## Abstract

The author presents a scheme that will help investors and financial advisers compare adaptive strategies for maintaining a robust financial plan over a time horizon. The article describes some methods for setting reasonable and attainable financial goals and suggests a schedule of investment allocation decisions designed to meet those goals. Monte-Carlo simulation provides data regarding the potential long-term outcomes associated with an adaptive plan. With this model, a number of different metrics for investment risk can be calculated to help the investor achieve an appropriate trade-off between minimizing investment risk and maximizing the probability of meeting or exceeding a portfolio target.

Mary is a financial adviser who enjoys a professional relationship with Jim and Sue, who wish to invest the money they are saving wisely. When she first met the couple, Mary helped them identify the assets in their portfolio and assess their value and quality. She also helped them to articulate some reasonable and attainable financial goals for the future. One of these goals is to accumulate money to help the couple pay for their daughter’s college tuition. Another goal is to earn and save enough to fund a long and comfortable retirement. Together, Mary, Sue, and Jim worked to determine how much of the portfolio to dedicate to each goal and how much of the family income to contribute to the portfolio each year, leaving enough for their day-to-day needs. For each goal, Mary determined which types of accounts to open, e.g., to take advantage of any available tax benefits. She also helped establish a plan for investing in a mix of asset classes (an “asset allocation”) so that expected investment growth, along with any planned contributions over time, would be sufficient to cover the payouts needed to achieve the goals. Now Mary dedicates much of her time with Jim and Sue to the tasks associated with maintaining the investment plan. Much of the groundwork they laid will not need to be retrod unless there are some unexpected events that require the couple to modify their goals, such as changes in the family’s employment or health situation, or if there is some big surprise regarding the performance of the assets in their portfolio.

For some of her clients, Mary recommends keeping to a plan that seeks to maximize the returns in the portfolio through a mix of investments that are at the upper threshold of their tolerance for risk. It is to be hoped that with such a strategy, the client will accumulate more than enough to satisfy his or her future needs. Maintaining such a plan entails periodically reviewing the assets in the accounts and rebalancing the asset allocation to best match the client’s evolving appetite for risk and return. As part of this review, it might also be opportune to buy and sell shares in individual securities and funds in the portfolio to take advantage of any tax benefits that are available and to fine-tune the composition of securities within the asset classes in accordance with any performance objectives sought by Mary and her adviser colleagues.

Investors like Jim and Sue, however, believe that the long-term financial goals that they set are reasonable, if not modest, and they are not willing to take on any unnecessary risk in an attempt to make more money than they believe they need in order to achieve those goals. To satisfy their needs, Mary creates a “goal-based” or “target-based” plan. For such plans, it is necessary for Mary to make sure that financial targets can be achieved using a combination of portfolio wealth, periodic contributions, and portfolio growth. A goal would be deemed unreasonable if it relies too heavily on asset growth, requiring an aggressive mix of investments that are beyond Jim and Sue’s tolerance for risk.

Once a reasonable goal is set, Mary establishes a *baseline schedule* of asset allocations that not only describes the investment plan for the immediate future, but shows how the returns sought (and associated level of investment risk taken) will change from one year to the next. This provides a sketch of the future that gives Jim and Sue some assurance that their goals are achievable and that there is a high likelihood that the target will be met within the time frame allotted. In addition, the schedule provides information that Mary could use to quantify for the couple the risks inherent in their plan and strategies to mitigate them over time (see Chhabra et al. [2008] and Das et al. [2011]).

Unpredictability and fluctuation in asset return rates over time are likely to cause the value of the investor’s portfolio to drift appreciably beyond what might have been expected in a target-based investment plan. For her periodic plan maintenance, Mary would need a strategy for how to respond when this happens. For example, if the portfolio value for a particular goal becomes somewhat higher than was expected, it would present an opportunity to reduce the level of risk in the portfolio’s asset mix, while still maintaining a reasonable likelihood that the goal will be met. Alternatively, a large drop in the portfolio value might compel her to suggest a more aggressive asset allocation, with a higher level of risk than planned, in order to meet the target. If Jim and Mary are not willing to do this, then Mary might have to suggest some modifications to the goal and help set up a revised plan to meet the new goal.

We use the term “adaptive strategy” to describe a set of rules governing when and how to modify the investment plan throughout the horizon in response to asset price fluctuation. In this article, we describe a simulation model that starts with a baseline investment plan and implements an adaptive strategy, assuming that yearly investment returns are normally distributed and the level of risk adopted in any year is functionally dependent on the return rate sought. Multiple runs of the simulation generate a distribution of estimated portfolio values that might be achieved during the “target” year. We would say that a plan is particularly robust if the distribution of these values has a small spread and the estimated probability of achieving or exceeding the target is high. Our comparisons consider two aspects to an adaptive strategy: the schedule of investment return rates to pursue now and in the future, and the periodic (e.g., yearly) activities performed by the adviser to maintain the investment plan.

**SETTING AN INVESTMENT SCHEDULE TO MEET A TARGET**

Financial advisers must have accurate data regarding available asset classes (e.g., stocks, bonds, cash equivalents, and alternative investments), their expected performance for the immediate future, and their correlations. From these data, it would be possible to evaluate a particular asset allocation and provide reasonable estimates for the overall mean and standard deviation of the percentage rate of return, adjusted for inflation (see Reilly and Brown [2009] and Scherer and Martin [2005], for example). Portfolio optimization models use this data to establish a functional relationship between the mean and the standard deviation of this random variable based on the efficient frontier, which identifies asset allocations that simultaneously minimize risk for a given level of return and maximize the average return for a given level of risk. The asset allocation options are illustrated in Exhibit 1.

For a given value of the mean return rate that the investor might like to pursue, such a chart would provide the framework for the asset allocation. The associated investment risk, as measured by the standard deviation of the random variable representing the yearly return, is usually a convex function of the mean, implying that any increase in the level of returns sought in a year requires that the investor accept a greater than proportional level of increased investment risk.

For illustrative purposes, suppose that Jim and Sue decide that they would like to pursue an asset allocation with an inflation-adjusted mean return rate of *r*. The associated risk is represented by the standard deviation, e.g., *SD*(*r*) = 12 *r*
^{2} + 0.25 *r* + 0.005. This particular risk function describes an environment in which it is necessary for investors to take on a small amount of risk, measured as a standard deviation of 0.5%, in order to keep up with inflation. The allocation would likely focus on bonds and cash equivalents. A more aggressive investment mix, with focus on stocks and high-yield alternative investments, has a mean after-inflation return rate that is close to 10% per year. It has the maximum level of investment risk that Mary would allow for Jim and Sue (having a standard deviation of over 15%). In our illustration, we assume that this functional relationship holds throughout the planning horizon, even though a different function may be used in different years. For example, a reasonable implementation for long-term planning might entail using one function for the first three to five years to represent the currently expected economic environment and a different function for the remaining years that represents longer-term (e.g., historical) trends in asset performance and correlation.

In order to develop an investment plan for a client having a specific goal, advisers like Mary need to determine the values for *X*
_{0}, the current value of the investment portfolio; *c _{t}
*, the expected contribution into the portfolio that Jim and Sue expect to make during year

*t;*and

*T*, the nominal target value for the portfolio by target year

*n.*We represent a multi-year investment plan as a sequence of decision variables for the mean return rates to be pursued over the next

*n*years:

*r*

_{1},

*r*

_{2}, …

*r*. From these decisions, associated values for the means and standard deviations can be determined for the random variables that represent the value of the portfolio over time,

_{n}*X*

_{1}…

*X*(see Cosares [2013]).

_{n}For each goal, the values of *r*
_{1}, *r*
_{2} … *r _{n}
* are constrained to be such that the mean value of the portfolio after

*n*years,

*E*(

*X*), is equal to the target value

_{n}*T*. The most important of these decision variables is the value for

*r*

_{1}, the rate of return to pursue in the coming year, since it dictates the actual investment decisions to be implemented immediately. Values for the remaining decision variables are important for target-based planning, but serve mostly as guidelines. As we shall see, the actual investment schedule is likely to be adjusted over time, depending on how assets actually perform and how the market for investments evolves. Obviously, the schedule would also change if it were necessary for Mary to generate a new plan in response any adjustments that Jim and Sue make to the goals.

We point out that we use a single target value in the model because the focus of this article is to measure adaptive strategies for the “accumulation phase” of investment planning, e.g., for retirement. During this phase, it is expected that asset price fluctuation would vex the adviser. Since we expect the investment strategy during the subsequent “spend-down” phase to be somewhat conservative, it will be inflation risk rather than investment risk that would have to be hedged (see Arnott et al. [2013]). We also point out that in practice, an adviser using the model for investment planning might set the target value *T* to be greater than what is actually needed to meet the client’s goal in order to assure a greater likelihood of success.

There are a number of different approaches to finding values for the variables representing the investment schedule (see, e.g., Bruynel [2005], Basu and Drew [2009], Graf [2011], and Cosares [2013]). We will focus on two in particular. The first is a “Level” schedule, where we set all of the *r _{i}
* to a single value,

*r**. The asset allocation is set to achieve this mean rate, and we assume that the investor will take on essentially the same level of risk and achieve roughly similar rates of return each year. The second is a decreasing “glide-path” schedule, where the investor seeks a higher return and takes on a higher level of investment risk during the immediate future, when it is assumed that there is more time to recover from potential losses. With this approach, it is hoped that the investor will achieve greater rates of portfolio growth in the early years, so that long-term targets might be met with asset allocations that have progressively smaller risk and more predictable returns as the target year approaches.

Goals having different time frames and/or differences in their relative importance to the investor must be treated differently (see Brunel [2011]). For shorter-term goals, there are fewer opportunities to recover from a loss, so the difference between a level and a glide-path schedule is not as pronounced as with longer-term goals. For Jim and Sue’s goal to set aside enough for their daughter’s upcoming college expenses, Mary would likely suggest a conservative, wealth-preserving investment strategy with predictable returns. In the absence of an unexpected calamity, there will be few adjustments needed to maintain the plan; the money that Jim and Sue have saved/earned for this goal will likely be available when it is needed.

The approach should be somewhat different with the longer-term goal. Suppose Jim and Sue’s retirement portfolio is currently worth about $200,000; they plan to make annual contributions of $15,000, increasing with inflation, for the next 30 years, and have determined that their retirement goals could be met with about $1,500,000 in today’s dollars. Exhibit 2 provides the two alternative investment schedules for this case.

It would be necessary for Jim and Sue’s investments to earn overall about 4% per year, over inflation, in order to meet the target. The investment risk associated with this rate is calculated to be a yearly standard deviation of about 3.4%. If this level of risk were not acceptable, then Mary would have to suggest some adjustments to the financial goal. We assume that this will not be necessary. The glide-path approach schedule, which is also deemed acceptable for consideration, front-loads much of the investment risk; a mean return rate of close to 6% will be pursued during the first year, allowing for much more conservative investments later on. The schedules in both cases are set so that the mean of the random variable *X _{n}
* is equal to the target of $1,500,000.

The standard deviation of the random variable *X _{n}
* measures the

*accumulated investment risk*associated with a schedule. This is a dollar value that represents the cumulative effect of subjecting the portfolio value to investment risk from one year to the next. It is determined from the following recursive relationship:

*SD*(

*X*) =

_{n}*SD*((

*1*+

*R*) *

_{n}*X*

_{n-1}), where

*R*is the random variable representing the actual return rate for year

_{t}*t*and

*SD*(

*X*

_{0}) is equal to 0. We assume that

*X*(the wealth accumulated by year

_{t}*t*), and

*R*

_{t+1}(the return rate during the subsequent year) are independent.

As is most often the case, the accumulated investment risk for the level schedule ($220,000 in this case) is less than that of the glide-path schedule ($223,000) (see Bruynel [2005]). The accumulated risk measure has two parts. The first, which we call *dollars-at-risk*, represents the total dollar value of the risk taken when the portfolio is invested according to the investment schedule from one year to the next. The second part represents the *correlation risk* associated with the evolving value of the portfolio. Glide-path schedules are obtained when one seeks to minimize the dollars-at-risk, i.e., by reducing the level of investment risk as the portfolio grows (see Cosares [2013]). For the case illustrated, the dollars-at-risk measure is $144,000. The value of the level schedule is larger ($156,000 in this case), because over time the increasing expected wealth in the portfolio would be continually subject to the same level of investment risk.

**SIMULATION-BASED MEASURES FOR RISK**

For each goal, we can run a Monte Carlo simulation model to observe the potential growth of the portfolio under a variety of randomly generated scenarios. For example, we first observe the performance of a baseline strategy, where Jim and Sue pursue the return rates suggested by the scheduling model without any modifications in reaction to asset performance. The plan maintenance activity each year is limited to rebalancing the asset allocation to align with the scheduled mean rate of return. We assume that yearly return rates are normally distributed with mean *r _{i}
* and standard deviation

*SD*(

*r*) and we collect the values of

_{i}*X*, the portfolio value at the end of the target year, associated with 5,000 randomly generated scenarios.

_{n}The distribution of portfolio values generated by the simulation allows for more precise measures of the risks associated with an investment strategy. For example, by calculating the standard deviation of the final portfolio values, we obtain what we call the *spread risk*, which measures how far beyond the target a strategy might vary. For this baseline strategy, the value is approximately equal to the accumulated investment risk anticipated in the scheduling model.

The distribution of final portfolio values also allows us to estimate the probability that any specific value will (or will not) be met. For example, the table below gives the proportion of scenarios in which *X _{n}
* meets or exceeds the target, within a margin of 5%. For this particular example, both schedules give rise to similar risk profiles.

Next, we use the simulation model to evaluate an adaptive strategy that Mary might apply as part of the plan maintenance, in order to mitigate one or more of these risks. For example, suppose that, after every year, Mary completely updates the scheduling model to reflect the actual value of the portfolio, rather than the expected value. This might give rise to a revised schedule that Jim and Sue would agree to adopt. The results from the simulation model that reflect the application of this strategy for the first 15 years of the planning horizon are given below. (Although the strategy will be applied every year, the information provided by the simulation for a longer time frame has little value over that of the baseline.) The third column in the chart gives the *average* of the accumulated investment risk values associated with the schedule of return rates actually pursued in each of the simulated scenarios.

Notice that the application of this straightforward adaptive strategy requires an overall increase in cumulative risk because of the potential fluctuations in the investment schedules. However, it reduces the spread risk and increases the likelihood of reaching the target. Notice also that the glide path appears to make more effective use of this particular adaptive strategy.

The simulation model provides details that allow for even deeper analysis. For example, if we were to isolate the scenarios in which the portfolio suffered a loss during the first year, we could evaluate how well an adaptive strategy allows for some form of recovery. The values of the risk metrics for our illustrative case are given in the table below.

Because the glide-path schedule suggests taking a greater risk during the first year, the percentage of applicable cases (16.2%) is larger than that for the level schedule (11.8%). The first two columns show that the adaptive schemes are particularly effective in recovering from an early loss, especially when the glide-path schedule is implemented. The third column shows that for these cases, it will be necessary to take on a marginally greater investment risk in order to do so.

The adaptive strategy illustrated above might be considered too sensitive to random fluctuations in asset performance because it responds every year by modifying the investment schedule, even if the deviation from expectation is small and likely to be of little consequence over the long term. As a matter of fact, after a modest unexpected gain or loss, experienced advisers like Mary often suggest that their clients “stay the course” and not react prematurely. Surely, Jim and Sue would not be happy if they were told in one year that they could reduce their investment risk, only to be told in a subsequent year that they have to become more aggressive. We can use the information provided by the simulation model to measure the extent to which an adaptive strategy is reactive (or over-reactive, as the case may be) by looking at the distribution of return rates pursued in each simulated scenario at a given time, say five years out. For example, the baseline strategy for the level schedule in our example always suggests the same return rate, but the rates among the scenarios for the adaptive strategy in that year have a standard deviation of 0.16% (with a mean of 4.08%). For the glide-path schedule, the distribution of rates pursued in year five has a standard deviation of 0.38%, with a mean of 5.24%.

Using an alternative adaptive strategy, Mary might elect to modify the investment schedule only when the projected value of the portfolio falls 5% or more below the target, or 10% or more above the target (it being considered appropriate to respond more aggressively to potential shortages than to potential overages). The simulation model representing implementation of this strategy provides the following results:

By dealing more moderately with potential shortages and allowing more gains to go without a response, the adaptive strategy has an improved rate of success, but with a greater overall risk profile. However, the spread in the distribution of return rates pursued in year five has decreased, indicating a more stable approach to dealing with minor asset price fluctuations. For the level schedule, the standard deviation is reduced to 0.09%, and for the glide path, the distribution of rates has a reduced standard deviation of 0.32%.

**SUMMARY**

There are multiple approaches to solving the mathematical programming formulation for the dynamic asset allocation problem. Each solution provides an alternative schedule for how much and when to take on investment risk to achieve some financial goal for the future. By calculating statistics for the random variable representing the projected wealth during a given target year, e.g., the standard deviation, we can compare alternative schedules and can measure the overall risk needed to meet a goal. The glide paths determined by the model provide value over “off-the-shelf” investment options, like target-date funds, because they take into consideration the investor’s specific target, his or her plan for portfolio contributions, and his or her preferences regarding the timing of the investment risk.

In practice, however, the lion’s share of the information provided by the scheduling model will be rendered irrelevant once time passes and the actual performance of the assets deviates from what was expected in the model. So any simulation that projects the future should take into consideration the adjustments to the schedule that will likely be implemented as the investment plan is maintained. We find that the actual risk that the investor takes on during the course of the planning horizon is likely to be greater than what was anticipated in the scheduling model, in particular when the portfolio suffers an early loss. In addition, we find that approaches that appear to have similar risk characteristics during the planning development phase are likely to differ after the plan is put into motion.

By providing a variety of simulation-based measures for the performance of some adaptive strategy, we hope to provide the means by which an adviser can fine-tune a strategy for his or clients that will give them faith that their planning needs over the long term will be satisfied. Information provided by the simulations could be integrated with other important customer relationship data, so that the adviser may track progress toward meeting the client’s financial planning needs and improve his or her client retention.

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