M O D E L I N G R I S K P R E M I A
Based on a Financial Market Model for the Netherlandsm u m ta z s h a f i q 1 0 0 0 2 1 4 4
A Master’s Thesis to obtain the degree in f i n a n c i a l e c o n o m e t r i c s
Carried out at
Supervisor: Prof. Dr. Ir. M.H. Vellekoop In-company supervisor: Ir. B.J.W. Kobus
A B S T R A C T
In this thesis, the financial market model for the Netherlands ofDraper (2014) is analyzed. The model is used to estimate risk premia for bonds and equity, and to determine the market prices of risk based on data relevant for the Netherlands. All model parameters are estimated by maximum likelihood using a simulated annealing algorithm. We find pa-rameter estimates that reasonably match the inflation and stock return expectations of the Commission Parameters (Langejan et al.,2014).
We find that the model implied expected inflation forecasts the real-ized inflation. Also, the nominal interest rate as implied by the model almost equals the three-month bond yield. Even when using only half of the dataset, the parameter estimates seem to describe the inflation and interest rate processes to reasonable extent. We conclude that the finan-cial market model is robust in the model parameters. This robustness applies to both the starting values chosen for the optimization, as well as the dataset used in the estimation. The parameter estimates presented in this thesis can be used to generate future scenarios for the economy. These can then be used by pension funds to determine the value of their pension obligations.
Keywords: Financial market model, interest rate process, stock risk
A C K N O W L E D G E M E N T S
I would like to take this opportunity to show my gratitude to a number of people who have made it possible for me to write my master’s thesis successfully. First of all, I would like to express my special thanks to my university supervisor Prof. Dr. Ir. Michel Vellekoop for his support and guidance during the entire process of writing my thesis. Our meetings were valuable and provided insight whenever I was struggling with my research. This stimulated me to go on, and for that, I am very grateful. Furthermore, I would like to thank my in-company supervisor Bertjan Kobus at Sprenkels and Verschuren for the valuable comments and dis-cussions on the subject matter. I would also like to thank Daan Kleinloog and Yvo Hauet for motivating me to continue when the end seemed so far away. I also want to thank Mark Verschuren for helping me in the programming process.
Special thanks go to Nick Draper who provided the dataset used in this thesis and was kind enough to provide insight in his market model. And lastly, I would like to take this moment to thank my family and friends who have supported me throughout the entire process.
Hereby, I proudly present my master thesis in conclusion to the Master’s in Financial Econometrics at the University of Amsterdam.
C O N T E N T S
1 i n t ro d u c t i o n 1
2 t h e f i n a n c i a l m a r k e t m o d e l 3
2.1 Previous studies . . . 3
2.2 The Financial Market . . . 4
2.3 An arbitrage-free market: parameter restrictions . . . 6
2.4 The term structure . . . 7
2.4.1 Nominal and inflation linked bonds . . . 7
2.4.2 The nominal term structure . . . 8
2.4.3 The real term structure . . . 9
2.4.4 Bond funds that implement constant duration . . . 11
2.5 The Ornstein-Uhlenbeck process . . . 12
3 e s t i m at i n g t h e f i n a n c i a l m a r k e t m o d e l : m e t h o d -o l -o g y a n d data 15 3.1 Exact discretization . . . 15 3.2 Estimation technique . . . 16 3.3 Data . . . 19 3.4 Estimation procedure . . . 21 3.4.1 Model setup . . . 21 3.4.2 Simulated annealing . . . 22 4 e s t i m at i o n r e s u lt s 25 4.1 Estimation results . . . 25 4.2 Risk premia . . . 29
4.3 Fit of the financial market model . . . 30
4.4 Robustness check . . . 32
5 c o n c l u s i o n 39
a t h e o r n s t e i n - u h l e n b e c k p ro c e s s 41 b s i m u l at e d a n n e a l i n g 43
1
I N T R O D U C T I O N
Dutch pension funds are bound by financial requirements set in the fi-nancial assessment framework (financieel toetsingskader, ftk). This frame-work is part of the Pension Act and is built around the principles of mar-ket valuation, risk-based capital requirements and transparency. Marmar-ket valuation implies that investments and pension obligations are valued in the same way. The technical provision is thus determined by discounting expected future cash flows at the current interest rate term structure. Sec-ondly, the capital requirements are determined in such a way that they are linked to the extent in which the pension fund is exposed to risks. A higher risk will therefore lead to higher capital requirements. Finally, transparency aims at identifying the financial position of a pension fund in a clear and objective way.
To achieve the above, the financial assessment framework uses param-eters that set bounds on the expected portfolio return and the expected wage and price inflation that pension funds may rely on when appreciat-ing their investments and determinappreciat-ing the value of pension contributions and benefits. These margins are imposed so that pension funds can create realistic expectations regarding the force of recovery in the event that the fund becomes underfunded. Underfunding refers to the situation where the market value of a pension fund’s assets is lower than the market value of the pension fund’s liabilities. The margins also help pension funds cre-ate realistic expectations regarding the indexation of pension liabilities. The relevant parameters are reassessed periodically by the Commission Parameters (Langejan et al.,2014) and include the risk premia for equity and bonds.
The equity risk premium can be defined as the excess return that an individual stock or the overall stock market provides over and above a risk-free rate. It can be seen as a price of risk as it compensates investors for taking on the risk of the equity market. There is a wide range of possi-ble equity risk premiums throughout history. Estimates for the long-run vary from 3% at one side (Campbell,2008), to 9.5% at the other
(Shack-man,2006). This wide range of possible risk premia reflects the inherent uncertainty associated with estimating future equity risk premiums.
The bond risk premium, on the other hand, is the yield compensation for the interest rate risk in government bonds. It compensates investors for taking on a greater risk when investing in long-term bonds relative to short-term bonds.Dimson et al. (2002) find a bond risk premium for the Netherlands of about 40 basis points for the period 1900-2000. The risk premium can, however, get quite large as the authors find a bond risk premium of 2.3% for France in the same period.
The objective of this thesis is to estimate the risk premia for bonds and equity, and to determine the market prices of risk. To achieve this, we need to describe the term structure of interest rates and the stock market. As the development of interest rates and stock returns depend on the inflation process, this needs to be modeled as well. We therefore need a model that accommodates these requirements and uses historical data on inflation, bond yields and stock returns.
A possible model for estimating the risk premia is the KNW-financial market model developed by Koijen et al. (2010). This model is closely related to Brennan and Xia (2002), Viceira and Campbell (2001) and Sangvinatsos and Wachter(2005). The model assumes a factor structure and accommodates time variation in bond risk premia. The equity risk premium, however, is assumed to be constant. The Commission Param-eters (Langejan et al.,2014) used the KNW-model model as a basis to estimate the aforementioned parameters. At the request of the Commis-sion Parameters,Draper (2014) estimated the KNW-model using Dutch data.
In this thesis we follow Koijen et al. (2010) and Draper (2014) to set up a financial market model for the Netherlands. We will calibrate this model using Dutch data and estimate the risk premia and the market prices of risk for both bonds and equity. The dataset used for this thesis is identical to the one used byDraper(2014). It covers a time period that spans from December 31st, 1972 up to and including December 31st, 2013 and consists of quarterly data on inflation, stock returns and six bond yields with different maturities. In doing so, the main question we want to answer in this thesis is whether the model parameters estimated by Draper are robust. In other words, will choosing a subset of the data give different estimation results?
This thesis also aims at making the estimation procedure of Draper (2014) more explicit. The remainder of this thesis is organized as follows. We elaborate on the financial market model in Chapter2. After this we will estimate the model parameters using quasi-maximum likelihood and describe the dataset in Chapter3. In Chapter4we present the estimation results. Ultimately, the conclusion follows in the last chapter.
2
T H E F I N A N C I A L M A R K E T M O D E L
The objective of this thesis is to estimate bond and equity risk premia and the corresponding prices of risk. To achieve this, the term structure of interest rates and the stock market need to be modeled. We also need to model the inflation process as it influences the development of inter-est rates and the stock return. In this chapter we describe our financial market model.
There are various studies related to the estimation of risk premia and market prices of risk. We start by discussing such models and motivat-ing our model choice in Section2.1. In Section 2.2, the processes for the state variables, the price index and the stock index are discussed. We will also introduce a pricing kernel that can be used to price all assets in the market. The drift and diffusion of this pricing kernel is driven by two underlying state variables. The market is assumed to be complete and arbitrage-free. This imposes restrictions on the equity and bond risk premia and this is further explained in Section2.3. In Section2.4we elab-orate on the nominal and real term structure of interest rates. The link between nominal and inflation linked bonds in the market is described and we discuss bond funds that implement constant duration. The com-pleted financial market model can be written in terms of a continuous multivariate Ornstein-Uhlenbeck process. This is shown in Section2.5.
2.1 p r e v i o u s s t u d i e s
A wide range of literature has been published on the estimation of equity and bond risk premia. These studies vary with estimation methods, but also with assumptions and constraints applied to the models in question.
Brennan and Xia(2002) andViceira and Campbell(2001), for example, estimate a two-factor term structure model for interest rates that fits both equity and bond returns. The factors are identified as the real interest rate and the expected rate of inflation, and both of these studies assume that risk premia on bonds are constant.
Fama and Bliss(1987), among others, show that the expected term pre-mia on long-term bonds vary over time. Moreover, it is possible to predict excess returns (term premia) on bonds using observable variables such as the forward rate. Although we might assume that the equity risk premium is constant over time, the assumption of constant bond risk premia has been proven to be quite unrealistic.Sangvinatsos and Wachter(2005) and Duffee(2002) propose a three-factor term structure model that allows for time-varying bond risk premia. Litterman and Scheinkman (1991) find that three factors explain the vast majority of bond price movements. The three-factor model proposed by Sangvinatsos and Wachter (2005) therefore seems a reasonable choice for our financial market model. How-ever, three factor models are computationally difficult to estimate due to the large number of parameters involved.
To make our model computationally easier, we would like to estimate a two-factor term structure model that allows for time-varying bond risk premia.Koijen et al. (2010) propose a model that has an identical factor structure as inBrennan and Xia(2002) andViceira and Campbell(2001) but accommodates time variation in bond risk premia. We therefore fol-low Koijen et al. (2010) and set up a two-factor term structure model, where the factors, or state variables, are identified as the real interest rate and expected inflation. This model is in the “essentially affine” class proposed by Duffee (2002). By “affine”, Duffee (2002) refers to models where zero-coupon bond yields, their physical dynamics, and their equiv-alent martingale dynamics are all affine functions of the underlying state vector.
2.2 t h e f i n a n c i a l m a r k e t
We start by assuming that the financial market is complete and arbitrage-free. This assumption ensures that every contingent claim can be repli-cated so every possible derivative claim can be hedged (Etheridge,2002), and that there are no arbitrage opportunities. More details on the no-arbitrage assumption are given in Section2.3. In the complete and arbitrage-free market, the portfolio that one can invest in consists of a stock index, long-term nominal and real bonds and a nominal money account.
Koijen et al.(2010) propose a two-factor model of the term structure that allows for time-varying bond risk premia, where the factors are iden-tified as the real interest rate and expected inflation. The dynamics and uncertainty of the term structure of interest rates are modeled using the two unobserved state variables, X1t and X2t. In order to accommodate the first-order autocorrelation in the interest rate, the state variables Xt= (X1t, X2t)0 are modeled to be mean-reverting around zero, that is,
dXt=−KXtdt+Σ0XdZt, (1)
where Σ0
X = [I2×202×2]and K is a 2 × 2 matrix normalized to be lower triangular. Z ∈ R4×1denotes a vector of independent Brownian motions
2.2 the financial market
driving the uncertainty in the financial market. A definition for a Brown-ian Motion can be found in Etheridge(2002, pg. 53). It is assumed that there are four sources of uncertainty in the market described by Z ∈ R4×1: the uncertainty regarding the real interest rate, the uncertainty regard-ing the instantaneous expected inflation, the uncertainty regardregard-ing the unexpected inflation and the uncertainty regarding the stock return.
We assume that the instantaneous real interest rate, r, is affine in both state variables, thereby preserving parallel relationships between the factors. More precisely,
rt=δ0r+δ01rXt, δ0r ∈R, δ1r ∈R2×1. (2) The instantaneous expected inflation, π, is also assumed to be affine in both factors,
πt=δ0π+δ1π0 Xt, δ0π ∈R, δ1π∈R2×1. (3) Here, δ0π can be interpreted as the expected long-run inflation. Any cor-relation between the real interest rate and the instantaneous expected inflation is captured in δ0
1r and δ1π0 . The expected inflation, π, determines the price index Π. We assume that the price index follows the process
dΠt
Πt =πtdt+σΠ0 dZt, σΠ∈R4×1, Π0 =1. (4) Here, σΠ is the volatility of the price index. Regarding the process for the stock index, S, we assume that
dSt
St = (Rt+ηs)dt+σ 0
SdZt, σS ∈R4×1, S0=1, (5) where ηs is the (constant) equity risk premium and σS is the volatility of the stock index. Rt denotes the instantaneous nominal interest rate process, which is determined in section 2.3. In this way, the drift term of the stock price process, Rt+ηS, is time dependent, even though the equity risk premium is constant.
The financial market model is completed by specifying an affine model for the term structure of interest rates. We accomplish this by specifying a pricing kernel that can be used to price all assets in the economy under the no-arbitrage assumption. The use of a pricing kernel to price assets is equivalent to other pricing methods such as risk-neutral valuation (for exampleCochrane(2005)) in complete markets, as is the case here, since all payoffs can be replicated using existing assets in the financial mar-ket. The pricing kernel is then uniquely defined. For incomplete markets, however, there exist infinitely many pricing kernels that lead to the same price of existing assets on the market, so that the pricing kernel is not unique.
We assume that the prices of risk are affine in the state variables Xt. Thus, the essentially affine model as proposed byDuffee(2002) is adopted.
More precisely, the pricing kernel or nominal stochastic discount factor φNt is given by dφNt φN t =−Rtdt −Λ0tdZt, (6)
with Λt the time-varying price of risk. The price of risk can be written in terms of the state variables Xtas follows.
Λt=Λ0+Λ1Xt, Λ0 ∈R4×1, Λ1 ∈R4×2. (7) The price of risk will depend on the risk aversion of investors and is driven by the four-dimensional vector Z of Brownian motions defined before. Koijen et al. (2010) argue that unexpected inflation risk cannot be identified by data on the nominal side of the economy alone. We therefore assume that there is no risk premium for unexpected inflation, thereby defining Λ0 and Λ1 as
Λ0= Λ01 Λ02 0 Λ04 and Λ1 = Λ11,1 Λ11,2 Λ12,1 Λ12,2 0 0 Λ14,1 Λ14,2 . (8)
Having completed the specification of our financial market, we can find a price for a nominal bond at time t with a maturity t+τ. As shown by Duffie and Kan(1996), nominal bond prices take the form
PN(Xt, t, t+τ) =exp(AN(τ) +BN(τ)0Xt), (9) so that the bond prices are exponentially affine in the state variables.
AN(τ) ∈ R and BN(τ) ∈ R2×1 solve a system of ordinary differential
equations and are given in Section 2.3. The corresponding zero-coupon bond yield is a closed-form solution of the above and can be written as
ytτ =−1 τ log P
N(Xt, t, t+τ) = (−A(τ)− B(τ)0Xt)
τ . (10)
This is one of the main advantages of an affine model (Piazzesi, 2010). Having closed-form solutions for bond yields is useful because if there are no closed-form solutions available, one must rely on Monte Carlo methods to price them, which are computationally costly.
2.3 a n a r b i t r ag e - f r e e m a r k e t : pa r a m e t e r r e s t r i c t i o n s As mentioned in the previous section, we assume that the market is complete and arbitrage-free. According to the Fundamental Theorem of Asset Pricing, a market is arbitrage-free if and only if there exists an equivalent martingale measure (Harrison and Kreps,1979). This means that a market is arbitrage free if there exists a vector of probabilities such that the price of a security equals its discounted expected payoff. This
2.4 the term structure
implies that the discounted price of the security must be a martingale. Therefore, if the financial market in this chapter is arbitrage-free, the discounted price of the stock index, φNS, is a martingale. Note that risk-neutral does not mean that the stock bears no risk. It means that we get conditional mean zero returns by multiplying the stock with the stochastic discount factor, thereby making it a martingale. The process for the discounted stock price can be written as
d(φNS) =SdφN +φNdS+dφNdS, which gives d(φNS) φNS = dφN φN + dS S + dφN φN dS S . (11)
Substituting equations (5) and (6) into the above gives d(φNS) φNS =−Rtdt −Λ 0 tdZt+ (Rt+ηs)dt+σS0dZt−Λ0tσSdt = (ηs−Λ0tσS)dt −(Λ0t− σ 0 S)dZt. (12) For φNS to be a martingale, the drift term (ηs−Λ0
tσS)in the equation above must equal zero. This imposes the restriction
ηs=Λ0tσS =σS0 Λ0+σS0Λ1Xt, (13) on the equity risk premium. This restriction in turn implies that σ0
SΛ0 = ηsand σS0Λ1 = 0 because the equity risk premium is constant over time. For the market to be arbitrage-free, similar restrictions must hold for bonds as well. These are discussed in the next section.
2.4 t h e t e r m s t ru c t u r e
In Section2.2, we gave an outline of the financial market and discussed that the nominal stochastic discount factor can be used to determine the term structure of interest rates. In this section, we first describe the link between nominal and inflation linked bonds. We will then discuss the nominal and real term structure of bonds. Afterward, bond funds that implement constant duration will be introduced.
2.4.1 Nominal and inflation linked bonds
Consider a nominal zero coupon bond with price PN and an inflation corrected real zero coupon bond with price PR. We previously discussed that the nominal stochastic discount factor was appropriate to discount all cashflows in the economy. The nominal bond can therefore be dis-counted using the nominal stochastic discount factor φN. The stochastic
discount factor for the real bond, however, must be corrected for inflation. The real stochastic discount factor is defined as
φR≡ φNΠ. (14)
Note that while we started out by defining a stochastic discount factor for nominal assets, we could also have started by defining a stochastic dis-count factor for real assets and using the relationship above. The process for φR is
dφR≡ΠdφN +φNdΠ+dφNdΠ. (15) Dividing by φRand substituting equations (4) and (6) gives
dφR φR = ΠdφN ΠφN + φNdΠ φNΠ + dφN φN dΠ Π = dφ N φN + dΠ Π + dφN φN dΠ Π =−Rtdt −Λ0tdZt+πtdt+σΠ0 dZt− σ0ΠΛtdt =−(Rt− πt+σΠ0 Λt)dt −(Λ0t− σ 0 Π)dZt =−rtdt −(Λ0t− σΠ0 )dZt, (16)
where rt=Rt− πt+σΠ0 Λt is the relationship between the real and nom-inal interest rate. The instantaneous nomnom-inal interest rate can therefore be written as Rt=rt+πt− σ0ΠΛt =δ0r+δ1r0 Xt+δ0π+δ1π0 Xt− σ 0 ΠΛ0− σ0ΠΛ1Xt = (δ0r+δ0π− σΠ0 Λ0) + (δ1r0 +δ 0 1π− σ 0 ΠΛ1)Xt ≡ R0+R01Xt, (17)
by substituting equations (2) and (3).
2.4.2 The nominal term structure
In this section, we derive the nominal term structure of interest rates by pricing nominal bonds. For this purpose, consider again the nominal zero coupon bond with price PN. Since nominal bonds are traded assets, the discounted price of this nominal bond must be a martingale for the market to be arbitrage-free.
Using integration by parts, the process for the discounted nominal zero coupon bond is given by
2.4 the term structure
We assume that bond prices are smooth functions of time and the state (X). That is, PN =PN(X, t). Applying Itô’s Lemma to PN gives
dPN =PtNdt+PXNt0dXt+12dXt0PXNtXt0dXt, (19) where PN t = ∂P N ∂t , PXNt = ∂PN ∂X and PXNtXt0 = ∂PN ∂X∂X0. Substituting equa-tion (1) gives dPN =PtNdt+PXNt0(−KXtdt+Σ0XdZt) +1 2(−KXtdt+Σ0XdZt)0PXNtXt0(−KXtdt+Σ 0 XdZt) =PtNdt+PXNt0(−KXtdt+Σ0XdZt) +12ΣXPXNtXt0Σ 0 Xdt. (20)
We substitute the above and equations (1) and (6) into the process for the discounted nominal zero coupon bond. Using the properties of stochastic differential equations gives
d(φNt PN) =PNφNt (−Rtdt −Λ0tdZt) +PXNt0(−KXtdt+Σ0XdZt) + 1 2ΣXPXNtXt0Σ 0 Xdt +φNt (−PtN0Σ0XΛt)dt. (21)
For the discounted bond price to be a martingale, the drift term must equal zero. More precisely,
PXN0(−KXt) + 1 2ΣXPXNtXt0Σ 0 X− PNRt− PN 0 X Σ 0 XΛt =0. (22) The solution for this equation is of the form (equation (9)):
PN(Xt, t, t+τ) =exp(AN(τ) +BN(τ)0Xt),
Substituting this into equation (22) and matching the coefficients of the constant term and the state variables eventually leads to a set of ordinary differential equations. These equations can be solved in closed form which gives, BN(τ) = (K0+Λ01ΣX)−1(exp(−(K0+Λ01ΣX)τ)− I2×2)R1, (23) AN(τ) = Z τ 0 (−R0−(Λ00ΣX)BN(s) + 1 2BN(s) 0Σ0 XΣXBN(s))ds. (24) 2.4.3 The real term structure
We now consider the inflation corrected real zero coupon bond with price PR. This real bond can be discounted with the real stochastic discount factor φR. In the previous subsection, the discounted value of the
nom-inal bond was derived in such a way that is was a martingale. In this subsection, the discounted value for the real bond is derived in a similar way. The process for the discounted real bond price is given by
d(φRPR) =PRdφR+φRdPR+dφRdPR, (25) which can be written as
d(φR) φR =P RdφR φR +dP R+dφR φR dP R. (26)
We assume that the real bond prices are also smooth functions of time and the state (X). That is, PR = PR(X, t). Applying Itô’s Lemma to PR gives dPR=PtRdt+PXRt0dXt+1 2dX 0 tPXRtXt0dXt, (27) where PR t = ∂P R ∂t , P R Xt = ∂PR ∂X and P R XtXt0 = ∂PR ∂X∂X0. Substituting
equa-tion (1) into the above gives
dPR =PtRdt+PXRt0(−KXtdt+Σ0XdZt) +1 2(−KXtdt+Σ0XdZt)0PXRtXt0(−KXtdt+Σ 0 XdZt) =PtRdt+PXRt0(−KXtdt+Σ0XdZt) + 1 2ΣXPXRtXt0Σ 0 Xdt. (28)
Now substituting the above and equations (1) and (16) into the process for the discounted real zero coupon bond, and using the properties of stochastic differential equations gives
d(φRPR) φR =P R(−rtdt −(Λ0 t− σ 0 Π)dZt) +PtRdt+PXRt0(−KXtdt+Σ0XdZt) + 1 2ΣXPXRtXt0Σ 0 Xdt − PR 0 X Σ 0 X(Λt−ΣΠ)dt. (29)
For the discounted bond price to be a martingale, the drift term must equal zero. More precisely,
−rtPR+PtR+PXR0(−KXt) +1 2ΣXPXRtXt0Σ 0 X− PR 0 X Σ0X(Λt−ΣΠ) =0. (30) The solution for this equation (Duffie and Kan,1996) is of the form:
PR(Xt, t, t+τ) =exp(AR(τ) +BR(τ)0Xt). (31) Substituting this into equation (30) and matching the coefficients of the constant term and the state variables leads to a set of ordinary differential
2.4 the term structure
equations. These equations can be solved in closed form which leads to similar equations for ARand BRas those for AN and BN in section 2.3.2.
2.4.4 Bond funds that implement constant duration
We will use yields of bonds with different duration to estimate the finan-cial market model. For this purpose, it is convenient to assume that there is a bond fund that has a portfolio that is constantly rebalanced to hold the maturity τ constant. In this section we followDraper(2014) to deter-mine the development of the return of such a bond portfolio. The bond fund is assumed to only invest in bonds with maturity τ. The bond index value of this constantly rebalancing bond fund is given in equation (9):
PFτ(X
t) =PN(Xt, t, τ) =exp(AN(τ) +BN(τ)0Xt).
By applying Itô’s Lemma to the equation above, the development of the bond index can be derived. Holding τ constant leads to
dPFτ =PFτ t dt+PXFτdXt+12dXt0PXFτtXt0dXt = (PFτ X dXt+ 1 2dX 0 tPXFτtXt0dXt) × PFτ PFτ =PFτBN(τ)0dX t+ 12PFτdXt0BN(τ)BN(τ)0dXt, (32)
where we obtain the second equality by substituting PN t = ∂P N ∂t = 0, PFτ X = ∂PFτ ∂X = ∂(exp(AN(τ)+BN(τ)0X t)) ∂X =PFτBN and ∂2(PFτ) ∂X∂X0 =PFτBNBN 0
into the equation. Premultiplying by PFτ
PFτ gives the third equality. Rear-ranging and substituting equation (1) gives
dPFτ PFτ =B N(τ)0dX t+12dXt0BN(τ)BN(τ) 0dX t = (−BN(τ)0KXt+12BN(τ)0ΣX0 ΣXBN(τ))dt+BN(τ)0Σ0XdZt. (33) Again, the restriction is imposed that the discounted bond price index must be a martingale, as the bond index is a tradeable asset. Therefore,
d(φNPFτ) φNPFτ = dφN φN + dPFτ PFτ + dφN φN dPFτ PFτ . (34)
For the bond price index to be a martingale, the drift term of this process must equal zero. This leads to the partial differential equation
−BN(τ)0KXt+1
2BN(τ) 0Σ0
Substituting this into equation (33) leads to the price dynamics equation dPFτ PFτ = (Rt+B N(τ)0Σ0 XΛt)dt+BN(τ) 0Σ0 XdZt. (36)
Equation (36) can be interpreted in the following way. The drift term con-sists of the nominal instantaneous interest rate Rt plus the risk exposure BN(τ)0Σ0
X multiplied by the market price of risk Λt. The risk exposure multiplied by the market price of risk is actually the time-varying bond risk premium. The dZt term is the Brownian Motion that drives the process.
2.5 t h e o r n s t e i n - u h l e n b e c k p ro c e s s
In the previous sections, we derived the financial market model. The model can be written as a (continuous) multivariate Ornstein-Uhlenbeck process of the form
dYt= (Θ0+Θ1Yt)dt+ΣYdZt. (37) The Ornstein-Uhlenbeck process is Gaussian and Markovian. A process is said to be Markovian if it satisfies the Markov property, that is, the condi-tional probability distribution of future states of the process, condicondi-tional on both past and present values, depends only upon the present state, and not on the events that preceded it. For a more detailed definition, we refer toEtheridge(2002, ch. 2). We can say that the Ornstein-Uhlenbeck is only dependent on the current value of the process, and not on the past. Moreover, if Θ1 is negative definite and the current value of the process is less than the (long-term) mean, the drift will be positive; if the current value of the process is greater than the (long-term) mean, the drift will be negative. In other words, the process is mean-reverting.
In our model, Yt∈R6×1 is such that
Yt0 =hXt lnΠ lnS lnPF0 lnPFτi. (38) Here, X is a vector containing the two state variables, Π is the price index and S is the stock index. PF 0 can be seen as the cash wealth index and is obtained by evaluating lnPFτ with τ = 0. In the case that τ = 0, AN(τ) =0 and BN(τ) =02×1, which leads to the following expression:
lnPF0 = (R
0+R01Xt)dt=Rtdt.
PF τ is the bond wealth index with duration τ. The stochastic differential equations in Y are derived by applying Itô’s Lemma in appendix A. The
2.5 the ornstein-uhlenbeck process
mean-reverting Ornstein-Uhlenbeck process increment dYtcan be written as d Xt lnΠ lnS lnPF0 lnPFτ = 0 δ0π−12σ0ΠσΠ R0+ηs−12σS0 σS R0 R0+BN(τ)0Σ0XΛ0−21BN(τ)0Σ0XΣXBN(τ) + −K 02×4 δ1π0 01×4 R01 01×4 R01 01×4 R10 +BN(τ)0Σ0XΛ1 01×4 Yt dt+ Σ0 X σ0Π σ0S 0 BN(τ)0Σ0X dZt. (39) As discussed previously, there are four sources of risk in this model. Two sources of risk are due to the state variables X1 and X2, the third is because of the price level Π and the fourth due to the stock price S.
Without loss of generality we impose the restriction that the elements in ΣY are stacked in such a way that it is a lower triangular matrix. The restriction is imposed because ΣY would otherwise be statistically unidentified. The restriction implies that the fourth element of σΠequals zero.
In this chapter we presented the financial market. We discussed the processes in the market and restrictions for the market to be arbitrage-free. In the next chapter, we will elaborate on the discretization and estimation of the model, using the Ornstein-Uhlenbeck process presented here.
3
E S T I M AT I N G T H E F I N A N C I A L M A R K E T M O D E L : M E T H O D O L O G Y A N D D ATA
In this chapter we present the estimation technique for our financial mar-ket model. We estimate the parameters that drive the processes in the market, and the equity and bond risk premia. The financial market model derived in the previous chapter is in continuous time. However, to make estimation and simulation of the model possible, we need a discrete time version of the model. We therefore discretize the model in Section 3.1. The discrete-time model can then be estimated in various ways. Koijen et al. (2010) propose the use of a Kalman filter whereas Draper (2014) uses (quasi-) maximum likelihood.
We describe the estimation procedure using maximum likelihood in Section3.2. The (quasi-)log-likelihood can be divided in three parts. The first part incorporates bond yields and is described byDuffee(2002). The second part takes inflation and stock return data into account (Sangv-inatsos and Wachter, 2005). Finally, the third part is a constant term related to the state variables (Draper,2014). We will use quarterly data on bond yields, inflation and stock returns to estimate the model parame-ters. The data is described in Section3.3. We then present the estimation procedure in matlab in Section3.4. A detailed analysis of the estimation results follows in the next chapter.
3.1 e x ac t d i s c r e t i z at i o n
As mentioned before, estimation of the parameters in the financial market model is only possible in discrete time. In this section, we discretize the financial market model following Sangvinatsos and Wachter (2005) and Koijen et al. (2010). For further information on the discretization technique, we refer toBergstrom(1984).
Consider the continuous time dynamics of Yt in equation (39) and assume that Θ1 is diagonalizable. The eigenvalue decomposition of Θ1
may be written as Θ1 =U DU−1, where D is a diagonal matrix. We can rewrite the continuous model in discrete time as a VAR(1)-model:
Yt+h=µ(h)+Γ(h)Yt+t+h, (40) where t+h ∼ N(0, Σ(h))for discrete-time parameters µ(h), Γ(h) and Σ(h) which are defined below.
Γ(h)=exp(Θ
1h) =exp(U DU−1h) =Uexp(Dh)U−1, (41) where exp(A) denotes the matrix exponential of the matrix A. For µ(h) we have µ(h)= " Z t+h t exp (Θ1[t+h − s])ds # Θ0=U F U−1Θ0. (42) Here, F is a diagonal matrix with elements Fii=hα(Diih), with
α(x) = exp(x)−1 x and α(0) =1. For Σ(h) we have Σ(h) =Z t+h t exp (Θ1[t+h − s])0ΣYΣ0Y exp(Θ1[t+h − s])ds =U V U0, (43) where V is a matrix with elements
Vij =
h
U−1ΣYΣ0Y(U−1)0i
ijhα([Dii+Djj]h).
The discrete model presented in this section will be a part of the quasi-maximum likelihood estimation technique described in the next section.
3.2 e s t i m at i o n t e c h n i q u e
For our discrete financial market model described before, we have several parameters, including the VAR parameters µ, Γ and Σ, and the price-of-risk parameters Λ0 and Λ1. To estimate these parameters simultaneously, we apply quasi-maximum likelihood as discussed by Duffee (2002) and extended by Sangvinatsos and Wachter (2005). As our model has an essentially affine structure, this estimation technique is straightforward. The log-likelihood specified here consists of three parts. For the first part, we followDuffee(2002) and assume that at each period-end t, t= 1, ..., T , yields on n bonds are measured without error. These bonds have fixed times to maturity τ1, ..., τn. From equation (10),
ytτ = (−A(τ)− B(τ) 0Xt)
3.2 estimation technique
We can use the above to determine the state variables X. In addition to the yields without measurement error, we assume that there are k other bonds whose yields are measured with serially uncorrelated, mean zero measurement error. The bonds have fixed times to maturity τn+1, ..., τn+k.
For these yields, the following holds: ˜yτt = (−A(τ)− B(τ) 0X t) τ +υ τ t, τ =τn+1, ..., τn+k, (45) where υt ∼ N(0, Στ)with υt0 = υτn+1 t , ..., υ τn+k t
. We assume that there is no correlation between the measurement errors of the yields, implying that the variance-covariance matrix Στ is diagonal. We can derive the implied state variables ˆXt from equation (44) by inverting yτt. Given ˆXt, the implied yields ˆyτ
t for the other k bonds are: ˆyτt = (−A(τ)− B(τ)
0Xtˆ )
τ , τ =τn+1, ..., τn+k This gives measurement error υτ
t = ˜ytτ− ˆyτt, for τ =τn+1, ..., τn+k.
To compute the first part of the quasi-log-likelihood, we assume that the measurement errors are Gaussian. The conditional distribution of the measurement error is given by
f(υt|0, Στ) = (2π |Στ|)−12 exp(−1
2υt(Στ) −1
υt0),
where |Στ|stands for the determinant of Στ. This gives the likelihood L1 = T Y t=1 f(υt|0, Στ) = T Y t=1 (2π |Στ|)−12 exp(−1 2υt(Στ)−1υt0), which in turn gives the first part of the log-likelihood,
L1 =log(L1) =− T 2log(|Στ|)− 1 2 T X t=1 υt(Στ)−1υt0, (46) after eliminating irrelevant terms.
For the second part of the log-likelihood, we extend the above to in-corporate inflation and stock returns in the estimation of the parameters (Sangvinatsos and Wachter, 2005). First, consider the model of equa-tion (40). This model holds for any choice of h > 0. Recall that this is a six-dimensional model because Yt ∈ R6×1 (equation (38)). To extend the quasi-log-likelihood, we will use the the first four equations from equation (40). This choice is substantiated by the fact that the last two rows describing the bond equations are linear combinations of the first four equations. Therefore these equations do not need to be taken into account in the estimation.
We will use quarterly data on inflation and stock returns to estimate our model parameters (see Section 3.3 for a description of the dataset). This gives h=1/4. We now define ˜Y0 =hXˆ, lnΠ, lnSi0. Here, ˆX are the
state variables that were solved from equation (44) and Π and S are the price index and stock index respectively. We now write the transformed discrete model as:
˜ Yt+1/4 =µ (1/4)+Γ(1/4)Yt˜ + t+1/4 ≡ µ+Γ ˜Yt+t+1/4, (47) where µ is a 4x1 vector, Γ is a 4x4 matrix and t+1/4 ∼ N(0, Σ) with
Σ=Σ(1/4) also a 4x4 matrix. Denote the measurement error by ˜:
˜t=Y˜t− µ −Γ ˜Yt−1/4, (48)
We again assume that the errors are Gaussian. The conditional distribu-tion of the measurement error ˜t at time t is given by
f(˜t|0, Σ) = (2π |Σ|)−12 exp(−1
2 ˜tΣ −1
˜0t). This gives the likelihood
L2= T Y t=1 f(˜t|0, Σ) = T Y t=1 (2π |Σ|)−12 exp(˜tΣ−1˜0 t). The second part of the log-likelihood is therefore
L2 =log(L2) =− T 2log(|Σ|)− 1 2 T X t=1 ˜t(Σ)−1˜0t. (49) The last part of the log-likelihood is a constant term related to the state variables (Draper,2014).
L3 =− T
2 log(|B|), (50)
with B0 = [B(τ
1), B(τ2), ..., B(τn)]from the yields without measurement error in equation (44). Using L1, L2 and L3, the quasi log-likelihood function is set up as follows
L=−1 2 Tlog(|Στ|) + T X t=1 υt(Στ)−1υt0 ! −1 2 Tlog(|Σ|) + T X t=1 ˜t(Σ)−1˜0t ! −1 2Tlog(|B|). (51)
The parameter estimates are obtained by maximizing this quasi log-likelihood function using matlab. That is,
ˆθ=arg max L(θ; υ, ), (52)
where ˆθ is a vector containing the parameter estimates. Section3.4gives details on the estimation technique in matlab. We do not only want to estimate the financial market model parameters but also their standard
3.3 data
errors. Under certain assumptions, the maximum likelihood estimator ˆθ is asymptotically normal, meaning that the distribution of the maximum likelihood estimator can be approximated by a multivariate normal dis-tribution with mean θ0 and covariance matrix
1
n (Var[OθL(θ0; υ, )]) −1
.
Here,OθL(θ; υ, )is the gradient of the log-likelihood, that is, the vector of first derivatives of the log-likelihood, evaluated at θ. As the true value θ0 is unknown, the covariance matrix is also unknown. To determine the standard errors of the maximum likelihood estimates, we need to esti-mate the covariance matrix. We can consistently estiesti-mate the covariance matrix using the outer product of gradients estimate. This is computed as ˆ V = 1 n T X t=1 OθLt(ˆθ; υ, )OθLt(ˆθ; υ, )0 !−1 , (53)
whereOθLt(ˆθ; υ, ) is the gradient of the log-likelihood in the maximum likelihood estimates ˆθ. The estimated standard errors can be found by taking the square roots of the diagonal elements of ˆV .
3.3 data
The dataset used for this thesis consists of data on inflation, stock returns and six yields with different maturities. The yields have three-month, one-year, two-one-year, three-one-year, five-year and ten-year maturities, respectively. For the yields with a maturity of three months, money market rates are used. For the remaining maturities, we will use long nominal yields on zero-coupon bonds. The dataset was provided to us by Draper (2014) and is identical to the set used in his research. It spans from December 31st, 1972 up to and including December 31st, 2013.
• Stock returns: For the stock returns, we use the MSCI index from Fact Set. The returns are in euros (Deutschemark before 1999) and are hedged for US dollar exposure. Using the MSCI index figures, quarterly log-returns figures are created.
• Money market rates: Three-month money market rates are taken from the Bundesbank website (www.bundesbank.de). For the pe-riod 1973:I to 1990:II, end-of-quarter money market rates reported by Frankfurt banks are taken. Thereafter, three-month Frankfurt Interbank Offered Rates (FIBOR) are included. On January 1st, 1999, the FIBOR merged into the Euribor. The Euribor is the benchmark giving an indication of the average rate at which banks lend unsecured funding in the Euro Interbank market for a given period, which in this case is three months. The money market rates are always quoted on a yearly basis.
1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Year
Log−inflation on year basis
(a)Inflation 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 −1 −0.5 0 0.5 1 Year
Log MSCI−return on year basis
(b) Stock return 19700 1975 1980 1985 1990 1995 2000 2005 2010 2015 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Year Nominal yields y = 3m yield y = 1y yield y = 2y yield y = 3y yield y = 5y yield y = 10y yield
(c) Nominal yields on bonds with different duration
Figure 1. Plots of the log-inflation, the log-MSCI-return and the nom-inal bond yields.The plots are on a yearly basis from September 30th, 1972 to
3.4 estimation procedure
• Long nominal yields: For the period 1973:I to 1987:III, zero-coupon yields with maturities of 1 to 15 years (from the Bundesbank web-site) based on government bonds were used (15 year rates start in June 1986). From 1987:IV on, zero-coupon rates are constructed from swap rates published by De Nederlandsche Bank (www.dnb.nl). No adjustments were made for possible differences in the credit risk of swaps, on the one hand, and German bonds, on the other. The biggest difference in yield between the two term structures (for the two-year yield) in 1987:IV was only 12 basis points. The long nom-inal yields are quoted on a yearly basis.
• Inflation: From 1972 until 1999, the German (Western German be-fore 1990) consumer price index figures published by the Interna-tional Financial Statistics of the InternaInterna-tional Monetary Fund are used. From 1999 on, the Harmonized Index of Consumer Prices for the euro area from the European Central Bank data website (http://sdw.ecb.europa.eu) is used. Using the consumer price index figures, quarterly log-inflation figures are determined by calculating the year-to-year log inflation from the consumer price index figures and dividing by 4.
Figure 1 gives plots, on a yearly basis, for the data used. We see that inflation has declined in the past decades. The same holds for the zero-coupon yields and the money market rates after 1980. The MSCI-return seems to revert around zero.
3.4 e s t i m at i o n p ro c e d u r e
This section describes the estimation procedure for the financial market model parameters. We first describe the model setup and summarize the general parameter restrictions in Section 3.4.1. The model is optimized using a simulated annealing algorithm. This optimization technique is described in Section 3.4.2.
3.4.1 Model setup
The dataset available to us spans from December 31st, 1972 up to and including December 31st, 2013, and includes inflation, stock returns and six bond yields with different maturities. As we have quarterly data, this gives a total of 165 data points.
We have two state variables, X1t and X2t (section2.1). We therefore need two bond yields without measurement error to solve equation (44) for those state variables. We assume that the bond yields without mea-surement error are those with maturities of 2 years and 5 years. A differ-ent choice for the yields without measuremdiffer-ent error is also possible. The bonds measured with error are then assumed to be those with maturities of 3 months, 1 year, 3 years and 10 years. These are used to determine υtin the first part of our log-likelihood (equation (46)).
In the second part of our log-likelihood we use the equations from the VAR-process in equation (47). The appropriate µ(h), Γ(h) and Σ(h) are
derived using the discretization technique in Section3.1.
For t=1, ..., 165, ln Πtis the cumulative logarithm of the quarterly in-flation. In a similar way, lnStis the cumulative logarithm of the quarterly MSCI-return. When creating the cumulative logarithm for inflation, we allow for seasonal adjustments by determining the year-to-year log infla-tion figures and dividing by 4. The errors ˜tare derived in equation (48). We lose one datapoint (December 31st, 1972) when deriving the error terms in this part of the log-likelihood. Therefore, we also drop the first datapoint from υt. This leads to a total of 164 data points for all time series considered. So, T =164 in equation (51).
The model has a total of 27 parameters. Recall that the value of the fourth element of σΠ is set to zero. Also recall that the third rows of Λ0 and Λ1 are zero, and that the elements of Λ0 and Λ1 are such that σS0Λ0 =ηs and σS0 Λ1 = 0. Thus, we only need to estimate the first two rows of Λ0 and Λ1. The fourth rows are implied by the model. No further restrictions are imposed on the model.
3.4.2 Simulated annealing
The market model parameters are estimated using the method of lated annealing in matlab. Rather than using matlab’s standard simu-lated annealing procedure, we follow Goffe et al.(1994) and construct a simulated annealing algorithm based on his research. The reason for this is that the simulated annealing algorithm in matlab is sealed, which makes it impossible to follow all the steps in the algorithm.
Simulated annealing originated in thermodynamics and models the physical process of heating a material and then slowly lowering the tem-perature to decrease defects, thus minimizing the system energy. The algorithm therefore minimizes an objective function. At each iteration of the simulated annealing algorithm, a new point is randomly generated. The distance of the new point from the current point, or the extent of the search, is based on a probability distribution with a scale proportional to the temperature.
Simulated annealing has several potential advantages over conventional algorithms (Goffe et al.,1994). First, it can move away from local minima. Second, the objective function does not need to be differentiable. Another benefit is that simulated annealing can identify corner solutions. The only drawback of simulated annealing it that it requires computational power, and can therefore be very time consuming.
As simulated annealing minimizes the objective function, we minimize the negative of the quasi-log-likelihood in equation (51) to obtain the parameter estimates. We refer to Appendix B for a pseudo-code of the simulated annealing algorithm and the essential parameter settings that were used in the optimization.
3.4 estimation procedure
The simulated annealing algorithm requires starting values for the pa-rameters in the model. We used a set of starting values that is different from the optimal parameter values ofDraper (2014), but was still within reasonable bounds for the parameters considered. This enables us to draw conclusions regarding the robustness of the parameter estimates with re-spect to the starting values.Draper(2014) does not mention the optimal value of the variance-covariance matrix Στ. As Στ is diagonal, we take as starting point a 4x4 matrix with value 1%2 on the diagonal. Note that the method of simulated annealing is a global optimization technique so that the chosen starting points do not affect the results as much as a lo-cal optimization technique would, as long as reasonable upper and lower bounds for the parameters are set.
4
E S T I M AT I O N R E S U LT S
With the model setup in the previous chapter, all of the parameters in the financial market model can be estimated. These estimates are discussed in Section4.1. Using the estimated parameters, we continue our research by discussing the market prices of risk and the risk premia in Section4.2. In Section4.3, we analyze the fit of the model. Finally, in Section4.4we present estimation results based on a subset of our data to be able to draw some conclusions about the robustness of the model.
4.1 e s t i m at i o n r e s u lt s
The financial market model parameters were estimated based on the tech-nique in Section 3.4. The results are given in Table 1. The parameters are expressed in annual terms. The standard errors are computed using the outer product of gradients estimate.
First, we find that the expected long-term inflation, δ0π, equals 1.84%. According to the CPB Netherlands Bureau for Economic Policy Analysis, the average inflation in the Netherlands is extraordinarily low, just like in the rest of Europe (CPB,2014). The average inflation was 11/4% in 2014 and is expected to rise to 11/2% in 2015. In the long run, the inflation is expected to rise to a level of 2% (Langejan et al.,2014). The estimate of 1.84% for δ0π is therefore a fair estimate for the long-term inflation.
Second, we find that the instantaneous nominal interest rate, Rt is decreasing in X1 and increasing in X2. R1(1) and R1(2) equal -1.49%
and 0.50% respectively. The expected long-term nominal interest rate R0 equals 2.46%. R0 should, in theory, equal the historical mean of the short-term interest rate. However, the estimated value for R0 is lower than the historical mean of the three-month bond yield, which is approximately 5%. This result is not unique. In fact, Duffee (2002) also finds values for R0 that are lower than the mean of the historical short-term interest rates.
Table 1
Parameter estimates of the financial market model
The financial market model is estimated by maximum likelihood using Dutch data on six bond yields, inflation and stock returns over the period from De-cember 31st, 1972 up to DeDe-cember 31st, 2013. The bond maturities used in the estimation are three months, one year, two years, three years, five years and ten years. The two-year and five-year yields are assumed to be measured without error. All other yields are assumed to be measured with error. The table gives the parameter estimates with the corresponding standard errors.
Process Parameter Estimate Standard Error
Expected inflation: δ0π 1.84% 2.70%
δ0π+δ1π0 Xt δ1π(1) -0.63% 0.10%
δ1π(2) 0.13% 0.25%
Nominal interest rate: R0 2.46% 5.85%
R0+R01Xt R1(1) -1.49% 0.22% R1(2) 0.50% 0.57% Dynamics factors: κ11 0.079 0.115 dXt =−KXtdt+Σ0XdZt κ21 -0.198 0.082 κ22 0.351 0.182 Dynamics inflation: σΠ(1) 0.02% 0.07% dΠt Πt =πtdt+σΠ0 dZt σΠ(2) -0.01% 0.06% σΠ(3) 0.61% 0.04%
Dynamics equity index: ηs 4.55% 3.74%
dSt St = (Rt+ηs)dt+σ 0 SdZt σS(1) -0.52% 1.44% σS(2) -0.76% 1.55% σS(3) -2.10% 1.51% σS(4) 16.60% 0.96% Prices of risk: Λ0(1) 0.398 0.326 Λt =Λ0+Λ1Xt Λ0(2) 0.047 0.264 Λ1(1,1) 0.156 0.158 Λ1(1,2) -0.383 0.038 Λ1(2,1) 0.096 0.075 Λ1(2,2) -0.090 0.125
Volatility measurement error: σ3/12 0.78% 0.03%
στ(τ = 3/12, 1, 3, 5) σ1 0.28% 0.01%
σ3 0.10% 0.02%
4.1 estimation results
Table 2
Correlations between asset returns
Correlations between stock returns and one-year, two-year, three-year, five-year and ten-year nominal bonds using the estimates in Table 1.
Stock 1-Year 2-Year 3-Year 5-Year 10-Year
Bond Bond Bond Bond Bond
Stock 1 1-Year Bond 0.024 1 2-Year Bond 0.032 0.986 1 3-Year Bond 0.038 0.950 0.989 1 5-Year Bond 0.046 0.854 0.928 0.973 1 10-Year Bond 0.053 0.673 0.787 0.870 0.960 1
We now consider the elements of σΠand σS. We find that the elements of σΠ are close to those estimated byDraper(2014). The same holds for σS. The first three elements of σS are negative. The fourth element, however, is positive and quite large. σS(4) is the loading on the fourth Brownian
motion which represents the risk associated with the stock return. The estimated value for σS(4) is therefore quite realistic.
Using the standard errors (s.e.) shown in Table1, we can derive confi-dence intervals for the parameter estimates ˆθj, where j = 1, ..., 27. The 95% confidence interval equals
h
ˆθj−1.96 × s.e.j; ˆθj+1.96 × s.e.ji.
If the confidence interval contains zero, we might conclude that the pa-rameter estimate considered is not significant. Looking at the standard errors in Table1, it becomes clear that not all estimates are significant. In particular, δ0π, R0 and ηS, which are important parameters in our model, are not significant. We also see that for the volatility of the stock return only the fourth element of σS is significant, and that for the volatilty of the price index, only the third element of σΠ is significant. In this sense, we might question the value of the results. The standard errors described here are similar to those found by Draper (2014). In his research, the parameters δ0π, R0 and ηS, among others, were also not significant.
Table2presents the correlations between the stock return and the one-year, two-one-year, three-one-year, five-year and ten-year nominal bonds in the financial model. We find that all correlations are positive, meaning that if returns of one asset go up or down, the other asset returns are likely to move in the same way. This result is consistent withSangvinatsos and Wachter(2005).
In Figure 2, we have plotted the time series of the realized inflation in the period from December 31st, 1972 up to December 31st, 2013, and the expected inflation series for the same period.
1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Year Inflation Expected inflation Realized inflation
Figure 2. Realized and expected inflation. The dotted line denotes
re-alized inflation from the data, see section 3.3 as well. The solid line denotes the expected inflation πt as implied by the model and parameter estimates in
Table1. 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 −2 0 2 4 6 8 10 12 14 Year
Interest rate in percentages
3−month yield Real interest rate Nominal interest rate
Figure 3. The nominal and real interest rateThe thick black line gives
the real interest rate as implied by the model and the parameter estimates in Table 1. The dotted line denotes the nominal interest rate as implied by the model. The nominal interest rate is compared to the three-month bond yield over the period from December 31st, 1972 up to December 31st, 2013.
The expected inflation is constructed from the parameters in Table1and the state variables X, and is derived from the relationship in equation (3)
πt=δ0π+δ1π0 Xt.
We can conclude from Figure2 that the expected inflation forecasts the realized inflation to some extent. Both time series generally follow the same pattern.Sangvinatsos and Wachter(2005) point out that this result is worth emphasizing as the state variables are determined using bond
4.2 risk premia
yields alone. We may conclude that bond yields hold information about future inflation.
We would also like to draw conclusions from the interest rates implied by the model. We estimated the parameters for the nominal interest rate process in equation (17). The parameters for the real interest rate process can be derived as follows
R0 =δ0r+δ0π− σ0ΠΛ0 ←→ δ0r =R0− δ0π+σΠ0 Λ0, R01 =δ1r0 +δ1π0 − σ0ΠΛ1 ←→ δ1r0 =R 0 1− δ 0 1π+σ 0 ΠΛ1.
In this way, we can write the real interest rate process as given in equa-tion (2). Figure3 presents the time series of the nominal interest rate Rt and the real interest rate rt as implied by the model and the parameter estimates, as well as the three-month money market rates from our data set. We see that the nominal interest rate is almost equal to the three-month money market rates. Furthermore, the real interest rate is positive for almost the whole sample, with exceptions starting in 2009. This is a reasonable property for interest rate time series as interest rates can be negative.
Having estimated the financial market model, there are a few general remarks to be made. The estimated parameter values in Table1are close to the estimates ofDraper(2014). In fact, the likelihoods based on both parameter sets are also almost similar: the likelihood based on Draper’s optimal parameter values is approximately 6525.28 whereas we find a likelihood value of approximately 6525.66. This is not an exceptional result as we used the same data set as Draper. The market prices of risk and the risk premia for both stocks and bonds have not been discussed yet. These will be discussed in the next section.
4.2 r i s k p r e m i a
We now turn to the prices of risk and implied risk premia. We find an equity risk premium ηS of 4.55%. Adding this risk premium to the ‘risk-free’ rate gives the stock return. It is safe to say that the risk-free rate approximately equals the expected nominal interest rate R0, leading to an equity return of R0+ηS. We therefore find an equity return of ap-proximately 7%, which is equal to the expectation of the Commission Parameters (Langejan et al.,2014).
We further find that the unconditional price of risk of X1 is larger than the price of risk of X2, that is, Λ0(1) > Λ0(2). The bond risk premium
implied by the model is given by the term BN(τ)0Σ0XΛ0,
from equation (39). The state variables are set to their unconditional expectation, that is, Xt = 02x1 for every t. The impact of the state variables X1 and X2 on the bond risk premia is governed by Λ1 through BN(τ), where again, τ is the maturity of the bond considered.
Table 3 Bond risk premia
The table presents the risk premia on one-year, two-year, three-year, five-year and ten-year bonds and their corresponding volatilities. The risk premia and the volatilities are expressed in annual terms.
Maturities Bond risk premia Volatilities
One-year 0.51% 1.33%
Two-year 0.94% 2.36%
Three-year 1.31% 3.27%
Five-year 1.92% 4.99%
Ten-year 3.08% 9.10%
Table3 presents the nominal bond risk premia in annual terms for τ = 1, 2, 3, 5 and 10 years, and their corresponding volatilities. The bond risk premia are close to the bond risk premia estimated byDraper(2014) and vary from 52 basis points for a one-year bond to over 3% for a ten-year bond. We also see that bond risk premia are higher for higher maturities which is exactly what we expect. Investors need to be compensated more for taking on investments in high maturity bonds compared to bonds with lower maturities.
4.3 f i t o f t h e f i n a n c i a l m a r k e t m o d e l
In this section, we look at the fit of the model by comparing the average yield and standard deviations of the bonds used in the estimation process, with their model implied values. This is done by simulating 5000 sample paths for the model implied yields for the same time period as our dataset using the VAR(1)-model in equation (40):
Yt+h=µ(h)+Γ(h)Yt+t+h,
where t+h ∼ N(0, Σ(h))for discrete-time parameters µ(h), Γ(h) and Σ(h). In doing so, we followSangvinatsos and Wachter(2005) andKoijen et al. (2010). At the start of our data set, that is, t= December 31st, 1972, Yt
is defined as
Yt=chol(ΣY˜ )× Z, (54)
where chol denotes the Cholesky decomposition and Z ∼ N(0, 1)is stan-dard normally distributed. ˜ΣY is not to be confused with ΣY in Sec-tion 2.5. Rather, ˜ΣY is obtained by using the general definition of vari-ances and matrix algebra, and is defined as
vec
˜ ΣY
=I −Γ(h)⊗Γ(h)−1×vecΣ(h) (55) In this way, a new starting point Ytis determined for every sample path by drawing from the normal distribution.
4.3 fit of the financial market model 0 1 2 3 4 5 6 7 8 9 10 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 Maturity (years) Average yield Data
Monte Carlo mean 97.5% Upperbound 2.5% Lowerbound
Figure 4. Average bond yields.The black line gives the mean of 5000 Monte
Carlo simulations of the average bond yields with maturities of three months, one year, two years, three years, five years and ten years. The bond yields are in annual terms and defined as in Equation (44). The blue line gives the mean of the bond yields based on our data from December 31st, 1972 till December 31st, 2013. 0 1 2 3 4 5 6 7 8 9 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Maturity (years) Yield volatility Data
Monte Carlo mean 97.5% Upperbound 2.5% Lowerbound
Figure 5. Volatilities of bond yieldsThe black line gives the mean of 5000
Monte Carlo simulations of the volatilities of the bond yields with maturities of three months, one year, two years, three years, five years and ten years. The bond yields are in annual terms and defined as in Equation (44). The blue line gives the mean of the volatilities of the bonds based on our data from December 31st, 1972 till December 31st, 2013.
We may write t+h as
t+h =chol(Σ(h))× ˙Z, (56) where ˙Z ∼ N(0, 1)is also standard normally distributed. For each Monte Carlo simulation, the model is developed for the sample period according to the VAR(1)-model above. The average bond yield for each maturity is
