The Impact of Parameter Variation in the ALM framework:
Empirical research on the Dutch pension industryJanuary 2013
Master Thesis Actuarial Science and Mathematical Finance
Kees Vermunt Student nr.: 5813611
University of Amsterdam
Supervisor: Prof.dr. R. Kaas Second assessor: Dr. L.J. van Gastel
Advisor: J.A. van Oord MSc
ABSTRACT
This thesis examines the impact of the parameter variation in the ALM framework. By studying the largest 32 Dutch pension funds, I found substantial variation in the economic parameters (e.g. expected asset returns and interest rates) used for their ALM studies. Subsequently, I simulated 15 year projections of the funding ratio and pension result using these parameters. I found a large dispersion in the output solely based on the difference in economic parameters. If these pension funds would optimize their asset allocation using these parameters, the optimal allocation to equity varies between 30 and 70%. These findings show that the outcome of ALM studies depend heavily on their subjective economic parameters.
CONTENTS PREFACE ... 3 1 INTRODUCTION ... 4 2 ALM FRAMEWORK ... 5 3 METHODOLOGY ... 6 ALM Model ... 6 3.1 Parameter variation ... 9 3.2 Continuity analysis ... 10
3.3 Optimal asset allocation ... 11
3.4 Control groups ... 12
4 RESULTS ... 13
4.1 Variation in economic parameters ... 13
4.2 Impact of variation in parameters on continuity analysis ... 16
4.3 Impact of variation in parameters on optimal asset allocation ... 19
4.4 Impact on grey and green fund ... 22
5 CONCLUSIONS ... 25
6 BIBLIOGRAPHY ... 27
7 APPENDICES ... 29
PREFACE
The writing of a thesis is impossible without the help of others. First of all, I would like to thank De Nederlandsche Bank for giving me the opportunity to use all resources I needed for my research and for allowing my supervisors at the Risk & ALM department to spend a considerable amount of their time on my thesis. My weekly meetings with Arco van Oord helped me in finding the right direction for my research as well as in polishing the end product. With the support of Henk Jan van Well, Lisanne Cock and Joost Duisters, I was able to construct the tool needed to perform the required computations. Discussions with other colleagues at Risk & ALM and Actuariaat & Risicobeheer provided me with very valuable additional insight in the world of retirement plans and asset management. I would like to thank my lecturers at the University of Amsterdam for preparing me for my research. I would like to give a special thanks to dr. Tamerus, who encouraged me in his lectures to study the (lack of) usefulness of ALM-studies. I am grateful for professor Kaas’ support of my decision to work on my thesis independently, and for his help at the finalization stage of the thesis. Finally, I would like to thank my family and friends for being the best distractors from my thesis writing that I could imagine.
Disclaimer
Views expressed in this thesis are my own and do not necessarily reflect the opinion of De Nederlandsche Bank.
1 INTRODUCTION
Asset-Liability Management (ALM) is an approach of managing risks on the balance sheet in an integrated way, so that the effects on assets, liabilities and the combination of both are taken into account. ALM studies always rely on various statistical and economic assumptions. The standard statistical modeling assumptions are among others the assumption of log-normality of asset returns and the assumptions on volatility conditions and on the correlations between the key variables in the model. Even more important are the economic assumptions, which are typically based on historical data or “expert” judgment. The future projections of, for example, the funding ratio are strongly dependent on the values used for economic parameters in the ALM model. In fact, the outcome of an ALM study may be more strongly affected by the economic parameter input than by the policy input to be evaluated.
Whereas there exists a substantial body of research on the effect of policy input on ALM outcomes, research on the effect of economic parameters input is lacking. This thesis focuses on the impact of economic parameters, e.g. interest rates, asset returns, and inflation. The main questions to be answered are:
1. What is the variation in parameters used by pension funds for their ALM studies? 2. What is the impact of parameter variation on the outcome of continuity analysis? 3. What is the impact of parameter variation on the optimal asset allocation?
The effect of parameter variation is studied with data from continuity analyses of the 32 largest funds by assets. I study the impact of variation in various types of parameters by adjusting these parameters in a standard ALM model. Subsequently, the effect on key output measures, such as projection of the funding ratio, pension result and optimal asset allocation, is calculated. The end goal is to check to what extent the output depends on the specified values for the economic input parameters.
The remaininder of the thesis is structured as follows. Chapter 2 discusses the relevant theoretical background on ALM. Chapter 3 presents the methodology used in this study and defines the mechanics of the ALM model that is used. Chapter 4 covers results and associated analyses. The last chapter presents the main conclusions and provides directions for future research.
2 ALM FRAMEWORK
This chapter describes the theoretical and historical background of ALM studies in the pension field and the role of the continuity analysis within the regulatory process. Luckner et al. (2003) wrote a paper on the history of asset-liability management for the Society of Actuaries. The volatile interest rates seen in the 1970’s were the first motivation for the concept of ALM. Due to a difference in duration, assets and liabilities respond differently to a change in the interest rates. This made the own funds of financial institutions exposed to interest rate risk. ALM is a process connecting asset management and liability management into one integrated system. The goal is to match assets and liabilities and to hedge, mitigate and prevent internal and external shocks on the balance sheet. The new development in ALM is measuring and dealing with liquidity and funding risk.
A pension fund uses ALM to measure, mitigate and control risks on the balance sheet. A pension fund is exposed to inflation and interest rate risk in the ALM framework. Inflation risk is due to the embedded options in the form of conditional guarantees. Interest rate risk is due to the duration gap between the very long-term liabilities and relatively shorter term assets. A decrease in interest rates causes a decrease in own funds, if this risk is not properly hedged. Other market risks like currency, equity and credit risk are also taken into account.
The ALM-studies help pension funds gain insight in the development of funding ratio and pension result given the policy instruments chosen. Policy instruments contain benefit policy, premium policy and investment policy. The fund can measure the effect of a change in policies on the projected outcomes. Another use of an ALM-study is in studying different economic scenarios and comparing the output of a policy combination over the different scenarios. These are two examples of useful applications of ALM-studies:
• A pension fund measures the effect of an increase in the asset allocation of equity investments and changing indexation policy from price inflation to loan inflation. Useful output could be median pension result or 2.5% percentile in funding ratio. • Compare a contract with nominal guaranteed benefits to a contract without
nominal guarantees over different extreme scenarios. Run fifteen years of projection for both policies over scenarios of deflation, inflation and prosperity. Compare the estimated probabilities of underfunding, pension result and median funding ratio to find the optimal asset portfolio.
3 METHODOLOGY ALM Model
This section describes the mechanics of the ALM model. A base model built and described by H.J. van Well (2011) was used for the ALM analysis. Additional adjustments were made to the model to meet the demands for this research. The following subsections describe how each part of the model functions.
Liabilities
For each projected year, all future cash flows are separately calculated for every type of benefit, age, and gender. All the cash flows are summed over each time period and discounted with the simulated zero curves. To calculate the cost effective premium, the new accrued rights over the following year to come are used. For the spousal pensions, males are 3 years older than females as suggested in Promislow (2011). The duration of the technical provision decreases from 21.5 to 18 over the fifteen year simulation. See Appendix B for a graphic view of the expected cash flows.
The premium consists of two components: market value of newly accrued pension right, and a charge for holding the regulatory requisite capital. The regulatory capital is calculated using the square root formula as described in the Financial Assessment Framework1. The cost-effective premiums are determined using the annual accrual of 2.25% and a franchise of €13,000.
The simulations are based on an average pay plan. Benefits cover of a funded old age pension and a risk basis spouse pension. Salaries increase every year by the wage inflation of the previous year and the franchise rises every year by the price inflation. Pension benefits and accrued rights are cut, if the funding ratio stays below the regulatory absolute minimum of 105% for three years. The pension rights are cut off so that the funding ratio is back at 105% in the fourth year. Accrued rights and benefits are subject to compensation for inflation by a mechanism based on the funding ratio as shown in Appendix C. If the funding ratio increases above 160%, extra indexation is distributed to compensate for the missed indexation in previous years. Pension result cannot increase above 100%. Administrative and other costs are zero, because costs are negligible compared to the technical provision.
Pension fund population
The fund demographics were chosen so that the influence on the output of the change in demographics is minimal. The average age of the policy holder has increased from 43 years to 48 years over the fifteen year simulation. Table 1 shows the funds population broken down by gender and type of policy.
1 Pensioenwet, chapter 6, article 125a-150, BWBR0020809
6
Table 1. Fund Demographics
Categories Male Female Total
Active 867 564 1,431
Deferred 370 241 611
Retired 120 75 195
Beneficiaries 35 35 70
All 1,392 915 2,307
Outflow is caused by withdrawal from the fund and by mortality. For every outflowing member, a new younger inflowing member is generated. This is modelled, in such a way that the change in average age over 15 years in minimized. Mortality is modeled with a deterministic generational mortality table published by the Actuarial Society of the Netherlands, which uses a longevity trend.
Interest rates
The nominal interest rates are simulated by the no-arbitrage method in line with a Hull-White model. Interest rates cannot be simulated by random drawings like stock returns only, because the interest rates depend on the rates in the previous year. The Hull-White model needs to be calibrated for two parameters. The mean-reversion parameter is set at 0.01 and the short-rate volatility at 0.002. The interest rate curve for the input is shown in Appendix D. Interest rate risk is not hedged, which is a big assumption. Short rates are simulated as follows:
𝑟(𝑡) = 𝛼µ + (1 − 𝛼)𝑟(𝑡 − 1) + 𝜃(𝑡) + 𝜎𝑟𝜀𝑟 (1) Where:
α : mean reversion parameter μ : long term interest rate
θ(t) : time dependent drift parameter 𝜎𝑟 : short-rate volatility
𝜀𝑟(t) : ~N(0,1), error term Inflation
Inflation is modelled with a simple Brownian Motion Process. This process creates random drawings from a normal distribution with a mean of 2% and an annual volatility of 1%. There are more precise models available to simulate inflation more realistically.
Equity and Real Estate
The ALM model uses three asset classes: equity, real estate and fixed income investments. Equity and real estate are modeled using a geometric Brownian motion
model. All the assets are rebalanced every year according to the preferred asset allocation. Yearly rates for returns on equity and real estate, wage inflation and price inflation are simulated by discrete geometric Brownian motions with a one year step size. The first step is to generate a random vector of standard normal distributed error terms. The second step is to multiply this vector by the Cholesky decomposition of the correlation matrix. Finally, this random vector is multiplied by the volatilities and then the mean is added.
Fixed Income
The bonds have a fixed duration and they are reinvested every year in a new risk-free bond with the same duration. Besides the return on the risk-free rate, a return on the credit spread is modeled. This spread return (S) is modeled with a jump-diffusion model (2) as first described by Merton (1976). This model is similar to Brownian motions, but with a probabily (κ=0.075) an extreme market shock occurs. This extreme market shock (jump) is normally distibuted with a mean of -0.4% and a volatility of 8.0%. This model fits better to the data, because it creates fatter tails and skewness in the random scenarios, just like found in the historical data. We also found mean reversion in historical spread returns, but since there is no economic relevance to this statisical property, this was left out of the model.
𝑆 = 𝑆𝑑𝑖𝑓𝑓𝑢𝑠𝑒 + 𝐼0,1∗ 𝑆𝑗𝑢𝑚𝑝 (2) Where: 𝑆𝑑𝑖𝑓𝑓𝑢𝑠𝑒 : ~N(0.8% , 3%) 𝐼0,1 : 1 with probability 0.075 : 0 with probability 0.925 𝑆𝑗𝑢𝑚𝑝 : ~N(-0.4% , 8%) Simulation runs
Due to the drawing of random vectors, an varation arises in the output. For example, the estimated median funding ratio is a function of random vectors for the shocks in inflation, interest rates and asset returns in each period for every run. Therefore, the number of runs (n) is determined, that minimizes the error sufficiently. It is evident that a more stable result will be achieved with an increasing number of runs. The 2.5% interval consists only of 1
40𝑛 observations. On the other hand ,one study takes 1
4𝑛 seconds, as where one
study examines one parameter set and one asset allocation. Table 2 shows the standard deviations of ten studies with the same parameters. With 2,500 runs per study, the error is sufficiently small compared to the extra time consumption. Therefore, this research chooses to run 2,500 simulations per study. This means the 2.5% interval has ±62 observations and one study takes approximately ten minutes. In order to eliminate the some of simulation error, the drawings were saved and used again.
Table 2. Error in funding ratio output (SD of 10 studies with the same parameters)
Sample size 2.5% interval Median
5 9.2% 7.7% 10 5.8% 3.9% 25 3.8% 3.7% 100 3.0% 2.2% 250 1.8% 1.4% 1000 1.3% 1.1% 2500 0.8% 0.5% 5000 0.4% 0.3% 3.1 Parameter variation
Pension funds have to report a continuity analysis of their fund every five years. In case of substantial policy changes or changes in the economic health of the fund, the regulator requires more frequent reporting. These reports contain information on the funds’ prediction expectations, standard deviations and correlations of economic parameters. The data on the variation of parameters used by pension funds for their continuity analysis will be examined. The parameters for asset return expectations are bounded by the regulatory maximum values as described in Appendix A. Funds are allowed to exceed the boundaries if they can give sufficient argumentation.
Section 4.1 describes a category of other investments. This asset class consists of alternative investments such as private equity, hedge funds and infrastructure. These assets were pooled in the results because all funds have distinguished these categories differently in their analysis. Pooling these assets classes creates more uniformity. The return and variance on other investments are calculated as a weighted average:
𝑅𝑜𝑡ℎ𝑒𝑟 = ∑ 𝑤𝑖 𝑖𝑅𝑖 (3) 𝜎𝑜𝑡ℎ𝑒𝑟2 = ∑ 𝑤𝑖 𝑖2𝜎𝑖2+∑ ∑ 𝑤𝑖 𝑗 𝑖𝑤𝑗𝜌𝑖𝑗𝜎𝑖𝜎𝑗 (4) Where: σ : volatility w : weight ρ : correlation
This asset class was not taken into account for the second and third part of the results since it consists of asset classes with different behavior in ALM studies. For example, hedge funds, infrastructure and private equity have very different return distributions.
3.2 Continuity analysis
This thesis focuses on a simulation exercise with a fifteen year projection. In most calculations, the pension policy is fixed and given. The difference between this thesis and scenario analysis is that the scenarios in this thesis are not extreme. It is a subjective matter, to determine what a “normal” economic scenario is. This thesis does not answer the question on what a “normal or average” economic scenario is. It will only look at the variation in what the pension industry sees as “normal” and the effects on the outcome of an ALM study.
By law, the Dutch pension funds are required to submit a standardized ALM-study report called a continuity analysis to the Dutch Central Bank every five years. This analysis shows the expected development of a fund regarding their assets and liabilities given their policy choices and prevailing economic scenarios. There are restrictions on the parameters that the fund can use. For the expected return, the parameter values are restricted by the maximum values, and the values for inflation are restricted by the minimum values. The exact values can be found in Appendix A. The goal of a continuity analysis is to check whether the fund is able to meet its goals in terms of expectation in a “normal” economic scenario. A static asset allocation of 40% bonds, 10% real estate and 50% equity was used for this study. These assets make up most of the asset portfolio of an average fund.The starting funding ratio is 100%. In the standard situation, the returns and volatilities are listed in Table 3.
Table 3. Annual returns and volatilities
Return Volatility Equity 7% 18% Real Estate 6% 11% Price Inflation 2% 1% Wage Inflation 3% 1% Credit Spread 0% 3%
To calculate the impact of parameter variation, a continuity analysis is conducted several times with a changed parameter. Impact is measured as change in the pension result and funding ratio after fifteen years. Pension result is defined as realized indexation divided by realized inflation. This output measures the quality of the pension from a policy holder’s point of view. The funding ratio measures the solvency of the fund and is a quality measure from the fund’s point of view. For these output results, the median result and 2.5%-percentile are being examined. These results provide a measure for return and risk when they are compared over different parameter sets. This thesis looks at the effect of a change in return and risk parameters for equity and fixed income. The next step is to compare the output of the 32 largest Dutch pension funds given their parameters. The same asset mix, pension policy and random drawings as in the previous calculations are
used. The only difference between the simulations is the return and volatility of the equity, fixed income, and real estate investments. The parameters used for each fund are the parameters found in the fund’s own continuity analysis reports.
3.3 Optimal asset allocation
ALM serves as an instrument in finding the optimal asset allocation. Campbell and Viceira (2003) used vector auto-regression to determine the dynamics of the returns of several assets. Subsequently, they determine an optimal portfolio for various types of investors. They rely solely on historical data to model the future return dynamics of assets. First, they found the hedge probabilities for inflation and liabilities in some assets. Second, they found the mean reverting properties in stocks and bonds return which gave time diversifying benefits for long-term investors. Hoevenaars et al. (2013) added the parameter uncertainty to the model. The benefit for time diversification disappears over long-term investment horizons. Due to the informative priors, the weight of short-term investments increases and equity decreases in the optimal portfolio.
To determine the optimal asset allocation (AA), the ALM model is run for different asset mixes. In order to obtain these results, a goal function is necessary. This is a function of return, risk and a risk aversion parameter. Siegman (2011) describes several utility functions to weigh return and risk given a risk aversion parameter. This thesis defines return as the median funding ratio (FR) multiplied by the median pension result (PR), and risk as the reciprocal of the product of the 2.5%-percentile of the FR and PR. The goal function is defined as follows:
𝑈(𝐴𝐴) = 𝐹𝑅𝑚𝑒𝑑𝑃𝑅𝑚𝑒𝑑− 𝜆 ∗ (𝐹𝑅2.5%𝑃𝑅2.5%)−1 (5) Where:
λ : risk aversion parameter
This function is not perfectly convex, but I chose this utility function, because of its intuitive easy understandability. The utility function is maximized over different asset allocations to find the optimal asset mix for a certain parameter set to demonstrate the dependency of the utility function upon its parameters. Secondly, the optimal asset mix is determined for every parameter set used by pension funds. This gives 32 optimal asset allocations, because the utility changes with different parameters.
Whereas 4.2 focused on the impact on the continuity analysis, section 4.3 focuses on the impact of a typical ALM study in finding the optimal asset allocation. The ALM study consists of running 2500 economic scenarios that provide outcomes for pension results and funding ratios. With these output variables, a given asset allocation, and economic parameter set and a risk appetite coefficient, a utility is calculated using the utility function (5). The optimal asset allocation is found by maximizing the utility over
different asset allocations. It is also possible to make 32 utility functions using the individual parameter sets of the funds in the sample. All these functions have maximum utility for a given λ. The maximum can be found for an optimal asset allocation to one of the investment classes.
3.4 Control groups
Control groups are used to test the dependency of the model upon the fund’s demographics. Section 4.4 describes results similar to section 4.2 and 4.3, but for a different fund demographics. The aim is to compare the results of section 4.2 and 4.3 to equal calculations with a green fund and a grey fund. For the green fund all the members aged 45 and over are taken out of the initial fund. The green fund has an average duration of 31 over the fifteen years. For the grey fund all the members aged 55 and younger are removed. The average duration of the grey fund is 10. The average duration of the standard fund used for all the previous calculations is 17.
4 RESULTS
4.1 Variation in economic parameters
This part describes the results from the comparison of the continuity analyses of Dutch pension funds over the last six years. Several tables are shown with return, volatility and correlation data such used by pension funds in their ALM studies. All these results use data that was provided by the 32 largest pension funds. In some cases, approximations were made to fit the data in fixed asset classes.
Table 4 shows the statistics for the annual returns of various asset classes. The first column shows the asset classes. The other columns contain the sample mean, and sample standard deviation (SD). Distribution is skewed, because of the maximum parameters set by the governmental legislation. There is a significant dispersion in the parameter choice by pension funds, especially for commodities and other investments. The effect of the variation in the latter is smaller due to a smaller allocation in the asset mix. The variation in the equity return parameter can have consequences for the outcome of a continuity analysis if the fund has invested a substantial amount in equity.
Table 4. Annual returns (Geometric averages)
Asset Class Mean SD
Fixed Income 4.0% 0.6%
Equity 6.7% 0.7%
Real Estate 6.1% 0.8%
Commodities 4.6% 1.3%
Other 6.1% 1.2%
Figure 1. Box plots of annual returns
Figure 1 shows the box plots for the previously mentioned returns. In the box plots the
2,0% 3,0% 4,0% 5,0% Fixed Income 4,5% 5,5% 6,5% 7,5% 8,5% Equity 4,0% 5,0% 6,0% 7,0% 8,0% Real Estate 1,0% 3,0% 5,0% 7,0% Commodities 2,0% 4,0% 6,0% 8,0% Other 13
four quartiles are separated in te usual way. This figure gives more information about the distribution of the annual return of the 32 funds.
Table 5 shows the distribution of the volatility parameters asumed by pension funds over different asset classes. Pension funds are not unanimous in their opinions on the volatilities of asset classes. This could possibly be caused by inconsistencies in the asset classes. The variation in real estate parameter may be caused by definition ambiguities. Some funds classify indirect real estate as equity while others define it as real estate. It is evident that a portfolio of mainly safe government bonds has a smaller volatility than a portfolio of junk bonds. Nevertheless, both of these portfolios are classified as fixed income portfolios. If funds have similar investment policies, dispersion in assumed volatilities is substantially large.
Table 5. Annual volatilities
Asset Class Mean SD [Min ; Max]
Fixed Income 7.0% 2.7% [3.2% ; 17%]
Equity 18% 1.8% [13% ; 21%]
Real Estate 11% 5.1% [6.0% ; 23%]
Commodities 21% 5.3% [6% ; 26%]
Other 14% 6.2% [4.1% ; 25%]
Table 6 displays the distribution of the asset allocation in the Dutch pension industry. On average a fund invests half of its assets in relatively less volatile fixed income investment. Some funds decide to invest more conservatively and invest up to seventy percent in bonds while other funds are less risk averse and invest up to seventy percent in risky assets. Furthermore, most parties invest a small amount in commodities, real estate and other risky assets.
Table 6. Strategic asset allocation
Asset Class Mean SD [Min ; Max]
Fixed Income 48% 11% [20% ; 72%]
Equity 34% 10% [11% ; 55%]
Real Estate 9.8% 5.3% [0% ; 27%]
Commodities 2.5% 2.2% [0% ; 7.0%]
Other 6.1% 5.7% [0% ; 20%]
Table 7 demonstrates variation in the correlations between different asset classes that are used by funds. Remarkable in this table is the dispersion in the results. Fo some asset combinations, funds do not even agree on the sign of the correlation. The correlation of equity with other asset classes shows substantial diversity. In all other asset classes, the equity shows negative as well as positive correlation coefficients. Dispersion in the
correlation of the “other investment” category can be caused by different content of the asset class; private equity and hedge fund have a different return distribution. The weights used for the calculation of regulatory capital requirements and premiums are shown in appendix E. In comparison to this table most of the correlations are underestimated by the pension fund. The risk of underestimating correlation is that one overestimates the diversification benefit between asset classes and therefore underestimates the total downside risk of their asset portfolio.
Table 7. Correlations Average
(Stdev)
Fixed Income Equity Real Estate Commodities Other
Fixed Income 1 - - - -
Equity 0.20 (.15) 1 - - -
Real Estate 0.01 (.13) 0.35 (.23) 1 - -
Commodities -0.20 (.10) 0.12(.23) 0.12 (.09) 1 - Other 0.18 (.18) 0.55 (.21) 0.27 (.14) 0.07 (.19) 1
Table 8 describes the relationship between the return and volatilities used in the calculations for the continuity analyses. The negative correlations imply that a fund with a low expected return for a certain asset assumes a low volatility for the same asset. This goes against the economic theory on asset pricing as described by Sharpe (1964) and Cochrane (2001). Funds with conservative parameters also tend to be more prudent on their risk-return trade off and vice versa.
Table 8. Correlation between returns and volatilities
Correlation Fixed Income -0.76*** Equity -0.29* Real Estate -0.02 Commodities -0.14 Other -0.28* Note. * = p < .1, ** = p < .05, *** = p < .01.
Table 9 tells something about the relations between the returns and asset allocation used in the calculations for the continuity analyses. A positive correlation indicates that a high return parameter for an asset is typically combined with a large allocation for this asset. The fixed income and commodities indicate a negative correlation whereas the risky assets show a positive correlation. This shows that pension funds tend to choose higher parameters for assets that are allocated in, or the fund invests more in certain assets ,because they have a more optimistic on this asset. Note that the correlations are quite
low; hence, the significance is not proven for most relations.
Table 9. Correlation between returns and asset allocation
Average Correlation Fixed Income -0.21 Equity 0.19 Real Estate 0.04 Commodities -0.10 Other 0.29* Note. * = p < .1, ** = p < .05, *** = p < .01.
Tables 10 and 11 shows the correlations between the returns and the date of the ALM-study used in the calculations for the continuity analyses. First, the assets returns have a positive cross-correlation. This suggests that the funds with high parameters for one asset return will likely have high parameters for the other asset returns. Second, the date of the analysis is negatively correlated with the asset return. There are two possible explanations for this phenomenon; risk-free rates decreased in the sample period or the funds became more prudent after the financial crisis.
Table 10. Correlation across asset returns
Correlation Fixed Income Equity Real Estate
Fixed Income 1.0
Equity 0.3* 1.0
Real Estate 0.3* 0.4** 1.0 Note. * = p < .1, ** = p < .05, *** = p < .01.
Table 11. Correlation asset returns and date of report
Correlation Date Date 1.0 Fixed Income -0.5*** Equity -0.4** Real Estate -0.3* Note. * = p < .1, ** = p < .05, *** = p < .01.
4.2 Impact of variation in parameters on continuity analysis
This section describes the impact of variation in parameters on the outcome of a continuity analysis. Table 12 shows the funding ratio and pension result after fifteen years for an increasing expected return on equity investments. The median result and 2.5%-percentile of the previously mentioned output are displayed. There is a strong
increase in all the output variables. An increase of one percent in equity returns causes an increase of approximately five percent in the median output and two percent in the 2.5% Value at Risk measures. This means that there is an increase in the expected return and a decrease in absolute risk, as a result of changing one input parameter.
Table 12. Funding ratios and pension results for increasing equity returns
𝑅𝑒𝑞𝑢𝑖𝑡𝑦 𝐹𝑅2.5% 𝐹𝑅𝑚𝑒𝑑 𝑃𝑅2.5% 𝑃𝑅𝑚𝑒𝑑 4.0% 86.2% 133.4% 37.1% 72.6% 5.0% 88.5% 139.0% 39.1% 76.1% 6.0% 90.8% 144.1% 41.3% 80.4% 7.0% 93.0% 150.1% 43.5% 85.0% 8.0% 95.2% 155.9% 45.8% 90.4%
Table 13 describes an output similar to Table 12, but for an increase in the average return on the credit spread. There is a stronger increase in the 2.5%-percentile funding ratio compared to the equity increase. The complete picture looks similar to Table 12 where there is a substantial increase in return and decrease in risk.
Table 13. Funding ratios and pension results for increasing credit spread returns
𝑅𝑠𝑝𝑟𝑒𝑎𝑑 𝐹𝑅2.5% 𝐹𝑅𝑚𝑒𝑑 𝑃𝑅2.5% 𝑃𝑅𝑚𝑒𝑑 0.0% 108.8% 139.0% 58.3% 74.9% 0.6% 112.4% 143.5% 60.4% 77.4% 1.2% 116.4% 147.8% 62.5% 79.6% 1.8% 120.2% 152.4% 64.4% 82.1% 2.4% 124.5% 156.6% 66.5% 84.7%
The upcoming two tables cover the output changes caused by changes in correlation input. It examines the effect of a change in the correlation between, equity and inflation, equity and fixed income, and equity and real estate.
Table 14 shows the 2.5% percentile and median pension result, when increasing the correlations between equity and fixed income and equity and inflation. An increase in the correlation between equity and fixed income causes a slight increase in risk and in return; a low correlation creates a diversification effect. An increase in the correlation between equity and inflation causes a decrease in the risk, because indexation is being financed with returns of equity. All the effects in Table 14 are small, because of the difference in volatility between the components. Concluding, the changes of correlation input have an slight effect on the pension result output, but not material.
Table 14. Pension results for increasing correlations
Equity – Fixed Income Equity - Inflation
𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑃𝑅2.5% 𝑃𝑅𝑚𝑒𝑑 𝑃𝑅2.5% 𝑃𝑅𝑚𝑒𝑑 0 65% 85% 61% 95% 0.2 65% 86% 62% 95% 0.4 64% 86% 62% 95% 0.6 63% 87% 63% 95% 0.8 63% 87% 65% 95%
Table 15 shows the 2.5% percentile and estimated median of the funding ratio and pension result over increasing correlations between equity and real estate. There is a decrease in the estimated medians and a bigger decrease in the 2.5% intervals. The effect in the 2.5% percentiles is bigger than in Table 14, because the volatilities of equity and real estate are more alike. There is a material diversification benefit if a fund chooses an advantageous correlation parameter. Therefore, a fund needs to properly justify their correlation parameter, because there is an effect on the final output.
Table 15. Output for increasing correlation between equity and real estate
𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝐹𝑅2.5% 𝐹𝑅𝑚𝑒𝑑 𝑃𝑅2.5% 𝑃𝑅𝑚𝑒𝑑 -0.1 105% 149% 54% 82% 0.1 104% 148% 52% 82% 0.3 102% 148% 51% 82% 0.5 100% 148% 49% 82% 0.7 98% 147% 48% 81%
Figure 2. Distribution of median pension results over 32 funds.
Figure 2 is a histogram of the distribution of the median pension result for the parameter sets of the 32 largest funds.. This figure shows the dispersion in the outcome of a continuity analysis solely based on return parameter variation. All the outcomes are
0 1 2 3 4 5 6 7 85% 90% 95% 100% F req u en cy PRMEDIAN 18
relatively high, due to the high expected returns and low interest rates and inflation. The median pension results are between 80 and 100%. So there is a dispersion of 20% purely due to variation in economic input.
Figure 3 shows the distribution of the 2.5%-percentile of the pension result for the 32 funds. These are based on the 2.5% Value at Risk of the pension result after 15 years. The results vary between 30 and 60%. There is dispersion in the outcomes. Most of the density in the distribution is around 45%. There is one outlier on the left side of the distribution.
Figure 3. Distribution of 2.5%-percentile pension results over 32 funds.
4.3 Impact of variation in parameters on optimal asset allocation Figure 4. Utility functions for increasing equity returns
0 2 4 6 8 10 12 30% 35% 40% 45% 50% 55% 60% F req u en cy PR2.5% -100% -50% 0% 50% 100% 150% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% U tility Equity Allocation Utility Functions, λ=0.75 8.5% 8.0% 7.5% 7.0% 6.5% 6.0% 5.5% 5.0% 4.5% 4.0% 19
In Figure 4, the utility functions are plotted. One line describes the utility for one parameter set and the increasing allocation to equity. Every line has different parameter for the expected return on equity investments. If only 30% of assets are allocated to equity, the utility functions are very close. Once there is more capital allocated to equity, the dispersion between the utility functions gets larger. The utility functions with high equity return parameter are increasing functions of the equity allocation. Utility functions with a low parameter (e.g. 4% or 4.5%) are decreasing function of the allocation to equity. Every utility function has different maximum. Concluding, based on the parameter for equity return, a fund has different optimal allocation to equity of its assets.
In Figure 5 the parameter for the expected return on the risk bearing part (credit spread) of the fixed income investments is varied. The function with the highest parameter is decreasing in the equity allocation, whereas the utility function with the lowest parameter is increasing. The optima of the utility functions are decreasing in the return parameter; the higher the parameter for expect return on fixed income, the lower the optimal asset allocation to equity. This shows the dependency of the ALM model on its parameters, because all parameters that were used are acceptable under legislation and can hypothetically be backed up by data and literature.
Figure 5. Utility functions for increasing fixed income returns
In Figure 6, utility functions are displayed with different correlations over varying asset allocations. Asset allocation to fixed income is 40% and equity is traded off with real estate. One line in the graph corresponds to one utility function with one given correlation parameter. Note that the for every asset allocation the highest utility is the one with the lowest correlation. This is in line with the results from Table 15, which showed the diversification benefits. The utility functions are decreasingly increasing in the asset allocation towards real estate. Besides the correlation, the other distribution properties
-80% -60% -40% -20% 0% 20% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% U tility Equity Allocation Utility Functions, λ=0.75 2,7% 2,4% 2,1% 1,8% 1,5% 1,2% 0,9% 0,6% 0,3% 0,0% 20
play a role in the utility functions. Real estate has a lower Sharpe Ratio. Thefore, this risk-averse fund (λ=0.75) prefers a large allocation of its asset to real estate. For λ’s lower than 0.5, all the utility function are decreasing in the asset allocation towards real estate.
Figure 6. Utility functions for increasing correlations
Figure 7 shows the distribution of the optimal allocation of the 32 funds. Pension policy and fund demography is identical. Based on funds’ thoughts on asset returns and volatilities, there is dispersion in the optimal allocation to equity. The allocation is capped between 30 and 75% (real estate is fixed at 10%) in this exercise, to restrict on realistic investment portfolios. This causes a distribution with a high density on the tails.
Figure 7. Distribution of optimal asset allocation over 32 funds.
A fund with higher parameters for equity will be at the right side of the distribution. Funds with lower parameters on equity and an optimistic view on fixed income will be at
-60% -40% -20% 0% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% U tility Real EstateAllocation Utility Functions λ=0.75 -10% 0% 10% 20% 30% 40% 50% 60% 70% 80% 0 2 4 6 8 10 12 14 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% F req u en cy
Optimal Equity Allocation, λ=0.75
the left side of the distribution. This proves again the dependency of the ALM study on its economic parameters. It seems advisable that pension funds look critically at their input parameters, realize the impact of their decision and to be cautious in managing a pension fund based on the outcome of the ALM studies.
4.4 Impact on grey and green fund
In Figure 8, the distribution of the 2.5%-percentile pension results is shown for the grey fund over 32 different parameter sets. The distribution has moved left a little bit, compared to Figure 3 in section 4.2, but the dispersion is still present. Approximately half of the funds have a 2.5% VaR of their pension result over fifteen years of 40%, but one fund is below 30% and three funds above 50%.
Figure 8. Distribution of 2.5%-percentile pension results (grey fund)
Figure 9 demonstrates the distribution of the optimal asset allocation for the grey fund using 32 different scenario sets; similar to the calculations for Figure 7. Since it is a very grey fund, a higher risk aversion parameter is chosen. There is dispersion in the optimal equity allocations. Compared to Figure 7, the density of the distributions has moved to lower allocations.
Figure 10 shows the histogram of the 2.5% intervals of the 32 parameter sets. The results look similar to Figure 3, but distribution has moved to right. The VaR measures for the pension result still exhibit variety. The pension result output varies between 35% and 55%. 0 2 4 6 8 10 12 14 16 18 25% 30% 35% 40% 45% 50% 55% 60% F req u en cy PR2.5% 22
Figure 9. Optimal asset allocation of 32 parameter sets (grey fund)
Figure 10. Distribution of 2.5%-percentile pension results (green fund)
Figure 11 displays the 32 optimal asset allocations for the green fund for every given parameter set. Since it is a very green fund, a lower risk aversion parameter is chosen. There is dispersion in the optimal equity allocations. The density of the distribution is concentrated on the two tails of the distribution. This is caused by the shape of most of the utility functions. Except for the middle observations, the utility functions are monotonic increasing or decreasing functions in the asset allocation. Therefore most of the parameter sets find their optimum in the boundary values of 30% and 75%. The hedging effect of a bond portfolio with a duration of 6 is minimal, when the liabilities have a duration of 31. The optimal asset mix is depending more on the risk-return tradeoff of the individual asset classes.
0 1 2 3 4 5 6 7 8 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% F req u en cy
Optimal Equity Allocation, λ=1
0 2 4 6 8 10 12 25% 30% 35% 40% 45% 50% 55% 60% F req u en cy PR2.5% 23
Figure 11. Optimal asset allocation of 32 parameter sets (green fund)
To conclude, the same dispersion in the output is found as in section 4.2 and 4.3. The results have different values for the grey and green funds, because risk and return is absorbed differently. Overall, the control groups show variation in the output. These results do not contradict the outcome of the previous sections.
0 2 4 6 8 10 12 14 16 18 20 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% F req u en cy
Optimal Equity Allocation, λ=0,5
5 CONCLUSIONS Summary of the findings
This thesis examines the impact of parameter variation in the ALM framework and answers the following main questions:
What is the variation in economic parameters used by pension funds for their ALM studies?
Researching the continuity analyses of the 32 largest Dutch pension funds, I found variation in the economic parameters (e.g. expected asset returns and interest rates) used for their ALM studies. This variation is found in all economic parameters: expected return, volatilities and correlations. Further examining the data, I found negative correlations between expected returns and expected volatilities as well as between expected return and the date of the report. Finally, there are positive relations between expected returns across different asset classes.
What is the impact of parameter variation on the outcome of continuity analysis?
Changing the expected returns, volatilities and correlations has an impact on the output of a continuity analysis. I simulated 15 year projections of the funding ratio and pension result using these parameters. I found a large dispersion in the output solely, because of the difference in economic parameters. Median pension results vary between 80 and 100% and 2.5%-percentile pension results vary between 30 and 60%, when using the parameters from the data set.
What is the impact of parameter variation on the optimal asset allocation?
The optimal asset allocation to equity varies from 30 to 75%. These findings show how heavily the results of an ALM study depend on its subjective economic parameters. There is impact of the variation in economic parameters on the outcome of a continuity analysis and on an ALM study to find the optimal asset portfolio. In the control groups, consisting of a green and a grey fund, the impact of input variation is different, but the output results show dispersion as well.
Limitations
There are some limitations to this research study. First, only one ALM model was looked at. That seems narrow, assuming that every pension fund uses a different ALM model. Second, the ALM model used was quite simple. No stochastic volatilities and correlation were implied. These could be used to model more realistic stress scenarios. Finally, some simplifying assumptions were made for calculations in this thesis, due to a lack of data or unclear data.
Recommendations
During this research the biggest problems faced were modeling credit spread and defining a utility function. The available literature was not sufficient on these subjects. The credit spread modeling is investigated from point of view of banks, but not in a pension ALM framework. The utility models in literature were outdated. This definitely needs more future research to improve the overall quality of the ALM models. Another point for future research is the search for objective economic parameters. From a regulatory point of view, it is important to restrict pension funds in the choice of their parameters. Academic research could help in finding proper measurements for the quality of parameters like prudence and robustness.
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7 APPENDICES
Appendix A. Maximum FTK parameter values
Maximum Average Geometrical Arithmetic
Fixed Income 4,5% 4,5%
Equity Mature Markets 7,0% 8,5%
Equity Emerging Markets 7,0% 8,5%
Real Estate (Direct) 6,0% 7,5%
Real Estate (Indirect) 7,0% 8,5%
Commodities 6,0% 7,5%
Private Equity 7,5% 9,0%
Hedge Funds 7,5% 9,0%
Minimum Geometrical Arithmetic
Price Inflation 2,0% 2,0%
Loan Inflation 3,0% 3,0%
Appendix B. Expected cash flows
0 5.000.000 10.000.000 15.000.000 20.000.000 25.000.000 0 10 20 30 40 50 60 70 80 90 100 t=0 t=15 29
Appendix C. Indexation ladder
Lower bound Upper Bound Percentage
0% 115,0% 0% 115% 120,0% 17% 120% 125,0% 33% 125% 130,0% 50% 130% 135,0% 70% 135% 140,0% 85% 140% ∞ 100%
Appendix D. Zero rates
Appendix E. Weights used for calculating regulatory capital requirements
Fixed Income Equity Real Estate Commodities Other
Fixed Income 1,00 0,00 0,50 0,50 0,00 Equity 0,00 1,00 0,50 0,50 0,75 Real Estate 0,50 0,50 1,00 0,50 0,50 Commodities 0,50 0,50 0,50 1,00 0,50 Other 0,00 0,75 0,50 0,50 1,00 0,0% 0,5% 1,0% 1,5% 2,0% 2,5% 3,0% 1 6 11 16 21 26 31 36 41 46 51 56 R ate Maturity 01-01-2013 30
