In our risk-reducing implementation, we want people to borrow to invest more when young and then invest less when older. The lifetime exposure to stocks is held constant. Compare the following two investment paths:
Year 1 Invest $1
Year 2 Invest $2
Year 3 Invest $3
Year 1 Invest $2
Year 2 Invest $2
Year 3 Invest $2
Our view is that option 2 is the safer bet.
That is a simple comparison, but I am still puzzled.
First, under option one, the investor faces more price risk, given that share prices will move between period two and period three. But price risk is not quantity risk, and a traditionally risk-averse investor, in the Arrow-Pratt sense, need not mind price risk on net. If you tell me I can buy into Microsoft at p = 50, or accept a lottery with 0.5 (p = 100), 0.5 (p = 0), I might prefer the latter, noting that we live in an equity premium scenario and sometimes share price understates fundamental value.
Second, under option one the investor buys less equity in the first period. Let's assume that same investor holds T-Bills instead, or perhaps avoids debt. There is nothing in the so-called "mutual fund theorem" which tells us what the T-Bills/equity ratio should be. Maybe Ayres and Nalbuff think it should be higher — and maybe they're right — but that's an argument distinct from the "smoothing purchases over time" argument.
Is there a "smoothing equity purchases over time" argument which does not collapse into the "it's better to have a longer total exposure to the market" argument? Or not? Do Ayres and Nalebuff think it is OK if the "smoothing" argument boils down to an "there is actually an equity premium, so seek greater total market exposure over time" argument. (For instance, what if option one is the non-smoothed, but highly exposed sequence "3, 2, 1" instead of the specified non-smoothed but lesser exposed "1, 2, 3"?
I am very willing to admit that the confusion here is mine, but confused I am. Here is my previous post on the book.
Addendum: Andrew Gelman comments.