Analysis of drawdowns and drawups in the US$ interest-rate market

Abstract

We investigate the statistical properties of drawdowns and drawups in interest rates (US$) using over 10 years' worth of daily data. We analyse the nature of the drawdowns in terms of length of runs, magnitude of the individual price moves and coincidence of their occurrence across the maturity spectrum. We document significant positive autocorrelation for several holding periods, pronounced term structure effects and an unexpectedly low degree of coincidence in the occurrence of drawdowns across the maturity spectrum (despite high correlation in daily moves). By drawing on previous work by Rebonato et al. (2005) we try to provide a coherent explanation for a complex set of empirical observations. An essential ingredient of this explanation appears to be the existence of at least two distinct types (normal and excited) of price dynamics, with different serial correlation properties. We concur with the results by Sornette and Johansen (Significance of log-periodic precursors to financial crashes. Quant. Finance, 2001, 1, 452–471) for different asset classes that very large drawdowns belong to the `undemocratic' case, and may therefore result from an amplification mechanism.

Acknowledgment

It is a special pleasure to thank Dr Kwiatkoski for insightful and useful comments.

Notes

§If there is no price (rate) change over two consecutive trading periods, the drawdown (drawup) is deemed not to have been interrupted. With this convention, given a time series of price (rate) changes, the difference in number of drawdowns and drawups is at most one.

¶Sornette (2004) gives a nice and simple example of a time series with zero serial autocorrelation that produces drawdowns very different from what would be produced if the increments were iid. Similar examples are provided in Robinson (1979), Hsieh (1989) and Osband (2002).

∥The studies conducted to date seem to find a different behaviour for the French CAC index (and possibly for the Italian index MIB)—whence the term `French exceptionality' used in this context.

†One often-quoted mechanism is the following. According to this description of the interest-rate market dynamics (IFR 2003), the US Government-sponsored mortgage Agencies (Fannie Mae and Freddie Mac) retain on their books a high volume of mortgages, that they hedge by entering, inter alia, pay-fixed swap positions. As rates fall, mortgage pre-payments increase, and this shortens the duration of the mortgage portfolio. As a consequence of this, the Agencies find themselves overhedged and to reduce their pay-fixed swap position they would have to enter receive-fixed swap positions. The demand for receiver swaps pushes rates lower, and this would cause further mortgage pre-payments, shorten the duration of the Agencies' mortgage portfolio, cause them to receive the fixed rate in swap transaction, etc. An increase in rates is supposed to produce the same phenomenon in reverse.

†As J&S recognize, the definition of what constitute an outlier is obviously model-dependent: what might appear as an `unexplainable' event given the assumption of a particular underlying distribution may become not at all exceptional if another distribution were used. One should therefore more correctly speak of `outliers' given a simple and parsimonious description of data that accounts for the majority of the observations. Some of the results presented below hold true irrespective of the distribution of returns in the limit of large drawdowns, as long as the return themselves are iid. Detection of outliers in the large-drawdown region is therefore in this case more model independent, but still relies on certain model features (identical distribution and independence) of the generating returns.

†Following Campbell et al. (1997) we define the weakest random walk hypothesis (RW3) as the hypothesis that price increments are uncorrelated. The assumption that increments are iid constitute RW2 and that increments are independent gives rise to RW1.

‡It is a pleasure to thank Dr Kwiatkowski for helpful discussions and invaluable assistance.

†To lighten the presentation, in the following we present the discussion referring to drawdowns. Mutatis mutandis the same considerations apply to drawups as well. Unless otherwise stated, the generic term `drawdown' therefore refers both to drawdowns and to drawups. The context should make clear if ever the term `drawdown' only refers to drawdowns.

†Helpful discussions with Dr Jan Kwiatkowski are gratefully acknowledged.

†When we increased M beyond 1000 or decreased the tolerance for z, we did not find noticeable differences in our results.

‡A C++ program was written and used to calculate drawdowns and drawups from the historical time series. It is available in Gaspari (2005).

§Convexity effects, due to discounting, slightly complicate the picture, but for the purposes of our analysis the statement is sufficiently correct.

†Clearly, the analysis need not be done numerically. For a given number of runs (say, N) and a null hypothesis (independent trials with probability p=0.5), one can specify an alternative hypothesis (e.g. p>0.5) and use the fact that one has observed, say, 2 runs of length 14 to reject the null hypothesis in favour of the former. For more complicated alternative hypotheses one must specify them very carefully and adjust the rejection region correspondingly. Alternatively, one could use a more general non-parametric test where one does not have to specify an alternative hypothesis. For instance, one can use the standard χ² `goodness of fit' test. In this case we know the probability (under H₀ ) of a run of length n, p(n)=0.5ⁿ , and hence the expected number of runs of length n, E[n]=Np(n). We can also count the actual number of occurrences, A(n). The χ² statistic measures the significance of the divergence of the expected from the observed numbers. One complication is that it becomes inaccurate if any of the expected number of runs is less than 5 or so. For N=358, this will be so for n>6 because E,[7]= 358/128<5, so we would have to group all runs greater than 6 together; call the corresponding expected number E,[7+] and count . Under H₀ the chi-squared statistic is then distributed as (i.e. chi squared with 6 degrees of freedom).

†We note that the results reported by Rebonato et al. (2005), who found a greater positive autocorrelation coefficient for lag 1 than we do, were conducted using percentage changes. This could be explained by the very wide range of values reached by interest rates over the period under study.

†In theory, this lack of independence could be eliminated by working with the probability density instead of the cumulative distribution. However, for discrete events, the results can depend strongly in the very tails we are interested in on the choice of the bin sizes and on the precise location of the bin boundaries. See Sornette (2004) on this point. Having tried both alternatives, we concur with Sornette that the lack of independence is the lesser of two evils.

‡Tests were performed on the chosen values of M and R. We found that increasing M or R beyond the values indicated above did not result in any noticeable difference in the resulting limit curves.

†This quantity is meant to provide a rough metric, because the volatility is clearly time-dependent, and the Gaussian assumption is not valid. The `volatility' σ is therefore used as a simple normalization factor.

‡If, for different maturities, two drawdowns start on slightly different dates, but cover largely overlapping periods, they have been deemed to `belong' to the same time bucket. There are only two cases when this happens, and for these cases the column on the left reports the range of starting dates.