Amsterdam School of Economics
Faculty of Economics and Business
M.Sc. Financial Econometrics
Master Thesis
Impact on Pension Funds of Model
Specification Risk of the Equity
Risk Premium
Author1: Supervisor UvA2:
Arjen Wildeboer Dr. S.A. Broda
Supervisor Willis Towers Watson3:
Dr. G.C.M. Siegelaer
August 11, 2016
Abstract
The equity risk premium is a premium that compensates for taking risk of investing in stocks. The Dutch Central Bank uses this premium in their market model to estimate the return on stocks and to create a set of economic scenarios for the next 60 years. In this paper a new market model is proposed, wherein the stock return no longer directly depends on the interest rate. This new market model is evaluated using maximum likelihood estimation, which results in new parameter values that describe the economy in the Netherlands. A set that contains new economic scenarios is created with these new parameters. This set is then used to rank 15 different investment strategies for an employee that is eligible for a pension plan. This reveals that the optimality of a pension investment strategy depends on the used model for the equity risk premium.
Statement of Originality
This document is written by Arjen Wildeboer, who declares to take full responsi-bility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
1 Introduction 3
2 Equity risk premium 5
2.1 Dependence on interest rates . . . 5
2.2 Literature review . . . 6
3 Model specifications 10 3.1 The capital market model . . . 10
3.1.1 Equity risk premium restriction . . . 12
3.1.2 Nominal rate restriction . . . 13
3.2 Ornstein-Uhlenbeck process . . . 14
3.3 Maximum likelihood estimation . . . 16
3.4 Simulated annealing . . . 17 4 Data 19 4.1 Bond yields . . . 19 4.2 Stock returns . . . 20 4.3 Inflation . . . 21 5 Results 22 6 Implications for pension funds 24 6.1 Uniform economic scenario simulation . . . 24
6.1.1 Updating scenarios . . . 24
6.1.2 DNB adjustments . . . 25
6.2 Lifecycles pension plan . . . 26
6.2.1 Ranking on utility . . . 28
7 Conclusion 31
Bibliography 33
Appendix A: Exact discretization 35
Appendix B: Descriptive statistics 37
1
Introduction
Every quarter of a year, the Dutch Central Bank (DNB) generates and publishes a scenario set that contains 2000 scenarios of the economy in the Netherlands. Each scenario describes a possible outcome for the return on stocks, the inflation, and the interest rate over the next 60 years. Pension funds use these scenarios to make a stochastic analysis of their feasibility studies, which gains insight into the financial prospects and risks of the fund.
Additionally, the scenarios are used to determine the possible future returns for the various investment strategies developed by the fund. By comparing the outcomes of those returns for each scenario and for each investment strategy, it is possible to determine the strategy that performs optimal over all scenarios. This outcome however thus depends on the scenarios that are in the scenario set of the DNB.
The model behind the generation of the scenarios is based on the KNW-model, developed by Koijen et al. (2010). In the original KNW-model, the processes for the return on stocks, the inflation, and the interest rates are determined by 25 observed variables and two latent factors. These three processes describe the key relations between the financial risks for pension funds, and with the scenarios describing those processes it is possible to gain insight into those risks.
The model was initially developed to estimate the three economic processes in the U.S. market. The results were then used to compare an investor who implemented an optimal pension strategy to an investor that did not hedge for the variation in the return on bonds. They found that the differences in total accrued pension between the two investors were only minor, and thus concluded that hedging the variation in the return on bonds has an insignificant effect.
Other papers have later used a modified version of the KNW-model to describe other economies than the U.S. market. The model was first used by Draper (2012) to derive a capital market model for the Netherlands. He compared his results with the results from Koijen et al. (2010) and, among other things, concluded that the equity risk premium, which is a premium that compensates for taking the relatively higher risk of investing in stocks, is approximately 1.9% higher in the Netherlands compared to the United States.
The same estimation procedure was repeated in 2014, however this time with a new data set that also contained data over 2013 (Draper, 2014). The result of this paper is a set of 27 parameters that describe the stock market, the interest rate and the inflation in the Netherlands, which are used by the DNB to create the economic scenarios.
The KNW-model assumes that the return in the stock market is determined by the risk-free interest rate plus the equity risk premium. This assumption however, as will be argued in the next chapter, is not that self-evident. Therefore a new market model for the Netherlands is made in this paper, in which the stock return
is no longer made up of the risk-free rate plus an equity risk premium. This new market model is then estimated using maximum likelihood estimation and the method of simulated annealing. The results coming from this estimation process are then used to generate a new scenario set. With this set, fifteen examples of investment strategies for building up a pension are compared with each other, such that the optimal investment strategy can be determined.
The outline of this thesis is set up as follows. First, the equity risk premium will be further specified in Chapter 2. This chapter is intended to gain more insight into the premium and it discusses various other papers that tried to estimate it. In Chapter 3, the new market model is derived and explained in detail. This market model resembles the KNW-model from Koijen et al. (2010), however a stock return that does not directly depend on the interest rates is used. Furthermore, this chapter describes the maximum likelihood estimation procedure that estimates the parameters in the market model. Chapter 4 will then briefly describe the data that is used in the estimation process. In Chapter 5 the results are given and discussed. Chapter 6 focuses on the implications of the results on pension funds and Chapter 7 will conclude.
2
Equity risk premium
When a pension fund wants to receive a certain yield on its investments, but does not want to be exposed to any risk, it could consider investing in a government debt instrument that is considered as almost default-free. Assuming that the instrument has no probability to default, the fund will not experience any risk in losing its investments. In return for investing the money in the instrument, the pension fund will receive the risk-free interest rate. Such an investment however does not result in a high yield, especially after the financial crisis in 2008 when the interest rates dropped and the 3-month interest rate even became negative in 2015.
Instead of investing in a government debt instrument and receiving the risk-free rate, an investor could consider to invest in equities. This may result in higher yields, however investing in equities is also associated with risk. Because an investor is exposed to this relatively higher risk when he invests in stocks, a premium is demanded above the risk-free rate. This market premium is called the equity risk premium (ERP).
In this way, the return on stocks can be described as
dSt
St
= (Rt+ ηS)dt , (2.1)
where Stis the stock index, Rtis the risk-free interest rate on fixed income
secu-rities, and ηS denotes the equity risk premium for the stock market. This formula
is used in most literature to describe the stock return (Duarte and Rosa, 2015) and is also implemented in the models by the Dutch Central Bank to describe the economy in the Netherlands (Langejan et al., 2014).
2.1
Dependence on interest rates
Equation (2.1) shows that the return on stocks depends on the interest rate. Several studies however focused their research on the correlation between the interest rate and the stock market, which resulted in different outcomes for the strength and direction of the relationship. While some studies, such as Scott (1997), showed that there is a negative correlation between stocks and interest rates, some other studies, including Flannery and James (1984), found a positive correlation. More recent studies however have shown a strong and significant changing correlation over time (Khrawish et al., 2010).
Furthermore, when the historic Bundesbank4 data of the short-term interest rate is observed in Figure 2.1, it is hard to determine what the short-term interest rate should be over a longer period. It might therefore be a better approach to think of the interest rate as a variable that is determined by policy. In this
0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 1973 1974 1976 1977 1979 1980 1982 1983 1985 1986 1988 1989 1991 1992 1994 1995 1997 1998 2000 2001 2003 2004 2006 2007 2009 2010 2012 2013 Yie ld Year
Figure 2.1: 3-month interest rate from 1973 to 2013.
way, the interest rate in Europe is set by the European Central Bank (ECB) and in the United States by the U.S. Federal Reserve. However, as already shown in Equation (2.1), the interest rate is used as a building block to determine the value of other risky assets. Simplified, the valuation of the assets begins with the interest rate and then the various risk premia for the specific assets are added. For example, the term premium is used for government bonds while for corporate bonds the credit risk premium is added too. Equities are valuated in the same way by starting with the real interest rate and adding the equity risk premium. In such a model, the real interest rate functions as a building block for all long-term investment returns in an economy. While cash and bonds are evidently related to each other, this is less obvious for equities. It might therefore be better to let the return on stocks be independent of the risk-free interest rate, which will be the main assumption in our market model described in Chapter 3.
2.2
Literature review
Many papers have tried to estimate the value of the ERP. As there is no general method to determine the ERP, those values can differ up to 6.5 percentage points. This can be seen in Table 2.1, which shows estimated values of the ERP with the corresponding paper. For example, Campbell et al. (2008) finds an ERP of 3.3% (for the world as a whole) while Shackman (2006) finds a value of 9.5%. This large bandwidth shows that estimation of the ERP goes along with high uncertainty.
Paper ERP Years of observation Mehra & Prescott (1985) 6.18% 1889-1978 Siegel (1992) 4.15% 1800-1990 Blanchard et al. (1993) 4.37% 1802-1992 Siegel (1999) 5.12% 1802-1998 Jagannathan et al. (2000) 4.84% 1930-1999 Welch (2000) 6.90% 1870-1998 Claus & Thomas (2001) 4.56% 1985-1999 Campbell (2002) 5.93% 1891-1999 Fama & French (2002) 4.44% 1872-2000 Canova & de Nicol´o (2003) 3.70% 1971-1999 Ibbotson & Chen (2003) 3.42% 1926-2000 Mehra (2003) 5.95% 1802-2000 Salomons and Grootveld (2003) 7.99% 1976-2002 Barro (2005) 7.16% 1880-2004 Siegel (2005) 5.68% 1802-2004 Digby et al. (2006) 8.14% 1910-2004 Dimson et al. (2006) 5.50% 1900-2005 Kyriacou et al. (2006) 5.95% 1871-2002 Shackman (2006) 9.50% 1970-2002 Ville (2006) 4.73% 1889-1978 De Santis (2007) 4.04% 1928-2004 Mehra (2007) 6.73% 1802-2004 Vivian (2007) 4.43% 1901-2004 Campbell (2008) 2.95% 1982-2006 Avdis & Wachter (2013) 5.10% 1927-2011 Dimson et al. (2014) 4.30% 1900-2013 Table 2.1: Overview of equity risk premia determined
by other papers over recent years.
These differences can be explained by the different approaches the papers in Table 2.1 used to estimate the ERP. Van Ewijk et al. (2012) points out four of the main factors that explain these differences.
First, there is a difference between the papers that use ex-post estimation and the papers that use ex-ante estimation. Ex-post studies base their estimation of the ERP on the historical difference between the stock index and the risk-free interest rate, while ex-ante estimations base their estimation on the future dividend discount model. The future dividend discount model assumes that the stock price today is equal to the sum of all expected future cash flows, discounted at an appropriate rate to take into account their riskiness and time value of money (Duarte and Rosa, 2015). This model follows from the idea that a certain stock should essentially have the same value as the value it pays off in current and future dividends. The corresponding equation for the model is given by:
St= Dt ρt +Et[Dt+1] ρt+1 +Et[Dt+2] ρt+2 +Et[Dt+3] ρt+3 + . . . (2.2)
flow k periods from time t, and ρt+k is the discount rate k periods from the
perspective of time t. The discount rate ρt+k can then be described by:
ρt+k= 1 + Rt+k+ ηS(k) , (2.3)
where ηS(k) is the equity risk premium implied for period k at time t. So by
substituting the prices for the stock, the risk-free rate and the expected future dividends into Equations (2.2) and (2.3), it is possible to derive the value for ηS,
the ERP. It is remarkable that most ex-ante studies, such as Blanchard et al. (1993), Loeys and Panigirtzoglou (2005) and Damodaran (2016), result in a lower ERP compared to ex-post studies, as can be seen in Table 2.1.
A second factor that explains the differences between the ERP estimations is the period over which the ERP is estimated. For example, Mehra and Prescott (1985) found a historic ERP of 6.18% over the period 1889-1978, while Siegel (1992) found an ERP of 4.15% when he shifted the measurement period of 1889-1978 to 1880-1990. In his paper, it is argued that this is the result of the low risk-free rate in the period 1889-1978.
The third aspect that affects the outcome is the economy wherein the mea-surements are performed. In fact, it is not necessarily the case that the ERP is the same over all countries. For instance, a higher outcome might be found for the ERP when the measurement is performed in the United States, due to the fact that the United States have had much more profitable stock markets. Dimson et al. (2015) found that the ERP of the whole world is 4.30%, which is indeed much lower than the 5.50% that were found when only data from the U.S. market was used (Dimson et al., 2006).
The last element that explains the differences in Table 2.1 is the reference rate that is used for the risk-free rate Rt in Equation (2.1). Most of the studies use
the return on fixed income securities for the risk-free rate. These securities can either be bills, which earn a short-term rate, or bonds, which earn a long-term rate. Some papers use bills because those are less sensitive to inflation, while others use bonds because their long-term rate fits better the long-term structure of stocks. Van Ewijk et al. (2012) finds a difference between estimating on bills and estimating on bonds of 0.81%, which approximately equals the value of of 0.80% that was found by Dimson et al. (2015).
When Dimson et al. (2015) analysed the economy in the Netherlands, they found an ERP of 4.3% relative to (short-term) treasury bills. If it is furthermore noted that the current short-term interest rate is approximately equal to 0.0%
5, Equation (2.1) would imply a short-term expected return on stocks of 4.3%.
However, Van Ewijk et al. (2012) stated that a decrease of 100 basis points in the interest rate results in an increase of 52 basis points in the ERP. Combining the estimate for the ERP of 4.3% with the conclusion that there is a negative
correlation between the interest rate and the ERP, and noting the current interest rate of 0.0%, the DNB assumes that an equity risk premium of 7% is reasonable for the Dutch economy. On the long-term the DNB expects that the risk-free rate increases to 3.5%, which results in a long-term ERP of 3.5%.
This value of 7% for the equity risk premium is also used by the DNB in their KNW-market model, which is developed by Koijen et al. (2010). This market model is used by the DNB to generate a uniform scebario set containing 2000 scenarios of the economy upcoming 60 years. Pension funds must use these specific scenarios in their feasibility tests, such that the funds can be monitored and compared by the DNB.
A new market model, wherein the return on stocks no longer depends on the uncertain interest rate, is specified in more detail in the next chapter. This new market model, which among other things results in a new value for the equity risk premium, will then be used to generate a new scenario set for the economy in the Netherlands.
3
Model specifications
In this chapter the used market model, which is based on the KNW-model de-veloped by Koijen et al. (2010), will be explained in further detail. First, the assumptions and parameter relations will be presented. Thereafter the estimation procedure will be explained, which includes maximum likelihood estimation and the method of simulated annealing.
3.1
The capital market model
The portfolio in the market contains a stock, the long-term nominal and real bonds, and a nominal money account. In the model the return on stocks, bonds, interest rates, and the inflation depend on observed factors and two unknown factors. In this way the market model contains the dynamics between the main financial risk factors which are used by pension funds.
The dynamics of the interest rate rtand the instantaneous expected inflation
πtin the market model are defined as follows:
rt = δ0r+ δ01rXt, (3.1)
πt = δ0π+ δ1π0 Xt. (3.2)
The interest rate rt as well as the instantaneous expected inflation πt is thus
as-sumed to be driven by the two unobserved state variables that are collected in the vector Xt. The dynamics in the two state variables will contain the
autocor-relation in the expected inflation and the interest rate. This allows modelling the correlation between the expected inflation and the interest rate with the param-eters δ1r0 and δ01π. Furthermore, Equation (3.1) shows that the long run interest
is denoted by δ0r, while δ01r(1) is the loading of the interest rate on the first state
variable and δ01r(2) is the loading of the interest rate on the second state variable. The same format is used for the expected inflation in Equation (3.2).
The two unobserved state variables in Xt follow a mean-reverting process
around zero:
dXt= (µ − KXt)dt + Σ0XdZt, (3.3)
where K is a 2 × 2 matrix and Σ0X = [I2×2|02×2]. Furthermore, Zt ∈ R4×1 is a
four dimensional vector of independent Brownian motions. These four Brownian motions function as the origin of four economic uncertainties in our financial market.
The first Brownian motion is the result of the uncertainty in the real inter-est rate. The second and third Brownian motions are used for the expected and unexpected inflation respectively. The difference between the expected and un-expected inflation is that only the unun-expected inflation should be seen as risk.
While expected inflation is priced into the market without shocks, the unexpected inflation is one of the sources of the volatility in the market model. The fourth Brownian motion functions as the origin of the uncertainty in the stock return.
The price index Πtis determined by the expected inflation πt:
dΠt
Πt
= πtdt + σ0ΠdZt, (3.4)
where σΠ ∈ R4×1 denotes the loading of inflation risk on the four Brownian
motions. The price index will start at value 1, such that Π0= 1.
The stock index St proceeds in a similar way. This development can be
de-scribed by two different models, as already explained in Chapter 2.1. The first model is used in the paper by Draper (2014), and is also applied in the calculations of the DNB. It assumes that the stock index depends on the nominal short rate Rtand the equity risk premium on stocks, ηS:
dSt
St
= (Rt+ ηS)dt + σS0dZt, (3.5)
where again σS ∈ R4×1 and S0= 1. In the same way as before, σS contains the
loading of stock risk on the four Brownian motions.
In this paper however, it will be assumed that the stock index only depends on the equity risk premium and the independent Brownian motions in Zt, as
explained in Chapter 2.1. This results in the following equation for the stock return:
dSt
St
= ηSdt + σS0dZt. (3.6)
Finally, the nominal discount factor BN
t is specified. This discount factor will give
the marginal utility ratio between consumption in the future and consumption today. This marginal utility ratio will be the same for everyone as complete markets are assumed. A complete market (or Arrow-Debreu market) means that there are no transaction costs and that every asset in the portfolio has a price. Due to this assumption, the discount factor BN
t will be an appropriate discount
factor for all cash flows (Cochrane, 2009). It is defined as
dBN t
BN t
= −(Rtdt + Γ0tdZt) , (3.7)
where Γtis the price of risk that varies over time. This price of risk is defined as
Γt= Γ0+ Γ1Xt, (3.8)
where Γt, Γ0 ∈ R4×1, and Γ1 is a 4 × 2 matrix. In this notation, Γ0 denotes the
long run average price of risk of the four Brownian motions, and Γ1is the loading
We follow the assumption of Koijen et al. (2010), which states that unexpected inflation risk can not be identified on the basis of data on the nominal side of the economy alone. This results in the third row of Γ1containing zeros, because the
price of unexpected inflation risk equals zero:
Γ1= Γ1(1,1) Γ1(1,2) Γ1(2,1) Γ1(2,2) Γ1(3,1) Γ1(3,2) Γ1(4,1) Γ1(4,2) = Γ1(1,1) Γ1(1,2) Γ1(2,1) Γ1(2,2) 0 0 Γ1(4,1) Γ1(4,2) . (3.9)
3.1.1 Equity risk premium restriction
The market equations for the price index, the stock index, and the stochastic discount factor in Equations (3.4), (3.6) and (3.7) respectively, can now be used to obtain some relations between the unknown parameters. First, it is noted that these three equations describe Itˆo processes. This makes it possible to use integration by parts in combination with the multiplication table, which results in a new restriction for the equity risk premium ηS.
We start with applying integration by parts on d(BNS):
d(BNS) = dBNS + dSBN + (dBN)(dS) . (3.10) After division by BNS this becomes
d(BNS) BNS = dBN BN + dS S + dBN BN · dS S . (3.11)
If now Equations (3.6) and (3.7) are substituted, which are the equations for the newly defined stock return and the discount factor respectively, this gives
d(BNS) BNS = (−Rtdt − Γ 0 tdZt) + (ηSdt + σS0dZt) − (Rtdt + Γ0tdZt) · (ηSdt + σ0SdZt) = (ηS− Rt)dt − (Γ0t− σS0 )dZt− RtηSdt2 (3.12) − (Γ0ηS+ σS0Rt)dtdZt− (Γ0tσS0)dZt2.
It is now possible to apply Itˆo’s multiplication table:
× dZt dt
dZt dt 0
dt 0 0
Table 3.1: Itˆo’s multiplication table.
This results in the following new expression for Equation (3.12):
d(BNS)
BNS = (ηS− Rt− Γ 0
tσS0)dt − (Γ0t− σS0)dZt. (3.13)
The next assumption is that the discounted stock price is a martingale:
E[BtNSt|Fs] = BsNSs. (3.14)
This implies that the expectation of Equation (3.13) is equal to zero. Therefore, in combination with the fact that Ztis a Brownian motion which has an expectation
of zero, the following restriction for the equity risk premium is found:
σ0SΓt+ Rt= ηS. (3.15)
Without reproducing the derivations, it is noted that Draper (2014) found the following restriction:
σS0Γt= ηS. (3.16)
When the two restrictions in Equations (3.15) and (3.16) are compared, it follows that with our new model, a new restriction for the equity risk premium is found. This is the result of using a stock return that does not depend on the interest rate.
3.1.2 Nominal rate restriction
The same steps will now be repeated for the fundamental pricing equation, which states that the real stochastic discount factor is a martingale:
E[BtNΠt|Fs] = BsNΠs. (3.17)
In the same way as Equation (3.12), we find:
d(BNΠ) BNΠ = dBN BN + dΠ Π + dBN BN · dΠ Π = −(Rt− πt+ σ0ΠΓt)dt − (Γ0t− σ 0 Π)dZt (3.18) = −rtdt − (Γ0t− σΠ0 )dZt.
Looking at the last two lines and using the multiplication table again, the following expression for Rtis found:
Rt = rt+ πt− σ0ΠΓt
= (δ0r+ δ0π− σ0ΠΓ0) + (δ1r+ δ1π− σΠ0 Γ1)Xt (3.19)
≡ R0+ R01Xt.
In this notation, R0 is the long run expected nominal short rate. Additionally,
R1(1) and R1(2) are the loadings of the nominal rate on the first and second state
variable, respectively.
3.2
Ornstein-Uhlenbeck process
Before estimation of the model, which is done by using returns over different discrete time periods, Equations (3.3) - (3.5) have to be discretized. Following Koijen et al. (2005), the market model is described as a multivariate Ornstein-Uhlenbeck process.
This is done by first applying Itˆo’s lemma to the log inflation:
d log Π = ∂ log Π ∂Π dΠ + 1 2 ∂2log Π ∂Π2 (dΠ)2 = 1 ΠdΠ − 1 2Π2(dΠ) 2 = (πdt + σ0ΠdZt) − 1 2[πtdt + σ 0 ΠdZt] 2 (3.20) = (πt− 1 2σ 0 ΠσΠ)dt + σ0ΠdZt = (δ0π+ δ1π0 Xt− 1 2σ 0 ΠσΠ)dt + σΠ0 dZt.
Note that Equation (3.4) is substituted into the third line and Equation (3.2) in the last line. The same procedure is repeated for the stock index S:
d log S = ∂ log S ∂S dS + 1 2 ∂2log S ∂S2 (dS)2 = 1 SdS − 1 2S2(dS) 2 = (ηSdt + σ0SdZt) − 1 2(ηSdt + σ 0 SdZt)2 (3.21) = (ηS− 1 2σ 0 SσS)dt + σS0dZt,
The model described by Equations (3.3), (3.21) and (3.22) can now finally be written in the following form:
d Xt log Πt log St = µ δ0π−12σ0ΠσΠ ηS−12σS0σS + −K 0 δ1π0 0 0 0 X log Π log S dt + Σ0X σΠ0 σ0S dZt ≡ (A0+ A1Υt)dt + ΣKdZt, (3.22)
where Υ0 = [X, log Π, log S]. This corresponds to an Ornstein-Uhlenbeck process. The following derivations between Equations (3.22) - (3.26) are taken from Muns (2015) and Koijen et al. (2005). These steps are derived and explained in more detail in Appendix A.
We start with using the following eigenvalue decomposition:
A1= U DU−1. (3.23)
After exact h-step discretization this gives the following VAR(1)-model:
Yt= γ + ΘYt−h+ t, t i.i.d. ∼ N (0, Σ) , (3.24) where we defined: Θ = exp (A1h) = U exp(Dh)U−1, γ = U F U−1A0, (3.25)
with F a diagonal matrix with elements Fii= hα(Diih) and α(x) = (exp(x)−1)/x
and α(0) = 1.
Finally, to complete the discrete model, Σ is defined as:
Σ = U V U0, (3.26)
where Vij = [U−1ΣYΣ0Y(U−1)0]ijhα([Dii+ Djj]h).
Now that the market model is discrete, it is possible to estimate the parameters in the model by using returns over different discrete time periods. This is done by letting the optimal parameters correspond to the global optimum of a maximum likelihood function. This process is described in more detail in the next section.
3.3
Maximum likelihood estimation
The first step in the estimation procedure is finding the values for the two state variables in Xt. This is done by using data of six different bond yields, which are
denoted by ytτ, where τ is the maturity of the bond. These are the bond yields
with 3-months, 1-year, 2-year, 3-year, 5-year and 10-year maturities.
First, only two out of the six yields are used, namely the ones with 2-year and 5-year maturities, which are denoted by y2
t and y5t respectively. It is assumed
that these two yields have no measurement error. This choice is argued by the desire to span a large part of the term structure without using the three-month yield, which shows idiosyncratic behavior Duffee (2002). By using the following equation for those two bond yields, the state vector Xtcan be found for given A
and B:
ytτ = (−A(τ ) − B(τ )0Xt)/τ . (3.27)
More specific, the state vector Xtat each time t can be derived from:
−2y2 t −5y5 t =A(2) A(5) +B(2) 0 B(5)0 Xt, (3.28)
where A(τ ) is the intercept and B(τ ) captures the time variation in the risk premia. These two functions depend on parameters in our model and these parameters will be calculated using maximum likelihood, as shown below.
For the remaining four yields in the data it is assumed that they do have a measurement error, which results in the following equations for those yields:
yτt = (−A(τ ) − B(τ )0Xt)/τ + υτt, υ0t∼ N (0, Σ
τ) . (3.29)
In this equation, υt0 = [υ0.25
t , υ1t, υ3t, υ10t ], such that it contains the errors of the
bond yields that do have a measurement error. This equation for the four yields is then extended with Equation (3.24) for equity and inflation. If the same steps as Duffee (2002) and Draper (2014) are followed, the following corresponding function for the log-likelihood is found:
log L = −0.5T (log |Σ| − T X t=1 t(Σ)−10t+ log |Σ τ| − T X t=1 υt(Στ)−1υt0+ log |B|) , (3.30)
where B0 = [B(2), B(5)]. This is the sum of the log-likelihood for the observed
time series of the yields, the stock returns, and inflation.
The optimisation procedure thus starts with determining the state-vector Xt
by using the yields with 2-year and 5-year maturities. This state-vector will then be used to determine the errors in Equation (3.24) at the other maturities. The
likelihoods for the stock returns, inflation and yields are then combined in Equa-tion (3.30), which is maximised to find the parameters from our market model that correspond to the lowest measurement errors. The optimization method called simulated annealing is used to make sure that a global optimum in the likelihood is found, instead of a local optimum. This method is explained in the next section.
3.4
Simulated annealing
The method of simulated annealing was invented by Kirtpatrick et al. (1983) and Cerny (1985) to be used in physics for simulating the process of cooling a solid until the solid is eventually frozen, which occurs when the state of the solid is at a minimum energy configuration.
This method can also be applied to the log-likelihood in Equation (3.30) from the previous section. As there is an enormous amount of possible values that the parameters in our market model can take, an unrealistic amount of time would be needed to calculate and compare the log-likelihoods of all those possibil-ities. Instead, the method of simulated annealing is used to determine the state which corresponds to the global optimum of the log-likelihood equation, in a much smaller amount of time.
The procedure starts with calculating the log-likelihood at an initial random configuration, denoted by state i. After the log-likelihood for the initial state i is calculated, which is denoted by log L(i), a neighbor state j of i is chosen at random. The following equation is then used to assign a probability to the outcome that the new state j will be accepted as the new optimal state:
PT(accept(j)) =
(
1 if log L(j) > log L(i)
exp log L(j)−log L(i)T
if log L(j) ≤ log L(i) (3.31)
It follows that the new state j will always be accepted as the new optimal state if that new state j gives a higher log-likelihood than the previous state i. However, even when the new state j has a lower log-likelihood, there is still a probability that the new state will be accepted as the new optimal state.
This probability depends on the temperature T , which originates from the ap-plication of simulated annealing in physics. We set this parameter T at 100, such that in the beginning of the procedure there is a higher probability of accepting a new state that has a lower log-likelihood than the previous state. Note that the term inside the exponent in Equation (3.31) is negative, such that a higher tem-perature corresponds to a larger probability to accept a lower log-likelihood and thus a larger probability to ‘jump’ out of a local maximum. This whole procedure of trying new states is iterated. As the optimization procedure repeats itself and new states are tested, the temperature is programmed such that it gradually de-creases. According to Corana et al. (1987), the optimal new temperature should
be around 85% of the old temperature after a certain amount of new states are tested. In our model this amount of new states is set to 30.
With this method of simulated annealing, it is more likely that a global max-imum is found. This is supported by the fact that the initial states are chosen at random, while repetitions of the procedure always resulted in the same optimal set of parameters for the market model. These results will be given after the next chapter, in which the used data is specified.
4
Data
For the estimation procedure, data is used from the Dutch economy between 1970 and 2013. This includes quarterly data for the stock returns, inflation data, and data for six yields. These are the same data series as Draper (2014) used, which allows for comparison between the results where possible. The descriptive statistics of the data in this chapter can be found in Appendix B.
4.1
Bond yields
The six yields that are used come from the Bundesbank6 and the Dutch Central
Bank7, and have maturities of 3-months, 1-year, 2-years, 3-years, 5-years and
10-years. As explained in Section 3.3, it is assumed that two of those six yields have no measurement error. These two yields are the ones with 2-year and 5-year maturities and are shown in Figure 4.1.
0 0,02 0,04 0,06 0,08 0,1 0,12 1973 1974 1976 1977 1979 1980 1982 1983 1985 1986 1988 1989 1991 1992 1994 1995 1997 1998 2000 2001 2003 2004 2006 2007 2009 2010 2012 2013 Yie ld Year
2-year maturity 5-year maturity
Figure 4.1: Nominal bond yields in the Netherlands with 2-year and 5-year maturity.
For the four yields with 3-month, 1-year, 3-year and 10-year maturities, it is assumed that they do have a measurement error. These data are also taken from the Bundesbank and the Dutch Central Bank and are shown in Figure 4.2.
6https://www.bundesbank.de/Navigation/EN/Statistics/Time_series_databases/ 7http://www.dnb.nl/en/statistics/statistics-dnb/
0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 1973 1974 1976 1977 1979 1980 1982 1983 1985 1986 1988 1989 1991 1992 1994 1995 1997 1998 2000 2001 2003 2004 2006 2007 2009 2010 2012 2013 Yie ld Year
3-month maturity 1-year maturity 3-year maturity 10-year maturity
Figure 4.2: Nominal bond yields in the Netherlands with maturities of 3-month, 1-year, 3-year and 10-year.
4.2
Stock returns
Data from the MSCI World Index8 is used to determine the stock returns. This index is a broad global equity benchmark that represents large and mid-cap equity performance across 23 countries, including the Netherlands. These stock returns are shown in Figure 4.3, which immediately shows the high volatility in the stock returns. -1.2 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013 Sto ck re tu rn s Year
Figure 4.3: MSCI world stock index.
4.3
Inflation
The final data that is needed to maximize the log-likelihood from the previous chapter is the inflation. Data from the European Central Bank9 is combined with data from the International Monetary Fund10, such that the inflation in the Eurozone is found over the period of 1970 to 2013. This inflation is shown in Figure 4.4. -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1973 1974 1976 1977 1979 1980 1982 1983 1985 1986 1988 1989 1991 1992 1994 1995 1997 1998 2000 2001 2003 2004 2006 2007 2009 2010 2012 2013 In flatio n ra te Year
Figure 4.4: European inflation between 1970 and 2013.
9http://sdw.ecb.europa.eu/ 10https://www.imf.org/en/Data
5
Results
The market model from Chapter 3, the associated discretization of the model, and the likelihood optimization is programmed in Matlab. The procedure of simulated annealing will result in a global maximum of the likelihood, which corresponds with the 23 parameters in our market model that best fit the data.
Although a different equation for the stock return is used, it is still possible to compare our results with the results from Draper (2014), as he used the underlying KNW-model and data. Those results are therefore also given as reference in Table 5.1. When the new results from the table are compared with the results from Draper (2014), it first follows that the maximum likelihood value decreased from 6525.7 to 6525.2. Although Draper (2014) found a higher likelihood, it can not be concluded that those results better describes the data. This is because both estimates are the global maximum of their own likelihood function, which differ from each other.
Furthermore, it follows that the long run expected inflation δ0π more or less
stays the same. The loading on the two state variables in Xt, denoted by δ1π(1)
and δ1π(2), changed. The long run nominal interest rate R0also drops from 2.40%
to 1.29%. The variables that describe the process for the state variables in Xt
remain more or less the same, which is also the case for the realized inflation process.
Most remarkable are the changes in the parameters for the stock return pro-cess. The equity risk premium increases from 4.52% to 7.06%, while the standard deviations more or less stay the same. However, it has to be stressed that when the two equity risk premia are compared, a difference is expected, due to the fact that the new formula for the stock process does not contain the nominal interest rate. This means that in the new model a part of the nominal interest rate is now captured in the equity risk premium, which explains the higher value that is found. It is therefore better to compare the two equity risk premia by looking at the long-run value for the two stock return processes. Equation (3.5) shows that in the model from Draper (2014) this is equal to R0 plus ηS, which is 6.92% when
the results from Table 5.1 are substituted. For the new market model, the long-run stock return is equal to solely ηS, as follows from Equation (3.5). This results
in a long-run stock return of 7.06% for the new market model that is developed in this paper. The difference of 0.14% between these two percentages is the result of using two different models from the market, and may have a large impact on the scenarios that have been generated using these new parameters.
Because pension funds use these scenarios to predict the future development of the economy and compare their investment strategies, this difference might have major implications for pension funds. Both the process of creating scenario sets and the impact on pension funds are explained in more detail in the next chapter.
(i ) (ii )
Parameter Estimate SD Estimate SD
Expected inflation: πt= δ0π+ δ01πXt
δ0π 1.81% (2.79%) 1.87% (2.25%)
δ1π(1) -0.63% (0.10%) 0.48% (0.18%)
δ2π(2) 0.14% (0.24%) 0.42% (0.15%)
Nominal interest rate: Rt= R0+ R01Xt
R0 2.40% (6.06%) 1.29% (0.32%)
R1(1) -1.48% (0.22%) 0.91% (0.40%)
R1(2) 0.53% (0.56%) 2.52% (4.91%)
State variables term structure: dXt= −KXtdt + Σ0XdZt
κ11 0.08 (0.11) 0.08 (0.19)
κ21 -0.19 (0.08) -0.20 (0.17)
κ22 0.35 (0.18) 0.35 (0.10)
Realized inflation process: dΠt
Πt = πtdt + σ 0 ΠdZt σΠ(1) 0.02% (0.07%) -0.02% (0.07%) σΠ(2) -0.01% (0.06%) -0.02% (0.06%) σΠ(3) 0.61% (0.04%) 0.61% (0.04%)
Stock return process:
dSt St = (Rt+ ηS)dt + σ 0 SdZt dSSt t = ηSdt + σ 0 SdZt ηS 4.52% (3.73%) 7.06% (3.90%) σS(1) -0.53% (1.44%) -0.30% (1.59%) σS(2) -0.76% (1.54%) 0.88% (1.52%) σS(3) -2.11% (1.51%) -2.09% (1.52%) σS(4) 16.59% (0.96%) 16.59% (0.95%) Prices of risk: Λt= Λ0+ Λ1Xt Λ0(1) 0.403 (0.333) -0.201 (0.312) Λ0(2) 0.039 (0.270) -0.347 (0.266) Λ1(1,1) 0.149 (0.156) 0.132 (0.218) Λ1(1,2) -0.381 (0.039) -0.080 (0.149) Λ1(2,1) 0.089 (0.075) 0.403 (0.182) Λ1(2,2) -0.083 (0.129) -0.068 (0.122) log L 6525.7 6525.2
Table 5.1: Results from the maximum likelihood estimation of (i ) Draper (2014) and (ii ) the new market model from Chapter 3. Note that the difference between the two models is the
6
Implications for pension funds
The ambitions, expectations and risks of a pension fund can be determined by cal-culating the expected returns for each investment strategy. The expected returns are calculated by using the scenarios from a scenario set. The DNB generates each quarter a uniform set with 2000 economic scenarios, which are estimates of the Dutch economy over a period of 60 years. By obligating all Dutch pension funds to use the same scenario set, it is possible for the DNB to monitor and compare the various pension funds with each other.
The scenario set of the DNB is based on the results from Draper (2014) and is kept up-to-date by adjusting some of the parameters that have been determined in the previous chapter. This process of updating the parameters is explained first, after which the expected returns will be calculated for fifteen example investment strategies. This will be done for three different scenario sets, namely one based on the model from this paper, one using the original parameters from Draper (2014) and one using the adjusted parameters from the DNB.
6.1
Uniform economic scenario simulation
6.1.1 Updating scenarios
The scenarios from the DNB are made with the KNW-model and the correspond-ing model parameters from Draper (2014), which are shown in Table 5.1. Some of these parameters are first adjusted, as will be explained in the next section. The scenarios from the DNB are freely accessible and published every quarter on their website11.
Each scenario shows the development of the price inflation, the risk-free rate, and the return on equity. Furthermore, it is assumed that the wage inflation is equal to the price inflation plus 0.5% for the real wage growth per year, which is in line with the assumption from the Advies Commissie Parameters who advise the DNB (Langejan et al., 2014). Ensuring that the scenario set will stay up-to-date with the economic circumstances, it is upup-to-dated every quarter of a year with new scenarios. As a result, the interest rate from the model corresponds with the realised interest rate in the economy.
Creating up-to-date scenarios is done by manually adjusting four of the 23 parameters in the KNW-model from Chapter 3. These four parameters are the two state variables in Xt at time t = 0 and the two constant parameters in the
price of risk, Λ0(1) and Λ0(2). The Advies Commissie Parameters advises the
DNB which four values should be used to stay in line with the current state of
the economy. This advice is based on minimizing the following equation: 50 X t=1 h (RStm− RSext t ) 2+ (RLm t − RL ext t ) 2i, (6.1)
where RS is the starting interest rate and RL is the interest rate after 10 years. The superscript m means that it is the average value over the 2000 scenarios, while the superscript ext means that the interest rate is determined using the Ultimate Forward Rate (UFR) from the actual economy. The values for Xt, Λ0(1),
and Λ0(2) are varied such that they minimize Equation 6.1. This will result in
a small difference between the interest rate structure from the KNW-model and the realised rate determined by the UFR, while the other parameters from the KNW-model are kept close to their estimated values from Table 5.1. The model parameters will then stay up to date with the actual state of the economy.
6.1.2 DNB adjustments
The DNB does not use exactly the same parameter values for generating the scenarios as the values that come out of the KNW-model from Draper (2014), which are shown in Table 5.1. This is because the DNB expects a price inflation equal to 2.0%, an expected return on bonds of 3.9 % and an expected return on equities of 7.0% with a volatility of 20.0% (Langejan et al., 2014).
They meet their expectations by again adjusting four of the 23 parameters. These four parameters are δ0π from Equation (3.2) for the expected price
infla-tion, Γ0(1) from Equation (3.8) for the expected return on bonds, and finally ηS
and σS(4) from Equation (3.5) to change the expected return on equities and its
volatility.
The adjustments for these four parameters are summarized in Table 6.1.
Parameter Wildeboer (2016) DNB adjustments DNB expectations
δ0π 1.87% 1.98% Exp. price inflation = 2.0%
Γ0(1) 0.201 0.242 Risk on bonds = 3.9%
ηS 7.06% 6.57% Exp. equity return = 7.0%
σS(4) 16.59% 17.69% S.E. equity return = 20.0%
Table 6.1: Parameter adjustments of the DNB.
With these DNB adjustments, a log-likelihood is found of 6450.7, which is substan-tially lower than the likelihoods found in Table 5.1. This is as expected, because the adjusted DNB parameters are not based on maximum likelihood estimation, while the previous results were. Therefore, it can be concluded that the model with the DNB adjustments give a poorer description of the data than the previous parameters. Despite this result, the DNB still uses the adjusted parameters to meet their economy expectations, which are shown on the right side of Table 6.1.
Together with the previous results, it is now possible to generate three different scenario sets. The first set is made using the 23 parameters estimated by Draper (2014), which are shown on the left side of Table 5.1. For clarity, this set will be called the Draper-set. The second set, called the DNB-set, is made with the same parameters, however four of those parameters are adjusted to the parameters from the DNB in Table 6.1. This is the set that pension funds also use in their risk analysis. The last scenario set is made with the 23 parameters from the right side of Table 5.1, which were the results of using the new market model that is described in Chapter 3. This will be called the Wildeboer-set.
The scenario sets are generated with the Java-based tool called the Tilburg Finance Tool12. In this tool, developed by the Tilburg University, the dynamics of the economy are estimated for the next 60 years. The tool is especially created for the KNW-model and uses the 23 parameters from Table 5.1 as input. Those 23 parameters can therefore be used to create scenarios for the dynamics for the stock return, inflation and interest rates. A modified Tilburg Finance Tool is developed for the results on the right side of Table 5.1, such that it corresponds with the modified model from Chapter 3.
6.2
Lifecycles pension plan
The generated scenarios are used to compare and rank the investment strategies in defined contribution pension schemes. A defined contribution (DC) scheme is a pension scheme wherein the employer pays a contractually agreed amount into each employee’s pension account. The money is invested according to an individual investment strategy, the so-called lifecycle pension plan.
In a lifecycle pension plan, every participant invests his money based on the lifecycle principle. This principle is a model where the risk and the return of a portfolio depends on the age of the investor, or more particular on the number of years that are left until the pension date. A participant could use such a model to accrue his pension.
For a young participant such a pension plan consists of more return seek-ing investments. These investments can be riskier because a young participant has many years left to compensate for a potential short-term depreciation of the investments. These younger participants could for example still change their pen-sion expectations or adjust their expenditure. This makes it possible to apply an aggressive investment strategy which aims for high returns. The aggressive invest-ment strategy goes along with higher risk, but as long as the young participant has high potential for their future earnings, this potential will compensate for the higher risk. These riskier investments in the pension plan are investments in the so-called return portfolio.
When the participant is older and the pension date is nearby, a considerable pension has been accrued in most cases. Such a participant has, in contrast to the young participant, fewer years to compensate for a possible sudden decline in the stock market. An older participant should thus have a portfolio with less volatility and risk. This results in a more defensive investment strategy for older participants. Older participants should therefore invest a larger fraction in the ’matching portfolio’, instead of risky equities. This portfolio, which could consist of bonds, interest rate swaps and inflation swaps, should hedge the interest rate risk and inflation risk for a desired percentage. A sudden decrease in the stock market or the interest rate would then minimally affect their pension. The closer the pension date gets, the less the participant invests in the return portfolio and the more he invests in the matching portfolio. In such a way, the older participant knows in the last phase of the lifecycle which pension he could expect after the pension date.
An example of a lifecycle is shown in Figure 6.1. It shows that a DC lifecycle keeps adjusting the composition of the total portfolio until the pension date, such that it keeps matching the preferred risk and return of a participant throughout the years. At a younger age the pension investments will mostly consist of the return portfolio, which is denoted in the figure by a high weight in the mixing strategy. At a higher age more money will be invested in the matching portfolio. It will thus have more risk and higher potential returns in the early years, and more certainty in the later years. In this way, the lifecycle as a whole has limited risk and could still acquire in a decent return. All the investments will be in the matching portfolio at the pension date, such that at this age the participant can buy his pension with his pension capital coming from his investments throughout the years. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 We ight in mixi n g stra teg y
Years until pension date
Return portfolio Matching portfolio
Willis Towers Watson provided 15 examples of realistic lifecycles that can be found in Appendix C. The matching portfolio in these lifecycles is divided into two more specific portfolios: the matching long portfolio and the matching short portfolio. The matching long portfolio consists of long-term bonds with maturities longer than 25 years. The amount of capital that is invested in the matching long portfolio gradually decreases from 15 years to 0 years before the pension date. In the same way, the amount of capital that is invested in the matching short portfolio gradually increases until the pension date, such that each linear combination of matching long and matching short represents the matching portfolio for each age cohort. The 15 lifecycles have in common that all investments at an age of 67 will be in the matching short portfolio, such that every participant can buy his pension with the available capital from the matching short portfolio at the pension date.
This differs from the newly introduced Dutch pension law, which makes it possible to have a variable and risk-bearing pension. This is done by allowing participants to have investments in the return portfolio post pension date. How-ever this is not taken into account in these provided lifecycles.
6.2.1 Ranking on utility
The optimal ratio between the two matching portfolios and the return portfolio can be determined for each year until the pension date. Although this in the end depends on whether the participant is risk loving, risk-neutral or risk-averse, it is possible to rule out some of the lifecycle models by using the three generated scenario sets from Section 6.1.2 in combination with a general utility function.
The scenario sets are used to estimate the return and the risk for each of the 15 lifecycles for the next 60 years. The return and the risk of each lifecycle are traded off by using the exponential utility function. With the outcomes of the utility function it is possible to compare and rank the 15 lifecycles. This utility function is given by:
u(c) =
(1−e−ac
a if a 6= 0
c if a = 0 (6.2)
In this formula, c denotes the economic variable that the participant prefers more of. In our case, it is the DC pension (built up using the lifecycle principle) as a percentage of the pensionable salary, which is the part of the salary over which the pension contributions are payed. Furthermore, a denotes the risk-preference of the consumer. Following Gordon et al. (1997), it is assumed that the participants are risk averse, which corresponds to a value for a greater than 0 (a < 0 corresponds to risk-loving participants and a = 0 corresponds to risk-neutral participants).
The Arrow-Pratt measure of absolute risk-aversion (ARA) is also calculated. This measure, developed by Arrow (1965) and Pratt (1992), measures the dollar
amount the participant will choose to hold in risky assets, given a certain wealth level denoted by c. The measure is given by the following equation:
ARA(c) = −u
00(c)
u(c) . (6.3)
If the utility function from Equation (6.2) is substituted into Equation (6.3), the following ARA measure is found:
ARA(c) = −−ae
−ac
e−ac = a . (6.4)
This shows that the ARA measure does not depend on the wealth c, thus there is a constant absolute risk-preference. This means that as the wealth c increases, which corresponds in this case to a higher DC pension as percentage of the pensionable salary, the participant will keep the same risk aversion.
For the participants it is assumed that their starting salary at an age of 25 is 25.000 Euro. The DC pension as a percentage of the pensionable salary is then substituted into Equation (6.2), such that for each scenario in every lifecycle, a corresponding utility is found. The optimal lifecycle is finally determined by taking the average of the utility over the 2000 scenarios. The lifecycle corresponding to the highest average utility is the optimal lifecycle. This process of determining the optimal lifecycle is repeated for each of the three scenario sets that are generated. The optimal lifecycle still depends on the (normalized) value for a, which de-notes the risk-preference of the participant. Nevertheless, it is possible to exclude the lifecycles that are suboptimal for all values of a, as can be seen in Figure 6.2. The three subfigures show for each generated scenario set the relation between the optimal lifecycle and the risk-preference a of the participant. For values of a close to zero, which means that the participant is less risk-averse and almost risk-neutral, the results show that the optimal lifecycle is Lifecycle 1. This is in line with the expectations, because Lifecycle 1 has the highest percentage in the return portfolio and thus has high risk, as can be seen in Appendix C. When the participant is not that risk-averse, he would thus prefer a lifecycle with high expected returns despite the higher risk.
However, as the participant becomes more risk-averse, i.e. the value for a increases, higher lifecycles become optimal. This is the result of the participant not wanting to be exposed to too much risk anymore, and thus prefers lifeycles that invest less in the return portfolio and aim for more certainty.
For a given risk-preference, Figure 6.2 shows that lower lifecycles are more preferred in the Wildeboer-set when compared to the Draper-set or the DNB-set. This reveals that the use of a different market model has a significant impact on the relative ranking of the lifecycles.
For the most risk-averse participants, the optimal lifecycles are Lifecycles 12, 11 and 10 for the Draper-set, the DNB-set and the Wildeboer-set, respectively. Fur-thermore, the results have in common that Lifecycles 13, 14 and 15 are never optimal investment strategies in any scenario set. These are the three most de-fensive investment strategies, as can be seen in Appendix C.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Op tim al li fe cy cle a (risk-aversion) (a) Wildeboer-set 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Op tim al li fe cy cle a (risk-aversion) (b) Draper-set 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Op tim al lif ecy cle a (risk-aversion) (c) DNB-set
Figure 6.2: Lifecycle with the highest average utility, for each of the three generated scenario sets.
7
Conclusion
All Dutch pension funds are obligated to use the economic scenario set published by the DNB to communicate about their ambitions, expectations and risks. Al-though the set contains 2000 different scenarios, all those scenarios are made with the same KNW-model. This model provides the analytical expressions for the expectations and variances of the stock returns, the interest rate and the inflation in the Netherlands. All Dutch pension funds thus use the assumptions made in one model.
One of the assumptions in the KNW-model is that the stock return depends on the risk-free interest rate. Because the influential role of the interest rate on the stock returns is not that self-evident, it is interesting to investigate what happens if this dependence is taken away. The goal of this paper is to develop such a new market model, wherein the stock return no longer depends on the risk-free interest rate. The same approach as Draper (2014) is followed, which results in new analytical expressions that describe the Dutch economy.
The parameters in the new model are estimated by using the method of max-imum likelihood, which among other things results in an equity risk premium of 7.06%. To conclude anything on the new parameters, a new scenario set that is based on the new market model is created. This set is then compared with the DNB-set and the Draper-set, which is the scenario set that is created with the parameters from Draper (2014). The sets are compared by ranking example lifecycle investment strategies from Willis Towers Watson for each of the three scenario sets.
It follows that the optimality of the lifecycles differs for each scenario set, and thus depends on the used model for the equity risk premium. When the new scenario set is compared with the DNB-set and the Draper-set, it can be concluded that lifecycles with a much lower percentage invested in stocks are preferred in the new scenario set. Remarkable is that the three sets have in common that the three most defensive lifecycles are sub-optimal for every participant, independent of the individual risk-preference of the participant. This can be translated into the important conclusion that those three lifecycles should be excluded from the defined contribution pension schemes. As a result, participants in those schemes would no longer invest using those sub-optimal lifecycles.
The DNB should thus be aware of the assumptions they make in their model and the implications those assumptions may have for their scenario set. Further research should focus on creating other modified models, as has been done in this paper. The DNB could then develop multiple scenario sets wherein the sets are based on different underlying models. The scenario sets would then be more diversified and pension funds would not be dependent on the assumptions made in only one model, namely the original KNW-model. An example of another alternative model is one wherein the stock return is determined by the long-term
interest rate plus the equity risk premium, instead of the short-term interest rate plus the equity risk premium.
In addition, when new scenario sets are used to rank the lifecycle investment strategies, further research should also take into account the new Dutch pension law. This new law allows participants to have a certain amount in the return portfolio at their pension date, which is not taken into account in the lifecycles from this paper. Such research could result in a new ranking of lifecycles, where also the returns on investment after the pension date are analysed.
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Appendix A: Exact discretization
In this section, the derivations between Equations (3.22) - (3.26) will be explained in more detail. With these equations it is possible to write the model as a discrete Ornstein-Uhlenbeck process. The derivations here presented are based on Koijen et al. (2005) and Muns (2015).
We start with applying Itˆo’s lemma to the function f (t, Yt) = e−A1tΥt, where Υt
is taken from Equation (3.22). This gives:
d(exp (−A1t)Υt) = exp (−A1t)(A0dt + ΣΥdZt)
= exp (−A1v)A0dt + exp (−A1t)ΣΥdZt, (A.1)
which implies that
exp (−A1t)Υt
= exp (−A1(t − h))Υt−h+
Z t
u=t−h
d(exp (−A1u)Υu) (A.2)
= exp (A1(h − t))Υt−h+ Z h−t v=−t exp (A1v)A0dv + Z h−t v=−t exp (A1v)ΣΥdZu.
If this is rewritten, we find that
Υt= [exp (A1h) − I]A−11 A0+ exp (A1h)Υt−h+
Z h
v=0
exp (A1v)ΣΥdZu. (A.3)
If we now use the Jordan matrix decomposition, which states that A1= U J U−1,
the following expressions for γ and Θ in Equation (3.24) is found:
Θ = U exp(J h)U−1 (A.4)
γ = U [exp(J h) − I]J−1U−1A0. (A.5)
The next step is to derive the disturbance covariance Σ from the integral in Equa-tion (A.3). For an nb× nbJordan block (denoted by Jb) in an n × n Jordan matrix
(denoted by J ), we have: exp(Jb) = eλb Xnb−1 d=0 ˜ M(d), Jb= λb 1 λb . .. . .. 1 λb , (A.6)
where each ˜M(d) is an n
b× nb diagonal-constant matrix (i.e. a Toeplitz matrix)
with entries ˜ Mij(d)= ( 1 (j−i)! if j = i + d 0 else (A.7)
If the previous two expressions are now applied to the whole Jordan matrix J , instead of only the Jordan block Jb, we find that
exp(J ) = exp(D)XnB
d=0M
(d), (A.8)
where D is a matrix with the same diagonal as the Jordan matrix J , nB = maxb(nb) − 1, and each M(d)is an n × n matrix with entries
˜ Mij(d)=
(1
k! if j = i + d and (i, j) is in a Jordan block of J
0 else (A.9)
If Equation (A.8) is applied to the earlier mentioned Jordan decomposition A1= U J U−1, we find the following disturbance covariance Σ in Equation (A.3):
Σ = Z h v=0 exp (A1v)ΣΥΣ0Υexp (A 0 1v)dv = Z h v=0
U exp (J v)U−1ΣΥ(U−1ΣΥ)0exp (J0v)dvU0
= U Z h v=0 exp (Dv) nB X a=0 M(a) ! U−1ΣΥ(U−1ΣΥ)0 nB X b=0 M(b) !0 exp(Dv)dvU0 ≡ U V U0, (A.10) where V = nB X a=0 nB X b=0 V(a,b), V(a,b) = Z h v=0
exp(Dv)M(a,b)exp(Dv)dv , M(a,b) = M(a)U−1ΣΥ(M(b)U−1ΣΥ)0.
The elements in V(a,b)are given by:
Vi,j(a,b)= (
Mi,j(a,b)h if Dii+ Djj = 0
Mi,j(a,b)(Dii+ Djj)−1[exp([Dii+ Djj]h) − In×n] if Dii+ Djj 6= 0
Note that for a diagonizable matrix we have J = D and nB = 0. This finally
shows that the expressions for Θ, γ, and Σ in Equations (3.23), (3.25), and (3.26) coincide with the expressions in Equations (A.4), (A.5) and (A.10), respectively.
Appendix B: Descriptive statistics
In this section, the descriptive statistics of the six bond yields, the inflation, and the stock return are given. These statistics quantitatively describe the main features of the used data from Chapter 4.
A summary of all the descriptive statistics are given in Table B.1. Among other things, it shows that the mean over the last 40 years of the bond yields is much higher than the bond yields nowadays. This can be explained by the relative high yields in the period from approximately 1973 to 2000, which is also shown in Figures 4.1 and 4.2. Furthermore, the variance of the stock return is much larger than the variances of the bond yields, which is expected due to the higher risk in stock markets. The higher kurtosis of 1.58 for the stock return reveals that a large part of this higher variance is the result of sporadic extreme deviations.
Mean S.E. Median S.D. Variance Kurtosis Skewness 3-month bond yield 4.89% 0.23% 4.45% 2.90% 0.08% 0.11 0.71 1-year bond yield 4.95% 0.20% 4.58% 2.63% 0.07% -0.40 0.43 2-year bond yield 5.15% 0.19% 4.80% 2.48% 0.06% -0.61 0.18 3-year bond yield 5.37% 0.19% 5.15% 2.40% 0.06% -0.69 0.03 5-year bond yield 5.70% 0.18% 5.61% 2.28% 0.05% -0.72 -0.12 10-year bond yield 6.16% 0.16% 6.37% 2.07% 0.04% -0.68 -0.23 Stock return 7.76% 2.26% 12.76% 32.88% 10.81% 1.58 -0.96 Inflation 2.90% 0.13% 2.47% 1.78% 0.03% -0.09 0.65
Table B.1: Descriptive statistics for the six bond yields, the stock return and the inflation.
Appendix C: Willis Towers Watson lifecycles (1-8)
This section shows the lifecycles investment strategies that are used in Chapter 6. These lifecycles are realistic examples and are provided by Willis Towers Watson.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 Wei gh t in mi xin g stra tegy
Years until pension date
Lifecycle 1
Return portfolio Matching long Matching short
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 Wei gh t in mi xin g stra tegy
Years until pension date
Lifecycle 2
Return portfolio Matching long Matching short
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 We ig h t in m ixi n g stra teg y
Years until pension date
Lifecycle 3
Return portfolio Matching long Matching short
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 We ight in mixi n g st rat egy
Years until pension date
Lifecycle 4
Return portfolio Matching long Matching short
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 Wei gh t in mi xin g stra tegy
Years until pension date
Lifecycle 5
Return portfolio Matching long Matching short
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 Wei gh t in mi xin g stra tegy
Years until pension date
Lifecycle 6
Return portfolio Matching long Matching short
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 Wei gh t in mi xin g stra tegy
Years until pension date
Lifecycle 7
Return portfolio Matching long Matching short
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 Wei gh t in mi xin g stra tegy
Years until pension date
Lifecycle 8
