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THE EMPIRICAL MODEL 36 = = ס סםםם סם ם סם ם סס סס ס ס tS. n SSR k kR n R R n R kk Rk kn n kk kn n r k p r empirical n t ,t i t ,,, ,n I Ii t ,,,n n stock price at 10 years, given 1 0, using both the TSE and S&P parameters. In both cases, over this long term, the left tail is substantially fatter for the RSLN model than for the LN model. This difference has important implications for longer-term actuarial applications. The probability function for the sojourn times can also be used to find unconditional moments of the stock price at any time . E[( ) ] E[E[( ) ]] E exp( ( ( ) ) ( ) 2 E exp ( ) ( ) exp 22 exp exp ( ) ( ) ( ) 22 Under the model of stock returns, we use the historic returns as the sample space for future returns, each being equally likely, sampling with replacement. That is, assume we have observations of the total stock return: Return on stocks in [ 1 ) 1 2 3 aswhere 1 Pr[ ] for 1 2 The empirical model assumes returns in successive periods are independent and identically distributed. It provides a simple method for simulation, though, obviously, analytical development is not possible. This distribution is useful as a simple, quick method to obtain simulated returns. It suffers from the same problems in representing the data as the LN model (which it closely resembles in distribution). Although we are sampling from the historical returns, by assuming independence we lose the autocorrelation in the data. The autocorrelation means that low returns kk nnn nn nn n n n r t r rt Խ Խ Խ ΘΙ Α ΄΅ ΄΅ ΂΂ ΃΃ ΂ ΃ ΂΃΂΂ ΃΃ ϪϪ ϪϪ ϪϪ Ϫ ␮␮␴␴ ␮␮ ␴␴ ␮ ␴ ␮␴ ␮␮␴␴ MODELING LONG-TERM STOCK RETURNS 0 2 22 12 12 22 22 2 12 2 12 2 22 222 212 212 0 Thenwemaysimulatefuturevaluesforstockreturnsforanyperiod[ r, – 1 r ) THESTABLEDISTRIBUTIONFAMILY 37 TheStableDistributionFamily םם Y F aX bX cX d . tend to be bunched together, giving a larger probability of very poor returns than we get from random sampling of individual historical returns. The autocorreleation is the source of fatter left tails in the accumulation factor distribution. Similarly, high returns also tend to be bunched together, giving fatter right tails. So the empirical model tends to be too thin- tailed, and the assumption of independence also means that volatility bunching is not modeled. One adaptation that would reintroduce some of the autocorrelation is to sample in blocks of several months at a time. The empirical method is used by some financial institutions for value- at-risk calculations, but these tend to be quite short-term applications. One particularly useful feature of the method, though, is the ease of constructing multivariate distributions. Suppose we are interested in a bivariate distribution of long-term interest rates and stock returns. These are not independent, but by sampling the pair from the same date using the empirical method, some of the relationship is automatically incorporated. We lose any lagged correlation, however. Stable distributions appear in some econometrics literature, for example, McCulloch (1996). Panneton (1999) and Finkelstein (1995) both used stable distributions for valuing maturity guarantees. One reason for their popularity is that stable distributions can be very fat-tailed, and are also easy to combine, as the sum of stable distributions is always another stable is a Levy process, then at any fixed time has a corresponding stable distribution. A distribution with distribution function is a stable distribution if for such that: (232) isclearlytrueforthenormaldistribution—thesumofanytwonormal random variables is also normal, and all normal random variables can be standardized to the same distribution. It is not true of, for example, the Poisson distribution. The sum of two independent, identically distributed Poisson random variables is also Poisson, but cannot be expressed in terms of the same Poisson parameter as the original distribution. It is not possible, in general, to describe stable distributions in terms of their probability or distribution functions, which require special functions. t t t ϳ Ͼ0 12 12 distribution.Stabledistributionsarerelatedto Levyprocesses ;if { } Y independent, indentically distributed X , X , X ,andforany a,b > 0,there exists cd > 0, (Weuse ~ heretomeanhavingthesamedistribution.)Thisrelationship GENERALSTOCHASTICVOLATILITYMODELS 38 ʦʦ / סס ס ס ס = = = = + = + Xeitctitzt,. c,, zt,. t characteristicexponent Y Y YY y , Itispossibletosummarizethefamilyintermsofthecharacteristicfunction, ()E[]exp(1sign()())(233) where0,(02],[11]and tanif1 ()(234) logif1 Theparameterisalocationparameter;thecomponentiscalledthe andisusedtoclassifydistributionswithinthestable family.Wesaythatadistributionis-stableifitisstablewithcharacteristic component.Thecase2correspondstothenormaldistribution and1istheCauchydistribution.TheinverseGaussiandistribution withinfinitevariance.If0,thenthedistributionissymmetric. Aswiththenormaldistribution,stabledistributionscanbeusedto describestochasticprocesses.Letbeastochasticprocess,suchasthe log-returnprocess.Ifhasindependentandstationaryincrements(for anytimeunit),thenisastableorLevyprocessandhasan-stable distribution. Stableprocesseshavebeenpopularformodelingfinancialprocesses becausetheycanbeveryfat-tailed,andbecauseoftheobviousattraction ofbeingabletoconvolutethedistribution.However,theyarenoteasyto use;estimationrequiresadvancedtechniquesanditisnoteasytosimulate astableprocess,althoughamethodisgiveninChambersetal.(1976), andsoftwareusingthatmethodisavailablefromNolan(2000).Themodel specificallydoesnotincorporateautocorrelationsarisingfromvolatility bunching,andthereforedoesnot,infact,fitthedatasetsinthesection ondataparticularlywell.Anexcellentsourceofexplanatoryandtechnical informationontheuseofstabledistributionsisgiveninNolan(1998); also,onhisWebsite(2000),Nolanprovidessoftwareforanalyzingstable distributions. Wecanallowvolatilitytovarystochasticallywithouttheregimeconstraints oftheRSLNmodel.Forexample,letand toassumearedistributedon(0).Forexample,wemightusea gammadistribution.Thesemodels,andmorecomplexvarieties,arehighly adaptable.However,ingeneral,itisverydifficulttoestimatetheparameters. iXt t t tt ttt t t tt tt t ͉͉ ͉͉ Ά· { } ΘΙ Ά Ϫ Ϫ ␣ ␲␣ ␲ ␴␴ ␴ ϪϪ ϾϪ Ϫ ␧ ␧ϱ ␾␥␤␣ ␣␤ ␣ ␣ ␣ ␥␣ ␣ ␣␣ ␣ ␤ ␣ ␮␴␴␴ MODELINGLONG-TERMSTOCKRETURNS 2 2 22 1 2 1 correspondsto ␣␤ = 1 / 2, = 1.For ␣ < 2,thedistributionisfat-tailed, a ( ␴␴ – ) + ␧ where ␧ and ␧ arerandominnovations.Itisconvenient Consumer Price Index Short Bond Yield Share Dividends Long Bond Yield Share Yield The Wilkie Model Structure FIGURE 2.11 THE WILKIE MODEL 39 Structure of the Wilkie investment model. The Wilkie Model multivariate The Wilkie model (Wilkie 1986, 1995) was developed over a number of years, with an early version applied to GMMBs in the MGWP Report (1980) and the full version first applied to insurance company solvency by the Faculty of Actuaries Solvency Working Party (1986). The Wilkie model differs in several fundamental ways from the models covered so far: It is a model, meaning that several related economic series are projected together. This is very useful for applications that require consistent projections of, for example, stock prices and inflation rates or fixed interest yields. The model is designed for long-term applications. Wilkie (1995) looks at 100-year projections, and suggests that it is ideally suited for appli- cations requiring projections more than 10 years ahead. The model is designed to be applied to annual data. Without changing the AR structure of the individual series, it cannot be easily adapted to more frequent data. Attempts to produce a continuous form for the model, by constructing a Brownian bridge between the end-year points (e.g., Chan 1998) add complexity. The annual frequency means that the model is not ideal for assessing hedging strategies, where it is important that stocks are bought and sold at intervals much shorter than the one-year time unit of the Wilkie model. The Wilkie model makes assumptions about the stochastic processes governing the evolution of a number of key economic variables. It has the cascade structure illustrated in Figure 2.11; this is not supposed to represent The Inflation Model 40 A of interest or inflation is the continuously compounded annualized rate. סם ם q yd cb a w wt t zt N tat zt . a causal development, but is related to the chronological processes. Each series incorporates some factor from connected series higher up the cascade, and each also incorporates a random component. The Wilkie model is widely used in the United Kingdom and elsewhere in actuarial applications by insurance companies, consultants, and academic researchers. It has been fitted to data from a number of different countries, including Canada and the United States. The Canadian data (1923 to 1993) were used for the figures for quantile reserves for segregated fund contracts in Boyle and Hardy (1996). The integrated structure of the Wilkie model has made it particularly useful for actuarial applications. For the purpose of valuing equity-linked liabilities, this is useful if, for example, we assume liabilities depend on stock prices while reserves are invested in bonds. Also, for managed funds it is possible to project the correlated returns on bonds and stocks. What is commonly called the Wilkie model is actually a collection of models. We give here the equations of the most commonly used form of the model. However, the interested reader is urged to read Wilkie’s excellent 1995 paper for more details and more model options (e.g., for the ARCH model of inflation). The notation can be confusing because there are many parameters and five integrated processes. The notation used here is derived from (but is not the same as) Wilkie (1995). The subscript refers to the inflation series, subscript to the dividend yield, to the dividend index process, to the long-term bond yield, and to the short-term bond yield series. The terms all indicate a mean, although it may be a mean of the log process, so is the mean of the inflation process modeled, which is the force of inflation process. The term indicates an AR parameter; is a (conditional) variance parameter; and is a weighting applied to the force of inflation within the other processes. For example, the share dividend yield process includes a term ( ), which is how the current force of inflation ( ( )) influences the current logarithm of the dividend yield (see equation 2.36). The random innovations are denoted by ( ), with a subscript denoting the series. These are all assumed to be independent (0,1) variables. AR(1)process: ()((1))()(235) 5 force q yq q qq qqqq qqq ϪϪ ␮ ␮ ␴ ␦ ␦ ␦␮␦ ␮␴ MODELING LONG-TERM STOCK RETURNS 5 Let ␦␦ ( tt )betheforceofinflationintheyear[ – 1 ,t ),then( t )followsan Share Prices and Dividends 41 The Wilkie Model סםם סם ס סם סם tt a a ztN Qt t,yt yt w t ynt . yn t a yn t z t . yn t z t yt e w t ynt . tynt wtMw Mu t t t Mu u u . yt e M w . a where ( ) is the force of inflation in the th year, is the mean force of inflation, assumed constant. is the parameter controlling the strength of the AR (or rather the weakness, since large implies weak autoregression)—that is, how strong is the pull back to the mean each year. is the standard deviation of the white noise term of the inflation model. ( ) is a (0,1) white noise series. sothat,if()isanindexofinflation,theultimatedistributionof arecorrelatedthroughtheAR. We model separately the dividend yield on stocks, and the force of dividend inflation. The share dividend yield in year ( ) is generated using: ( ) exp ( ) ( ) (2 36) where ( ) ( 1) ( ) (2 37) So ( ) is an AR(1) process, independent of the inflation process, ( ) being a Normal(0,1) white noise series. Clearly E[ ( )] E[exp( ( ))] E[exp( ( ))] (2 38) because ( ) and ( ) are independent. E[exp( ( ))] is ( ), where ( ) is the moment generating function of ( ). For large , the moment generating function of ( ) is ( ) exp( ( ) 2) (2 39) So E[ ( )] ( ) exp (2 40) 2(1 ) q q q q q q q qq yq y yyy y yq qyqy q q qq y qy yn y ΋ Ά· ΄΅ ΂΃ ␮ ␦ ␦ ␦ ␮ Ϫ Ϫ ␦ ␮ ␴ ␦␮ ␴ ␦ ␦␦ ␦ ␦ ␮␴ ␴ ␮ y q q q y 22 22 2 2 Theultimatedistributionfortheforceofinflationis N ( ␮␴ , / (1 – a )), Q ( t ) / Qt ( – 1)isLN.However,unliketheLNmodel,successiveyears Long-Term and Short-Term Bond Yields 42 סם ם םם ם סם ס = ם ס סם סם סםם t tw t w tdzt bzt zt tdt d t . t zt Dt Dt e Pt Pt pyt Pt Dt py t . Pt c t real cn t cm t ct cmt cnt cm t d t d cm t cn t a cn t y z t z t The force of dividend growth, ( ), is generated from the following relationship: ( ) DM( ) (1 ) ( ) ( 1) ( 1) () where DM( ) ( ) (1 )DM( 1) (2 41) The force of dividend then comprises: A weighted average of current and past inflation—the total weight the th year before . A dividend yield effect where a fall in the dividend yield is associated Aninfluencefromthepreviousyear’swhitenoiseterm. A white noise term where ( ) is a Normal(0,1) white noise sequence. The force of dividend can be used to construct an index of dividends, () ( 1) A price index for shares, ( ), can be constructed from the dividend index and the dividend yield, ( ) each year ( ) can be summarized in the gross rolled up yield, () () () 10 (1) The yield on long-term bonds, ( ), is split into a part, ( ), and an inflation-linked part, ( ), so that () () () where () () (1 ) ( 1) and ( ) exp( ( 1) ( ) ( )) d qyyy dd d dddd dd q dd q ddd d y d t cq c cc cyy cc ␶ ␦ ϪϪϪ ϪϪ Ϫϫ Ϫ Ϫ ϪϪ Ϫ ␦ ␦␦␴␮␴ ␴ ␦ ␶ ␦ ␮␴␴ MODELING LONG-TERM STOCK RETURNS d () assignedtothecurrent ␦ (tw)beingd + (1 – w).Theweightattached dd fortheforceofinflationintopastforcesofinflationis wd (1 – d ) withariseinthedividendindex,andviceversa(i.e., d < 0). D ( t ) / yt ().Theoverallreturnonshares Other Series Parameters 43 The Wilkie Model ס סם ם ם bt ct bt ct bdt bd t a bd t b z t z t The inflation part of the model is a weighted moving-average model. The real part is essentially an autoregressive model of order one (i.e., AR(1)), with a contribution from the dividend yield. The yield on short-term bonds, ( ), is calculated as a multiple of the long-term rate ( ), so that ( ) ( ) exp( ( )) where ( ) ( ( 1) ) ( ) ( ) These equations state that the model for the log of the ratio between the long-term and the short-term rates is AR(1), with an added term allowing for a contribution from the long-term residual term. Wilkie (1995) also describes integrated models for wage inflation, property, bonds linked to an inflation index (“index-linked stocks”), and exchange rates. The paper also presents and investigates alternative models, including ARCH models in place of the AR models used, transfer functions, and a vector autoregression model. The parameters suggested in Wilkie (1995) for Canada and the United States are given in Table 2.2. Note that figures for the short-term interest rate for the United States are not available. These parameters were fitted using 1923 to 1993 data for the Canadian figures, and data from 1926 to 1989 for the United States. To run the Wilkie model, one can start the simulations at neutral values of the parameters. These are the stationary values we would obtain if all the residuals were zero. Alternatively, we can start the model at the current date and let the past data determine the initial parameter values. For general purposes, it is convenient to start the simulations at the neutral values of the parameters so that the results are not distorted by the particular nature of the current investment conditions. If new contracts are to be written for some time ahead, the figures using neutral Wilkie starting parameters are close to the average figures that would be obtained at different dates using formerly current starting values. However, for strategic decisions that are designed for immediate implementation it is appropriate to use the contemporary data for starting values for the series. ccc bb b bb Ϫ ϪϪ ␮␮␴␴ TABLE2.2 SomeCommentsontheWilkieModel 44 q q q y y y y d d d d d d c c c c c b b b b ParametersforWilkiemodel,Canadaand UnitedStates,fromWilkie(1995). 0.0340.030 0.640.65 0.0320.035 1.170.50 0.70.7 0.03750.0430 0.190.21 0.191.00 0.260.38 0.00100.0155 0.110.35 0.580.50 0.070.09 0.0400.058 0.950.96 0.03700.0265 0.100.07 0.1850.210 0.26 0.38 0.73 0.21 ParameterCanadaU.S. InflationModel DividendYield DividendGrowth Long-TermInterestRates Short-TermInterestRates TheWilkiemodelhasbeensubjecttoauniquelevelofscrutiny.Many companiesemploytheirownmodels,butfewissuesufficientdetailfor independentvalidationandtesting.Themostvigorouscriticismofthe WilkiemodelhascomefromHuber(1997).Huber’sworkisconcerned with: –– ␮ ␴ ␮ ␴ ␮ ␴ ␮ ␴ ␮ ␴ a w a w d y b d a y a c MODELING LONG-TERM STOCK RETURNS VECTOR AUTOREGRESSION 45 Vector Autoregression q Evidence of a permanent change in the nature of economic time series in Western nations around the second world war is not allowed for. This criticism applies to all stationary time-series models of investment, but nonstationary models can have even more serious problems in generating impossible scenarios with explosive volatility, for example. It is useful to be aware of the limitations of all models—to be aware, for example, that in the event of a major world conflagration the predicted distributions from any stationary model may well be incorrect. On the other hand, in such circumstances this may not be our first worry. The inconsistency of the Wilkie model with some economic theories, such as the efficient market hypothesis. Note, however, that the Wilkie model is very close to a random walk model over short terms, and the random walk model is consistent with the efficient market hypoth- esis. Huber himself points out that there is significant debate among economists about the applicability of the efficient market hypothesis over long time periods, and the Wilkie approach is not out of line with those of other econometricians. The problem of “data mining,” by which Huber means that a statistical time-series approach, which finds a model to match the available data, cannot then use the same data to test the model. Thus, with only one data series available, all non-theory-based time-series modeling is rejected. One way around the problem is to use part of the available data to fit the model, and the rest to test the fit. The problem for a complex model with many parameters is that data are already scarce. This argument is, as Huber noted, not specifically or even accurately aimed at the Wilkie model. The Wilkie model is substantially theory driven, informed by standard statistical time-series analysis. Huber’s work is not intended to limit actuaries to a deterministic methodology, although it has often been quoted in support of that view. However, it is certainly important that actuaries make themselves aware of the provenance, characteristics, and limitations of the models they use. The Wilkie model is an example of a vector AR approach to modeling financial series. The vector represents the various economic series. The cascade structure makes parameter estimation easier and, perhaps, makes the model more transparent. The more general vector AR is to use an AR( ) structure for a vector of relevant financial series, with correlations between the series captured in a variance-covariance matrix. [...]... kj LRT p 2 3 3 4 4 4 5 6 8 12 929.8 930.0 933.8 935.0 945 .2 939.1 939.1 953 .4 953.8 962.7 923.5 920.6 9 24. 4 922.5 932.7 926.6 923 .4 9 34. 6 928.7 925.1 927.8 927.0 930.8 931.0 941 .2 935.1 9 34. 1 947 .4 945 .8 950.7 Ͻ 10Ϫ8 Ͻ 10Ϫ8 Ͻ 10Ϫ8 Ͻ 10Ϫ8 0.0003 Ͻ 10Ϫ6 Ͻ 10Ϫ6 0.98 0.01 We use likelihood ratio test to compare the models discussed above, and a few that are not dealt with in detail above The following models... percentiles for the longer accumulation factors where the data is sparse (Further details are given in Appendix C of SFTF (2000).) In Table 4. 2, the range of values for the available left-tail percentiles are given The 2.27 percentile for the one-year return is based on the worst result of 43 nonoverlapping periods of annual returns; 2.27% = 1 44 The 4. 55 percent result is the second smallest The final column... RSLN 1΋ 44 ‫%72.2 ס‬ 2΋ 44 ‫%55 .4 ס‬ 4 44 ‫%90.9 ס‬ 1΋ 9 ‫%11.11 ס‬ 1΋ 5 ‫%00.02 ס‬ (0.61, 0.82) (0.76, 0.85) (0.85, 0.92) (0.98, 1 .41 ) (1.60, 2.59) 0. 74 0.82 0.89 1.05 1.88 The calibration points used by the CIA Task Force were found by extrapolating from the available data This was done by looking at a number of different models that appeared to fit well where there is more data, and using these models... (4. 2) z0.25 Ϫ ␮ ‫ ס‬Ϫ0.6 745 ␴ (4. 3) 3 , z0.25 ‫ ס‬Ϫ0.6 745 ␴ ‫␮ ם‬ (4. 4) ⌽ 3 , ΂ ΃ Similarly, z0.1 = – 1.2816␴ + ␮ We equate these with the empirical percentiles to get: 3 , ␮ ‫8610.0 ס‬ and ␴ ‫7 940 .0 ס‬ (4. 5) Now these are quite different values to those found by using maximum likelihood (␮ = 0.0081 and ␴ = 0. 045 1), or by matching moments The reason is that by choosing to match the 10th and 25th percentiles,... RSLN-2 RSAR-2 RSLN-3 Parameters kj logL lj SBC lj Ϫ 1 kj log n 2 AIC lj Ϫ k j LRT p 2 3 3 4 4 4 5 6 8 12 885.7 887 .4 889 .4 889 .4 912.2 896.2 900.2 922.7 923.0 925.9 879 .4 878.0 880.0 876.9 899.7 883.7 8 84. 5 903.9 898.7 888.3 883.7 8 84. 4 886 .4 885 .4 908.2 892.2 895.2 916.7 915.0 913.9 Ͻ 10Ϫ8 Ͻ 10Ϫ8 Ͻ 10Ϫ8 Ͻ 10Ϫ8 Ͻ 10 4 Ͻ 10Ϫ8 Ͻ 10Ϫ8 0.82 0.38 S&P 500 (1956–1999 Monthly Total Returns) Model j LN AR(1) ARCH... Hamilton (19 94) points out that the likelihood ratio test is not a valid test for the number of regimes in a regime-switching model The results of the likelihood ratio tests, then, should be viewed with caution In Table 3.6, the final column gives the p-value for a likelihood ratio test of the RSLN model against each of the other models listed For models with fewer than six parameters, the null hypothesis... (3.11) (3.12) The maximum likelihood estimates for ␮ and ␴ are found by setting the partial derivatives equal to 0 for parameter estimates, signified by ˆ This gives ¯ ˆ ␮‫ס‬y (3.13) and ˆ ␴‫ס‬ Ί ˆ Α n‫( 1ס‬yt Ϫ ␮ )2 t n (3. 14) So the MLE for the mean of the log-returns is the mean of the log-data The n MLE for the variance is nϪ1 s2 y The estimator for ␮ is unbiased for all sample sizes The estimator... quantiles of the empirical distribution Say we decide to use the 10th and 25th percentiles of the empirical and lognormal distributions The 10th percentile of the log-return for the TSE 300 monthly data from 1956 to 2001 is – 0. 046 82 and the 25th percentile is – 0.01667 We equate these empirical percentiles with the model percentiles The model 25th percentile is z0.25 where z0.25 Ϫ ␮ ‫52 ס‬ ␴ ΂ ΃ (4. 2) z0.25... over the four values of equation 3. 34, with ␳t = 1, 2 and ␳tϪ1 = 1, 2, the sum is f (yt ͉ytϪ1 , ytϪ2 , , y1 , ␪ ), which is the contribution of the tth value in the series to the likelihood function To start the recursion, we need a value (given ␪ ) for p(␳0 ), which we can find from the invariant distribution of the regime-switching Markov chain The invariant distribution ␲ = (␲1 , ␲2 ) is the unconditional... data Since the observations are independent, the likelihood function, which is the joint probability density function (pdf) for the data, is simply the product of the individual density functions It is unlikely, looking at the three values, that the ␮ parameter for the model is, for I 47 48 MAXIMUM LIKELIHOOD ESTIMATION FOR STOCK RETURN MODELS example, 2.0 This is confirmed by calculating the likelihood . M w . a where ( ) is the force of inflation in the th year, is the mean force of inflation, assumed constant. is the parameter controlling the strength of the AR (or rather the weakness, since large. 13) and (ˆ) ˆ (3 14) So the MLE for the mean of the log-returns is the mean of the log-data. The MLE for the variance is . The estimator for is unbiased for all sample sizes. The estimator for is asymptotically. start the model at the current date and let the past data determine the initial parameter values. For general purposes, it is convenient to start the simulations at the neutral values of the parameters