
20 year (new) 
Oct93 
Sep96 
10/1/93 
9/30/96 
30 year 
Feb77 
Sep96 
2/15/1977 
9/30/96 
EXHIBIT 2 n Theoretical convexity bias vs. maturity, 80 bp annual volatility
EXHIBIT 3 n Model vs. actual yields: basis point errors
EXHIBIT 4 n Model basis point errors over time
EXHIBIT 5 n Long and short rates, monthly estimates 19531996
EXHIBIT 6 n Mean reversion coefficient, monthly estimates 19531996
EXHIBIT 7 n Implicit volatility, monthly estimates 19531996
EXHIBIT 8 n Mean reversion coefficient, daily estimates 19771996
EXHIBIT 9 n Implicit volatility, daily estimates 19771996
EXHIBIT 10 n Implicit volatility regimes
Period 
Regime 
Annual volatility 
195457 
off 
 
195865 
on 
~70 bp 
196667 
off 
 
196872 
on 
~100 bp 
197379 
(mostly) off 
 
198083 
on 
~80 bp 
198493 
off 
 
199496 
(mostly) on 
~80 bp 
EXHIBIT 11 n Implicit volatility vs. historical volatility