## Monday, March 10, 2008

### Equity Allocation: A New Approach

In an earlier post I was looking at what fraction of your portfolio should be in stocks. I also listed Larry Swedroe’s table of time horizon vs. stock percentage from his book “Rational Investing in Irrational Times”.

His table basically says to put everything in stocks if you won’t need the money for 20 or more years. The stock percentage then drops steadily to zero when you are three years from needing the money. I’ve been looking for some justification for this advice.

The answer comes from considering the utility of money. The basic idea of utility is that the wealthier you are, the less an additional dollar is worth to you.

An Example

Suppose that if you invested your entire portfolio in risk-free investments, you would have \$1 million when you retire. A game show host then makes you the following offer. You can just take the \$1 million or you can toss a coin to get either \$800,000 or \$2 million. Would you take the sure \$1 million or would you take the chance?

What I really want to know is how much lower (or higher) than \$800,000 you would be willing to go before the two options looked equally attractive to you. This gives an idea of your level of risk aversion.

Risk Aversion

I developed a model of risk aversion based on the idea that underperforming the risk-free return causes pain, and the answer to the coin-flip question determines the amount of pain. More details on this model are at the end of this post.

In applying this model, I assumed that your pain level is always relative to what you could get with risk-free investments. So, if your investments perform extremely well for a few years, and risk-free investing the rest of the way would give you \$1.5 million, you would consider it painful to underperform \$1.5 million even though a few years earlier the pain was relative to the \$1 million level.

Similarly, if your investments perform poorly for a few years, and risk-free investing would give you \$750,000, you would consider it painful to underperform \$750,000.

The Results

I did the computations assuming the following. You consider a coin flip between \$800,000 and \$2 million to be equally desirable as a sure \$1 million. Also, you consider a coin flip between \$700,000 and \$4 million to be equally desirable as a sure \$1 million. This is a high degree of pain for underperformance. I also used the stock returns and volatility from the paper Portfolio Optimization by John Norstad (2002-09-11).

Here is the resulting table of time horizon vs. stock percentages. The percentages are rounded to the nearest 5% because any further precision is pointless.

1 year: 25%
2 years: 30%
3 years: 35%
4 years: 40%
5 years: 50%
6 years: 55%
7 years: 65%
8 years: 75%
9 years: 85%
10 years: 95%
11 years or more: 100%

Don’t get too attached to this table, though. The results are very sensitive to the answers to the coin-flip question. Consider the following two cases.

Case 1: You consider the coin flips (\$900,000 or \$2 million) and (\$850,000 or \$4 million) to be equally desirable as a sure \$1 million. In this case, you would be less than half in stocks even 30 years from needing the money.

Case 2: You consider the coin flips (\$700,000 or \$2 million) and (\$550,000 or \$4 million) to be equally desirable as a sure \$1 million. In this case, you would be 100% invested in stocks even during the last year before needing the money.

In conclusion, I doubt that there will ever be any single answer to the equity allocation question. I tend to lean toward staying fully invested in stocks until I’m about 3 years from needing the money, but I may change my mind if I find a convincing argument to take a different approach.

The Model (lots of math)

I came up with a series of points on a graph that reflected the pain of underperformance, and then found an equation to fit this curve. The challenge was to find an equation that made sense even in extreme cases, but also made the math easy enough to work with when applying the model.

Here are the parameters and the equation:

X – The final dollar amount after the returns on investments.
T – A threshold dollar amount below which the investor starts to feel pain.
a – A parameter for the incremental pain once underperformance is deep.
b – A parameter for determining what is considered to be deep underperformance.
P – Pain function.

P(X) = (X/T)^(a(1-(X/T)^b))

The idea here is that when optimizing expected compound returns, any return X that is below threshold T is changed to the smaller value X*P(X) to reflect the investor’s pain. I don’t know if anyone has used this particular pain function before.

The function P may not look very simple, but when we operate in the log domain it permits fairly easy integration when multiplied by the normal probability density function. All optimization is done in the log domain because we are optimizing compound returns while using the pain function. If we let p=ln(P), x=ln(X), and t=ln(T), then we get

p(x) = a(x-t)(1-e^(b(x-t))),

and if x is below the threshold t, we replace x with x+p(x). Note that p(x) is negative when x is less than t.

The parameters a and b were computed so that the function P(X) would match the coin-flip answers. For the table I listed above, these parameters were a=4.74 and b=2.64.

When I applied this model, the threshold of pain was always set at the risk-free return over the remaining years of investing. I began by finding the optimal stock percentage for the last year before the money is needed. Then I optimised the case where there are 2 years left to go knowing what would happen in the last year. I continued this for 40 years worth of results.