Simple Rules of Thumb for Investing

Perhaps less well known than Markowitz's modern portfolio theory (MPT) is the subsequent work of Merton and Samuelson. This is a shame because while MPT only concerns itself with optimizing investing in a single time period, Merton's portfolio model concerns itself with optimizing over time, where it is possible to change asset allocation and consumption in response to portfolio performance. This is far closer to the problem faced by most investors. Unfortunately the math involved is quite complex. I've been trying to derive some very simple rules of thumb for stock/bond asset allocation and consumption planning using Merton's portfolio model and the current returns environment as a guide. Here is what I came up with:

stocks percent of portfolio = 50% with future income treated like bonds = 50% * (1 + If / W)
(but not to exceed your risk tolerance)

annual retirement portfolio consumption = W / e

both evaluated each year, where:

If = future income (expected value of future defined benefits and retirement savings contributions without any discounting for simplicity; will often be equal to annual Social Security multiplied by life expectancy)
W = portfolio size
e = remaining life expectancy

There is a lot of simplification to get this far, but empirically these rules appear to stand up well to simulated testing. The certainty equivalent retirement income (CE) is the constant amount of income received over retirement that an individual would be neutral to receiving compared to the uneven uncertain amount they expect to receive using the given strategy. For one reasonable accumulation/decumulation life cycle scenario (male born 1950; accumulating \$500/year starting at age 25 growing by 7% real/year; retiring age 65 Social Security \$15,000; coefficient of relative risk aversion 4; subjective discount rate 3%; historical stock/bond data from Shiller 1927-2013 massaged to 4.5%/0.5% real 20%/10% stdev with returns chosen at random (no autocorrelations)) the CE was:

```                               CE     CE less \$15,000 Social Security
age in bonds/4% rule         \$20,964       \$5,964
simple rules                 \$25,390      \$10,390 (74% better)
SDP (theoretical maximum)    \$25,676      \$10,676 (79% better)```

In other words for this scenario the simple rules delivered close to the theoretical maximum, and substantially more than age in bonds/4% rule (for the age in bonds rule future income was not treated like bonds).

The rest of this article explains how the two rules were derived.

Model assumptions:

stocks: 4.5% real geometric, 6.5% real arithmetic, 20% standard deviation
bonds: 0.5% real geometric, 1.0% real arithmetic, 10% standard deviation
covariance: 0.0, i.e. independent

No taxes are assumed, such as for savings in an IRA or 401k. Leverage is not assumed to be available.

Consider for a moment a constant relative risk averse utility function with a coefficient of relative risk aversion, gamma, of 4. Merton provides a simple formula for the optimal stock proportion assuming bonds are risk free:

total stocks percent = (stock return - bond return) / (stock stdev ^ 2 * gamma)
= (6.5% - 1.0%) / (20% ^ 2 * 4)
= 34%

Note that it is the difference between stock and bond returns that matters. As a result, provided they are roughly the same for both stocks and bonds, management expenses don't enter into the picture.

Two problems. Bonds aren't risk free. This is especially true of bond funds which try to maintain a constant maturity. Computing the optimal asset allocation in this case requires the use of matrix algebra, as shown by this R script. The script solves for the risk free rate at which the risk free holdings are zero, so that stocks plus bonds equals 100%. Second gamma might not be 4. Running the script with different values for gamma produces the following table:

```gamma  total stocks percent
1        130%
2         75%
3         57%
4         48%
5         42%```

What does gamma mean? A dollar at consumption level C/2 is 2^gamma times more valuable than consumption of a dollar at C. Most investors probably have a gamma of 3 or 4. If you have exceeded your required and desired consumption levels, and are spending on philanthropic projects gamma might be 0.5 or even lower. For most investors though a total stocks percent of 50% would seem reasonable.

Most investor's already hold substantial virtual bond like assets in the form of expected Social Security payments and even future planned retirement savings contributions. These offset the need to hold bonds in the investment portfolio, increasing the amount of stocks that need to be held. Setting:

stocks percent of portfolio = 50% * (1 + If / W)

results in:

total stocks percent = stocks percent of portfolio * W / (W + If)
= 50% * (W + If) / (W + If)
= 50%

as required.

For many investors stocks percent of portfolio is likely to significantly exceed their risk tolerance, and quite possibly exceed 100%. Indeed if If > W, which will be true for most investors until at least close to retirement age, then stocks percent of portfolio > 100%. Note that the If / W term could explain rising glide paths late in retirement when the planned life expectancy has been exceeded.

The ability to short is limited. If shorting is not allowed this has the effect of increasing stocks when stocks is less than 100% to compensate. Conversely if future income is less volatile than bonds this will reduce stocks. Also in an ideal world future income should be discounted over time at about the bond rate reducing stocks. On balance 50% seems like a reasonable rule of thumb, but there are clearly limits to it.

Consumption is trickier. Merton provides a formula for optimal consumption but it depends upon life expectancy being fixed. I attempted to modify it to handle a stochastic lifespan:

annual retirement portfolio consumption = nu * (W + Ifa * e * ef) / (1 - exp(- nu * e * ef)) - Ifa

where:

Ifa = annual future expected income
ef = a life expectancy scaling factor to account for stochastic life span (1.5 was found to be a good value)
nu = 0.00942 for the model when gamma is 4 and no subjective time discounting is being performed

I am being lazy in not discounting estimated future income. I can get away with this because the discount rate is small. If nu is small enough, and life expectancy is fixed, the optimal consumption is W / e. In reality we should consume more because the portfolio experiences growth over time (ignoring withdrawals), but we should consume less to set aside funds for the possibility that we live longer than our life expectancy. On balance it is thought these two largely cancel out and the simple rule, W / e, seems to work quite well.

A more sophisticated approach of using the complex formula was tried, but while it performed 3% better for large portfolios it performed 4% worse for small portfolios (where there was a shorting constraint). Lacking a consistent advantage for the complex formula, I elected to use the simple formula, W / e. (Using VPW is another possibility. If you get the parameters right, it performed 1% better than the simple formula for affluent and constrained investors alike. W / max(e, 8) was also found to be a contender: 3% better for the affluent, 1% better for the constrained.)

These two rules are very simple, but they have strong theoretical underpinnings, and they work well empirically. Like all rules of thumb there are limits. These limits are likely to be hit when gamma is not a constant but drops as consumption becomes satiated, if real interest rates ever rise significantly and we are no longer able to ignore the need for time discounting of future income, or there is interest in leaving a bequest. In these cases it may be better to perform stochastic dynamic programming (SDP) to compute the optimal asset allocation and consumption.