A key decision when designing an investment portfolio is the allocation between equities and bonds. Equities are the primary contributor to the total portfolio risk.

An intuitive view is that younger investors can afford to take more risk than older ones, so the allocation to equities should decline with time. Is this correct? Constant or time-varying equity allocation?

We can choose a constant value for the equity allocation (e.g., 60%). Even though the portfolio grows, continuous rebalancing keeps the equities proportion constant. An alternative is to allow the proportion of equities to change with the investor’s age. This gives rise to what is commonly referred to as a glide path, where the decline may be linear or curved (see Chart 1).

## Is a glide path better than a constant equity allocation?

To answer this question, we have to be precise about what we mean by “better.” We assume the investor’s objective is to reach a cumulative wealth target with minimum risk, where risk is the spread of possible outcomes as measured by the standard deviation. Consider two investor portfolios, A and B, as illustrated in Figure 1.

Portfolio A has, on average, accumulated wealth of \$200,000, with a standard deviation of \$40,000. Over the same investment period, Portfolio B has the same accumulated wealth but a lower standard deviation of \$30,000. Portfolio B is preferred over Portfolio A because it achieves the same wealth with greater certainty.

Both of these portfolios have the same risk, but Portfolio A has, on average, lower accumulated wealth than Portfolio B. So, Portfolio B is also preferred because it achieves a higher accumulated wealth with no greater risk.

So, we define “better” as a glide path portfolio being better than constant equity allocation if they both have the same cumulative wealth but the glide path portfolio is lower risk.

Given that risk increases when equity allocation increases, you could also say that a glide path portfolio is better than a constant equity allocation portfolio if both have the same risk but the glide path portfolio has a higher cumulative wealth.

## Linear glide path with a lump sum investment

Suppose the equity allocation is given by p(t) such that:

Where t is the age of the investor in years

For example, a 35-year-old would have an asset allocation of 65% equity and an 85-year-old would have an asset allocation of 15% equity.

Consider the example of a 35-year-old investor who has a single lump sum of \$100,000 to invest. Say the investor has an inheritance he wants to pass to his children. According to the above formula, he reduces his equity allocation every year until age 85. Using a statistical simulator, we’ve computed the expected wealth and the risk as measured by the standard deviation. Chart 2 shows the expected returns and standard deviations as a function of the portfolio equity allocation.

We’ve also calculated that the constant equity allocation yields the same cumulative wealth as the glide path (for the math, see bit.ly/2hlnKyZ). For a single lump sum investment, this is the average equity allocation over the period, which in our case is simply (65% + 15%) / 2 = 40%.

We compare the decreasing equity glide path and the constant equity allocation in Table 1, below.

## Table 1: Decreasing equity glide path and constant equity allocation

Cumulative average wealth Standard deviation
Decreasing equity (A) \$ 723,386 \$ 259,414
Constant 40% equity (B) \$ 722,303 \$ 244,293
Difference, (B–A)/A -0.15% -5.82%

Source: PWL Capital, using Returns2 from Dimensional Fund Advisors

As expected, the terminal wealth is similar—and within modelling error—but the range of distribution of that wealth, as represented by the standard deviation, is less for the constant equity allocation. An investor concerned about reaching a terminal wealth would likely prefer the constant equity asset allocation, where the likelihood of achieving the goal is greater.

Suppose we reversed the direction of the glide path so that the investor started with a low equity allocation that increased with age:

In this case, the equity allocation at age 35 is 15% and the equity allocation at age 85 is 65%, as illustrated in Chart 3. One would think equities should decline with age, so we might anticipate this to be a poor strategy.

The results are shown in Table 2, below.

## Table 2

Cumulative average wealth Standard deviation
Decreasing equity (A) \$ 723,386 \$ 259,414
Constant equity (B) \$ 722,599 \$ 257,245
Difference, (B–A)/A -0.04% -0.08%

Source: PWL Capital, using Returns2 from Dimensional Fund Advisors

Increasing equity allocation with time is, within modelling error, no different from decreasing equity allocation over time, and both are worse than a constant 40% equity allocation. These results are a reminder that compounded wealth is a series of annual returns and the sequence of those returns has no impact on the final result. As shown below, this situation changes when there are contributions during the period.

A more general mathematical analysis shows we can always choose a constant equity allocation equal to the average equity allocation, which has the same expected wealth—but with lower risk than using a glide path strategy.

## Linear glide path with regular savings

We have considered the impact of asset allocation only on lump sum investing. Next, we consider the more realistic scenario where investors are saving regularly.

Consider the example of an investor who starts with nothing, but saves \$10,000 annually from age 35 to age 65: a savings period of 30 years. The equity allocation declines with age. The average equity allocation over the savings period is now 50% (see Table 3).

## Table 3

Cumulative average wealth Standard deviation
Decreasing equity (A) \$ 619,258 \$ 120,944
Constant 50% equity (B) \$ 635,996 \$ 137,228
Difference, (B–A)/A 2.70% 13.42%

Source: PWL Capital, using Returns2 from Dimensional Fund Advisors

A constant equity allocation that is also the average equity allocation is no longer a good substitute for the glide path portfolio. The cumulative wealth is now dependent on the timing of the savings. In our example, we estimate the appropriate constant equity that yields the same cumulative wealth as the glide path to be 46% (again, for the math, see bit.ly/2hlnKyZ).

In Table 4, below, we compare the results and note that there is no significant difference between the standard deviation of a constant equity allocation of 46% and the glide path portfolio.

## Table 4

Cumulative average wealth Standard deviation
Decreasing equity (A) \$ 619,258 \$ 120,944
Constant 46% equity (B) \$ 615,591 \$ 121,207
Difference, (B–A)/A – 0.59% – 0.22%

Source: PWL Capital using Returns2 from Dimensional Fund Advisors

For completeness, we repeat the analysis with regular savings and a glide path that increases with time. The equity allocation is 35% at age 35, increasing to 65% at age 65. The constant equity allocation, generating the same cumulative wealth, is estimated at 55%. As expected, the risk is also the same, within modelling error (see Table 5, below).

## Table 5

Cumulative average wealth Standard deviation
Decreasing equity (A) \$ 659,117 \$ 160,818
Constant 55% equity (B) \$ 663,794 \$ 159,302
Difference, (B–A)/A 0.71% –0.94%

Source: PWL Capital, using Returns2 from Dimensional Fund Advisors

## Similar outcomes

While glide paths may appear to offer a better approach to accumulating wealth, they can be replaced with a constant equity allocation with very similar outcomes as measured by cumulative wealth and the standard deviation of the cumulative wealth. For lump sum investing, the best constant equity allocation is the simple mean, and the risk is lower than using a glide path.

For regular savings with a declining equity allocation with age, the best constant equity allocation is slightly less than the simple mean. If the slope of the glide path is reversed, so that the equity allocation increases with age, then the best constant equity allocation is slightly higher than the simple mean. Changing the slope of the glide path has only a modest impact on the choice of constant equity allocation.

## Implications for target date funds

Target date funds (or lifecycle funds) use the idea of a glide path, so that the equity allocation decreases with age, and are popular in the U.S. (with US\$763 billion in assets), mainly within defined contribution savings plans. Their presence is growing in Canada with offerings from Fidelity (Clearpath), Blackrock (LifePath) and Vanguard (Target Retirement). They offer the attraction of simplicity: the investor only has to decide their retirement date and put all their savings in the fund with the same target date. The fund manager takes care of the rest.

This simplicity can be a real advantage if the alternative is that the investor makes a poor choice of individual funds or, worse, no choice at all. Despite the apparent sophistication of target date funds, a simple balanced fund with a constant equity allocation can perform as well before fees.

Target date funds are typically a fund of funds with higher fees. For example, the Fidelity target date funds typically have management fees (excluding advisor fees) of 1% to 1.15%. By comparison, the DFA Global 60% Equity, 40% Fixed Income fund (excluding advisor fees) has a cost of 0.42%. We estimated a 0.5% additional fee cost on our investor would be \$53,510, resulting in 8.65% less cumulative wealth, if he saved \$10,000 annually for 30 years.

Our conclusion is that target date funds offer little or no advantage over adhering to a fixed asset allocation if the goal is building maximum wealth with minimum risk before fees. After fees, they are poor value. In the U.S., target date funds are Qualified Default Investment Alternatives, which means that DC plan contributors who do not specify an investment vehicle for their contribution (and possibly their employer’s) can find themselves defaulted into target date funds. The concern is that such a default allows the fund providers to take more money in fees without offering any value beyond what could be achieved with a lower-cost balanced fund with a constant equity allocation.

## A victory for constant asset allocation?

If an investor is focused solely on achieving a target wealth while minimizing the risk of missing that target, then we conclude that a constant equity asset allocation is likely to be preferable to a glide path strategy, after fees.

A more fundamental question is whether prescribing the asset allocation, while ignoring other useful information, is the best way of achieving a wealth target. Examples of information that should be included as time progresses are:

1. An investor’s wealth, since it influences their subsequent choice of asset allocation. If, for example, the market is so favourable that the wealth target is achieved early, why would the investor continue to risk equity exposure? This suggests that equity allocation should be a function of current wealth, which is dependent on the investor’s experience in the market.
2. The cost of meeting a future obligation, which depends on interest rates. Falling rates means the cost of meeting a future liability rises. If the investor is saving to match a future liability (e.g., a pension requirement) then the end wealth also needs to change to match the changing liability as interest rates change

Recent studies suggest that adaptive schemes using both the time to the target wealth and the current portfolio value can achieve the same cumulative wealth as a constant equity allocation portfolio with significantly lower cumulative and downside risk. Glide paths are here to stay, but as an output rather than an input to portfolio design.

by Graham Westmacott, portfolio manager, PWL Capital Inc.