This is a Sunday feature looking back at selected articles from the early days of this blog before readership had ramped up. Enjoy.
Most commentators advise investors to shift money from equities to fixed income as they age, and this makes sense. We may disagree on the exact asset allocation percentages and exactly how soon before (or after) retirement to start lightening up on equities, but it seems clear enough that the average 40-year old should have more in equities than the average 80-year old.
However, there is a body of academic work that argues that investors should maintain a constant asset allocation regardless of their age. This work is based on what is known as constant relative-risk aversion (CRRA). I’ll show the problems with CRRA in an example below.
One consequence of the CRRA assumption is that the optimal asset allocation percentages remain constant regardless of the length of the investor’s investment horizon. Paul A. Samuelson advocates this view in his keynote address to “The Future of Life-Cycle Saving and Investing” conference.
To show the problem with CRRA, I’ll consider a case where it gives silly results. To do this, I’ll have to pick a particular constant level of risk aversion. Morningstar uses the CRRA assumption in their star rating system for mutual funds. I’ll base my analysis on the risk aversion level chosen by Morningstar.
A Fun Example
A beer company decides to promote their new brand of beer sold in cases of 24 by roaming around to different bars giving away $24 to people who are drinking the new brand. The prize crew enters a bar and descends on a table where two friends, average Avery and rich Richard, are sitting drinking the new brand of beer.
As both men accept their $24 prizes, Avery is thrilled, but Richard is only mildly pleased. Avery comments “I guess a rich guy like you doesn’t get very excited by a lousy $24. How much bigger would your prize have to be to make you as excited as I am?”
The answer to this question depends on how much money Avery and Richard have. Suppose that Avery’s net worth is $120,000, and Richard’s is $6 million. According to the CRRA assumption, to make Richard as excited as Avery, his prize would have to be $340 million! This is obviously ridiculous.
Just for fun, let’s make some seemingly inconsequential changes. Each man calculates his net worth more accurately, and Avery’s comes out $20 higher than he first thought, and Richard’s comes out $2400.10 higher than he first thought. Now, the CRRA says that Richard’s prize must be $11 trillion!
You may find it hard to believe that CRRA could get these answers so horribly wrong, but that is what the math says. In fact, if Avery’s prize were $25, then according to CRRA there would be no amount you could give to Richard to make him as happy as Avery.
Is This a Fair Test? Yes, it is.
Most theories are based on approximations and simplifying assumptions. These theories can be shown to be wrong in extreme or unimportant cases. However, my example is directly relevant to long-term investing. CRRA gives answers that are close to reality for small risks. But, for the large dollar amounts involved in long-term investing, CRRA gives ridiculous answers.
Matthew Rabin has an interesting paper where he discusses the problems of extrapolating from people’s attitudes about small risks to making decisions about large risks.
In truth, CRRA is more of a mathematical convenience than a valid theory. With CRRA, how much money you have and how much money you want to have aren’t relevant when deciding on your asset allocation. Without CRRA, Morningstar would not be able to give star ratings for mutual funds without knowing your net worth and your future needs for money.
So, if anyone tries to tell you to maintain the same asset allocation for your whole life, remember that this advice is based on a faulty theory.