Do barrier options add value? An Asset Liability Management Study for
an Average-‐wage Dutch Defined Benefit Pension Fund
Master Thesis Finance Puck de Ridder (s2034921)
June 24, 2014
Abstract
Pension funds worldwide face great pressure on their funding ratios due to changes in their environment. The financial crisis, which started in 2008, still has a tremendous impact on the portfolios of pension funds. Pension funds can reduce market risk by investing higher percentages in bonds. However, this strategy is far from optimal. The use of put options is already widely implemented, but this strategy comes along with paying high premiums. The question arises whether barrier options can add value to the portfolio of a pension fund, since they are known for having lower premiums in comparison to put options with the same strike price. These lower premiums are caused by the path dependency of barrier options. An asset liability management study for Dutch defined benefit pension funds is described to look at the effects of different portfolios on the funding ratio. With the help of vector auto regression, economic and financial scenarios are constructed to simulate the future. The model shows results that are in contrast with the underlying ideas of the portfolios; the use of options can lead to higher chances of underfunding. Based on the results of our model it is hard to make a definite conclusion on the use of barrier options in the portfolios of pension funds and further research is necessary.
JEL codes: G11, G13
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1. Introduction and background
1.1 Research topic
The use of pension funds is a phenomenon of the 20th century, which started in the United Kingdom, the
United States and The Netherlands (Blake, 1999). Pension funds are created for the purpose of regulating pensions of their participants: employees, sleeping participants and retirees. Employees financially contribute to the fund during their working life in return for benefits when they retire. Martin and Grundy (1987) state their objective as:
“to invest the scheme monies in such a way as to ensure that the scheme will always have
resources to meet its liabilities to pay benefits as and when they fall due at all times in the future; and, in so doing, to take account of the risk factors inherent in any investment situation”.
Pension funds worldwide face difficulties to cope with the great pressure on their funding ratios, which is due to changes in their environment. Increases in life expectancy, changing accounting rules, contribution reductions, low interest rates and very poor equity market returns have led to a steel fall in funding ratios (Capelleveen et al., 2004). Many funds crossed the lower boundary of the minimum funding ratio of 105% required by the Financial Assessment Framework. Recently, many pension funds in The Netherlands had to make a recovery plan to solve their funding ratio problem. Several obvious solutions have been suggested: one is to increase the contributions of the sponsors and the other is to reduce pensions. However, both options can only be used to a certain extent; they raise the costs for the sponsor or reduce benefits for the participants. These negative effects raise the question whether there is another possible solution for underfunding. Capelleveen et al. (2004) already mention the complexity of investment management for pension fund and state that it should be worthwhile to investigate whether pension fund’s health can be improved by changing the way pension funds invest.
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(Hoevenaars et al., 2008). A pension plan manager has to find the right balance between this risk and the potential positive return on investments. Equity returns are known to be more volatile than the returns of bonds. In rising markets the gains will be large, but in bad performing markets the losses will be great as well. Put options could be a solution for the potential large losses; they limit the downside risk, but the upside potential is still available. Put options establish a floor in the equity portfolio. However, important to understand is that this strategy is accompanied with an option premium. This put option strategy seems to work well in a crisis, but standard put options come with high premiums. In stable and rising markets these premiums will probably not payoff because the chance that the options become worthless at the expiration date is greater than in a crisis. Since the market has historically been known to spend years at a time in a stable and rising market, the expensive premiums create a cumulative drag on the overall portfolio value and the funding ratio of the pension fund (Bonsee et al., 2012). Therefore, the question arises whether the strategic asset allocation of a pension fund can be improved by adding barrier options. Barrier options are options with payoffs depending on whether the underlying asset price hits a certain threshold during the maturity of the option. They are attractive to some market participants because they are less expensive than the corresponding regular options (Hull, 2008). To fully understand the benefits and pitfalls of this product for pension funds, it has to be incorporated in an asset and liability management framework.
1.2 Literature review
The main goal of this thesis is to examine whether barrier options can add value in strategic asset allocation of Dutch defined benefit pension funds. An asset liability management (ALM) model is the key tool of this analysis. The main research question is: Can barrier options add value to the strategic asset
allocation of Dutch defined benefit pension funds? The strategic asset allocation problem is approached
with the help of scenarios of the macroeconomic and financial environment. These scenarios, generated using a vector autoregressive (VAR) model, serve as input for the ALM model.
1.2.1 Asset liability management models
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funds. Prominent examples of research on ALM for pension funds are the studies of Boender (1995), Boender et al. (1998) and Ziemba and Mulvey (1998).
The paper of Kingsland (1982) is considered to be one of the classic papers concerning an ALM approach. In this paper Kingsland stated the following:
“The dynamic behaviour of a pension plan is clearly dominated by rules and methodology which are discontinuous and nonlinear functions of its financial condition. The task of developing a closed-‐form solution to evaluate the potential state of a pension plan following a series of stochastic investment and inflation experiences would be extremely difficult, if not possible. To date, the only approach that has proven feasible is the application of Monte Carlo simulation, wherein an investment and inflation scenario is generated by random draws based on the expected probability distribution of year to year investment and inflation behaviour. In order to develop an accurate assessment of the range of potential uncertainties, it is necessary to repeat this simulation process by generating dozens or hundreds possible scenarios, consistent with statistical expectations.”
Since then, science has enormously moved its borders; however, the core of the statement is still valid. Since the mid-‐1980s the use of strategic programming models has been widely advocated to be used by pension funds (Consigli and Dempster, 1998 and Mulvey et al., 2000). Ziemba and Mulvey (1998) provide a good number of applications in ALM, where mostly stochastic programming models are used. The hybrid simulation/optimization scenario model developed by Boender (1997) also still represents the starting point of many ALM studies in The Netherlands.
According to Drijver et al. (2000) the main goal of ALM, in general, is to find acceptable investment and contribution policies that trade-‐off risks and returns such that the solvency position of the fund is sufficient during the planning horizon.
1.2.2 Strategic asset allocation
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variability in returns of a typical fund across time is explained by policy, about 40 percent of the variation of returns among funds is explained by policy and on average about 100 percent of the return level is explained by the part of the total return that comes from the asset allocation policy (active return is explained by the remainder).
The paper of Ibbotson and Kaplan (2000) shows the importance of strategic asset allocation, however, there is still uncertainty about strategic asset allocation. Modern finance theory is often thought to have started with the mean-‐variance analysis of Markowitz (1952). Markowitz showed how investors should pick assets if they care about the mean and variance of portfolio returns over a single period only. Canner et al. (1997) mention the asset allocation puzzle, which discusses the idea that conservative investors when compared to aggressive investors are typically encouraged to hold more bonds, relative to stocks. Campbell and Viceira (2001) argue in their book that optimal portfolios for long-‐term investors need not be the same as for short-‐term investors. Long-‐term investors, who value wealth not for its own sake but for the standard of living that they can support, may judge risks very differently from short-‐term investors.
Financial managers of pension funds still choose to allocate their wealth to traditional asset classes such as stocks and bonds. Recently, other asset classes including commodities, currencies, derivatives, hedge funds, private equity and real estate have gained popularity. Hoevenaars et al. (2008) discusses the alternative asset classes and their value for long-‐term investors. They look at investors that invest in stocks, government bonds, corporate bonds, T-‐bills, listed real estate, commodities and hedge funds. They conclude that these alternative asset classes add value for long-‐term investors, though there is a difference between the strategic portfolio of asset-‐only and asset-‐liability investors.
1.2.3 Models
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(1980) strongly advocates the use of VAR time series to generate the scenarios of the economic environment.
1.3 Thesis outline
In the second chapter a brief introduction on pension funds and ALM is provided. First, the basic idea of pension funds and their core business will be explained. Also the different types of pension funds will be mentioned and the risks these different funds face. Finally, the concept of ALM will be introduced. An introduction on the building blocks of ALM follows and in the end the application of ALM will be discussed. For ALM economic scenarios are generated, so in chapter 3 the idea of economic scenarios will be discussed. Different models can be used to generate these scenarios; in line with Hoevenaars et al. (2003) the VAR model is used in this research. This model will be discussed in the second section of chapter 3. Finally the parameters of this model are estimated in chapter 4.
In chapter 5 the designed model will be used to perform a small ALM study for different pension funds where the scenarios serve as input. The type of pension fund used is introduced and the assumptions made for the model are discussed. The age distribution, age expectancy, status of the participants and wage development will be discussed first. After that the factors that influence the liabilities and the assets of a pension fund will be explained and assumptions are described. The results of the small ALM study follow in chapter 6. In chapter 7, the results are discussed and possible drawbacks of the research methodology are shown. The paper finishes with conclusions and recommendations.
2. Pension funds and ALM
This chapter introduces the idea of pension funds and provides background information on the funds. This introduction will show that funds need to trade-‐off risk and return. ALM is often used to gain insight into this trade-‐off between risk and return. ALM will be discussed in section 2.2.
2.1 Pension funds
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“to invest the scheme monies in such a way as to ensure that the scheme will have resources available to meet its liabilities to pay benefits as and when they fall due at all times in the future; and, in so doing, to take account of the risk factors inherent in any investment situation”.
Retirees receive a pension annuity, which is dependent on the structure of the pension scheme. There are different pension schemes, which are differentiated by the set of rules relating to the benefits of the members of the fund. Three main types are acknowledged and the obligations accumulating under each of these schemes constitute the liabilities of the scheme’s sponsor or manager (Blake, 1999). The first type is the defined contribution (DC) scheme that uses the full value of the fund’s assets to determine the amount of pension. The DC pension depends on the value of this fund only, which might be high or low depending on the success of the fund manager, the investment mix, the results of the financial markets and the interests at the moment of retiring of the participant. This system is well known in the United States. In The Netherlands, the defined benefit (DB) system is most often used. This second type pension fund calculates the benefits in relation to factors such as final or average salary, the length of pensionable service and the age of the member, rather than to the value of the assets in the fund as such. The third system is the targeted money purchase (TMP) scheme, which aims to use a defined contribution scheme to target a particular pension at retirement, but which also benefits from any upside potential in the value of the fund’s assets. The TMP scheme ensures a minimum pension level, but there is not a maximum pension level.
Off course, just as other portfolios, pension funds face many risks. There are risks specifically for pensions funds: risk of death, early retirement and longevity. On the other hand, they face interest risk, credit risk and market risk. These risks are important for pension plans, since pension funds need to fulfill the requirement of their funding ratio. In this way they will be able to keep paying their liabilities in the future. Pension managers have to trade-‐off these risks against returns continuously. ALM considers risks faced by the assets and liabilities and tries to make the best trade-‐off between risks and returns.
2.2 Asset Liability Management
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al. (2000) the main goal of ALM, in general, is to find acceptable investment and contribution policies that trade-‐off risks and returns such that the solvency position of the fund is sufficient during the planning horizon. Swaiger et al. (2010) mention the central decision problem in ALM as the construction of a portfolio of fixed-‐income securities for a pension fund, which takes into account the future outflows (liabilities) of the pension scheme and a set of other constraints. They also determine the optimum trade-‐off between paid premiums and deviations between assets and liabilities. ALM studies are well acknowledged in strategic asset allocation policies for pension funds. Following Leibowitz et al. (1994) ALM in this research is approached from a funding ratio perspective. The funding ratio is the main aspect that pension funds have to manage and control; risk of underfunding is one of the greatest concerns of a pension plan. Funding ratio is measured as the ratio of the value of the assets and liabilities.
As mentioned before, pension funds face many risks. ALM analysis takes these uncertainties into account, consisting of both the future economic and financial environment and the pension liabilities that result from uncertain demographic developments. An ALM model typically consists of a number of building blocks or modules, each related to different aspects of fund management decision: external economy, actuarial liability structure, asset structure, policy instruments and objective function (Capelleveen, 2004). Figure 2.1 shows the formulation of a general ALM problem graphically (Steehouwer, 2005).
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3. Economic scenarios
This chapter start with explaining economic and financial scenarios and how and when these scenarios can be used. In section 3.2 the VAR model is discussed. This model is used to generate the economic and financial scenarios.
3.1 Economic and financial scenarios
Macroeconomic and financial variables from the ALM problem are the most important risk and return factors for a pension fund. The risk drivers are modelled in scenarios and serve as input for our ALM model. Brauers and Weber (1988) define a scenario as:
“a future environment, considering possible development of relevant interdependent factors of the environment”.
Instead of focus on a single future development, a large number of scenarios of economic and financial variables are generated. A large set of scenarios is assumed to be a reasonable representation of the uncertain future; the model assumption is made that one of these paths will materialize. Uncertainty is still preserved in that the decision maker does not know which scenario describes the true future state of the world (Dert, 1995). Generating economic scenarios is also called scenario analysis, stochastic simulation or Monte Carlo simulation. Scenario analysis is not using extrapolation of the past and does not expect past observations to be still valid in the future. It tries to consider possible developments and turning points, which may be connected to the past. The interrelations between the variables are taken into account as well.
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The relation between scenario generation and ALM modelling is well described in figure 3.2 (Dert, 1995). Economic scenarios together with an investment policy are fed into a model that states all relations between the policy, scenarios and the relevant output (funding ratio in our case). Along the lines in the diagram, ALM and scenario analysis enable decision makers to evaluate and compare the risk and return consequences of different policies. Thereby they can arrive at both more efficient and more effective strategic policies.
Gallo (2009) mentions two reasons why scenario analysis is often preferred over alternative approaches. First, scenario analysis offers the flexibility to model complex interactions and relations within and between the parameters of an ALM problem. The second reason for the popularity of scenarios is that it offers great possibilities for learning about the problem under investigation.
3.2 VAR
There are several ways to generate future scenarios. Hoevenaars et al. (2003) mention independent drawings from a normal distribution, VAR model, cascade approach, stochastic differential equation approach and risk-‐neutral simulation. All these approaches aim to explain the most important challenges in macroeconomics and financial markets, in example they try to explain relations between money, interest rates, prices and output. Traditionally, these challenges have been solved using structural models that imposed a priori restrictions on the intercorrelations of the data. In the 1980s, Sims (1980) made a new approach popular, the VAR models. VAR models in economics are used as a forecasting method using historical data. Hoevenaars et al. (2003) describe the idea of using VAR for scenario generation as that a draw is done from the probability distribution of the error terms given the economic
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variables of the last period, such that the historical correlations are taken into account and the value of the economic variables for the next period are computed. The model is one of the most successful, flexible and easy to use models for the analysis of multivariate time series. It has proven to be specifically useful for describing the dynamic behaviour of economic and financial time series and for forecasting. Forecasts from VAR models are quite flexible because they can be made conditional on the potential future paths of specified variables in the model.
The variables are modelled together in a multi-‐equation time series model, a VAR model in the chosen methodology. The model includes autocorrelation (correlation of a variable through time) of variables and cross correlation (correlation between variables). Additionally, correlation between variables is also based on the state of the economy, in example the macroeconomic and financial variables that are included in the model (Hoevenaars et al., 2003). Describing the joint behaviour of the yield curve and macroeconomic and financial variables is important for bonds pricing, investment decisions, public policy and liabilities.
4. The model for the ALM problem
In this chapter the parameters of the designed model will be estimated. In the first section, Hoevenaars (2008) is used as guideline for building an arbitrage-‐free VAR model. It is not realistic to have arbitrage opportunities in the model, since they give investors the opportunity for a so-‐called ‘free lunch’. A well-‐ known example to solve the issue of arbitrage opportunities is to model the term structure of interest rate with a stochastic discount factor (SDF), also called a deflator or pricing kernel. In the chosen methodology, with generating individual scenarios, the issue of arbitrage opportunities is not relevant since the model does not make use of an optimization strategy that takes advantage of arbitrage opportunities. In section 4.2, the simulation method of the VAR is described. The results are shown in section 4.3. The last section describes the method used to value the derivatives in the portfolios.
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4.1 VAR (1) modelFollowing Campbell and Viceira (2005), the return dynamics are described by a first order VAR model. We use this simple, flexible statistical model to describe the dynamic behaviour of asset returns and macroeconomic variables. The simple structure of a VAR model of order one allows for a straightforward interpretation of the model parameters. The model starts with a set asset classes that can enter the portfolio and of which returns are modelled. It adds a set of variables that are relevant for the simulation of those returns because of the underlying relations. These variables are most often referred to as state variables.
The first order VAR for monthly data is described as
𝑧!!!= 𝑣 + 𝐵𝑧!+ 𝑢! (1)
where 𝑣 is a (nx1) vector of the constant terms and B is a (nxn) vector containing the VAR coefficients. 𝑧! is a (nx1) vector of n state variables. The designed model includes sixteen state variables: interest swap rates with maturities of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20 and 30 years, the MSCI return, the corresponding dividend yields and inflation. These state variables are somewhat different than other models in the literature use. Campbell and Viceira (2005) use four different predictive variables: the nominal 3-‐months interest rate, the dividend-‐price ratio, the term spread and the credit spread. 𝑢! is a (nx1) vector containing the error terms of the regression equations.
4.2 Simulation
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4.3 Data and estimation results
For the ALM study monthly data from January 1999 to December 2013 are used. European interest swap rates with maturities of 1 to 10 years, 15 years, 20 years and 30 years are used. Price inflation (not seasonally adjusted) of Germany is used for the representation of price inflation and Morgan Stanley Capital International (MSCI) world total return index, with the accompanying dividend yields reflect returns and dividend yields of stocks. All data are retrieved from Datastream1. The MSCI world index is
used as stock portfolio since it reflects international investment opportunities. In figure 4.1 the plots of the interest rates and the dividend yield are given.
The figures show that the swap rates are highly correlated and the long-‐term rates are in general higher than the short-‐term rates. The monthly inflation and returns are plotted in figure 4.2. The data satisfy the stationarity condition, in example all eigenvalues of matrix B are smaller than 1. In table 4.1 the descriptive statistics and the eigenvalues of the variables are given.
Variable μ σ Eigenvalue
1 year swap rate 0.224 0.116 0.979764 2 year swap rate 0.238 0.113 0.956777 3 year swap rate 0.253 0.110 0.956777 4 year swap rate 0.267 0.107 0.923559 5 year swap rate 0.280 0.103 0.830596 6 year swap rate 0.292 0.100 0.692410 7 year swap rate 0.302 0.100 0.692410 8 year swap rate 0.311 0.095 0.562512 9 year swap rate 0.312 0.094 0.461694 10 year swap rate 0.325 0.092 0.355323 15 year swap rate 0.347 0.087 0.353232 20 year swap rate 0.355 0.087 0.275632 30 year swap rate 0.355 0.090 0.266221 Inflation 0.134 0.320 0.114044 MSCI Return 0.494 4.651 0.114044 Dividend Yield 0.186 0.046 0.015383
Table 4.1 Descriptive statistics where μ (in percentage) is the mean on monthly basis and σ (in percentage) the standard deviation of the sample. The last column represents the eigenvalues of the variables.
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Figure 4.1 Monthly values of (a) swap rates and (b) monthly dividend yield of the MSCI world total return index.
.000 .001 .002 .003 .004 .005 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13
1 year swap rate 2 year swap rate 3 year swap rate 4 year swap rate 5 year swap rate 6 year swap rate 7 year swap rate 8 year swap rate 9 year swap rate 10 year swap rate
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Figure 4.2 Values of (a) monthly price inflation and (b) the monthly returns on the MSCI.
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The coefficients, which are used to simulate future values, are different than the VAR estimates. Multiple regressions are used to get regression equations with significant coefficients (at the 5% significance level) only. In table 4.2 the regression equations are given. The p-‐values of the coefficients are shown between brackets. According to the VAR and the following regression analyses applied, almost all swap rates are predicted by the return of the MSCI index. The 6 and 9 years swap rates, as well as the dividend yields are often predictors of future values of the swap rates. Inflation is predicted by its own lag-‐value and a constant. The MSCI world index returns, does not have a significant predictor and is thus calculated as a random draw from the error distribution. The last variable included in the VAR, the dividend yield, is predicted by the returns of MSCI world index, its own lag-‐value and a constant. The simulation for the MSCI return is adjusted since the random draw only is in conflict with the average long-‐term return of approximately 6% per year. Therefore, the mean of the sample (0.494% on monthly basis) is added to the scenarios of the MSCI return. The simulation of the one-‐year swap yield also leads to some serious problems, over time it will gradually decrease to around -‐0.2 per month. To overcome this problem we simulated the one-‐year swap rate by deducting the difference of the means of the one-‐year and the two-‐ years swap rates from the simulated two-‐years swap rates. Since the swap rates are highly correlated we prefer this method above adjusting the regression equation from the VAR model. The adjustments take place after all the other scenarios are calculated to prevent that these other scenarios are influenced by
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Regression equations from OLS
y1t+1 1.015𝑦! ! 0.0000 − 0.057𝑦 !! 0.0362 + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛! 0.0000 − 0.177𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑! 0.0000 + 0.000(0.0000) + 𝜀! y2 t+1 2.524𝑦𝑡9(0.0000) − 1.663𝑦𝑡30(0.0000) − 0.537𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑𝑡(0.0000) + 0.001(0.0003) + 𝜀𝑡 y3 t+1 2.249𝑦𝑡 9(0.0000) − 1.316𝑦 𝑡 30(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛 𝑡(0.0195) − 0.412𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑𝑡(0.0000) + 0.001 + 𝜀𝑡 y4 t+1 2.170𝑦𝑡 6(0.0000) − 1.236𝑦 𝑡 9(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛 𝑡(0.0005) − 0.164𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑𝑡(0.0000) + 0.001(0.0001) + 𝜀𝑡 y5 t+1 1.598𝑦𝑡 6 0.0000 − 0.648𝑦 𝑡 9 0.0000 + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛 𝑡 0.0004 − 0.138𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑𝑡(0.0001) + 0.000(0.0009) + 𝜀𝑡 y6 t+1 1.769𝑦𝑡 9(0.0000) − 0.813𝑦 𝑡 20(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛 𝑡(0.0008) − 0.156𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑𝑡(0.0000) + 0.000(0.0027) + 𝜀𝑡 y7 t+1 0.941𝑦!!(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛!(0.0005) − 0.135𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑!(0.0000) + 0.001(0.0000) + 𝜀! y8 t+1 0.912𝑦! !(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛 !(0.0005) − 0.157𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑!(0.0000) + 0.001(0.000) + 𝜀! y9 t+1 −0.002𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛!(0.0323) − 1.350𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑!(0.0000) + 0.006(0.000) + 𝜀! y10 t+1 0.477𝑦𝑡6(0.0000) + 0.518𝑦𝑡20(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛𝑡(0.0067) + 𝜀𝑡 y15 t+1 0.139𝑦𝑡6(0.0000) + 0.858𝑦𝑡20(0.000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛𝑡 (0.0178) + 𝜀𝑡 y20 t+1 0.997𝑦! !"(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛 ! (0.0046) + 𝜀! y30 t+1 0.301𝑦𝑡6(0.0000) − 2.564𝑦𝑡15(0.0000) + 3.250𝑦𝑡20(0.0000) + 0.001𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛𝑡(0.0032) + 𝜀𝑡 Inflation −0.331𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛! + 0.002(0.000) + 𝜀! MSCI return 𝜀! Dividend yield −0.002𝑀𝑆𝐶𝐼 𝑟𝑒𝑡𝑢𝑟𝑛!(0.0000) + 0.987𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑!(0.0000) + 0.000(0.0032) + 𝜀! Table 4.2 Regression equations for the parameters of the VAR model. All coefficients are significant at the 5% level. The p-‐values of the coefficients are shown between brackets.
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4.4 Black-‐Scholes formulaThe prices of derivatives cannot be generated using the VAR model, but prices can be calculated with the use of the VAR output. The relation between the variables is specified by the pricing formulas and the scenarios serve as input for the formulas. The value of put options will be calculated using the well-‐ known Black-‐Scholes formula explained in Hull (2008)
𝑝 = 𝐾𝑒!!"𝑁 −𝑑 ! − 𝑆!𝑒!!"𝑁 −𝑑! (2) where d!= !" !! ! ! !!!!! ! ! ! ! ! (3) and d!= !" !! ! ! !!!!!!! ! ! ! = d!− σ T (4) The formulas describe the value of a put option, with strike price K, risk-‐free rate r, current price of the underlying stock S0, dividend yield q, time to maturity T and volatility σ.
Barrier options are options with payoffs depending on whether the underlying asset price hits a certain threshold during the maturity of the option and cannot be priced using the ‘standard’ Black-‐Scholes formula since path dependency is involved. Hull (2008) describes different types of barrier options that regularly trade in the over-‐the-‐counter market and they can be classified as either knock-‐out options or knock-‐in options. A knock-‐out option ceases to exist when the underlying asset price hits the threshold; a knock-‐in option comes into existence only when the underlying assets price hits the threshold. We use the down-‐and-‐in put option in our investment portfolio. This barrier option has the same pay-‐off as a standard put option when the barrier has been hit and otherwise it is worthless at the expiration date. When the barrier is greater than the strike price, the down-‐and-‐in put has the same value as a normal put option. When the barrier is less than the strike price, the formulas described in the book of Hull (2008) are used to value the barrier options.
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The Black-‐Sholes formula, described in Hull (2008), for a down-‐and-‐in put with a barrier level that is lower than the strike price is
p!" = −S!N −x! e!!"+ Ke!!"N −x!+ σ T + S!e!!" H S! !" N y − N y! − Ke!!" H S! !"!! [N y − σ T − N y!− σ T ] (5) where λ =!!!!! ! ! !! , (6) y =!" (! ! !!!) ! ! + λσ T, (7) x!=!" (! ! !) ! ! + λσ T (8) and y!=!" (! !! ) ! ! + λσ T (9) A new variable comes along in these formulas, H, which is the barrier the stock has to reach before the option comes into existence. The formula described above can be used, when the underlying is monitored continuously. In the designed model the stock price is monitored only once a month, so the formulas need to be adjusted for discrete monitoring. Broadie et al. (1997) came up with a solution, which will be applied in the model. In their adjusted formula, they replace the barrier level by 𝐻𝑒!!.!"#$! !/!, where m is the number of times the asset price is observed.
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5. Pension fund ALM study
In this chapter, simulations of stock returns, bond returns, inflation and interest rates serve as input for an ALM study of an average Dutch defined benefit pension fund. As explained in chapter 2, there are many forms of pension funds, but this research focusses on the defined benefit structure since almost 90% of the pension funds in The Netherlands use this structure2. In section 5.1 the pension fund, which is
used for this research, is described. Sections 5.2 and 5.3 discuss the made assumptions for the ALM model.
In line with Leibowitz et al. (1994), the ALM problem is approached from a funding ratio perspective. The funding ratio is the main aspect that pension funds have to manage and control; risk of underfunding is one of the greatest concerns of a pension plan. The funding ratio is measured as the ratio of the value of the assets and the liabilities.
The planning horizon of most pension funds stretches out for decades; as a result of the long-‐term commitment to pay benefits to the retirees (Kouwenberg, 2001). For the purpose of the ALM study, the planning horizon is split into sub periods of one month. The sub periods of one month are important for the pricing formulas of the barrier options. These normal formulas are adjusted for the discrete observation of the underlying.
5.1 Average-‐wage Dutch defined benefit pension fund
For the ALM framework, an average-‐wage Dutch defined benefit pension fund is modelled, which aims for full indexation of the pension rights. Final pension benefits depend on the average wage that is earned during the participant’s career and is built up out of 2.05%3 of the participant’s pensionable wage
for each year of service. So when someone retires at the age of 65 and this person started working at 25, this person has built up pension rights of 82% (40x2.05%) of his/her average salary. In 2015 the maximal contribution rate will decrease to 1.875%. The rationale behind this decrease is that people have a longer period to build up their pension because the retirement age has increased by two years (from 65 to 67 years). The data of participants is collected from De Nederlandse Bank4. The data used are from
pension funds related to companies in the same branch of industry. Of all pension funds, around 80% are included in this type of pension funds.
2 This information is gathered from the Pensioenthermometer. On the website of the Pensioenthermometer,
www.pensioenthermometer.nl, Aon Hewitt tracks the funding ratio of the average Dutch pension fund.
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5.1.1. Age distribution and status of the participants
Data from De Nederlandse Bank for the age distribution and status of the participants are available for the year 2012. For the purpose of this research it is assumed that no significant changes have taken place since then and therefore the data of 2012 is used for the age distribution and participant status. The participant status can be split into three different categories: active, sleeping and retired participants. Figure 5.1 shows the distribution of the participants. It is remarkable that there are retired participants before the pensionable age of 65. It is possible that these participants receive disability support pension or partner’s pension, since these payments fall in the same category as normal pension payments. Around the age of 65, a shift can be seen from active and sleeping participants to retired participants. This shift is quite logic, since the legal pension age was 65 for a long term. In The Netherlands a shift is taking place for the legal pensionable age. The pensionable age will incrementally increase to 66 years in 2019 and to 67 years in 2023. For the calculations the ‘old’ pensionable age of 65 years is used. The life expectancy for the individuals from Het Centraal Bureau voor de Statistiek5 is used for the remaining life
expectancy of the participants, which is necessary for calculating the liabilities. For simplicity, the data are rounded to the nearest integer.
5 StatLine is the electronic Databank of Het Centraal Bureau voor de Statistiek.
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For the purpose of calculating the liabilities some assumptions are made. For the age category of <20 and 20-‐25, it is assumed that the individuals have been working since their 18th birthday and for the
other age categories it is assumed that the individuals have been working from the age of 236. For the
pension age, it is assumed that everybody that has not reach the age of 65 yet, will retire at the age of 65. The transitional arrangements are not taken into account in these calculations. For everybody that already passed that age and is not retired yet, it is assumed that they will retire directly, so at their current age. For calculating the liabilities of the sleeping participants, the age group is taken where the median of the sleeping individuals is included as the age that they will change from active to sleeping participants. For the retirees in the pension fund, the same assumptions are used as for the active participants, in example the employees of the first two age groups started working when they were 18 and the other age groups started working at the age of 23. The assumption is made that the people who are younger than 65 retired at their current age and whoever is older than 65 retired at the age of 65. Figure 5.3 shows the total liabilities of the fund. The zigzag pattern is probably the result of the simplified assumptions.
6 The ages of 18 and 23 for starting working are based on the distribution of the participants of the pension fund
and that the most common age where you can start building up your pension. Legally this age is 21 years old, but in many pension funds you can participate at a younger age. Since 21 years falls in the age category 20-‐25 years, we took the mean (rounded to the nearest integer) of this age group as starting age.
0 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100 Male Female R e m a in in g l if e e xp e ct a n cy Age
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0 20,000 40,000 60,000 80,000 100,000 5 10 15 20 25 30 35 40 45 50 55 60 65 L ia b ili ti e s (i n m ln € )Years from now
5.1.2. Wage
During the career of individuals their wage will be different, depending on the job and due to compensation for price inflation. For the wage of employees the same average wage is used for everyone in the fund, being the median family income in The Netherlands for 2014 from Het Centraal
Planbureau7. For 2014 the value of the median family income is estimated at €34,500 per year.
5.2 Liabilities
Economic variables and actuarial predications drive the liability side, whereas economics variables and sentiment drive financial markets and security prices (Ziemba, 2003). Based on Hoevenaars (2008), this research focuses on three factors that change the liabilities each year: actuarial factors, interest rates and inflation.
5.2.1 Actuarial factors
The main concern with actuarial factors is the level and length of nominal future cash flows, which are earned by the employees. The evolution of the liabilities is influenced by mortality, hiring and firing decisions and disability. To generate scenarios for the uncertain development of the liabilities and the benefit payments, future values of the earned rights should be determined. An important first step is to
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