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The Intertemporal Capital Asset PricingModel with returns that follow Poissonjump–diffusion processesEric Bentzen a & Peter Sellin ba Institute of Operations Management, Copenhagen Business School,Denmarkb Research Department, Sveriges Riksbank, Stockholm, SwedenPublished online: 23 Mar 2012.

To cite this article: Eric Bentzen & Peter Sellin (2003) The Intertemporal Capital Asset Pricing Model withreturns that follow Poisson jump–diffusion processes, The European Journal of Finance, 9:2, 105-124, DOI:10.1080/13518470110099704

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The European Journal of Finance 9, 105-124 (2003) 11 Routledge\. r.,lor' -"Group

The Intertemporal Capital Asset PricingModel with returns that follow Poissonjump-diffusion processesERIC BENTZEN1 and PETER SELLlN2

'Institute of Operations Management, Copenhagen Business School, Denmark2Research Department, Sveriges Riksbank, Stockholm, Sweden

Capital market equilibrium Is derived In a model where asset returns follow a mixedPoisson jump-dlffuslon process. In the resulting modified Capital Asset Pricing Model(CAPM) expected returns are still linear In beta, but In addition premia have to be paidto compensate the Investor for taking on jump risk. When jump risk Is dlverslfiable In themarket portfollo the model Is reduced to the standard CAPM. Jumps are found to beprevalent In the dally returns of the market Indices In the 18 countries Investigated. Acontinuous return process does not give an adequate description of the market returnsIn any of the countries Investigated.

Keywords: asset pricing, CAPM, )ump-diffuslon, Poisson process

1. INTRODUCTION

Amodel of the dynamics of stock prices that seems to be gaining In popularityIs the jump-dlffuslon (or mixed Polsson-Gausslan) process. The reason can beascribed partly to its realism in modelling price changes as being generated bythe arrival of two types of Information. The first type Is the usual flow of newsthat gives rise to frequent and relatively small price changes, while the secondtype is the rare news event that causes the price of an asset to jump.Nimalendran (1994) suggested using the jump-dlffuslon model to conduct'event studies'. Kille et at. (1999) use the methodology to study the effects of the1994 Mexican crisis on US bank stock returns. Fisher (1999) investigates the expost Jump probabilities around the crash of 1987. He finds that the most likelyJump dates prior to the crash are connected to a disproportionate amount ofnews relating to trade Imbalances, level of the dollar, and financing of US debt.A case can also be made for the jump-diffuslon model on purely statisticalgrounds. Including jumps In the price process leads to leptokurtosls In thedistribution of returns of the magnitude that are typically observed In actualfinancial time series. Other alternative models do not succeed In matching theempirical distribution of returns to the same extent (see Akgiray and Booth(1987) for a comparison with a mixture of normal distribution).

Early empirical evidence of Jumps In Individual stock returns were found byPress (1967), Oldfield et al. (1977), Ball and Torous (1983, 1985) and Akgiray and

The European Journal of FinancelSSN 1351-847X prlnt/lSSN 1466-4364 online © 2003 Taylor & Francis Ltd

http://www.tandf.co.uk/joumalsDOl: 10.1080/13518470110099704

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106 E. Bentzen and P. Sellin

Booth (1986, 1987). In a well~lversified portfolio these jumps could of coursewash out, which would Imply that they would not have to be priced. Jarrow andRosenfeld (1984) test for the existence of jump risk In a number of US marketIndices. They find evidence of jumps In dally returns, but not In weekly returns.Jorion (1988) finds evidence of jumps In the weekly returns on the CRSP value­weighted Index, even after having taken account of ARCH effects. A more recentstudy of Jumps and ARCH effects Is Brorsen and Yang (1994). Kim et al. (1994)test for the existence of a common jump component In a multivariate setting.They find that the component stocks of the Major Market Index (MMI) containa common jump component. Feng and Smith (1997) find that trading profits canbe generated by employing technical analysis when returns are modelled as apure diffusion process. However, If jumps (and time-varying risk premia) areIncluded In the model the profit opportunities Virtually disappear, giving addi­tional support for Including jump risk In asset pricing models.

It Is In the option pricing literature that Interest In considering differentstochastic processes for asset prices (of the underlying) has been the greatest,starting with the classical papers by Cox and Ross (1976) and Merton (1976).The valuation of options becomes more difficult when prices follow processesthat Include jumps because It will no longer be possible to form a risk-freeportfolio, as in the Black-Scholes approach to options pricing. The llterature onoptions pricing with an underlying jump-diffuslon process has therefore beenpreoccupied with finding ways around this problem. Merton (1976) assumesthat Jump risk Is unsystematic, I.e. uncorrelated with the return on the marketportofolio, and therefore is not priced. Unfortunately Merton's model does notgenerate option prices that are much different from Black-Scholes (Ball andTorous, 1985). Models where jump risk Is systematic seem to be required Inorder to Improve upon Black-Scholes option prices (see, for example, Bates(1991) and Kim et al. (1994)).

Some recent papers that price options on an underlying jump-dlffuslonprocess Includes Scott (1991), Trautmann and Belnert (1999), Lelsen (1999),Bates (2000), and Kou (2000). Scott (1991) considers a jump-dlffuslon modelwith stochastic volatllity and Interest rates. Trautmann and Belnert (1999) applyBates' (1991) equlllbrium option valuation model with systematic jump risk toprice German options. L1esen (1999) uses a randomized trinomial model, wherea Poisson process Is the driving process, to price barrier options. Bates (2000)uses option prices to test whether the underlying Is better modelled as astochastic volatlllty model or a stochastic volatlllty/Jump-diffusion model. Hefinds that the model Including jumps leads to more plausible parameterestimates. Kou (2000) derives option prices In a model where the underlying Isa jump-dlffusion with (the logarithm of) the Jump amplitudes drawn from adouble exponential distribution.

The empirical evidence suggests that Jump risk should be considered whenconstructing an asset prlclng model. Ho et al. (1996) extend the continuous-timeAPT framework, developed In Chamberlain (1988), by considering jump riskand stochastic volatllity. They make a direct assumption about the form of thestochastic discount factor, which Includes Jump risk. Jarrow and Rosenfeld(1984) give sufficient conditions for the ICAPM, developed In Merton (1973), to

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The lCAPM with Poisson jump-diffusion returns 107

(1)

hold for asset prices that have discontinuous sample paths. The sufficientcondition is that jump risk be diversifiable in the market portfolio. However,they do not derive the capital market equilibrium for the case when jump risk isnot dlversifiable. This will be the purpose of the present paper. The strategy willbe to add enough restrictions to the model to be able to obtain an explicitsolution. We have not found any studies of the existence of Jumps in non-USstock returns, except for Germany (Belnert and Trautmann, 1991), so we alsoexamine if there are Jumps In the market indices of 18 countries, as well as in aworld stock market Index.

The plan of the paper Is as follows. In the next section the equlllbrium pricingrelationships are derived for our Intertemporal Jump-risk Capital Asset PricingModel (JACPM). Section 3 tests the hypothesis that there Is undlversifiable jumprisk In wel1-diverslfied portfolios. Section 4 concludes. Appendixes A-{:, containsmathematical derivations of results presented In the main text.

2. ASSET PRICING WITH JUMP RISK

We consider an economy of the type developed In Merton (1973) and modifiedby Jarrow and Rosenfeld (1984). It Is a pure exchange economy with one good,which serves as numeraire. The underlying assumptions are:

1. There are n risky assets and one risk-free asset. All assets are marketableand perfectly divisible. There are no taxes, transactions costs, or restric­tions on short sales.

2. Investors take prices as given.3. Trading takes place continuously In time at equlllbrium prices.4. There Is a risk-free rate of Interest, r, for borrowing and lending.5. Investors have homogenous expectations about asset prices, which

satisfy the stochastic processes

dPI

_PI - lL,dt + u,dZI + f,d Y - ~e,dt, i = I, 2, ..., n

where PI Is the price of asset i, ILl represents the Instantaneous expectedrate of return (InclUding the Jump), Zt Is a Wiener process, ut Is theInstantaneous standard deviation of the rate of return conditional on aJump In prices not occurring, Y Is a Poisson process with parameter A, £1

Is the stochastic Jump amplitude with expected value equal to et; and Zi' Y,and £1 are assumed to be Independent. The last two terms In (1) togetherrepresent the unexpected rate of return connected with the rare event.

6. Investors maximize their von Neumann-Morgenstern expected utility ofllfetlme consumption functions,

E, fOU(C(s), s)ds (2)

where E, Is the conditional expectations operator given the Informationavailable at time t and C(s) Is the rate of consumption. Investors have

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108 E. Bentzen and P. Sellin

Instantaneous utlllty functions that exhibit Constant Relative Risk Aversion(CRRA),

ct-y - 1U(C(f), f) = 1 exp(- pf), y > 0

-'Y(3)

where y Is the Arrow-Pratt measure of relative risk aversion and p Is theutility rate of time preference.

7. The jump amplitudes are nonstochastlc, f j & el , i = 1, 2, .•., n.8. All Investors have the same relative risk aversion, y.

Since we are considering a multivariate model, the following matrix notationwlll be useful:

w = (n X 1) vector of portfolio shares (wT Is the transpose of w);v = (n x 1) vector of excess rate of return, I.e. v~ = ILl - r for asset i;e = (n x 1) vector of jump amplitudes;n = (n X n) covariance matrix of (diffusion part of) returns.

Assumptions 1-4 are standard. In assumption 5, In addition to the diffusioncomponent, we let the returns be affected by a rare event that can cause theprices to jump. The probability of a Jump caused by the event In the timeInterval dt Is Adt, where A Is a contant. When the event occurs, there Is anInstantaneous jump In the return on asset I of size fl' For a homogeneousPoisson counting process with Intensity A the Interarrlval times, I.e. the timeInterval between two successive events, are Independently and Identicallydistributed. This may not be totally realistic for some events. For example, wewould expect the probablllty of a devaluation to be smaller Just after adevaluation has occurred. However, we can look at the Jump process as ageneric rare event, In which case homogeneity wlll be less of a problem; I.e. onetype of rare event, which Is Independent of the first event having taken place.For example, a devaluation could be followed by a strike In the steel Industry.Other examples of the type of rare events we have In mind are stricter environ­mental legislation, raised energy taxes, Inventions, a defaulting bank, or someother news that typically wilt affect more than one company.

The specification In assumption 5 Is slightly different from the one adopted byJarrow and Rosenfeld (1984), which originated In Press (1967). They let eachprice process have Its own Independent Jump component. We have chosen tolook at a rare event as something that affects more than one stock althougheach stock may be affected In a different way. Assumption 6 on preferences Ismore restrictive than In Jarrow and Rosenfeld's model. They merely assume atwice differentiable, strictly Increasing and strictly concave Instantaneous utllltyfunction. We need to make the assumption of CRRA In order to get an explicitsolution.

Under assumptions 1-6, the Investor chooses a portfolio rule, w = {w/(s)17.t,and a consumption rule, C(s), so as to maximize (2) subject to

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The lCAPM with Poisson jump-diffusion returns

N (dP )dW = LW,W -F - rdt + (rW - C)dtlsi I

109

(4)

(5)

where W is the investor's wealth. The way the risk-free asset enters into theequation for the change of wealth over time ensures that the portfolio shareswill sum to one. (See Merton, 1990, p. 127. A description of the construction ofthe portfolio shares Is also contained In Appendix A.) The Investor's problem Issolved by the use of dynamic programming (see Merton, 1990, Chapter 5). Asolution is found in terms of the maximum value (or Indirect utility) function,

JOY. t) = max tE, 1-V(C(s) , S)dS}IC,w);' t

In the empirical part of the paper (Section 3 below) we assume that €, is drawnfrom a lognormal distribution. But to facl1ltate the theoretical Interpretation weuse assumption 7 and consider the special case of nonstochastic Jump ampli­tudes (the general case Is treated In AppendiX C). Assumption 8 says thatinvestors have the same risk aversion. Thus, Investors will differ only withrespect to their wealth and utl1lty rate of time preference, which, as we will see,will not affect their portfolio choice.

The optimal portfolio rule is Implicitly given by

(6)

as derived In AppendiX B. This portfolio rule can be compared to the similar onederived in Merton (1990, p. 147). However, Merton assumes that jumps can onlyoccur in the return on the bond, which Is then not risk-free (note also that InMerton's notation the CRRA coefficient is 1 - 'Yand not 'Y). The portfolio rule Isclearly nonlinear. This means that there Is no easy way of aggregating portfolios.However, from assumption 8 It Is clear that all Investors will hold the sameportfolio In accordance with (6). Hence, all Investors wlll have to hold themarket portfolio, wm• If we.substltute the market clearing equation W = wm Into(6), we can derive the 'securlty market lines' for the Intertemporal Jump-riskCapital Asset Pricing Model (JCAPM), as described In Appendix B. The equilib­rium pricing relationship for asset; will be

v, - ~ = Pi(vm - g,,;) (7)

where P, = Ulm/~ Is the covariance (conditional on no jump) between asset iand the market portfolio, divided by the variance (condltlonal on no jump) ofthe market portfolio; vm = w~v Is the excess rate of return on the marketportfolio; g, = -Aei((1 + em)l-y - 1] Is the risk premium for jumps in the returnon assets; = 1, 2, ..., n, m; em =w~e Is the Jump In the return on the marketportfolio.

Equation 7 states that the JCAPM holds for expected rates of return that havebeen adJusted by the risk premia for Jump risk, ~ (i = 1, 2, ..., n, m). The risk

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110 E. Bentzen and P. Sellin

premia may be positive or negative, since the Jump amplitudes may be negativeor positive. If Jump risk Is dlverslfiable In the market portfolio, em = 0, then e, =o (i = I, 2, ..., n, m) and the standard ICAPM will hold, thus confirming theresult In Jarrow and Rosenfeld (1984). In the case of logarithmic utility ('Y = 1)we also get ~ = 0 (i = I, 2, ..., n, m) and the JCAPM again reduces to thestandard ICAPM.

Note that two assets with the same beta can have different expected rates ofreturn because of different expected Jumps In their prices. For example,consider an economy with 'Y> 1. A rare event that has a negative effect on bothasset i and the market portfolio (e, < 0 and em < 0) wltl result In a positive riskpremium for asset i a, > 0). If the same event has a positive effect on the returnto asset j (eJ > 0) It will result In a negative risk premium for asset j (tj < 0).Thus, the required rate of return will be higher on asset i than on asset j evenIf they have the same beta value (fJ, = Pi).

The risk premia can also be written In terms of the Indirect utility functionas

_ J..{W(1 + em)' f) - J..{~ f) _e, - J~~ t) "e" i-I, 2, ..., n, m (8)

where the first factor Is the relative Jump In the marginal utility of real wealth Ifa rare event occurs. An event that results In a negative Jump In the return on themarket portfolio wlll cause a positive relative Jump In the marginal utlllty ofwealth. For e, > 0 we than get €, > 0, I.e. the usefulness of asset i as a hedgeagainst the rare event results In a lower reqUired rate of return.

Let us next say something about the practlcallmpllcatlons of (7). Our resultsIndicate that If one Is to use the CAPM In evaluating the required rate of returnfor a project one should attempt to take the risk premia for Jump risk Intoaccount. Questions of the following type should be asked. What Is the prob­ablllty of, for example, a devaluation and what effect would It have on the returnto the project and on the return to the market portfollo? The answers should bestated In terms of expected return to the proJect, "e" and to the marketportfollo, "em, respectively. The CAPM can then be modified along the lines of(7) but computing e, and €m and making an assessment of the coefficlent ofrelative risk aversion, 'Y.

Of course, If lump risk Is diversifiable In the market portfolio we need notworry about that risk as long as we hold the market portfolio. This leads to theempirical part of the paper, which wl1l shed some light on this question that Iscruclal to the relevance of the model.

3. IS THEIR JUMP RISK IN THE MARKET PORTFOLIO?

We first discuss how to test whether Jump risk Is dlverslfiable In the marketportfolio. Next follows a description of the data. The maximum likelihoodestimates and the likelihood ratio test results are then presented. Due to knownproblems with the likelihood estimation for this type of model the results froman alternative estimation method are also presented.

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The lCAPM with Poisson jump-diffusion returns 111

3.1 Test procedureTo Investigate If there is Jump risk in the market portfolio let us look at themarket portfolio dynamics. The market portfolio consists of the marketweighted values, and equals M = 'i7- l wmiPj • Using (1), the return on the marketportfolio can be written as

dM"M = LWmi[JL,dt + u,dZ, + E,dY - Ae,dt] (9)

I-I

If Jump risk Is diverslfiable the condition for the ICAPM to hold can be stated as(Jarrow and Rosenfeld, 1984)

"LWmi[g,d17i + E,dY - Aejdt] = 01"1

(10)

where g,d17i = u,dZ, - f,dl/1, I.e. dZ, has been divided up into a common factor, dl/1,and residuals, d17,. Condition (10) says that the market portfolio shares {Wmi17.1must be such that the stochastic components of the returns from the assets areeliminated, except for the common factor dl/1.

The hypothesis to be tested is the condition In (10) and if accepted we havethe return on the market portfolio,

dMM = JLdt + fdl/1 (11)

with (9) with drift JL = "i..7.IWmiJLI and standard deviation f = 'i7.1wmif,. If Jump riskis not dlverslfiable we have the alternative market return process

dMM = JLdt + fdl/1 + roY (12)

where f Is the standard deviation conditional on no Jump and dl/1 and dYareIndependent Wiener and Poisson processes.

The log-likelihood function corresponding to (11) is

(13)

where T is the number of returns, h is the increment of time between observa­tions, and m, = M,/M'_I'

Now let Jumps arrive according to a Poisson process with mean number ofJumps equal to A > O. As in the previous literature, we assume that the Jumpsize, E, Is a sequence of independently and Identically distributed lognormalrandom variables with parameters (0, ll). The Jump size and the Poissonprocess are Independent (see Karlin and Taylor, 1981). Generally, the processcan be described as

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112

In rot = p. + adZ +~E.

"

E. Bentzen and P. Sellin

(14)

The jump sizes, c", which all are Identically lognormally distributed, give thefollowing law for the sum: I"E. - T(nO, n~. With the law of Adenoted by P(A),the final process can be described as P(A)T(jL + nO, cr + nOZ), with mean timebetween jumps given by E(T) = A-1T(p. + nO, cr + n~.

We can write the log-likelihood function corresponding to (12) as (Basawa andRao, 1980)

nIn Lu = -nAh - '2ln(217')

(15)

Alikelihood ratio test given by LR = -2 (In Le - In LJ can be used to test thehypothesis that jump risk Is dlverslfiable, with likelihood Le, versus the altern­ative that jump risk Is not dlverslfiable, with likelihood Lu' The LR has a X­distribution with degrees of freedom equal to the difference In the number ofparameters between the two models (In our case 5 - 2 = 3 d.f.). As pointed outby Ball and Torous (1985), the Infinite sum In (15) has to be truncated so thatsufficient accuracy Is achieved. The actual truncation depends on the value of A.The estimation was carried out In double precision and the Infinity sum wastruncated at J= 10.

3.2 Description of dataThe empirical tests were performed on value-weighted Indices from 18 countries(and a 'World' Index). The Indices used were conected from Morgan StanleyCapital Market Indices. The data consist of dally observations from January1985 to December 1999, which means that for each Index we have 3914 observa­tions. Weekend returns are treated as overnight returns. A technical descriptionof the Indices can be found In Morgan Stanley (1986).

In Table 1 simple summary statistics are given for each Index. Besides means,standard deviations, skewness, and kurtosis, we report the number of observa­tions with respect to sigma limits. In order to shed some light on the possibleeffect that the events In October 1987 may have on the analysis, the results for1987 are reported separately. Panel A of Table 1 contains summary statistics forthe period for 1985-1986, 1988-99, while Panel B contains statistics for 1987only.

Table 1should be read as follows. It Is seen that the world Index has a positivedally return of 0.0518%, a standard deviation of 0.7009%, a negative skewness of-0.2229, and a kurtosis of 3.8217. Compared with a normal distribution we havean empirical distribution with negative skewness and a kurtosis which Impliesthat the distribution Is much more peaked (and has fatter tails) than the normal.A normal distribution has a skewness of zero and a kurtosis of three. We found

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The lCAPM with Poisson jump-diffusion returns 113

Table 1. Summary statistics of daily Morgan Stanley Capital Market Indices of 18 countries,1985-1999

Mean Std Less than More than

Indices (%) (%) Skewness Kurtosis ::t2*Std :t 2*Std

Panel A: Summary statistics 1985-1986, 1988-1999World 0.0518 0.7009 -0.2229 3.8217 3464 189Austrlia 0.0341 1.1454 -0.2569 3.3169 3469 184Austria 0.0485 1.2389 -0.2017 7.6941 3481 172Belgium 0.0657 1.0200 0.1820 5.6107 3483 170Denmark 0.0574 1.1468 -0.2829 8.7698 3478 175France 0.0771 1.1426 -0.2712 3.6408 3477 176Germany 0.0735 1.2713 -0.5862 7.9970 3478 175Italy 0.0648 1.4760 -0.3797 4.5881 3470 183Japan 0.0327 1.4156 0.3918 4.8774 3454 199Netherlands 0.0656 0.9822 -0.1964 3.1884 3468 185Norway 0.0393 1.3561 -0.3113 5.9120 3469 184Spain 0.0615 1.2681 -0.2605 5.5872 3481 172Sweden 0.0845 1.3207 0.0164 5.1793 3488 165Switzerland 0.0731 1.0715 -0.2746 4.0941 3459 194UK 0.0485 1.0214 -0.0021 2.0535 3466 187Canada 0.0327 0.8239 -0.6689 7.2108 3471 182Hong Kong 0.0687 1.6669 -1.1171 25.4012 3487 166Singapore 0.0399 1.3050 -0.2082 13.1808 3484 169USA 0.0604 0.8680 -0.5476 6.1106 3462 191

Panel B: Summary statistics 1987World 0.0513 1.3508 -2.3311 25.0290 256 5Austrlia 0.0250 2.5413 -5.4338 53.4370 251 10Austria 0.016 1.2197 -0.7628 4.1518 248 13Belgium 0.0161 1.4610 -1.1477 11.1999 248 13Denmark 0.0412 1.3678 -1.8768 17.8768 250 11France -0.0623 1.6347 -0.8418 7.3617 249 12Germany -0.1152 1.7411 -0.6481 4.7665 248 13Italy -0.0975 1.3826 -0.5887 2.1948 250 11Japan 0.1355 2.0011 -2.5762 30.1590 251 10Netherlands 0.0132 1.7904 -0.8588 13.2461 249 12Norway 0.0137 2.5474 -3.6384 37.6041 251 10Spain 0.1084 1.8838 -0.5697 6.4745 252 9Sweden 0.0015 1.6568 -1.0828 6.0034 246 15Switzerland -0.0434 1.5984 -2.2825 14.3334 248 13UK 0.1051 1.7713 -2.8193 20.6310 253 8Canada 0.0422 1.5937 -1.8527 18.3505 253 8Hong Kong -0.0285 2.9934 -8.6634 109.6533 258 3Singapore 0.0032 2.6248 -4.4162 45.3670 254 7USA 0.0023 2.0723 -5.1611 57.9661 254 7

The summary statistics are reported in percentage terms. The final two columns count the number ofreturns less than (more than) two times the standard deviation.

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114 E. Bentzen and P. Sellin

3464 observations with a return less than two times the standard deviation and189 observations with a return above two times the standard deviation. Lookingat 1987, in panel B, It Is seen that the dally mean in the world index Is 0.0513%,the standard deviation Is 1.3508%, with a skewness of -2.3311, and a kurtosis at22.029. We found 256 returns less than two times the standard deviation and fivereturns above two times the standard deviation.

The computed standard deviations are quite large. With these large standarddeviations nothing significant could be said about the mean. This Is the case forboth data sets.

The computed skweness should be compared with a normal distribution,where the skewness Is zero. For the period excluding 1987 most Indices arefound to be negative. Only three Indices have a positive skewness, Belgium(0.18), Japan (0.39), and Sweden (0.01). looking only at 1987 It Is seen that allindices have a negative skewness. It is also seen that very large values ofskewness are reported In 1987. Hong Kong has a positive skewness (0.3918) forthe period that excludes 1987, but shows negative skewness for 1987 (-2.5762).The largest negative skewness is found In 1987 for Hong Kong (-8.6634).

The kurtosis measures indicate that the return distributions for some indicesare peaked. If we look at 1987 alone the values of the kurtosis are very large, thelargest being for Hong Kong (109.6533). Three distributions are less peaked In1987, Austria (4.1519), Germany (4.7665) and Italy (2.1948).

If we let a possible Jump be Identified by a return whose absolute value isabove two times the standard deviation, there are several possible Jumps In theIndices according to the last column In the table. Not all of these jumps belongto 1987, but a relatively large number do. It seems reasonable to make theempirical Investigation In two parts. One which Includes 1987 and a secondwhich excludes 1987 and then compare the two Investigations.

3.3 Estimation of parameters

The parameters In the model have been estimated using the data describedin the previous section. Table 2 covers the period from 1985 to 1999 and theestimated parameters from the two stochastic processes (11) and (12) aredisplayed together with the corresponding standard errors and the llkellhoodratio tests. l

In Table 2 it is seen that the estimates of the Poisson jump process have amean number of jumps, A, that are all significantly different from zero. Thissuggest the existence of Infrequent discrete movements. The mean number ofJumps per day, which can also be Interpreted as the probability of a Jump perday, for Austria (0.292), Italy (0.467), Japan (0.594), Hong Kong (0.261) and USA(0.677) are large numbers compared with those from other studies. Beckers(1981) finds a 12% probability of a Jump per day, while Ho et al. (1996) find a 20%probability for the period 1984-1986 and a 10% probability for the period1987-1989. Both studies use US data. For the US index we find a 67% probability

I The estimations of the maximum likelihood estimators have been carried out using the sub­routine VF13AD from Harwell (1989). Standard errors are obtained from the diagonal of the Hessian(Rao, 1965).

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The lCAPM with Poisson jump-diffusion returns 115

Table 2. Estimates of the mixed Poisson Jump-diffusion model and the simple Wiener

process model of daily returns to the stock market indices of 18 countries, 1985-1999

Weiner-Poisson jump process Weiner process1985-1999 JL CT A e ~ P- CT LA. test

World 0.082 0.503 0.263 -0.104 1.035 0.052 0.761 36460(0.008) (0.003) (0.031) (0.003) (0.316) (0.001) (0.009)

Australia 0.085 0.997 0.063 -0.698 2.806 0.033 1.286 34251(0.023) (0.022) (0.013) (0.051) (0.855) (0.001) (0.015)

Austria 0.046 0.804 0.029 0.023 1.702 0.045 1.238 33096(0.011) (0.005) (0.217) (0.010) (2.080) (0.001) (0.014)

Belgium 0.076 0.803 0.166 -0.047 1.627 0.062 1.055 35248(0.012) (0.006) (0.087) (0.012) (1.298) (0.001) (0.012)

Denmark 0.081 0.810 0.229 -0.080 1.660 0.056 1.163 34009(0.013) (0.006) (0.056) (0.006) (0.643) (0.001) (0.013)

France 0.118 0.896 0.201 -0.214 1.669 0.068 1.182 34546(0.014) (0.006) (0.026) (0.008) (0.579) (0.001) (0.013)

Germany 0.126 0.952 0.183 -0.311 2.022 0.061 1.308 33671(0.014) (0.006) (0.016) (0.010) (0.516) (0.001) (0.015)

Italy 0.094 0.949 0.467 -0.062 1.613 0.054 1.470 31868(0.014) (0.006) (0.094) (0.006) (0.593) (0.001) (0.017)

Japan 0.009 0.751 0.594 0.068 1.574 0.040 1.462 30301(0.012) (0.004) (0.071) (0.002) (0.436) (0.001) (0.017)

Netherlands 0.105 0.776 0.141 -0.269 1.821 0.062 1.055 35137(0.012) (0.004) (0.014) (0.009) (0.509) (0.001) (0.012)

Norway 0.067 0.965 0.215 -0.091 2.225 0.038 1.465 32093(0.016) (0.006) (0.053) (0.008) (0.546) (0.001) (0.017)

Spain 0.092 0.987 0.135 -0.138 2.338 0.065 1.318 33711(0.016) (0.008) (0.034) (0.010) (0.847) (0.001) (0.015)

Sweden 0.111 1.025 0.134 -0.174 2.345 0.079 1.346 33703(0.016) (0.008) (0.030) (0.010) (0.867) (0.001) (0.015)

Switzerland 0.122 0.831 0.159 -0.220 1.702 0.065 1.114 34939(0.013) (0.006) (0.019) (0.009) (0.512) (0.001) (0.013)

UK 0.088 0.883 0.109 -0.273 1.817 0.052 1.087 35567(0.013) (0.006) (0.015) (0.016) (0.722) (0.001) (0.012)

Canada 0.069 0.571 0.211 -0.152 1.404 0.033 0.896 35205(0.009) (0.003) (0.019) (0.004) (0.299) (0.001) (0.010)

Hong Kong 0.124 0.949 0.261 -0.48 2.535 0.062 1.786 29347(0.015) (0.005) (0.029) (0.005) (0.329) (0.001) (0.020)

Singapore 0.061 0.790 0.214 -0.070 2.342 0.037 1.431 31006(0.013) (0.004) (0.039) (0.005) (0.337) (0.001) (0.016)

USA 0.070 0.437 0.677 -0.006 0.955 0.057 9.994 31725(0.007) (0.002) (0.187) (0.003) (0.368) (0.001) (0.011)

Equations 12 and 13 are estimated using daily returns, where the mean rate of return is given by IL,

while tr Is the standard deviation of the return, AIs the mean number of jumps per day, (J and 6 are the

expected value and standard deviation of the logarithm of the jump amplitude. The estimated

parameters IL, tr, 8 and 6 are reported In percentage terms. Standard errors are reported in parenthesis

below the point estimates (also In percentage terms). The final column gives the likelihood ratio test

for the hypothesis that Jump risk Is diversifiable versus the hypothesis that it is not.

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116 E. Bentzen and P. Sellin

Table 3. Estimates of the mixed Poisson Jump-diffuslon model and the simple Wiener

process model of daily returns to the stock market Indices of 18 countries, 1985-1986,

1988-1999

Weiner-Poisson jump process Weiner process

1985-1999 p. U' A 6 8 p. U' LA. test

World 0.080 0.436 0.512 -0.050 0.754 0.052 0.701 34057

(0.007) (0.003) (0.063) (0.002) (0.314) (0.001) (0.008)

Australia 0.082 0.822 0.370 -0.111 1.287 0.034 1.145 32008

(0.013) (0.006) (0.059) (0.004) (0.579) (0.001) (0.013)

Austria 0.045 0.804 0.291 0.036 1.707 0.048 1.239 30879

(0.016) (0.005) (0.218) (0.023) (3.615) (0.001) (0.015)

Belgium 0.072 0.785 0.199 -0.004 1.419 0.066 1.020 33164

(0.012) (0.006) (0.510) (0.024) (2.139) (0.001) (0.012)

Denmark 0.086 0.778 0.308 -0.074 1.455 0.057 1.147 31632

(0.013) (0.005) (0.068) (0.005) (0.596) (0.001) (0.013)

France 0.131 0.837 0.344 -0.139 1.301 0.077 1.143 32176

(0.013) (0.006) (0.049) (0.005) (0.573) (0.001) (0.013)

Germany 0.130 0.912 0.246 -0.199 1.721 0.073 1.271 33671

(0.014) (0.006) (0.031) (0.008) (0.533) (0.001) (0.015)

Italy 0.098 0.926 0.506 -0.044 1.585 0.065 1.476 29525

(0.014) (0.006) (0.123) (0.006) (0.634) (0.001) (0.017)

Japan -0.009 0.714 0.680 0.071 1.456 0.033 1.416 28337

(0.012) (0.004) (0.067) (0.001) (0.336) (0.001) (0.017)

Netherlands 0.113 0.705 0.283 -0.151 1.266 0.066 0.982 33031

(0.011) (0.005) (0.033) (0.004) (0.483) (0.001) (0.011)

Norway 0.045 0.876 0.365 0.008 1.674 0.039 1.356 30266

(0.013) (0.006) (0.570) (0.016) (1.092) (0.001) (0.016)

Spain 0.086 0.959 0.144 -0.118 2.158 0.062 1.268 31755(0.016) (0.008) (0.043) (0.011) (0.982) (0.001) (0.015)

Sweden 0.105 1.038 0.126 -0.094 2.265 0.084 1.321 31743(0.017) (0.009) (0.056) (0.018) (1.510) (0.001) (0.015)

Switzerland 0.115 1.192 0.013 -3.907 1.169 0.073 1.071 32459(0.020) (0.019) (0.001) (0.517) (0.434) (0.001) (0.013)

UK 0.085 0.787 0.314 -0.098 1.150 0.049 1.021 33166(0.012) (0.006) (0.066) (0.005) (0.824) (0.001) (0.012)

Canada 0.068 0.544 0.269 -0.120 1.159 0.033 0.824 33576(O.OOg) (0.003) (0.027) (0.003) (0.325) (0.001) (0.010)

Hong Kong 0.116 0.927 0.278 -0.123 2.460 0.069 1.667 28079(0.015) (0.005) (0.035) (0.005) (0.345) (0.001) (0.020)

Singapore 0.047 0.734 0.278 -0.040 1.960 0.040 1.305 29630(0.012) (0.004) (0.319) (0.014) (0.553) (0.001) (0.015)

USA 0.062 0.385 0.854 -0.003 0.821 0.060 0.868 30387(0.006) (0.002) (0.184) (0.003) (0.545) (0.001) (0.010)

Equations 12 and 13 are estimated using daily returns. where the mean rate of return is given by IJ.,

while fT Is the standard deviation of the return, A Is the mean number of jumps per day, 6and 8are the

expected value and standard deviation 01 the logarithm 01 the jump amplitude. The estimated

parameters J.t. fT, 6and 8are reported In percentage terms. Standard errors are reported In parenthesis

below the point estimates (also In percentage terms). The final column gIves the likelihood ratio test

for the hypothesis that jump risk Is diversifiable versus the hypothesis that It Is not.

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The ICAPM with Poisson jump-diffusion returns 117

for the period 1985-1999. In the next section we show that this difference is dueto the differences in the estimation methods used.

Estimates of the jump size show an expected positive value for two indices,but for most indices a negative jump size is found. The estimated standarddeviations for the jump size are all very large, so nothing unambiguous could beread from these numbers. The estimates for the diffusion part shows all positivevalues for the mean. But again the estimated standard deviations are very large.The simple Wiener process shows positive means in all indices, but also withvery large standard deviations.

The likelihood ratio tests are all significant compared with a K with threedegrees of freedom, which clearly indicate the existence of jumps in the Indices.So while the estimated values for the probabilities of jumps are larger than inprevious studies, it is clear that the model with a jump component gives a muchbetter fit to the data. The two estimated models show that It Is not possible todiversify jump risk. With a computed likelihood ratio test that Is significant at alllevels, we are able to reject the hypothesis that jump risk is diversifiable in themarket portfolio.

As indicated in the simple data description In Table I, it is possible that theturbulence during the single year 1987 could be a major cause of the jumps inthe indices. In Table 3 we have re-estimated the two models, excluding the datafor 1987. The largest mean numbers of jumps are found in USA (0.854), Japan(0.68) and UK (0.314). The smallest number of jumps is found in Switzerland(0.013). The jump size in Austria (0.036) and Japan (0.071) are both positive. Thejump size in all other indices are negative. This means that when a jump occursIn most indices one would expect this with a negative sign. The negative jumpsize in Switzerland (-3.907) is the largest negative jump size estimated for anyIndex. But it is not outweighed by a large positive drift (0.115) In the diffusionpart of the model. All Indices show a positlve drift in the diffusion part, i.e. apositive trend Is found In all indices. The simple Wiener process does not giveany significant results. The means are all positive but not significantly differentfrom zero.

The likelihood ratio tests are all significant compared with a K with threedegrees of freedom. This result shows that even for the period that excludes1987, It Is not possible to diversify jump risk. We find a significant number ofjumps In all Indices.

It Is also Interesting to compare the Jump component's variance as a share ofthe total variance, I.e. Ad/(eil + A8~, for the two periods. If we look at the WorldIndex for the period excluding 1987 the share Is 60%, and If we include 1987 thejump component's share of the total variance falls slightly to 58.7%. Thus, thesingle year 1987 does not appear to be Important for our overall results.

3.4 An alternative estimation method

The estimation of the mixed jump-diffuslon model is subject to serious estima­tion problems. From inspection of Equation 15 it Is clear that it is not easy toidentify a jump. The likelihood function (and its derivatives) are highly non­linear and contain an Infinite sum. This has led some researchers, starting withPress (1967), to employ a version of the method of moments known as the

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118 E. Bentzen and P. Sellin

cumulant matching method as an alternative to the maximum likelihoodmethod.

We wl11 folJow Beckers (1981) In assuming that the mean jump amplitude Iszero. which Implies that our theoretical model reduces to the standard lCAPM.This gives us one less parameter to estimate. Even though the mean jumpamplitudes are significant In all cases. as can be seen In Table 2. they are small Inmost cases and we should obtain more accurate estimates of the Jump prob­abilities by using Beckers' approach. The remaining parameters are estimatedas

p. = K1

cr = K2 - K2J3K6

fjZ = KrJ5K.

A = 25K~/3Kl

where K, Is the ith cumulant of the distribution.The estimated parameters are presented In Table 4. The jump components

share of total variance Is 30% for the World Index. The probablllty of a jump perday ranges from a low of 0.002945 (USA) to a high of 0.081231 (Sweden) and Is

Table 4. Methods of moments parameter estimates 1985-1999

1985-1999 JL u A 8

Wortd 0.0518 0.6348 0.0128 3.7380Australia 0.0335 1.1134 0.0032 11.4292Austrta 0.0453 1.0225 0.0292 4.0885Belgium 0.0624 0.8434 0.0381 3.2637Denmark 0.0563 0.9029 0.0370 3.8193France 0.0678 0.9032 0.0670 2.9567Germany 0.0609 1.0783 0.0293 4.3421Italy 0.0540 1.1312 0.0678 3.6110Japan 0.0396 1.2616 0.0141 6.2231Netherlands 0.0621 0.8424 0.0308 3.6393Norway 0.0375 1.2868 0.0052 9.6931Spain 0.0647 0.9555 0.0726 3.3754Sweden 0.0789 0.9750 0.0812 3.2649Switzertand 0.0654 0.9076 0.0332 3.5675UK 0.0523 0.9640 0.0099 5.0874Canada 0.0333 0.6997 0.0223 3.7473Hong Kong 0.0622 1.4549 0.0044 15.6677Singapore 0.0374 1.1777 0.0068 9.8586USA 0.0566 0.8459 0.0029 9.6862

Equation 14 Is estimated using daily returns for computing the moments and cumulants. 1£Is the mean rate of return, q Is the standard deviation of the return, " Is the mean number oflumps per day, and ~ Is the standard deviation of the logarithm of the jump amplitUde. Theestimated parameters 1£, q and 8 are reported In percentage terms.

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The ICAPM with Poisson jump-diffusion returns 119

on average 0.031. These results are more In line with the jump probabilitiesfound In Beckers (1981) and Ho et al. (1996). Of course this estimation methodIs not free of problems. Beckers (1981) and Ball and Torous (1983) obtainseveral negative variance estimates using this method. Beckers then Imposesthe restrictions that all 47 stocks investigated must have the same mean numberof jumps, after which all variance estimates become positive. We do not havethat problem with our country Indices, perhaps because they are broad indicesand not the price of Individual stocks. A further drawback with this method isthat standard errors of the estimates are not readily available.

4. CONCLUSION

This paper has Investigated the Intertemporal Capital Asset Pricing Model withreturns that follow mixed jump-diffusion processes. We added just the numberof restrictions as needed to give an explicit solution for capital market equilib­rium. It was shown that the CAPM relation holds only after expected returnshave been adjusted for jump risk premia, except In two special cases. The firstcase Is when jump risk is dlverslfiable In the market portfolio. The second caseIs when Investors have logarithmic utility.

In estimating the model we used Indices from 18 countries covering theperiod 1985 to 1999. We come to the same conclusions as Jarrow and Rosenfeld(1984) did for US data for an earlier period. namely that (1) the market portfoliocontains a Jump component, and (2) risk Is not diversifiable In the marketportfolio.

Our study also Illustrates the problems Inherent In estimating a mixed jump­diffusion model. Maximum likelihood and the cumulant matching method givevery different results when estimating the mean number of Jumps. However.both methods agree that the Jump component's share of total variance Is hlgh­30% according to the method of moments estimates and 60% for the maximumlikelihood estimates (59% If 1987 Is excluded from the sample).

APPENDIXES

A Construction of portfolio sharesLet N, be the number of company i shares the Investor holds (i = 1,2, ..., n). Ifthe Investor's only Income comes from his portfolio of shares and a risk-freeasset (i = 0). his wealth wlll evolve according to

n n

dW = 'LN,dP, - Cdt = 'LN,dP, + NodPo - Cdt'-0 '-I

where Cdt Is the flow of consumption. We can rewrite this as

n n

d W ~ N,P, dP, ( ~ N,P,)=W LJ - - - W 1 - LJ - nlt - CdtW P W'-1' I-I

(16)

(17)

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120 E. Bentzen and P. Sellin

where r Is the return on the risk-free asset. Defining the portfollo share w, = N,P,!I¥, we get

n dP ( n)dW = WLw,y - W 1 - LW, rdt - Cdt

'-1' '-I

= iw,w (rlJ' - rdt) + (rW - C)dt'-I '

which Is Equation 4 In the main text.

B Deriving the equilibrium with fixed Jump size

We begin by defining the maximum value function

Jel¥, t) = max {Et J.-V(C(s) , S)dS}.IC.w);" t

(18)

(19)

The Hamllton-Jacobl-Bellman OIJB) equation for this problem Is (see Malllarisand Brock, 1982: Chapter 2, Sectlon 12),

0= max(1 - 'Y)-IC'-,. + J,(lY, f) + Jw(Jf. t)(WwT(v - Ae) + Wr - CIC.w

+ ~J~J¥, t)W2wTfiw + f"A(J(W + WwT E(a), f) - J(l-Y. f)lf(a)da} (20)

where Jx Is the partial derivative with respect to argument x, I(a) Is the densityfunction for the Jump amplitudes, and the other notation Is the following:

w = en x 1) vector of portfolio shares (wT Is the transpose of w);v = (n x 1) vector of excess rate of return, I.e. v~ = IL, - r for asset i;E = en x 1) vector of Jump amplltudes;e = (n x 1) vector of expected Jump amplitudes;n = (n x n) covariance matrix of asset rates of return.

Under assumption 7 the Jump amplitudes are nonstochastlc, E = e. We seek asolution to the following set of equations, where a hat denotes an optimalcontrol:

o= t(l - 'Y)-IC1-,. + J,(lY, f) + Jw(lY, t)[WwT(v - Ae) + Wr - C)

+ ~J~l¥, I)W2WTnw + A[J(W + WwTe, I) - J(J-Y, t)]} (21)

a!:} = c-,. - Jw<1¥, t) = 0 (22)

:~ =Jw<1¥, 1)(1' - Ae) + J~J-Y, t)wnw + Jw(W + WwTe, I)e =0 (23)

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121

(24)

The ICAPM with Poisson jump-diffusion returns

A solution is conjectured to be

J(W') = pW'-Yexp(-pI)

where p is a constant. Plugging in the proposed solution we get

p = _1_ {! (p _ h - A[(1 + wTe)l-Y - I D}-Y (25). 1-""

which will be constant If the optimal portfolio, W, is constant over time. Thesolution to the investor's problem is well-defined If the following growth con­dition holds,

p> h + A [(1 + wTe)I-Y - 1] (26)

where h is defined as

(27)

We find that the Investor's optimal portfolio rule is Implicitly given by thefollowing first-order condition:

(28)

Since there are no tlme-dependent variables In this equation we conclude thatthe optimal portfolio wlll be constant through time. This Is consistent with theassumption that p is a constant and thus the conjectured solution In (24) IsIndeed valid. Note that for A = 0 the optimal portfolio in (28) reduces to theusuailCAPM optimal portfolio, w = ,,-In-Iv.

Using assumption 81n the main text, we can substitute W = wm and rearrange(28) to get

II = ,.Gwm + A[(1 + w~e)l-Y - l]e (29)

Premultlplying (29) by w~ we get

(30)

(32)

Simplifying the notation,

JIm = ,,~ + ~m (31)

where the notation can be found after Equation 7 in the main text. Combining(29) and (31) we get

1v = O! (JIm - ~,JnWm + ~

m

>From this equation we get the scalar forms (7) in the main text.

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122 Eo Bentzen and P. Sellin

C Deriving the equilibrium with stochastic Jump size

The HJB equation Is the same as In the problem with a fixed Jump size. However,the solution will now have to be valid for the following set of equations:

0= {(I - ,.rICS-" + J,(W; t) + JwCW; t)[WwT(v - Ae) + Wr - C]

+ ~J~W; t)W2wToW+ fAA[J(W + WWT €(a), I) - JOY. I)]f(a)da} (33)

a~} = C-" - J~ t) = 0

t! = J~ t)(v - Ae) +J~ t)WOw

+ AfA J(W + WWT€(a), I)€(a)f(a)da = 0

(34)

(35)

where a tilde denotes an optimal control. The solution should be of the sameform as In (24). Plugging In that solution, we get

p= 1 : ,. {; (p - h - AL[(l + wT€(a))l-Y - l]f(a)da)ry(36)

The Investor's optimal portfolio Is given by

w - .; [0-1(1' - eA) - A£[(l + wT €(a))l-,.n- I€(a)f(a)da] = 0 (37)

In exactly the same way as for the case with a fixed Jump size, we can derive theJCAPM relation,

•where the risk premia for the Individual assets are now given by

gl = A [el + £[(1 + wT €(a))I-,.O-lf/a)f(a)da]

while the risk premium on the market portfoUo Is given by

gm = A [w~e + fA[(1 + wT€(a))I-"O-lw~€(a)f(a)da]

(38)

(39)

(40)

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The ICAPM with Poisson jump-diffusion returns 123

ACKNOWLEDGEMENTS

We wish to thank Marten Blix, Stephen Satchell, Lars E.O. Svensson, IngridWerner, and the participants at seminars at the Institute for Internatio~alEcon­omic Studies at Stockholm University, and European Finance Association Meetingin Copenhagan, as well as two anonymous referees for valuable comments.

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