Ian Ayres writes (source, with further explanation, here):
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:
Option 1
Year 1 Invest $1
Year 2 Invest $2
Year 3 Invest $3
Option 2
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.
When I was young I “invested” in a house, which back then got a big tax subsidy in Britain, proved a fine inflation hedge, and provided a magnificent view over a village green to a castle, while being only a ten minute bike ride from the beach. When I was much older I “invested” in a pension, which got a bigger tax break than when I was young, because my marginal tax rate was higher, and because I would be tieing up the money for a shorter spell, being nearer to retirement. My guess is that such considerations often outweigh notional optimisation of the investment trajectory.
The only difference between the scenarios is a dollar invested in year 1 versus year 3. Assuming the investor had to borrow a dollar in year one, this boils down to whether the investment will yield a two-year return to cover the borrowing costs.
I largely agree with Tyler on this.
It is more than just comparing the expected 2 year return on stocks vs. the borrowing cost. It involves a whole pattern of risk that this last decade in general and the last 3 years in particular underscores. When talking about borrowing for stocks, we mean “margin.” Suppose that each period is a decade, so that we are talking about a worker in his 30s, 40s and 50s who is saving for a retirement in his 60s and beyond. This would mean a 30 year old with no savings would begin to invest say $5,000 per year in equities and then buy an additional $5,000 per year on margin. Not only does that change the risk path of investment, but it also introduces some serious liquidity issues. So, a 30 year old in 1990 would have done great with that strategy in period 1, but a 30 year old in 2000 would have not only lost most of his money (after paying margin interest), but he would have received several margin calls. “Yes, honey, we will have to stiff the mortgage company and the daycare this month, because I just got a margin call and need to send in some more cash.” The margin call forced sales would have decimated the returns. Also, following such a strategy involves assumptions about the risk pattern of employment, namely that the investor has a stable job with a stable income flow.
Personally, I think that risks in investment returns and in employment stability means that it is not optimal to consumption smooth. People are probably better off living less lavishly and saving and investing more (without margin) in younger years, at the risk of oversaving and having substantial consumption increases in later years. Such a strategy (one which I have personally followed) increases the probability of solid retirement funds, even if investment markets and/or employment markets perform worse than expected. Worrying about paying a mortgage or replacing a car in your 50s, because you cut it too close is no way to live your life, and if it happens you can’t turn back the clock.
Why not test the proposition in a worst case scenario? Here’s a tool where you can do just that using the historic data for the S&P 500, which also adjusts for inflation. Try it with Ayres’ or your own data, selecting June 1932 (or other period of your choice) as the end date for your investment holding period.
Maybe I’m missing something, but the time diversification argument is really just coming from some basic math in probability theory—here’s a model that’s hopefully not oversimplified. Anyone, feel free to correct misstatements.
Each January 1, you take some money out of your zero interest checking account and invest it in the stock market. Each December 31, you cash out. You do this for three years. The percentage returns for those three years are X1, X2 and X3, assumed to be independent with the same mean and standard deviation.
If you invest $2 each January 1 for three years, you’ll have total return of 2X1 + 2X2 + 2X3 at the end of the third year. The mean total return will be 6 * mean annual return.
Let s be the standard deviation of each year’s return. The variance of your total return will be (4+4+4) * s-squared = 12 s-squared.
If you invest in the pattern 1, 2,3, the mean will also be 6 * mean annual return. The variance will be (1 + 4 + 9) * s-squared= 14 s-squared.
For a given total exposure, the variance will be smallest if the investment pattern is as diversified as possible, with equal exposure each year. (This can be proven from Var (Y) = E(Y^2) – (E(Y))^2, where Y is the exposure.)
So, the pattern 1,2,3 and the pattern 3,2,1 are both worse than 2,2,2. You can dress up the model with more details, but this time diversification concept still affects the final result. It does not depend on varying total exposure to the market.
JRP, doesn’t your model and the dollar cost averaging theory depend on the stock market having random variance like a slot machine? If the “variance” reflects underlying values, then it doesn’t work, only if there is “variance” for no reason.
In your world, volatility harvesting and just buying every time the market goes down should also be profitable.
A traditional Arrow-Pratt risk averter need not mind price risk if he can *adjust* quantities after prices are revealed but there is no adjustment here by assumption.
Tyler, I was puzzled by your previous post and remain puzzled by this one. The book addresses the argument that this is just about historic returns in detail. The key insight is that even if you are worried that the past performance of stocks vs. bonds might not hold up in the future, that should affect only the proportion of your total portfolio over time that is made up of stocks. Whatever that proportion is, it still makes sense to try to diversify across time. The book also includes numerous simulations, both over actual data (in multiple countries) and with randomized methods, that show that you can use this strategy to achieve the same expected return and yet have a lower variance of returns, or to target the same variance of returns and receive a higher expected return.
Meanwhile, Gelman’s comment is silly. There are thousands of books by economists on specialized topics. To say that it is not worth writing a book that points out that the standard investment advice that everyone receives is wrong seems very odd. I suppose it’s true that rich kids are more likely to read it than middle class kids, but it’s not a zero sum game, and anyone can benefit from this.
OH! WAIT! I get it now!
I misread the conclusion of the notes I’d just posted. The conclusion, correctly read, *does* support the frontloading strategy. It does not say you should *add* the same amount to your investment each year; it says you should have the same total amount invested each year.
More precisely: Invest some amount the first year, observe your return, then scoop out profits or add to your investment so that you have the same total investment the second year as the first.
So—I now believe the notes I posted are correct, and that they accord with the intuition I quoted.
I would tend to disagree with borrowing to invest in non-useful assets. I do tend to agree with the sentiment that it is better to push the edge while young and attempt to invest as much as possible even while earnings are still low.
However, attempting to dollar-cost-average over a long time horizion is difficult since earning power does increase and thus you should be investing more in periods two and three versus period one.
When it comes to retirement accounts it is even more important to try and smooth out investment to the point of legal investment maximums since once the year is over that opportunity to invest the shortfall in a tax advantaged way is lost forever.
A more likely scenario is actually a reverse of example 1; a large investment in a small percentage of the periods with steady investments in the other periods. Specifically, I am describing situations such as a bonus or inheritance where the person has a lot of cash sitting around and can either A) slowly invest that cash into equities/bonds over the course of the next year+ or B) can invest all of it immediately. Under what conditions is each option preferable.
Steven Landsburg, you conclusion that it is best to have the same amount invested each year, seems plausible but impractical to implement for two reasons:
1. Lack of access to that kind of capital early in life, or to invest more when markets perform poorly.
2. How do you know what the amount to have invested in any (every) given year is?
What about a system that dollar cost averaged based on the Tobin Q ratio? The Q ratio being the ratio between stock market value and replacement cost. When the q is high, invest less, and when the q is low, lever up. This means you will miss out on bubbles like 1997-2000, but you’ll also miss out on the bubble bursting.
Why is Ian Ayres (law/econ background) getting attention for belaboring a straightforward investing problem? There is nothing new here.
There could be a few reasons why a person might invest more like Option 1 rather than Option 2. It is simply a lack of access to funds. Gradually as you get older, your income goes up, and so you would probably have more money to invest.
But there could be another reason. It could simply be inflation. A dollar invested today may be equivalent to 2 dollars invested 5 years later, and 3 dollars invested 10 years later. So what you might want to look at is the real rate of return, rather than the nominal rate of return.
Guys its very nice to read all your script
its like a movie all ups and downs its very nice conclusion of reading all this investment with care… is nice and it give good return investing in good stock and THE THEORIES OF INVESTMENT < theory of how much to invest OF OUR SAVING > ??? i think 50% IN REALITY 20% IN STOCKS 20% IN FIX DEPOSITS AND 10% IS IN HAND AS A CASH EVERY YEAR NO MATTER HOW OLD YOU ARE GIVES A GREAT RETURN AND GOOD SECURITY FOR FUTURE
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