testing of retirement portfolio sustainability

8/11/2019 TESTING OF RETIREMENT PORTFOLIO SUSTAINABILITY

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EKONOMIKA IR VADYBA: 2011. 16 ECONOMICS AND MANAGEMENT: 2011. 16

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TESTING OF RETIREMENT PORTFOLIO SUSTAINABILITY:

SIMULATION VS ANALYSIS WITH HISTORICAL DATA

Aura Klimaviien

Vilnius Gediminas Technical University, Lithuania, [email protected]

Abstract

The article examines the problem of determining asset allocation of sustainable retirement portfolio.While principles of modern portfolio theory are perfectly suitable in retirement portfolio accumulation stage,they may not be applicable in the decumulation stage of human cycle when withdrawals from retirement

portfolio are being made to fund retirement expenditure. The lack of widely accepted methodology to solvethe puzzle of asset allocation of retirement portfolio during retirement period has been noticed. This research

focuses on three different methods to determine asset allocation of retirement portfolio (heuristics, multiplehorizon approach and stochastic optimization) for a retiree and incorporates both stochastic simulation and

analysis with historical data to evaluate the sustainability of retirement portfolios formed using differentmethods under investigation. The research results are presented and the important aspects of both stochastic

simulation and analysis with historical data are highlighted.Keywords: sustainable retirement portfolio, methods to determine asset allocation of retirement

portfolio, stochastic simulation, probability of retirement portfolio ruin, historical data.JEL Classification:G11.

Introduction

Global and domestic economic instability, increasing life expectancy, falling birth rates pose serious

challenges for the state pension system, increase population insecurity for retirement income, therefore, the

issues of retirement planning and accumulation of retirement funds are becoming more relevant both in the

world and Lithuania. Inevitably the responsibility for financial well-being in retirement will fall on

households and individuals, thus retirement planning techniques will be given increasing attention. The

financial crisis has shown how important it is to protect the people savings for retirement. While various

measures can be undertaken to accommodate individual needs and goals of retirees, the paper emphasizes the

role of smart portfolio formation in extending the sustainability of retirement savings and increasing theportfolio return.

Economic-social processes, evolving computing technologies and access to information leads to the

need to review currently employed methods to determine asset allocation of retirement portfolio and search

for new ones. Retirement portfolio sustainability is relevant to both working stage of human cycle and

retirement period, still this paper seeks to contribute to finding answers to the questions of retirement

portfolio allocation focusing on the retirement period only. The scientific problem may be defined as search

of verified methods to determine asset allocation of sustainable retirement portfolio.

The objective of the paper is to test the sustainability of retirement portfolio with historical market

data. To achieve this objective the retirement portfolios are formed applying three different methods to

determine asset allocation for a retirement portfolio, including the heuristic, the multiple horizon approach

and stochastic optimization methods, then stochastic simulation models are developed to calculate the

probabilities of depleting the retirement portfolio earlier than the planned retirement horizon (often termedportfolio ruin). Further, the analyzed portfolio methods are explored testing the portfolio sustainability with

historical stock and bond market data. The nature of the research indicates the novelty of this paper.

Despite the growing scientists attention to the issue of retirement portfolio sustainability, there is a

lack of one widely accepted methodology of determining asset allocation of retirement portfolio during

retirement period. The sustainability of retirement funds has been addressed by many studies but they usually

focus on calculating the maximum withdrawal rate instead of optimal asset allocation of retirement portfolio.

Traditionally the researches test and report the combinations of hypothetical asset allocations and withdrawal

rates for fixed retirement planning horizon. Relatively fewer have examined the sustainability of withdrawals

over the uncertain retirement life span (Milevsky, Robinson 2005), even fewer authors have studied the

optimal asset allocation in retirement portfolios (Ho et al., 1994; Milevsky et al.,1997; Stout, Mitchell, 2006;

Stout, 2008; Klimaviien, 2010B). The wider review of former researches in this field is documented by

Klimaviien(2010A).


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The complex analysis combining both stochastic simulation and testing with historical data to test the

sustainability of retirement portfolios formed employing pre-selected methods (heuristics, multiple horizon

approach, stochastic simulation) was originally performed and introduced by the author of this paper.

Methods Applied to Determine Asset Allocation of Retirement Portfolio

Retirement planning is a process used to help individuals establish retirement goals, gatherinformation and determine a strategy to fund retirement goals. To meet financial goals during retirement,

individuals normally need periodic cash withdrawals from their retirement portfolio to fund their desired

standard of living. Achieving adequate returns on the retirement portfolio is a critical component of the

success of retirement planning. The better the investment portfolio performs, the longer the portfolio can

sustain these periodic withdrawals in retirement. Therefore, asset allocation strategy and good portfolio

returns are the essential determinants of success for a retirement plan and the sustainability of retirement

portfolio. The optimal asset allocation problem is a serious task that is still missing for clear solving

methodology. Three known methods applied to determine asset allocation of retirement portfolio were

selected for further analysis:

1. Heuristic method is an approximate method to determine asset allocation of retirement portfolio

traditionally defined as 100 minus age should be invested in equities.

2.

Multiple horizon approach presented by Cordell (2005) views retirement planning as a strategy offunding a serial of future liabilities. Multiple horizon approach treats each annual retirement

withdrawal as a separate portfolio with its specific horizon.

3. Stochastic optimization identifies the asset allocation that minimizes the probability of exhausting

the retirement portfolio from constant withdrawals over the retirement life span.

This paper specifically ignores the purchasing life annuities as option and focus on smart retirement

portfolio management as a mean to cope with the core longevity risk defined as risk that one will outlive

ones money.

Stochastic simulation of retirement portfolios

To model the portfolios formed applying each method under investigation the dynamic stochastic

simulation techniques (GoldSim v. 10.10.) were employed to calculate the portfolio ruin probability, i.e.probability that the retirement portfolio will be exhausted earlier that planned retirement horizon. Each

simulation generates 20 000 realizations and the portfolio ruin probability calculated by dividing the number

of realizations when portfolio value hits zero by the total number of realizations.

It is assumed that a person retires being 65 years old, thus the fixed retirement planning horizon

including the cushion of 5 years respectively is 25 years. A person withdraws a fixed percentage of initial

portfolio amount annually, hereafter called withdrawal rate. 5 % withdrawal rates is used. Initial portfolio

amount is 1 mln. Litas.

The retirement portfolio contains two asset classes stocks and US intermediate-term treasury bonds.

Simulated market returns are based on annual data and one year wealth relatives (1+annual real return) are

log normally distributed. Real annual returns are being employed in order to avoid the modeling of inflation.

Model parameters are evaluated analyzing stock (S&P index) and bond (1926-1995 year data from Center

for Research in Security Prices, 1996-1999 - Lehman Bros. intermediate-term Treasury index, 2000-2009 -DFA Five-Year Government I index) historical market data for 1926-2009 period. The arithmetic mean

inflation-adjusted returns to large-cap stocks and intermediate-term U.S. government bonds over the years

1926 to 2006 are 8.65% and 2.20%, respectively, and the standard deviations of the inflation-adjusted returns

are 20.52% and 7.18%, respectively. Historical correlation between stocks and bonds returns is maintained

and equal to 0,16. Autocorrelation is ignored. Latin Hypercube Sampling method is employed to model

distribution of stochastic variables.

Transaction costs and taxes are ignored in the calculations. Borrowing is not allowed in the models.

A person starts each year by withdrawing the annual fixed amount and the remaining portfolio invests

based on target asset allocation into analyzed two asset classes. One-year portfolio return is being calculated:

Rt,i = Dt,AAt,i + (1 Dt,A)Ot,i (1)

here Rt,i portfolio return in year tand realization i ; Dt,A allocation to stocks inyear t; At,i real stockreturn in year tand realizationi; Ot,i real bond retun in year tand realizationi.


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Portfolio value at the end of t period is being calculated:

Vt,i = (Vt-1,i I V0)(1+Rt,i) (2)

here Vt,i portfolio value at the end of year t in realization i; Vt-1,i portfolio value at the end of year t in

realization i;I fixed annual withdrawal rate in percents; V0 portfolio value at the beginning of retirement

period.

Simulation model contains one state variable EQUAL TO 0, always checking if portfolio value hits0 and able to get values true or false. Thus:

Vt,i = (Vt-1,i I V0)(1+Rt,i), if EQUAL TO 0 = false otherwise Vt,i = 0 (3)

Portfolio is being rebalanced on a yearly basis. Conceptual model of retirement portfolio is provided in

Figure 1.

Retirement portfolio

Stock and bond return Annual withdrawals

Portfolio rebalancing

Target porfolio allocation Portfolio ruin

No

Yes

Porfolio

value

>=0?

Retirement portfolio

Stock and bond return Annual withdrawals

Portfolio rebalancing

Target porfolio allocation Portfolio ruin

No

Yes

Porfolio

value

>=0?

Figure 1.Retirement portfolio model

Stochastic optimization aims at finding asset allocation of retirement portfolio consisting of two asset

classes ensuring minimum portfolio ruin probability over the fixed retirement planning horizon for a

specified withdrawal rate and taking into account stochastic market returns. A model created for stochasticoptimization purpose combines static optimization and dynamic stochastic simulation techniques: a static

simulation model contains other submodel dynamic stochastic simulation model because in case of

optimization of a probabilistic system the objective function to be optimized cannot be a single deterministic

output. Rather, it must be a statistic (e.g., the mean or 50th percentile). Optimized variable is allocation to

stocks in the retirement portfolio.

Stochastically optimal asset allocations are provided in Table 1.

Table 1.Results of stochastic optimization

Withdrawal rate, % Optimal allocation to stocks, % Optimal allocation to bonds, %

5,0 56,45 43,55

Stochastically optimal allocations are further applied in simulations to calculate the portfolio ruin

probability, the standard deviation of portfolio ruin probability, the average ratio of ending and initial

portfolio value and the standard deviation of the ratio. These rates are being calculated for portfolios formed

applying heuristics and multiple horizon approach and the results are compared in Table 2.

Table 2.Results of stochastic simulation

Simulation resultsHeuristicmethod

Multiple horizonapproach

Stochasticoptimization

Average portfolio ruin probability, % 20,91 20,69 14,83

Standard deviation of portfolio ruin probability, % 40,66 40,51 35,54

Average ending and initial portfolio value ratio 0,50 0,56 1,34

Standard deviation ending and initial portfolio value ratio 0,55 0,63 1,60


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The table 2 shows that stochastic optimization ensures lower average portfolio ruin probabilities for 65

years old retiree with fixed retirement planning horizon of 25 years. Portfolios formed using heuristics and

multiple horizon approach performs similarly. Stochastic optimization also allows expecting higher portfolio

ending value that could serve as cushion for extra retirement years or as a bequest for ones heirs. Former

research shows that stochastic optimization outperforms other methods for wide range of withdrawal rates

(Klimaviien, 2010A).

Retirement Portfolio Sustainability Testing with Historical Data

Empirical testing of methods to determine asset allocation of retirement portfolio is highly

complicated due to long retirement period and may last for decades. Therefore testing with historical stock

and bond market data was conducted. The paper uses actual real annual return for stock and intermediate

term bonds from 1926 to 2009. The research aims at finding how the portfolios formed applying the methods

under investigation would have performed historically. The same age at retirement of 65 years was chosen

for analysis. For each method 40 historical portfolios are being monitored. Each portfolio is presumed to

have a 25-year time horizon.

In contrast to heuristic method, multiple horizon approach and stochastic optimization requires

historical data for certain period in order to calculate expected return and standard deviation. Thus the

historical data from 1926 to 1945 was used to form retirement portfolio at year 1946. This portfolio extends25 years to 1970. The sustainability of this portfolio is evaluated using real annual return for stocks and

intermediate bonds from 1946 to 1970. The last 40th

portfolio starts at 1985 and extends to 2009. Historical

data available for evaluation of parameters is complemented each year with new actual data.

Asset allocation of retirement portfolios formed applying heuristics are the same during each 25-year

period under investigation. They all start with 35 % (i.e. 100-65) allocation to stocks and ends with 10 % of

portfolio in stocks while decreasing allocation to stocks on a yearly basis by 1 ppt.

Asset allocation of retirement portfolios formed employing multiple horizon approach are different for

each 25-year period under investigation because it uses prior historic period geometric mean returns as

expected return input variables to derive the asset allocation strategies for retirement portfolios. Figure 2

depicts the dynamics of geometric mean real returns for stocks and bonds as expected return for asset classes

in the portfolio. On the x-axis a year when certain retirement portfolio starts is shown. On the y-axis a

corresponding geometric mean real return for stocks and bonds calculated using data from 1926 to previousyear is shown. For example, expected return for stocks and bonds input variables in multiple horizon

approach for portfolio starting in 1946 is 7 % and 3,2 % respectively. Portfolio allocations for 40 historical

portfolios formed applying multiple horizon approach are similar, i.e. portfolios start with average 50 %

allocation to stocks and ends with 100 % allocation to bonds.

0,0%1,0%

2,0%3,0%4,0%5,0%

6,0%7,0%8,0%

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Geometricmean

Stocks Bonds

Figure 2.Expected real stock and bond return

Retirement portfolios formed employing stochastic optimization are also different for each 25-yearperiod under investigation because of different historical data used to evaluate model parameters each


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portfolio starting year. Retirement portfolios are rebalanced to determined stochastically optimal asset

allocation on a yearly basis. Stochastically optimal allocations to stocks in the historical retirement portfolios

are shown in Figure 3.

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10,00

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,%

Figure 3.Optimal allocation to stocks in the retirement portfolio

In total, 120 historical retirement portfolios formed applying three methods are being monitored.

Figure 4 depicts the comparison of portfolio sustainability according to different methods

investigated. The chart discontinues at 25 years point because the planned retirement horizon of 25 years was

chosen. Obviously, the retirement portfolios formed applying three different methods would have been

sustainable historically in most of the cases when 5 % withdrawal rate is used. Portfolios formed starting

year 1959 till 1974 would have been exhausted earlier than planned retirement horizon of 25 years. It can be

noticed that portfolios formed employing multiple horizon approach outperform the portfolios formed using

heuristics in some cases, nevertheless the stochastic optimization would lead to better portfolio sustainability.This confirms the conclusion made based on stochastic simulation results.

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liosustainability,years

Heuristics Multiple Horizon Approach Stochastic Optimization

Figure 4.Comparison of portfolio sustainability

The insufficient sustainability of retirement portfolios formed from 1959 to 1973 was mainly caused

by the sequence of market cycles. It is known that the sequence of returns has a serious effect on ending

value of the investment. The effect is even more visible when higher withdrawal rate is applicable. The

market cycle history documented by Harris (2009) is shown in Table 3. Portfolios formed before year 1959benefited from growing stock market. Later portfolios had opportunity to grow but the recession significantly

diminished the value of portfolios. Portfolios formed stating 1974 to 1981 suffered the market decline during


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their first years but the strong growth during later years was sufficient to ensure portfolio sustainability at the

requested level.

Table 3.Market cycles

Bear market Bull market

Period Duration,years

Nominalreturn

Real return Period Duration,years

Nominalreturn

Real return

1906-1921 15 -220% -6,30% 1921-1929 8 18,10% 18,80%

1929-1949 20 -2,60% -4,20% 1949-1966 17 11,60% 9,70%

1966-1982 16 1,80% -4,70% 1982-2000 18 14,80% 11,10%

Figure 4 doesnt completely reveal the impact of growing market on portfolio performance as the

analysis ceases at point of 25 years. Figure 5 shows the ending value of portfolios after 25 years. The higher

the value of retirement portfolio, the longer it may be useful to make withdrawals and fund retirement needs.

0

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Figure 5.Portfolio value at the end of 25 years retirement period

Figure 5 confirms that stochastically optimal portfolios outperform the other two methods under

investigation in terms of ending portfolio value. The higher ending portfolio value means that higher

portfolio growth might be determined by the higher allocation to stocks than in portfolios formed using other

methods. In addition, the higher the higher portfolio ending value ensures funds for extra years, i. e. longer

period than retirement planning horizon, and may serve as bequest for ones heirs.

It should be noted that ending values of portfolios formed applying heuristics and multiple horizon

approach after 1974 are closed to each other but portfolios starting in 1980 and 1981 formed employing

heuristics outperforms retirement portfolios formed by multiple horizon approach. This was caused by the

heavier allocation to stocks during the last decade in the portfolios formed applying heuristics while the

portfolios formed applying multiple horizon contain only bonds during the last six years of retirement period

under investigation.

This research employs dynamic stochastic simulation model to find the optimal asset allocation of

retirement portfolio for fixed retirement planning horizon. Nevertheless 65 years old person can expect to

live average 16 years more, this research assumes that a person certainly lives 25 years, thus in reality the

value of retirement portfolio left for ones heirs might be even higher.

In comparison to stochastic simulation results shown in table 2 the historical simulation showed that

40 % of all historic retirement portfolios formed applying heuristics were exhausted before the end of

retirement period, and respectively 35 % when portfolio was formed employing multiple horizon approach

and 27,5 % when portfolio was stochastically optimal.


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Conclusions

The sustainability of retirement portfolio is a major concern for each retiree who intends to fund part

of retirement spending by periodically withdrawing certain amount from her portfolio. Due to long

retirement period and complex mathematical solutions necessary to evaluate the expected performance of

retirement portfolio, the lack of one widely accepted methodology is clearly observed. While some scientists

test some hypothetical portfolio allocations and withdrawal rates using historical data, the others usestochastic simulation to explore expected retirement portfolio performance paths. This research attempted

not only to incorporate both research methods; it also covered three essential methods applied to determine

asset allocation and enabled to make conclusions about methods application in practice. The research results

showed that stochastic optimization ensures the highest retirement portfolio sustainability compared to

heuristics and multiple horizon approach. In addition, retirement portfolios formed employing stochastic

optimization method guarantees higher ending portfolio value that can be used to finance increased

expenditure or can be left as bequest. Testing with historical data also confirmed that stochastic optimization

while providing minimal probability of portfolio ruin and higher portfolio sustainability can not ensure

success in the every case.

While stochastic simulation enables to monitor a huge amount of possible retirement portfolio

performance trajectories, the model is highly dependent on assumptions that determine the return and

variance of asset classes the portfolio consists of. The reality might be far away from those assumed rulesthat govern the behavior of retirement portfolio system (model).

The analysis with historical data puts the retirement portfolio into certain constrains and there are less

chances that history may repeat in the same order. Nevertheless it allows seeing how portfolio would have

performed under the real circumstances. Historical testing of retirement portfolio sustainability confirmed

that portfolio performance is highly affected by market cycles thus portfolio formation methods should be

explored taking into account market cycles and their sequence. The current market stage should be evaluated

and market cycle forecast for retirement period should be created while setting the parameters of stochastic

retirement portfolio model. This should help to evaluate thoroughly the factors influencing the asset

allocation of retirement portfolio and make the right investment decisions. The major advantage of historical

testing is the possibility to find out how market cycles affect the retirement portfolio and further develop

strategies effectively responding to the changes in the market.

References

1. Cordell, D. M. (2005). A multiple-horizon approach to asset allocation in retirement portfolio. Journal of FinancialPlanning, 8(5), 3439.

2. Harris, J. (2009). Market Cycles and Safe Withdrawal Rates. Journal of Financial Planning, 22(9), 3848.

3. Ho, K., Milevsky, M. A.; & Robinson, C. (1994). Asset allocation, life expectancy, and shortfall. FinancialServices Review, 3(2), 109126.

4. Klimaviien, A. (2010) [A]. Determination of asset allocation for a sustainable retirement portfolio using dynamicstochastic simulation. Social Science Studies, 3(7), 59-79. (in Lithuanian)

5. Klimaviien, A. (2010) [B]. Using dynamic stochastic simulation to determine asset allocation of sustainableretirement portfolio for a stochastic lifetime. Business: Theory and Practice, 11(4), 381-386. (in Lithuanian)

6.

Milevsky, M. A., Ho, K., & Robinson C. (1997). Asset allocation via the conditional first exit time or how to avoidoutliving your money. Review of Quantitative Finance and Accounting, 9, 5370.

7. Milevsky, M. E., & Robinson, C. (2005). A sustainable spending rate without simulation. Financial AnalystsJournal, 61, 89100.

8. Stout, R. G. (2008). Stochastic optimization of retirement portfolio asset allocations and withdrawals. FinancialServices Review, 17(1), 115.

9. Stout, R. G., & Mitchell, J. B. (2006). Dynamic retirement withdrawal planning. Financial Services Review, 15(2),117131.

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