A Savings Plan With Targeted Contributions

INTRODUCTION

We consider a simple savings problem where an individual puts aside funds in order to meet a certain liability at a given date in the future. The individual may contribute to a medium-term investment and savings vehicle, for example, to meet school or college fees for his children in 5 years time. Another example is where the individual wishes to purchase property. He may save for, say 5 years, and then withdraw some or all of his investment and use it as a deposit or down payment as part of a home loan. A longer-term example is a retirement or pension fund, where the individual saves out of labor income to provide a lump sum at retirement. This may then be used to buy an annuity.

In most problems of this kind, a financial planner or adviser will assist the individual to determine how much he should save and invest every month (say), depending on his household finances. Two related decisions must be made: how much the monthly contribution will be and where the savings will be invested. In this article, we do not consider the second decision, that is, asset allocation, although we discuss this briefly later. We assume instead that the individual saves in a variable-rate bank account or in a low-risk mutual fund.

A very risk-averse investor could save in a term-deposit bank account or in a zero-coupon bond, yielding a fixed interest rate. However, it could be that a medium-term fixed-income investment like this is unavailable or pays unattractive rates. A less risk-averse investor, such as a young worker aspiring to home ownership, may instead wish to invest in a stock market fund to try and achieve the largest possible future down payment as part of a home loan. If his stock market investment performs poorly after 5 years, this may mean that a smaller home loan, or a more expensive loan with a higher debt-to-equity ratio, will be available to him. On the other hand, a worker who is close to retirement age is usually advised to invest in less risky, bond-like, assets so as to secure a comfortable retirement.

In standard economic theory, this savings problem can be cast as a consumption–investment problem, with consumption (or saving) and asset allocation as decision variables, and with a utility or loss function as an objective criterion that is optimized dynamically. Even though idealized assumptions may be made, this approach can lead to decision rules that are simple to implement and that can be used to benchmark performance. In practice, however, financial planners do not employ stochastic dynamic optimization to provide retail advice to individual savers. They will suggest instead that the individual sets aside a given amount every month, or possibly a fixed percentage of income, and will also offer suggestions regarding asset allocation. Dynamic optimization is not used regularly by financial planners at a retail level because of the complications caused by real-world features such as taxes, transaction costs, and investment charges, because of imperfect knowledge about asset return distributions, and because it is difficult to capture individuals' varying financial circumstances and requirements.

One example is in the deterministic lifestyling strategies commonly employed in target-date funds and in pension planning (Shiller, 2005; Blake, Cairns, and Dowd, 2001). Advisers typically suggest a fixed monthly contribution based on a range of assumed investment rates of return (Employee Benefits Security Administration, 2006). Another example is work-related savings vehicles, such as an employer-sponsored pension plan, where a fixed proportion of salary is determined (McGill et al., 2004). The new Florida public sector pension plan described by Lachance, Mitchell, and Smetters (2003) requires that individuals save a uniform 9 percent of pay. The Actuarial Foundation and WISER (2004) suggest a rule of thumb of saving 15 percent of pay toward retirement.

Our approach in this article is to try to improve upon the conventional fixed-contribution approach. We suggest a flexible targeted-contribution savings plan that has variable contributions that depend on current and historic investment performance. We investigate whether this new plan performs robustly, in terms of meeting an investment target at a given horizon, under a wide range of investment conditions. We show, by means of simulations, that this plan is less risky than the conventional fixed-contribution plan. Our proposed savings plan is based on a method used in industrial process control (e.g., Box and Luceño, 1995). The method has also been proposed in the econophysics literature (Gandolfi, Sabatini, and Rossolini, 2007) and is applied in the pensions literature to defined benefit pension funding with deterministic economic scenarios (Owadally, 2003). The innovation in this article is to apply this method to a savings problem with a simple stochastic investment environment.

In the next section, we describe the targeted-contribution plan. We then run stochastic simulations with serially independent normally distributed log-returns to compare the performance of this plan over a 5-year saving period with a conventional plan. We measure the risk that end-of-period wealth falls short of the target liability using both the standard deviation and 95th percentile. The sensitivity of our proposed plan to the financial planner's assumptions is verified by repeating the simulations with different parameter values. We also run simulations with constraints imposed on the flexible contributions, with non-Gaussian investment returns incorporating jumps, and with serially correlated returns. Finally, we consider an application in pension funding with a long investment horizon and with a bootstrap stochastic asset model using 59 years of investment data on equities and bonds.

FIXED- AND TARGETED-CONTRIBUTION PLANS

We set up and compare two savings plans here. The first plan that we consider is a conventional savings plan where a level contribution is paid regularly. We refer to this as a fixed-contribution plan. Under this plan, an individual decides on a target fund value F to be reached at a time horizon T (in months). A financial planner makes an estimate of the future monthly rate of return. This estimate is typically based on the planner's statistical or stochastic model of asset returns, and on an informed view of macroeconomic conditions or market performance. We denote this estimate or return assumption by i_A. The individual contributes a constant amount C at the beginning of every month, which is calculated by amortizing F at rate i_A over T months, that is, the stream of monthly contributions in advance accumulate at rate i_A to F:

(1)

If the fund is invested in risky securities, then the actual rate of return on the fund, denoted by i_t in month (t − 1, t), is random. The corresponding log-return is δ_t = ln (1 + i_t). Let the value of the fund at time t be F_t, with F₀ = 0. Then,

(2)

The value F_T of the fund at time T is random and, except by chance, will differ from the desired fund target F. A terminal or final deficit will therefore emerge at time T:

(3)

The risk for the individual saver is that there is a large deficit at time T. (A surplus is just a negative deficit here.)

The second plan that we consider is a targeted-contribution plan. This is based on the pension funding method discussed by Owadally (2003) and adapted from industrial process control (Box and Luceño, 1995). Although we do not consider the asset allocation problem in this article, it is worth noting that Gandolfi, Sabatini and Rossolini (2007) propose a similar method for tactical asset allocation in an investment portfolio.

Under the targeted-contribution method, the individual agrees to vary his monthly contribution payment C_t for month (t, t + 1) as follows

(4)

where λ₁ and λ₂ are variables to be specified, subject to the constraints λ₁ > 0 and λ₂ > 0. We assume that D_t = 0 for t < 0. Here D_t represents the notional deficit at time t in the fund, relative to what the value of the fund would be if the individual followed the fixed-contribution plan and if the fund earned the anticipated return i_A every month.

(5)

Equation (4) therefore requires that, at time t, an overpayment at a rate of λ₁ is made based on the notional deficit D_t. If actual returns on the fund are persistently lower than the anticipated return i_A, then deficits will recur systematically. Therefore, Equation (4) also imposes a further overpayment, at a rate of λ₂, based on the cumulative sum of deficits up to time t. If there is no persisting deficit, that is, positive and negative deficits cancel each other out on average, then the additional overpayment represented by the third term on the right-hand side of Equation (4) is on average zero.

We do not directly address the optimal choice of parameters λ₁ and λ₂ here. Instead we assume a set of practical parameter values and investigate the targeted-contribution plan under various conditions such as the presence of parameter estimation errors, persistence and jumps in investment returns, investor cash flow constraints, and long-term retirement planning objectives. Our aim is to establish robustness of the targeted-contribution approach, and we leave formal optimization to future work.

Practical values for λ₁ and λ₂ in Equation (4) are likely to be small because a saver may be unable or unwilling to make additional contributions, and tax advantages may also mean that he is unwilling to reduce his contributions. In the following, our base parameter values are , as this represents paying a deficit over approximately 5 months (ignoring interest), and λ₂ = 0.01 as a charge or tax of 1 percent per month thereafter on the deficit in any month. These parameter values are comparable to values employed by Box and Luceño (1995) and Owadally (2003) in other contexts. In a later section, we consider other values for λ₁ and λ₂ and we also discuss parameter choice by reference to an ARMA time series process.

RESULTS OF STOCHASTIC SIMULATIONS

Stochastic simulations are performed to compare the terminal deficits under the two plans described in the preceding section. An investment horizon of T = 60 months and a target fund of F = 100 are assumed, along with values λ₁ = 0.2 and λ₂ = 0.01 in the targeted-contribution plan in Equation (4). A set of 10,000 simulations was carried out throughout, with frequent checks for convergence using 5,000 and 20,000 simulations, and with a fixed seed for the random number generator to minimize sampling error when comparing results.

The monthly logarithmic return δ_t = ln (1 + i_t) on the fund is simulated as an independent and identically distributed (i.i.d.) sequence of normally distributed random numbers with mean 0.3792 percent and standard deviation 2 percent. The mean arithmetic return is thus 0.4 percent per month. These are representative values for a relatively conservative investment fund.1

A financial planner has to estimate future investment return and make an assumption i_A as to the return on the fund. Since he does not have perfect foresight, he may over- or underestimate future average investment return. For example, if i_A = 0.7 percent, the planner overestimates the mean return by 0.3 percent per month. Henceforth, we define the estimation error as the excess of i_A over the average return on the fund. When i_A = 0.7 percent, the planner's estimation error is therefore 0.3 percent per month.

Histograms of Terminal Deficit

Figure 1 shows the distributions of the terminal deficit under the two plans, for i_A = 0.4 percent and 0.7 percent. Since the accumulated fund has a right-skewed distribution, the distribution of the deficit is left-skewed. We make two observations.

Observation 1

The fixed-contribution plan is riskier than the targeted-contribution plan (where we measure risk in terms of the fund falling short of the target at the end-date).

The terminal deficit is more dispersed under the fixed-contribution plan (top panels in Figure 1) than under the targeted-contribution plan (bottom panels in Figure 1). The fixed-contribution plan is riskier than the targeted-contribution plan in that end-wealth is more volatile and has a heavier-tailed distribution in the former case than in the latter case.

Observation 2

Risk in the fixed-contribution plan is sensitive to the error in the financial planner's estimate of future investment return, whereas risk in the targeted-contribution plan is relatively insensitive to this. (Again, we measure risk in terms of the shortfall or deficit from the target at the end.)

When the return assumption is 0.7 percent, rather than 0.4 percent, the adviser's return assumption overstates the mean return on the fund and we anticipate that the plans will, on average, fall short of the target. The distribution of the terminal deficit does indeed shift to the right when the return assumption changes from 0.4 percent to 0.7 percent in the fixed-contribution plan. However, it remains centered around zero in the targeted-contribution plan. If the planner's return assumption turns out to be overoptimistic, the risk of a shortfall is greater in the fixed-contribution plan than in the targeted-contribution plan.

DISCUSSION OF RESULTS AND SENSITIVITY ANALYSIS

Terminal Wealth and Interim Consumption

The targeted-contribution savings plan shifts uncertainty from terminal wealth to intermediate contributions. For a short-term savings plan, or one where contribution is a small proportion of income, the variability in contributions may have a negligible impact on the saver and on the saver's consumption during the saving period. On the other hand, for a long-term savings plan, or one where the contribution is a large proportion of income, large contribution requirements over a long period could be unaffordable.

To investigate this, we repeat the simulations of the earlier section with an unbiased estimate i_A = 0.4 percent. First, we calculate the variance of the fund and contribution at intermediate time steps.2 The results are displayed in Table 3. The contribution variance over time in the fourth column is significantly smaller than the variance in terminal wealth (at time t = 60) for either the fixed-contribution plan (88.363) or the targeted-contribution plan (10.095). Much more of the “total variance” is allocated to end-wealth than to interim contributions.

Table 3. Variance of Fund and Contribution Over Time Under Fixed-Contribution Plan (FC) and Targeted-Contribution Plan (TC)

Time (Months)	Var Fund		Var Contribution TC
	FC	TC
0	0	0	0
10	0.3581	0.1764	0.0089
20	2.7890	0.8225	0.0443
30	9.7172	2.0260	0.1102
40	23.9019	3.8799	0.2104
50	48.9548	6.5508	0.3520
60	88.3625	10.0954	0.5393

• Note: Contribution under FC has zero variance. F = 100, , i_A = 0.4%, λ₁ = 0.2, and λ₂ = 0.01.

Second, we recognize that individuals may have cash flow constraints. We repeat the earlier simulations except that we use a constrained contribution in the targeted-contribution plan

(6)

where C_t is as in Equation (4), C is the planned contribution in the fixed-contribution savings plan, and u provides an upper constraint relative to C, with u ≥ 1. The larger u is, the less budget constrained the saver is, that is, the more he is able to afford contributions that are larger than planned if investment returns are unfavorable.

We anticipate that, the more budget constrained the saver is, the more increases in contribution will be limited, and the less effective the targeted-contribution plan will be in compensating for lower than anticipated investment returns. Table 4 does indeed show that the lower u is, the more end-wealth risk increases in the targeted-contribution plan: the volatility and 95th percentile of terminal deficits in the targeted-contribution plan increase with decreasing u.

Table 4. Mean, Standard Deviation, and 95th Percentile of Terminal Deficits Under Fixed-Contribution Plan (FC) and Targeted-Contribution Plan (TC), for Various Upper Constraint Factor u

	Mean Deficit		Standard Deviation of Deficit		95th Percentile of Deficit
	FC	TC	FC	TC	FC	TC
No constraint	−0.036	−0.003	9.458	3.195	14.591	5.155
u= 2	−0.036	−0.015	9.458	3.249	14.591	5.185
u= 1.75	−0.036	0.044	9.458	3.310	14.591	5.465
u= 1.5	−0.036	0.270	9.458	3.530	14.591	6.189
u= 1.25	−0.036	1.072	9.458	4.215	14.591	8.560
u= 1.1	−0.036	2.451	9.458	5.153	14.591	11.806
u= 1.05	−0.036	3.313	9.458	5.631	14.591	13.397

• Note: FC is independent of u but statistics are shown for comparison. F = 100, , i_A = 0.4%, λ₁ = 0.2, and λ₂ = 0.01.

The upper constraint u does not affect the fixed-contribution plan, of course, but statistics are shown in Table 4 for the sake of comparison. We observe from Table 4 that the targeted-contribution plan, even when constrained, performs better than the fixed-contribution plan in terms of achieving a lower end-wealth risk. It is also of practical interest to note that, even if contributions in the targeted-contribution plan are constrained to be no more than 110 percent of the planned contribution under the conventional fixed-contribution plan (and no less than zero), the standard deviation of the terminal deficit is reduced by almost half.

Constraining contributions as in Equation (6) may be realistic to the extent that individuals are able to budget and form a view of an upper limit as to how much they can save every month. Nevertheless, the analysis in this article is limited in that risk is measured in terms of terminal wealth only. We note, as a direction for future research, that one could consider utility over both end-wealth and the interim consumption stream. Labor income risk then becomes relevant (Campbell and Viceira, 2002). If labor income is inversely correlated with investment return, a hedging effect is induced that may reduce combined risk in terms of both consumption and end-wealth.

The intermediate consumption pattern also matters in conventional fixed-contribution plans, in fact. A prudent financial adviser will underestimate future investment return (i.e., he will end up with negative values of the estimation error in Table 1) so as to minimize the 95th percentile of deficits. However, this can also result in large surpluses, in the individual making larger contribution payments than he can afford, and in a consequent loss in utility of consumption during the saving period. Again, we note this here as an item for further research.

Asset Return Model and Asset Allocation

In the earlier sections, we assumed that asset returns were normally distributed and serially independent. We investigate the relative performance of the fixed and targeted-contribution plans under two alternative investment return models here.

First, we consider leptokurtic asset returns by incorporating “jumps” in returns through a discrete-time version of a Lévy process. The jumps represent market corrections and crashes. We assume that the log-return δ_t = ln (1 + i_t) on the fund in month (t − 1, t) is given by

(7)

where {ε_t}, {κ_t} and {x_t} are independent sequences of random variables. {ε_t} is an i.i.d. sequence of zero-mean Gaussian random variables. We assume that μ = 0.00372 and Var ε_t = 0.02² so that μ + ε_t gives the normally distributed returns of the earlier sections (where the mean arithmetic return was 0.4 percent per month). {κ_t} is an i.i.d. sequence of Bernoulli random variables with probability distribution . Here p is small and is the probability of a “rare event,” such as a market correction, in a given month. Finally, {x_t} is an i.i.d. sequence of Gaussian random variables with mean μ_x and standard deviation σ_x. Given that a market correction occurs in month (t − 1, t), the size of the market correction is represented by x_t.

We repeat the work of the earlier section, with unconstrained contributions and with λ₁ = 0.2, λ₂ = 0.01 and i_A = 0.4 percent. Table 5 shows that the volatility and 95th percentile of terminal deficits are lower in the targeted-contribution plan than in the fixed-contribution plan, for various values of p, μ_x, and σ_x. The targeted-contribution plan is effective at reducing end-wealth risk compared to the fixed-contribution plan, even when asset returns have fat-tailed nonnormal distributions.

Table 5. Mean, Standard Deviation, and 95th Percentile of Terminal Deficits Under Fixed-Contribution Plan (FC) and Targeted-Contribution Plan (TC), for Various Parameters of Discrete-Time Jumps in Asset Returns, with λ₁ = 0.2, λ₂ = 0.01, i_A = 0.4%

p	μx	σx	Mean Deficit		Standard Deviation of Deficit		95th Percentile of Deficit
			FC	TC	FC	TC	FC	TC
No jump			−0.036	−0.003	9.458	3.195	14.591	5.155
1/60	−7%	0.1	3.234	0.203	11.210	3.838	21.414	6.052
2/60	−7%	0.1	6.381	0.342	12.450	4.407	26.322	7.323
3/60	−7%	0.1	9.273	0.502	13.394	4.870	31.183	8.990
1/60	−7%	0.1	3.234	0.203	11.210	3.838	21.414	6.052
1/60	−10%	0.1	4.629	0.282	11.509	4.039	23.750	6.437
1/60	−13%	0.1	5.958	0.358	11.963	4.296	25.889	6.916
1/60	−7%	0.1	3.234	0.203	11.210	3.838	21.414	6.052
1/60	−7%	0.2	2.489	0.188	15.102	4.990	26.140	6.547
1/60	−7%	0.3	1.206	0.148	20.914	6.659	31.584	7.473

In Table 5, p = 1/60 means that on average one jump occurs in a 5-year period. As p increases, market corrections become more frequent. As μ_x becomes more negative, the severity of downward market corrections increases. As σ_x increases, the volatility of market corrections increases.3 Table 5 shows that if a financial planner does not allow for jumps in asset returns, both the fixed and targeted contribution plans will experience greater end-wealth risk as either the frequency or severity or volatility of market corrections increases. However, end-wealth risk remains lower in the targeted-contribution plan compared to the fixed-contribution plan.

Next, we consider investment returns that are serially correlated (and normally distributed), as might be the case if the fund is invested in cash and fixed-income securities. We assume that the log-return δ_t on the fund in month (t − 1, t) follows an autoregressive process of order 1, AR(1)

(8)

where |α | < 1 and {ε_t} and μ are as defined earlier (Equation (7)). When α = 0, this gives serially independent returns with a mean arithmetic return of 0.4 percent per month.

Table 6 shows that the volatility and 95th percentile of terminal deficits are smaller in the targeted-contribution plan than in the fixed-contribution plan, for various values of parameter α. The lag 1 autocorrelation in {δ_t} increases as the autoregressive parameter α increases. (The variance of {δ_t} also varies with α.)

Table 6. Mean, Standard Deviation, and 95th Percentile of Terminal Deficits Under Fixed-Contribution Plan (FC) and Targeted-Contribution Plan (TC), for Various Values of AR(1) Parameter α, with λ₁ = 0.2, λ₂ = 0.01, i_A = 0.4%

α	Mean Deficit		Standard Deviation of Deficit		95th Percentile of Deficit
	FC	TC	FC	TC	FC	TC
−0.6	0.341	0.014	6.000	2.385	9.795	3.875
−0.4	0.270	0.011	6.795	2.516	10.998	4.070
−0.2	0.155	0.060	7.894	2.784	12.436	4.503
0	−0.036	−0.003	9.458	3.195	14.591	5.155
0.2	−0.386	−0.018	11.836	3.814	17.663	6.074
0.4	−1.135	−0.049	15.882	4.797	22.338	7.539
0.6	−3.246	−0.120	25.425	6.555	30.232	10.039

The targeted-contribution plan therefore appears to be less risky than the fixed-contribution plan, even with investment returns that exhibit nonnormality and serial correlation.

It is also worth highlighting that the asset allocation decision has been ignored in this article, for simplicity. As set out in the introduction, we assume that the individual is saving for a medium term of 5 years in a vehicle such as a variable-rate savings bank account or in a low-risk mutual fund. Asset allocation becomes more important with longer-term savings, and this is explored in detail by Campbell and Viceira (2002) among others. Asset allocation depends on the individual's coefficient of risk aversion and elasticity of intertemporal substitution of consumption, as well as on his labor income risk and investment horizon, and on issues such as taxation and transaction costs, etc. In the insurance and actuarial literature, asset allocation is discussed by Vigna and Haberman (2001), Taylor (2002), Owadally and Haberman (2004), Battocchio and Menoncin (2004), Cairns, Blake, and Dowd (2006), Emms and Haberman (2008), among others.

The asset allocation decision can be incorporated in future research. It is possible to model lifestyling portfolios, such as life-cycle or target-date funds (Shiller, 2005; Blake, Cairns, and Dowd, 2001). The method employed to adjust contributions can also be used to adjust asset allocation, as in Gandolfi, Sabatini, and Rossolini (2007).

AN APPLICATION TO PENSION FUNDING

In this section, we consider a long investment horizon and we use a bootstrap stochastic asset model resampled over 59 years of equity and bond return data. We implement the targeted-contribution method in the funding of a defined-benefit pension plan and investigate whether it leads to a better long-term funding position compared to a typical actuarial funding method. It is assumed that an employer sponsors the pension plan and pays contributions into a pension fund to provide retirement benefits to its employees.

We use the model pension plan described by Owadally (2003), except that he uses deterministic economic scenarios whereas we use a stochastic asset model. A key assumption is that employees' salaries, and also their pensions in retirement, grow in line with price inflation. All the monetary quantities that appear below are therefore be expressed in real terms, that is, net of price inflation.

Pension liabilities have a simplified structure. Employees join at the age of 20. The age profile of the pension plan membership is constant, as is the number of new entrants to the plan every year. A pension, indexed with price inflation, is paid when employees retire at the age of 65. Mortality follows a standard actuarial life table (English Life Table No. 12 for males). Payroll is constant in real terms and the real yearly benefit outgo is normalized to 1. The actuarial liability is calculated at 16.94 (in real terms) using the projected unit credit method and a real annual discount rate of 4 percent (McGill et al., 2004, p. 665).

On the asset side, the pension fund is assumed to be invested in UK equities and long-term Treasury bonds (gilts) only. We use the bootstrap method to simulate real returns on these assets (Efron and Tibshirani, 1993). We resample real annual returns from the equity and gilt indices calculated by Barclays Capital (2009) from 1950 to 2008, with income reinvested. The returns are net of price inflation, as given by the cost of living index also calculated by Barclays Capital (2009). The pension fund portfolio is rebalanced every year and a 60:40 equity:gilt ratio is maintained.

Annual actuarial valuations take place and the sponsor's contributions are varied to make up any deficit in the plan. The deficit, also known as an unfunded liability, is the excess of the actuarial liability over assets, and a surplus is just a negative deficit. We assume that there is no deficit initially, for example, from setting up the plan or from recent improvements to employee benefits. That is, the initial unfunded liability is zero. Alternatively, any initial unfunded liability could be amortized by means of a separate schedule of payments from the plan sponsor. (see, e.g., McGill et al., 2004, p. 620).

The pension plan actuary calculates actuarial gains and losses at yearly valuations of the pension plan. A gain (loss) occurs if experience is more (less) favorable than was assumed at the previous valuation (McGill et al., 2004, p. 621). We assume here that the only source of gains and losses is the investment performance of the pension fund. That is, all other economic and demographic factors turn out to be as assumed.

The typical actuarial practice is to amortize investment gains or losses over 5 years (McGill et al., 2004, p. 686), which is also assumed here. If investment returns are consistently lower than the investment return assumption i_A made by the actuary at each actuarial valuation, then investment losses will occur and will build up into a deficit. Higher employer contributions are then required as these losses are amortized. Our aim is to compare the amortization method with the targeted-contribution method given by Equation (4) (with a time unit of 1 year as opposed to 1 month).

Figure 2 shows the distribution of deficits after 40 years when the investment return assumption i_A overstates the long-term average real return on the fund by 3 percent per annum. (Using the data from Barclays Capital, (2009, the pension fund makes an average annual real return of 6.15 percent, so i_A = 9.15 percent here. Parameters λ₁ and λ₂ in Equation (4) are equal to 0.5 and 0.03, respectively. A 40-year horizon is appropriate because employees typically have a 40- to 50-year working lifetime.) The spread of deficits, and the probability of large deficits, appear to be smaller when the targeted-contribution method is used than when amortization is used.

Table 7 displays various statistics for the deficits under both the amortization and the targeted-contribution method, for different errors in the investment return assumption i_A. The standard deviation and 95th percentile of the deficit show that the targeted-contribution method is less risky than the amortization method in the sense that large deficits are less likely under the former. The targeted-contribution method is also less sensitive to errors in the investment return assumption, as the 95th percentile and standard deviation do not change considerably for different investment return assumptions. This mirrors earlier results.

Various commentators (e.g., Watson Wyatt Worldwide, 2009) have noted a shift from equities to bonds in pension fund asset allocation over the past few years. We repeated the simulations above for equity:gilt allocations of 50:50 and 40:60. The conclusions are essentially unchanged; that is, the targeted-contribution method results in smaller deficits and is less sensitive to the return assumption.

The average values of the pension fund assets and contribution over 40 years are shown in Table 8 for a 40:60 equity:gilt portfolio, which has a long-term average real return of 4.82 percent, and a real return assumption i_A of 5.82 percent (so that the error in i_A is 1 percent). Because the investment return assumption i_A is too optimistic, both the amortization and targeted-contribution methods start off with a low contribution from the employer initially. As deficits mount in the pension fund, the targeted-contribution method demands larger contributions more quickly than the amortization method. After 40 years, the fund levels off at an average value of 16.92 in the targeted-contribution case and 16.45 in the amortization case, compared to the actuarial liability value of 16.94. The average deficit is therefore almost zero with the targeted-contribution method, whereas it is never completely removed under amortization and a higher contribution is permanently required on average.

Note that under both methods, the pension fund is in balance after 40 years. In the amortization case, a fund of 16.45, with contribution income of 0.24 and benefit outgo of 1, accumulates at an investment return of 4.82 percent to 16.45 again after a year. In the targeted-contribution case, a fund of 16.91, with contribution income of 0.22 and benefit outgo of 1, accumulates at an investment return of 4.82 percent to 16.91 again after a year. However, the pension fund is in balance with a nearly zero average deficit in the latter case.

On the Barclays Capital (2009) data, autocorrelations on real equity returns up to a lag of 10 years were not significant at 5 percent. Autocorrelations on real gilt returns up to lag 10 were not significant at 5 percent, except at lag 5. This appears to have little physical meaning. Nevertheless, we also incorporated serial correlation in the simulations by resampling using blocks of 3 years and 5 years (Efron and Tibshirani, 1993). Again, the qualitative conclusions above are unchanged. (The detailed results are not included here to save space and are available from the authors.)

CONCLUSION

We considered a savings plan where contributions are received and invested for the purpose of meeting a given liability in the future. An example could be an investment and savings vehicle being used to meet school or college fees in 5 years time. In practice, financial advisers recommend a fixed and level regular contribution, based on an assumed investment return on the fund. Since investment returns are random and will be different from the assumed investment return, the fund usually undershoots or overshoots the target fund objective, leading to a deficit or surplus, respectively.

We investigated an alternative savings plan, based on a method borrowed from industrial process control and econophysics. Under this method, contributions are flexible and are adjusted and targeted systematically so that the fund objective is met. We used stochastic simulations to measure the risk that a terminal deficit occurs, that is, that end-of-period wealth falls short of the target fund. The targeted-contribution method appears to be less risky than the fixed-contribution method, in the sense that the volatility and 95th percentile of the deficit are smaller in the former case. In particular, the size of deficits in the targeted-contribution plan is less sensitive to errors in the financial planner's estimate of future investment return than in the fixed-contribution plan. These results appear to be robust under a wide range of parameter values, with and without constraints on contributions, with asset returns that are serially correlated, and with returns that incorporate large but rare market corrections and exhibit nonnormality. Finally, we applied the targeted-contribution method to a defined-benefit pension plan, with a long investment horizon and using bootstrap resampling from actual equity and bond market data. The standard deviation, mean square and 95th percentile of the deficit in the pension fund were again lower with the targeted-contribution funding method than with a conventional pension funding method.

A practical implication of this research for financial planners is that savings plans can be designed that are more flexible for individuals. A suitable design, such as the one described in this article, should be not only flexible but also targeted in a systematic way to achieve robust outcomes for individual investors.

This research can be extended in several directions, as discussed in the article. First, the asset allocation decision should be explored. Lifestyling portfolios can be readily included (Shiller, 2005). The method employed to adjust contributions can also be used to adjust asset allocation, as in Gandolfi, Sabatini, and Rossolini (2007). Second, the investment objective considered in this article was simple and related to the end-of-period wealth only. If the targeted-contribution plan is used for longer-term savings, such as retirement plans, then the utility of both wealth at retirement and consumption during the accumulation phase must be considered. Third, the choice of parameters for the targeted-contribution plan was discussed but further work, to optimize these parameters and with back-testing on historical data, is required. Fourth, practical considerations such as taxes and transaction costs can be included. Finally, the distribution of the deficit and the evolution of contributions in the targeted-contribution method are also of interest.