Evaluating stochastic discount factors from term structure models

rk has ate ve likelihood-based time-series analysis is then ricing of ob ed po M) Journal of Empirical Finance 16 (2009) 852–861 Contents lists available at ScienceDirect Journal of Empirical Finance j ourna l homepage: www.e lsev ie r.com/ locate / jempf in assumptions need to be made on the distribution of the data. Heretofore most applications of the SDF approach have been in discrete time. However an SDF is guaranteed to exist in any model, either in discrete or continuous time, in which arbitrage is not allowed. The purpose of this paper is to examine to what extent the SDF approach can be useful in estimating and testing continuous-time term structure models. Typically term structure models fully specify the dynamics of the state vector under the true probability measure and so likelihood-based inference is (in principle) feasible. However in order to apply likelihood-based techniques the researcher must know the exact functional relationship between asset prices and the state vector. In many cases this is not known, as when dealing approach. The SDF approach has gain Generalized Method of Moments (GM ☆ This paper has benefitted from useful comments b of California at Berkeley, the University of Illinois at U Arizona, the University of Utah, Brigham Young Univer gratefully acknowledge financial support provided by ⁎ 15217 Isleview Dr. Chesterfield, MO 63017, USA. E-mail address: farnsworth@wustl.edu. 0927-5398/$ – see front matter © 2009 Elsevier B.V. doi:10.1016/j.jempfin.2009.06.004 pularity partially because it lends itself well to straightforward testing, based on the as in Hansen and Singleton (1982). An advantage of this approach is that very few Singleton (2003) for a survey). By contrast, recent work in equity p by a given model as a known function focused on models for which the prices of some set of claims, usually zero-coupon ctor so the paths of the state variables can be inferred from bond prices. Typically applied to the paths of this state vector in order to estimate parameters (see Dai and models has been focused on constructing the stochastic discount factor (SDF) implied servable variables and applying it to a set of returns. I will call this approach the SDF Term structure models Monte Carlo method GMM estimator 1. Introduction Most empirical term structure wo bonds, are known functions of the st Evaluating stochastic discount factors from term structure models☆ Heber K. Farnsworth⁎ Olin Business School Washington University in St. Loius, USA a r t i c l e i n f o a b s t r a c t Article history: Accepted 16 June 2009 Available online 27 June 2009 This paper examines the feasibility of applying the stochastic discount factor methodology to fixed-income data usingmodern term structuremodels. Using this approach the researcher can examine returns on bond portfolios whose exact composition is unknown, as is often the case. This paper proposes an observable proxy for the SDF from continuous-time models and documents via Monte Carlo methods the properties of the GMM estimator based on using this proxy. © 2009 Elsevier B.V. All rights reserved. JEL classification: G12 C1 C5 Keywords: Stochastic discount factors y my chairman, Wayne Ferson, as well as Richard Bass. Comments by seminar participants at the University rbana-Champagne, Ohio State University, Columbia University, Arizona State University, the University o sity and the University of Southern California are also acknowledged. All remaining errors are my own. I also the Thomas Foster as well as the Samuel Stroum fellowhips at the University of Washington. All rights reserved. f with r (2006 In a approa least t restric Wh may d feasib Duffie I fi (QML) there a restric 2. Met the tim implie 853H.K. Farnsworth / Journal of Empirical Finance 16 (2009) 852–861 piðtÞ where r(t) is the instantaneous risk-free rate (see, for example, chapter 6 of Duffie (1996)). A stochastic discount factor process is a scalar processm(t) having the property thatm(t)pi(t) is a martingale for all i. Therefore we must have mðtÞpiðtÞ = Et ½mðTÞpiðTÞ� or Et mðTÞ mðtÞ Riðt; TÞ � � = 1 ð1Þ where Ri(t,T) is the gross return on asset i between time t and T. By applying Ito's formula we can verify thatm(t) must satisfy dmðtÞ mðtÞ = −rðtÞdt−ΛðtÞ ′dWðtÞ: ð2Þ Solving the SDE (2) yields the following expression for the stochastic discount factor. mðTÞ mðtÞ = exp −∫ T t rðsÞds− 1 2 ∫Tt ΛðsÞ′ΛðsÞds−∫Tt ΛðsÞ′dWðsÞ � � ð3Þ In order to test the stochastic discount factor for a particular model wemust be able to constructm(T)/m(t) from observed data. An examination of Eq. (3) suggests that in general it will not be possible to recover the SDF directly from discretely sampled data. Therefore we must construct an SDF proxy which is based on the observed data. An affine model of the term structure is one in which bond prices satisfy Pðt; sÞ = expðAðs−tÞ−Bðs−tÞ′XðtÞÞ nsider a continuous-time economy inwhich the uncertainty is driven by a d-dimensional BrownianMotionW(t). Let pi(t) be e t price of the ith non-dividend paying security and let σi(t) be the vector of sensitivities toW(t). The absence of arbitrage s that there exists some vector process Λ(t) such that p(t) solves dpiðtÞ = rðtÞdt + σiðtÞ′ΛðtÞdt + σiðtÞ′dWðtÞ To understand the issues involved in applying the SDF methodology in continuous time one must understand what the SDF from a continuous-time model looks like. Co ally likelihood-based approaches do not provide an overall test of model fit. In the SDF approach there are over-identifying tions which can be tested so a given model can be rejected by the data. ether these considerations override the advantages of the likelihood-based methods is an open question and the answer epend on the particular model and data set under consideration. In this paper I endeavor to show that the SDF approach is a le and somewhat attractive alternative. To be concrete I will restrict attention to models which fall into the affine class of and Kan (1996). This class of model is well-known in the literature and can be estimated using either approach. nd that parameters are estimated with more accuracy using the SDF approach compared with a commonly used alternative . This may be due to the fact that the SDF approach allows the use of more data as mentioned above. However I find that re biases in the SDF approach which seem to be responsible for throwing off the distribution of the test for over-identifying tions. So the approach must be used with caution. hods additional parameters. Fin wn difficulties. ddition, since no model fits the data exactly some joint distribution of pricing errors must be assumed in a likelihood-based ch. But this increases the number of parameters that need to be estimated. The SDF approach is robust to pricing errors, at hose which are orthogonal to the SDF, so there is no need to specify a statistical model for these errors or estimate any generally used in the term structure literature. There are other advantages to the SDF approach over likelihood-based approaches. The first is the difficulty of evaluating the likelihood of the state vector. Except for a few special cases the likelihood of the state vector from a continuous-time model is typically not known in closed form. Numerical techniques to evaluate this likelihood are computationally expensive and present their o eturns on portfolios of bonds. The SDF approach allows including such returns in the analysis, as was done by Ferson et al. ) in their study of bond mutual funds. Such an analysis is not possible using the likelihood-based approaches that are where X(t) is the state vector and A and B are deterministic functions of time. In an affine model the state vector, X(t), is assumed to solve1 In No skepti paper. data o In is kno difficu 15 yea distrib Th daily y bonds will be beyond 854 H.K. Farnsworth / Journal of Empirical Finance 16 (2009) 852–861 are only considering affine models with state variable that have continuous sample paths. Affine jump diffusion models exist but examining these is the scope of this paper. 1 We order to evaluate the performance of the SDF approachwemust examine estimates using this approachwhen the truemodel wn. To do this we will simulate from a realistic model of the term structure that has multiple factors but is one which is lt to estimate using the time-series approach because it is non-Gaussian and has no closed-form likelihood.Wewill simulate rs worth of data and then estimate parameters. I will repeat this two hundred times to get an idea of the sampling ution of the estimators. ere is fairly broad agreement that three factors are adequate to describe the dynamics of U.S. yield curves. We will simulate ields on three bonds that we will use to solve for the three factors. In addition we will simulate monthly returns on coupon of varying maturities. These monthly returns will be polluted with pricing errors so the covariance matrix of these returns full rank even though there are more than three bond returns. estimation. For instance, Ferson et al. (2006) use these approximations to study monthly returns on bond mutual funds with daily data on the state variable proxies. In the next section we will use the approximations above at a daily frequency, assuming that the bonds we observe daily are observed without error, and then take products of daily SDFs to obtain monthly SDFs which wewill then apply to monthly returns on portfolios. 3. Finite sample properties talking to bond traders one discovers that the reported daily prices on the less liquid bonds are viewed with considerable cism. However month-end numbers are consideredmore reliable. The extent towhich this is true is beyond the scope of this However it is the case that researchers typically have at their disposal daily series on bonds of a fewmaturities andmonthly n a larger set of bonds and portfolios of bonds. The SDF approach allows the researcher to utilize both types of data in one to 1 day since daily is a common frequency of observation that researchers have access too. One of the peculiarities of fixed-incomemarkets is that there are a few bondswhich are very liquid and the others aremuch less so. In ∫t + Δt λ ′ SðuÞλdu≈λ′SðtÞλΔ ð8Þ ∫t + Δt λ ′ ffiffiffiffiffiffiffiffiffiffi SðuÞ p dWðuÞ≈λ′∑−1½Xðt + ΔÞ−XðtÞ−KΘΔ + KXðtÞΔ� ð9Þ what follows I will use M(t,T) to denote this proxy to the true SDF, m(T)/m(t). w clearly the quality of these approximations will be better for small Δ. In our simulation results we shall take Δ to be equal XðsÞ = XðtÞ + ∫stKðΘ−XðuÞÞdu + ∫st∑ ffiffiffiffiffiffiffiffiffiffi SðuÞ p dWðuÞ ð4Þ where S(t) is a diagonal matrix whose diagonal is given by α+β′X(t). The short rate r(t) is also assumed to satisfy r(t)=δ0+δ′1X(t). The market price of risk process, Λ(t) is assumed to be given by ffiffiffiffiffiffiffiffiffi SðtÞp λ for some vector λ so we have mðTÞ mðtÞ = exp −∫ T t ðδ0 + δ′1XðuÞÞdu− 1 2 ∫Tt λ ′ SðuÞλdu−∫Tt λ′ ffiffiffiffiffiffiffiffiffiffi SðuÞ p dWðuÞ � � ð5Þ The difficulty is that this continuous-time process is not observed continuously so some discretization is necessary. There is a large literature on approximating diffusions via discrete processes using normally distributed random numbers as proxies for Brownian increments. Our aim is somewhat the reverse. Assuming that we have identified state variables we can only observe their paths at discrete points. We must approximate the terms in the SDF expression using these discrete observations. If we did observe this process continuously thenwe could easily compute the integrals with respect to time in the SDF formula above. In that case we could obtain the stochastic integral using Eq. (4) λ′∑−1½XðTÞ−XðtÞ−∫Tt KðΘ−XðuÞÞdu� = ∫Tt λ′ ffiffiffiffiffiffiffiffiffiffi SðuÞ p dWðuÞ: ð6Þ Now suppose we only observe the state vector at discrete intervals of length Δ. Then we could approximate the integrals with respect to time by a (left) Riemann sum. ∫t + Δt rðuÞdu≈ðδ0 + δ′1XðtÞÞΔ ð7Þ model Single In For each path of the state variables I calculated two kinds of price data: (1) daily yields on three zero-coupon bonds with matur accord 10, 15, norma 3.2. Es In that th what k Sin discou where α(τ)=A(τ)/τ and b(τ)=B(τ)/τ. By stacking these equations for the 3 τs we obtain the following system of equations. Th 855H.K. Farnsworth / Journal of Empirical Finance 16 (2009) 852–861 variables for each date. XðtÞ = b−1ðYðtÞ + aÞ ð11Þ The SDF proxy is formed by the econometrician from these daily observations of the state variable. YðtÞ = −a + bXðtÞ: e estimation routine uses the current guess of the parameters to form a and b and solves for the implied levels of the state ities of 6 months, 3 years, and 20 years and (2) monthly returns on 6 coupon bonds. The daily data is simulated exactly ing to the model. The 6 coupon bonds were simulated to be at par at the beginning of each month with maturities of 2, 5, 7, and 20 years. At the end of each month the bonds are re-priced to compute a return. Then pricing errors were added as IID l noise with mean zero and standard deviation of 3 basis points per month. timation the estimation stage of the Monte Carlo the econometrician is assumed to know that the daily data has no pricing error and e monthly data has error. But the econometrician does not know either how much error is in the monthly returns or even ind of bond or portfolio the monthly returns correspond to. ce the yields of the three bonds observed daily are assumed to be observed without error we have that the yield of the nt bond with time-to-maturity τ, yτ satisfies. yðtÞτ = −aðτÞ + bðτÞ′XðtÞ XðtÞ = rðtÞ θðtÞ vðtÞ 4 5; K = κ −κ 00 ν 0 0 0 μ 4 5; Θ = θθ v 4 5 ∑= 1 σrθ σrv σθr 1 0 0 0 1 2 4 3 5; α = 0ζ2 0 2 4 3 5; β = 0 0 00 0 0 1 0 η2 2 4 3 5 where we note that α and β, defined as above, give us SðtÞ = vðtÞ 0 0 0 ζ2 0 0 0 η2vðtÞ 2 4 3 5: 3.1. Simulation strategy Since this model has no closed-form density it is also difficult to sample from. However the process which determines the conditional variance, v(t) is Markovian and in fact of CIR form. We can sample paths of this state variable which have exactly the right conditional distributions using well-knownmethods (see Glasserman, 2004 for details). I sampled 4 points per day and then simulated the other factors as Gaussian conditional on the path of this conditional volatility process. Ideally wewould have liked to initialize each path by drawing from the unconditional distribution of the state variables. However this unconditional distribution isnot knownanalyticallyand so sampling from it directly is not feasible. So instead I started eachpath at the unconditional mean of the process and simulated 25 years of data. Then I threw away the first 10 years so that the remaining portion of each path is started at a random point drawn from a distributionwhich ought to be close to the unconditional distribution. drðtÞ = κðθðtÞ−rðtÞÞdt + vðtÞdWrðtÞ… + σrθζdWθðtÞ + σrvη vðtÞdWvðtÞ dθðtÞ = νð−θ−θðtÞÞdt + σθr ffiffiffiffiffiffiffiffi vðtÞp dWrðtÞ + ζdWθðtÞ dvðtÞ = μð−v−vðtÞÞdt + η ffiffiffiffiffiffiffiffivðtÞp dWvðtÞ ð10Þ terms of the general framework of affine models we have 2 3 2 3 2 3 its “central tendency” θ(t), and its volatility v(t). ffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffip 1 with one process which drives the conditional variances of yields. The parameter values will be those estimated in Dai and ton (2000) using the parameterizationwhich the authors call their “preferred”model. The state vector is the short rate, r(t), The model we will use is from the A (3) branch of affine models. This notation means that the model is a three factor affine where condit i.e. where Be E ∑ d i=0 Xðt + ði + 1ÞΔÞ−Θ−expð−KΔÞðXðt + iΔÞ−ΘÞ " # = 0 ð14Þ E ∑ d i=0 Xðt + ði + 1ÞΔÞ−Θ−expð−KΔÞðXðt + iΔÞ−ΘÞ ! ⊗XðtÞ " # = 0 ð15Þ where exp is the matrix exponential and d is the number of trading days in a month (assumed to be 21 in the simulation). This gives us 12 addition moment conditions. Combining Eqs. (12), (13), and (14) gives us 36 moments to use in estimating 13 parameters. The estimation employs the so-called two-step GMM technique. In the first step an identity matrix is used as the weighting matrix. In the second step a new weighting matrix is calculated using the first-stage estimates which are then used in a second- 2 For instance the scores of the likelihood of the transition density of the state variables are arguably the best moments to match. But for this affine model the scores are not known in closed form. Table 1 Monte Carlo results using SDF method. 5% 25% Median True Mean 75% 95% θ ̅ v ̅ σrθ σrv σθr ξ2 η2 λr λθ λv Means 856 H.K. Farnsworth / Journal of Empirical Finance 16 (2009) 852–861 We could derive moment conditions directly from this equation but notice that these moment conditions are assumed to hold at a daily frequency rather than a monthly frequency like the pricing moments. The only complication here is in calculating the covariance of these daily moments with the monthly moments to form the weighting matrix. So instead I defined the following monthly moment conditions ional mean is known in closed form. E½Xðt + ΔÞ jXðtÞ� = Θ−expð−KΔÞðXðtÞ−ΘÞ the state variables. Some interesting moments are not known in closed form and so cannot be used. But for all affine models the condit E½Mðt; TÞðRcðt; TÞ−Rf ðtÞÞ� = 0 ð12Þ Rf is the return on a 1-period risk-free asset observed at time t. Using the 6 coupon bond returns gives us 6 moment ions. Additional moment conditions were obtained by imposing orthogonality with the beginning of month values of X(t), E½Mðt; TÞðRcðt; TÞ−Rf ðtÞÞ⊗XðtÞ� = 0 ð13Þ ⨂ is the Kronecker product. This gives us an additional 18 moments. sides the pricing moments above there are moment conditions we can impose based on the assumed stochastic process for 2 where Rc(t,T) is the gross return on the coupon bond from t to T (one month). In order to minimize the impact of any bias that the discretization introduces into the mean of our SDF proxy we instead use the following moment condition E½Mðt; TÞRcðt; TÞ−1� = 0 Since coupon bond is a portfolio of zero-coupon bonds the gross return on a coupon bond is the gross return on a portfolio of zeros. Using these portfolioweights and takingweighted averages of equations like Eq. (1) we see that the same equation holds for coupon bonds. By taking unconditional expectations and plugging in our proxy we obtain the following 0.176 0.285 0.361 0.365 0.350 0.405 0.504 0.049 0.068 0.078 0.083 0.077 0.084 0.110 0.007 0.012 0.015 0.015 0.015 0.018 0.025 −4.437 −3.573 −2.777 −3.420 −1.434 −1.755 7.033 3.213 4.682 6.103 4.270 15.806 8.130 15.035 −0.134 −0.112 −0.098 −0.094 −0.097 −0.081 −0.057 4.8e−5 1.5e−5 2.1e−4 2.0e−4 3.0e−4 3.1e−4 8.1e−4 0.000 0.001 0.005 0.008 0.024 0.023 0.122 8.510 10.355 12.337 9.320 14.552 16.630 28.032 4.113 25.055 36.591 31.700 43.829 50.844 142.893 −16.961 −5.191 −0.731 −0.344 −22.073 2.290 17.552 and percentiles of estimates compared with true values. κ 5.494 15.788 16.535 17.400 15.311 17.122 18.122 ν 0.206 0.220 0.227 0.226 0.225 0.232 0.261 μ step es using of the In estima to affin vector To where Table 2 Monte Carlo results using quasi-maximum likelihood. 857H.K. Farnsworth / Journal of Empirical Finance 16 (2009) 852–861 as noted above. The conditional variance is a bit more complex. But recall that for integrals with respect to Brownian motion we have that Vart ∫ s tgðuÞdWðuÞ h i = Et ð∫stgðuÞdWðuÞÞ 2 h i = Et ∫ s tgðuÞg′ðuÞdu h i : Using this fact we can write the conditional variance as Va rtðXðsÞÞ = Et ∫stΦðu−tÞ∑SðuÞ∑′Φðu−tÞ′du h i = ∫stΦðu−tÞ∑Et ½SðuÞ�∑′Φðu−tÞ′du which Howev means Sin into th this is ignore daily o On known instan one w this w SDF fr 3 Kee ed in the table are 5th, 25th, 75th, and 95th percentiles of the estimated parameter values to give an idea of the distribution estimates. order to gauge the accuracy of the SDF approach we need to compare to some other technique. Exact maximum likelihood tio