ELSEVIER Mathematics and Computers in Simulation 43 (1997) 475-480

MATHEMATICS AND

COMPUTERS IN SIMULATION

Short-term interest rate models and generation of interest rate scenarios

Y.K. Tse*

Department of Economics and Statistics, National University of Singapore, Singapore

Abstract

This paper investigates the stochastic behaviour of the short-term interest rates. The lognormal model, the stable Paretian model and the continuous time mean reversion model are considered. The parameters of the models are estimated using 17 years of weekly data. Our results show that the lognormal and the stable Paretian models are likely to give rise to unreasonably large interest rate values even for horizon of five years. In comparison, the mean reversion model appears to provide more realistic results than the other two models.

1. Introduction

In this paper we examine the stochastic behaviour of short-term interest rates. We consider the following interest rate models: the lognormal distribution, the stable Paretian distribution and a continuous time model that allows for mean reversion. The parameters of these models are estimated using historical data. The estimated models may be used as the basis for forming interest rate assumptions in cash-flow testing and valuation of contingent claims. We examine some aspects of the sensitivity of the generated scenarios to the choice of these models. The objective is to investigate if scenarios generated by these models are generally 'reasonable.'

The plan of this paper is as follows. Section 2 reports the estimation results for the stable Paretian distribution and the continuous time mean reversion model. In Section 3 we compare the scenarios generated by the models based on some Monte Carlo experiments. Finally some concluding remarks are given in Section 4.

2. Data and estimation results

The data used in this study were extracted from various issues of The Economist. Weekly data were collected for the three-month money market rates from 1977 January through 1994 April, totalling 884

* Corresponding author.

0378-4754/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S 0 3 7 8 - 4 7 5 4 ( 9 7 ) 0 0 0 3 4 - 7

476 Y.K. Tse/Mathematics and Computers in Simulation 43 (1997) 475-480

observations. The finding that the logarithm of security price ratios has empirical distributions with high kurtosis has been well documented. The stable Paretian distribution has been proposed to describe such data. Recently, Klein [6] applied this distribution to fit the 30-year Treasury yield data and argued in support of its use. However, Klein's analysis was restricted to the symmetric stable Paretian family. To extend his analysis we examine the hypothesis that the differenced logarithmic short-term interest rates are independently and identically distributed as a (possibly asymmetric) stable Paretian variable.

The full stable Paretian distribution that allows for asymmetry is characterized by four parameters. We follow the definition proposed by Zolotarev [l l] and adopted by McCulloch [7], in which the

J . .

logarithmic characteristic function of a stable Paretian distribution has the form:

O ( u ) = l o g E ( e i"x)

{ iu~5-]cu] '~[1 - i/3sign(u)tan(~l] c~ ¢ 1 (1) = i u ~ 5 - I c u l [ 1 + i / 32s ign (u ) log tu l J c~ = 1

where X is a stable Paretian variable, u is the parameter of the characteristic function, i 2 = -1 and c~,/3, ~5 and c are, respectively, the characteristic exponent, the skewness parameter, the location parameter and the scale parameter. The normal distribution is a special member of the family with c~ = 2, and is the only stable Paretian distribution for which the variance exists. When c~ < 2, absolute moments of order less than c~ exist while those of order greater than or equal to c~ do not. To estimate the parameters Fama and Roll [4,5] suggested a fractile method based on the ordered statistics. 1 This method was improved by McCulloch [7], who took account of asymmetry corrected the bias in the Fama-Roll results and considered a broader range of c~. The results below were obtained using McCulloch's method.

An important property of the stable Paretian distribution is that it is invariant under addition. That is, a sum of independently and identically distributed stable Paretian variables with characteristic exponent c~ is again stable Paretian with the same exponent. In what follows we consider two methods of forming sums of observations. First, we arrange the differenced logarithmic series in chronological order and sum up adjacent observations k terms at a time to form a new series. We define this method as the chronological sampling scheme. Second, we randomize the differenced logarithmic series and sum up adjacent observations as above. This method is defined as the randomized sampling scheme. As the summed series must have sufficient observations for estimation, the value of k is constrained. We consider k = 2, 4 and 8. The estimates of c~, and fl are given in Table 1.

The results show that c~ of the chronological series are smaller than those of the randomized series. The values of /3 are generally small. To give an indication of the significance of the asymmetry parameters, we performed a Monte Carlo experiment. We estimated the 95% intervals of estimates of/3, denoted /3", of samples from the stable Paretian distributions with c~ = & and /3 = 0 . 2 The results, summarized in the fifth column of Table 2, show that ~ are all within the estimated 95% intervals. Thus, there is no evidence of asymmetry. Another Monte Carlo experiment was conducted to estimate the 95% intervals of the estimates of c~, denoted by c~*, of samples from the stable Paretian distributions

IAs the density function of a stable Paretian distribution in general does not exist in closed form, estimation by the MLE is intractable.

2Random numbers of the stable Paretian distribution were generated using the method suggested by Chambers, Mallows and Stuck [2]. All simulation results reported in Table 1 were based on Monte Carlo experiments with 1000 replications.

Y.K. Tse /Mathematics and Computers in Simulation 43 (1997) 475-480

Table 1 Estimates of the characteristic exponent (c~) and the skewness parameter (/3) of the stable Paretian distribution

477

k N Parameters Simulated 95% interval

Chronological sampling l 884 1.1908 -0 .0222 ( -0 .2004, 0.1875) (1.0931, 1.2997) 2 442 1.2488 -0 .0669 ( -0 .2729, 0.2782) (1.1088, 1.4069) 4 221 1.3420 -0 .1245 ( -0 .3835, 0.3767) (1.1123, 1.5797) 8 110 1.3139 0.1433 ( -0 .5578, 0.6372) (0.9745, 1.7462)

Randomized sampling 2 442 1.3530 -0 .1656 ( -0 .3093, 0.2875) (1.2062, 1.5429) 4 221 1.4302 0.0120 ( -0 .4806, 0.4303) (1.2158, 1.7265) 8 110 1.4433 0.1173 ( -0 .8629, 0.7810) (1.1205, 1.9931)

Note: k is the sum size and N is the number of observations.

Table 2 Simulation results of accumulated value and average yield

Percentile Accumulated value of $1 ($) Average yield(%)

Lognormal Mean reversion Stable Paretian Lognormal Mean reversion Stable Paretian

Panel A: Initial interest rate = 5.0% 1.0 1.13 1.30 I. 16 2.41 5.37 3.01 5.0 1.15 1.32 1.21 2.88 5.68 3.88

10.0 1.17 1.33 1.23 3.24 5.88 4.25 50.0 1.28 1.38 1.29 5.12 6.70 5.18 90.0 1.49 1.47 1.36 8.35 8.06 6.42 95.0 1.60 1.52 1.42 9.80 8.73 7.36 99.0 1.87 1.65 2.65 13.35 10.47 21.53

Panel B: Initial interest rate = 8.2% 1.0 1.21 1.34 1.26 3.97 6.07 4.80 5.0 1.26 1.37 1.36 4.80 6.46 6.39

10.0 1.31 1.39 1.40 5.51 6.80 6.93 50.0 1.51 1.47 1.51 8.62 8.10 8.54 90.0 1.95 1.65 1.68 14.27 10.50 10.95 95.0 2.15 1.73 1.82 16.52 11.64 12.78 99.0 2.60 1.95 2.90 21.02 14.29 23.75

Panel C: Initial interest rate - 11.0% 1.0 1.29 1.38 1.34 5.30 6.63 6.19 5.0 1.37 1.41 1.51 6.51 7.15 8.60

10.0 1.43 1.44 1.57 7.46 7.52 9.38 50.0 1.74 1.55 1.72 11.70 9.15 11.46 90.0 2.49 1.78 2.00 20.06 12.17 14.91 95.0 2.82 1.89 2.22 23.01 13.55 17.26 99.0 3.51 2.28 5.70 28.58 17.87 41.64

Note: The results are based on 1000 simulated sample paths over five-year horizon with monthly interest rate changes.

478 YK. Tse/Mathematics and Computers in Simulation 43 (1997) 475-480

with c~ = & and/3 =/3. The results are summarized in the last column of Table 1. While these results do not provide significance tests for the hypothesis that the characteristic exponents are invariant under addition, they do suggest that the differences in d are small compared to the estimated intervals. 3 To test whether these results support the lognormal hypothesis, we generated 1000 random samples from the lognormal distributions. The 5th percentiles of 6~ based on sample sizes of 884, 442, 221 and 110 are, respectively, 1.803, 1.733, 1.673 and 1.580. These values are higher than the reported values of c~ and show that the hypothesis c~ = 2 is rejected. In summary, there is evidence that the differenced logarithmic interest rates follow a symmetric stable Paretian distribution with o~ < 2.

Various stochastic models in the continuous time framework have been proposed for the short-term interest rate with mean reversion. In this paper, we consider the following mean reversion process:

dr = ~ ( # - r )d t + GrTdZ, (2)

where dZ are independent Wiener increments, n is the speed-of-adjustment coefficient, # is the longo run interest rate level and 7 is the parameter that determines the sensitivity of volatility with respect to the interest rate level. Note that Eq. (2) includes the following models as special cases: (i) the Vasicek [10] model with 9' = 0, (ii) the Cox-Ingersoll-Ross (CIR) [3] model with "y = 0.5, and (iii) the Brennan-Schwartz (BS) [1] model with 7 = I.

We discretize Eq. (2) to obtain the following approximation:

rt+l -- r, = ec(# -- rt) 4- et+,, (3)

where et+l is assumed to satisfy E ( e t + l ) = 0 and E(~L1 ) = ~rert 2"r, for t = 1 . . . n . To avoid misspe- cification due to nonnormality we estimate the parameters of Eq. (3) using the robust generalized method of moments (GMM) approach. Using four-weekly data we obtain the following estimates of the mean reversion process: ~ = 0.0304(0.0271), /2 = 8.1420(1.7100), ~ = 0.0112(0.0069) and

= 1.8677 ( 0 . 2 8 4 8 ) . 4 Thus, the estimated speed-of-adjustment coefficient is small and statistically insignificant. This result, however, is in line with other studies (e.g. [8,9]). The estimate of the elasticity parameter is quite large, indicating that the volatility is very sensitive to the level of the interest rates. Using X 2 tests of model restrictions, we find that the three special cases of the mean reversion process discussed above are all rejected.

3. Monte Carlo comparison of the models

We have considered the covariance stationary mean reversion process and the nonstationary stable Paretian models. In this section we perform some comparisons of the generated interest rate scenarios based on the estimated interest rate models. In addition to the stable Paretian and the mean reversion models we also consider the lognormal model (i.e. the differenced logarithmic interest rates are independently and identically distributed as normal variates with mean zero), which has been widely used in the actuarial literature.

3As the sample sizes of the different estimates are different and the samples are not independent, formal significance test is difficult.

4Thc use of four-weekly data is to obtain estimates appropriate for the base interval of the simulation in the Section 3. Figures in the parentheses following the reported parameter estimates are the estimated standard errors.

YK. Tse/Mathematics and Computers in Simulation 43 (1997) 475~180 479

The parameters of the interest rate models depend on the f requency of the interest rate changes. In the simulation study below, we consider monthly interest rate changes. For the lognormal distribution, the standard deviation of the differenced monthly logari thmic interest rate is 0.0813. For the stable Paretian model we use the es t imated parameters of the chronological sampling scheme with k ---- 4. Condit ional on a fixed initial interest rate we generate 1000 sample paths over 5-year horizon. The interest rate paths are compared based on the accumulated value of one dollar and the average annualized yield over the f ive-year duration. These quantities are important for the analysis of the investment risks of financial products such as a guaranteed investment contract or a single p remium life insurance contract. To study the effects o f the initial interest rate we consider the following three values of the initial interest rate: 5.0, 8.2 and 11.0%. Quantiles of the distributions of the accumulated values and average

yields are summar ized in Table 2. It can be seen that the medians of the lognormal and the stable Paretian models are quite close. This

is true for all initial interest rates. When the initial interest rate is 5%, the median of the mean reversion model is the highest. When the initial interest rate is 11%, however, the median of the mean reversion model becomes the lowest. This is due to the effects of mean reversion. As for the lognormal and the stable Paretian models , the medians are close to the initial interest rate values. The lognormal model gives the highest 95th percenti les for all cases. For the 99th percentile, however, the stable Paretian

model produces more ext reme results.

4. Conclusions

In this paper we investigate the stochastic behaviour of the short- term interest rates. We examine three short- term interest rate models, namely, the lognormal model , the stable Paretian model and the mean reversion model. The characteristic exponent of the stable Paretian distribution is significantly less than two and appears to be invariant under addition. Our simulation study shows that the lognormal and stable Paretian models are likely to generate unrealistic scenarios even for horizon of five years. Overall we favour the use of the mean reversion model as the effect of mean reversion prevents the

interest rates f rom reaching unreasonably high levels.

References

[ 1 ] M.J. Brennan and E.S. Schwartz, A continuous time approach to the pricing of bonds, Journal of Banking and Finance 3 (1979) 133-155.

[2] J.M. Chambers, C.L. Mallows and B.W. Stuck, A method for simulating stable random variables, Journal of the American Statistical Association 71 (1976) 340-344.

[3] J.C. Cox, J.E. Ingersoll and S.A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985) 385-407.

[4] E.E Fama and R. Roll, Some properties of symmetric stable distributions, Journal of the American Statistical Association 63 (1968) 817-836.

[5] E.F. Fama and R. Roll, Parameter estimates for symmetric stable distributions, Journal of the American Statistical Association 66 (1971) 331-338.

[6] G.E. Klein, The sensitivity of cash-flow analysis to the choice of statistical model for interest rate changes, Transactions of the Society of Actuaries 45 (1993) 79-124.

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[7] J.H. McCulloch, Simple consistent estimators of stable distribution parameters, Communications in Statistics: Simulation and Computation 15 (1986) 1109-1136.

[8] J.E Ogden, An analysis of yield curve notes, Journal of Finance 42 (1987) 99-110. [9] Y.K. Tse, MLE of some continuous time financial models: Some Monte Carlo results, Mathematics and Computers in

Simulation 33 (1992) 575-580. [10] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics 5 (1977) 177-188. [11] V.M. Zolotarev, Mellin-Stieltjes transforms in probability theory, Theory of Probability and Its Applications 2 (1957)

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