* Crossposted from my blog *

### Summary

In this post I review the 2010 book “Lifecycle Investing” by Ian Ayres and Barry Nalebuff. (Amazon link here; no commission received.) They argue that a large subset of investors should adopt a (currently) unconventional strategy: One’s future retirement contributions should effectively be treated as bonds in one’s retirement portfolio that cannot be efficiently sold; therefore, early in life one should balance these low-volatility assets by gaining exposure to volatile high-return equities that will generically exceed 100% of one’s liquid retirement assets, necessitating some form of borrowing.

“Lifecycle Investing” was recommended to me by a friend who said the book “is extremely worth reading…like learning about index funds for the first time…Like worth paying >1% of your lifetime income to read if that was needed to get access to the ideas…potentially a lot more”. Ayres and Nalebuff lived up to this recommendation. Eventually, I expect the basic ideas, which are simple, to become so widespread and obvious that it will be hard to remember that it required an insight.

In part, what makes the main argument so compelling is that (as shown in the next section), it is closely related to an elegant explanation for something we all knew to be true — you should increase the bond-stock ratio of your portfolio as you get older — yet previously had bad justifications for. It also gives new actionable, non-obvious, and potentially very important advice (buy equities on margin when young) that is appropriately tempered by real-world frictions. And, most importantly, it means I personally feel less bad about already being nearly 100% in stocks when I picked up the book.

My main concerns, which are shared by other reviewers and which are only partially addressed by the authors, are:

- Future income streams might be more like stocks than bonds for the large majority of people.
- Implementing a safe buy-and-hold leveraged strategy in the real world could be even more of a headache, and incur more hidden costs, than the authors suppose.
- If interest rates go up enough and expected stock returns go down enough, the whole thing could be rendered moot.

By far the best review of this book I’ve found after a bit of Googling is the one by Fredrick Vars, a law professor at the University of Alabama: [PDF]. Read that. I wrote most of my review before Vars, and he anticipated almost all of my concerns while offering illuminating details on some of the legal aspects.

### A key puzzle

One way to frame the insight, slightly different than as presented in the book, is as arising out of a solution to a basic puzzle.

The vast majority of financial advisors agree that retirement investments should have a higher percentage of volatile assets (stocks, essentially) when the person is young and less when they are old. This is often justified by the argument that volatile returns can be averaged out over the years, but, taken naively, this is flat out wrong. As Alex Tabarrok puts it^{a }

Many people think that uncertainty washes out when you buy and hold for a long period of time. Not so, that is the fallacy of time diversification. Although the average return becomes more certain with more periods you don’t get the average return you get the total payoff and that becomes more uncertain with more periods.

More quantitatively: When a principal is invested over years in a fund with a given annual expected return and volatility (standard deviation) , the average of the yearly returns becomes more certain for more years and approaches for the usual central-limit reasons. However, your payout is not the average return! Rather, your payout is the compounded amount^{b }

and the uncertainty of that does *not* go down with more time…*even in percentage terms*. That is, the ratio of the standard deviation in payout to the mean payout, , goes *up* the larger the number of years that the principal is invested.

Sometimes when confronted with this mathematical reality people backtrack to a justification like this: If you are young and you take a large downturn, you can adapt to this by absorbing the loss over many years of slightly smaller future consumption (adaptation), but if you are older you must drastically cut back, so the hit to your utility is larger. This is a true but fairly minor consideration. Even if we knew we would be unable to adapt our consumption (say because it was dominated by fixed costs), it would *still* be much better to be long on stocks when young and less when old.

Another response is to point out that, although absolute uncertainty in stock performance goes up over time, the odds of beating bonds *also* keeps going up. That is, on any given day the odds that stocks outperform bonds is maybe only a bit better than a coin flip, but as the time horizon grows, the odds get progressively better.^{c } This is true, but some thought shows it’s not a good argument. In short, even if the chance of doing worse than bonds keeps falling, the distribution of scenarios where you lose to bonds could get more and more extreme; when you do worse, maybe you do *much* worse. (For an extensive explanation, see the section “Probability of Shortfall” in the John Norstad’s “Risk and Time“, which Tabarrok above linked to as “fallacy of time diversification”.) This, it turns out, is not true — we see below that stocks do in fact get safer over time — but the possibility of extreme distributions shows why the probability-of-beating-bonds-goes-up-over-time argument is unsound.

### Puzzle resolution

To neatly resolve this puzzle, the authors make a strong simplifying assumption. (Importantly, the main idea is robust to relaxing this assumption somewhat,^{d } but for now let’s accept it in its idealized form.)

The main assumption is that the portion of your future income that you will be saving for retirement (e.g., your stream of future 401(k) contributions) can be predicted with relative confidence and are financially equivalent to today holding a sequence of bonds that pay off on a regular schedule in the future (but cannot be sold). **When we consider how our retirement portfolio today should be split between bonds and stocks, we should include the net present value of our future contributions.** That is the main idea.

Under some not-unreasonable simplifying assumptions, Samuelson and Merton showed long ago^{e } that if, counterfactually, you had to live off an initial lump sum of wealth, then the optimal way to invest that sum would be to *maintain a constant split* between assets of different volatility (e.g., 40% stocks and 60% bonds), with the appropriate split determined by your personal risk tolerance. However, even though you won’t magically receive your future retirement contributions as a lump sum in real life, it follows that if those contributions were perfectly predictable, and if you could borrow money at the risk-free rate, then you should borrow against your future contributions, converting them to their net present value, and *keep the same constant fraction of the money in the stock market*. Starting today.

Crucially, when you are young your liquid retirement portfolio (the sum of your meager contribution up to that point, plus a bit of accumulated interest) is dwarfed by your expected future contributions. Even if you invest 100% of your retirement account into stocks you are insufficiently exposed to the stock market. In order to get sufficient stock exposure, you should borrow lots of money at the risk-free rate and put it in the stock market. It is only as you get older, when the ratio between your retirement account and the present value of future earnings increases, that you should move more and more of your (visible) retirement account into regular bonds.

The resolution of the puzzle is that the optimal portfolio (in the idealized case) only *looks* like it’s stock-heavy early in life because you’re forgetting about your stream of future retirement contributions (a portion of your future salary), which, the authors claim, is essentially like a bond that can’t be traded.

(If the above concept isn’t immediately compelling to you, my introduction has failed. Close this blog and just go read the first couple chapters of their book.)

### But what about practicalities?

Most of the book is devoted to fleshing out and defending the implications of this idea for the real world where there are a variety of complications, most notably that you cannot borrow unlimited amounts at the risk-free rate. Nevertheless, the authors conclude that when many people are young they should buy equities on margin (i.e., with borrowed money) up to 2:1 leverage, at least if they have access to low enough interests rates to make it worthwhile.

The organization of chapters are as follows:

- Basic idea and motivation
- Theory. Outline of lifecycle strategy.
- Comparison of lifecycle strategy with conventional strategies on US historical data
- Responses to various objections
- Implications for older investors, inheritances, and trusts
- Contraindications – who shouldn’t use the strategy
- Risk tolerance and details
- Mechanics of implementing the strategy
- Macroimplications: What if everyone did it? How do we bring that about?

In general the authors compare their lifecycle investing strategy to two conventional strategies: the “birthday rule” (aka an “age-in-bonds rule“), where the investor allocates a percentage of their portfolio to stocks given by 100 (or 110) minus their age, and the “constant percentage rule”, where the investor keeps a constant fraction of their portfolio in stocks.

In Chapter 3, the authors argue that the lifecycle strategy consistently beats conventional strategies when (a) holding fixed expected return and minimizing variance, (b) holding fixed variance and maximizing expected return, (c) holding fixed very bad (first percentile) returns while maximizing expected return. If you look at a hypothetical ensemble of investors on historical data, one retiring during each year between 1914 and 2010 (when the book was published), every single investor would have been had more at retirement by adopting the lifecycle strategy, and generally by an enormous 50% or more. Here’s the total return of the investors vs. retirement year depending on whether they following a lifecycle strategy, birthday rule, or the constant percentage rule:

And here are the quantiles:

Although they rely on historical simulations for this, it’s really grounded in a very simple theoretical idea: your liquid retirement portfolio is extremely small when you’re young, so for any plausible level of risk aversion, you are better off leveraging equities initially.

Chapter 4 considers more testing variations: international stocks returns, Monte Carlo simulations with historically anomalous stock performance, higher interest rates, etc. They also show the strategy can easily be modified to incorporate (possibly EMH-violating) beliefs about one’s ability to time the market. (The authors use Robert Shiller’s theory of cyclically adjusted price-to-earnings ratio, which they neither endorse nor reject.)

In Chapter 7, the authors draw on the work of Samuelson and Merton to address the key question: what is the constant fraction in stocks that you should be targeting anyways? Assuming assumptions, the optimal “Samuelson share” to have invested in stocks is

The variables above are defined as follows.

- : The equity premium, i.e., the difference in the expected rate of return between stocks and (perfectly safe) bonds.
- : The equity volitility, i.e., the standard deviation of the annual equity rate of return.
- : The relative risk aversion, a measure of an individual investor’s trade-off between risk and reward.

The authors give reasons to be wary of taking this formula too seriously, especially because it’s not so easy to know what you should choose (discussed more below). However, it is very notable that as equity volatility increases — say, because the world is gripped by a global pandemic — the appropriate amount of the portfolio to have exposed to the stock market drops drastically. The authors suggest using the VIX to estimate the equity volatility, and appropriately rebalancing your portfolio when that metric changes. Continuously hitting the correct Samuelson share without shooting yourself in the foot looks hard, in practice, which the authors admit. Still, there is so much to gain from leverage that it’s very likely you can collect a good chunk of the upside even with a conservative and careful approach.

### Criticism

#### Risk tolerance intuition

The first general point of caution tempers (but definitely does not eliminate) the suggestion to invest in equities on margin: one’s risk tolerance is not an easy thing to elicit. To a large extent we do this by imagining various outcomes, deciding which outcomes we would prefer, and then inferring (with regularity assumptions) what our risk tolerance must be. Therefore, it would likely be a mistake to immediately take whatever risk tolerance you previously thought you had *as deployed in conventional investment strategies* and then follow the advice in this book. After introspection, I’ve sorta decided that although I am still less risk averse than the general population, I’m more risk averse than I thought because I was following the intuition (which I can now justify better) that I should be heavy in stocks at my age. The authors address the general difficulty of someone identifying their own risk tolerance (e.g., how dependent it is on framing effects), but they do not discuss how your beliefs about your risk tolerance might be entangled with what investment strategy you have previously been using.

However, this bears repeating: For every level of risk tolerance, there exists a form of this strategy that beats (both in expectation and risk) the best conventional strategy. The fact that, when young, you are buying stocks on margin makes it tempting to interpret this strategy is only good when one is not very risk averse or when the stock market has a good century. But for any time-homogeneous view you have on what stocks will do in the future, there is a version of this strategy that is better than a conventional strategy. (A large fraction of casual critics seem to miss this point.) The authors muddy this central feature a bit because, on my reading, they are a bit less risk averse than the average person. The book would have been more pointed if they had erred toward risk aversion in their various examples of the lifecycle strategy.

#### Retirement as a rainy day fund

The second point of caution is gestured at in the criticism by Nobel winner Paul Samuelson^{f }. (He was also a mentor of the authors.) The costs of going *truly* bust would be catastrophic:

The ideas that I have been criticizing do not shrivel up and die. They always come back… Recently I received an abstract for a paper in which a Yale economist and a Yale law school professor advise the world that when you are young and you have many years ahead of you, you should borrow heavily, invest in stocks on margin, and make a lot of money. I want to remind them, with a well-chosen counterexample: I always quote from Warren Buffett (that wise, wise man from Nebraska) that in order to succeed, you must first survive. People who leverage heavily when they are very young do not realize that the sky is the limit of what they could lose and from that point on, they would be knocked out of the game.

The authors respond to these sorts of concerns by emphasizing that (1) the risk of losing everything is highest when you are very young, which is exactly when the amount you have in your retirement account is very small, and (2) they are recommending adding leverage to your retirement account, not all your assets. If you expect the total of your retirement contributions to be roughly $1 million by the time you retire, losing $20,000 and zeroing out your retirement account when you are 25 is not catastrophic (and is still a rare outcome under their strategy). You should still have a rainy day fund, and you’ll just earn more money in the future.

However, I don’t think this response seriously grapples with the best concrete form of the wary intuition many people have to their strategy. I think the main problem is that most people are implicitly using their retirement account not just as a place to save for retirement assuming a normal healthy life, but also as a rainy day fund for a variety of bad events. In the US, 12% of people are disabled; I don’t know how much you can push down those odds knowing you are healthy at a given time, but it seems like you need to allow for a ~3% chance you are partially or totally disabled at some point. Although people buy disability insurance, they also know that if they ever needed to tap into their retirement account they could, possibly with a modest tax penalty. (Likewise for other unforeseen crises.)

Another way to say this: your future earning are substantially more likely to fail than the US government, so they cannot be idealized as a bond. By purchasing the right insurance, keeping enough in savings account, etc., I’m sure there’s a way to hedge against this, and I’m confident the core ideas in this book survive this necessary hedging. But I would have liked the authors to discuss how to do that in *at least* as much concrete detail as they describe the mechanics of how to invest on margin.^{g } If people have been relying on the conventional strategy and have consequently been implicitly enjoying a form of buffer/insurance, it is paramount to highlight this and find a substitute before moving on to an unconventional strategy that lacks that buffer.

Now, if we only had to insure against tail risks, that would be fine, but there is an extreme version of this issue that has the potential to undermine the entire idea: why is my future income stream like a bond rather than a stock? I have a ton of uncertainty about how my income will increase in the future. Indeed, personally, I think I trust the steady growth of the stock market more! The authors do advise against adopting their strategy if your future income stream is highly correlated with the market (e.g., you’re a banker), but they don’t get very quantitative, and they don’t say much about what do if that stream is highly volatile but not very correlated with stocks. (Sure, if it’s uncorrelated then you’re want to match your “investment” in your future income stream with some actual investment in stocks for diversification, but how much should this overall high volatility change the strategy?^{h })

### So did I immediately go out and lever my portfolio, or what?

It will take some time before I have mulled this around enough to even start assessing whether I should be investing with significant leverage. It seems pretty plausible to me that my future income is much more uncertain than a bond, although that’s something I’ll need to meditate on.

I, like the authors, really wish there was a mutual fund that automatically implemented this strategy, like target-date funds do for (strategies similar to) the birthday rule. At the very least it would induce pointed discussion about the benefits and risks of the strategy. Unfortunately, a decade after this book was released there is no such option and, as the authors admit in the book, concretely implementing the strategy yourself in the real world can be a headache.

However, because of this book I can at least feel less guilty for being overwhelmingly in equities. After finishing this book I finally exchanged much of my remaining Vanguard 2050 target-date funds, which contain bonds, for pure equity index funds. I had been keeping them around in part because going 100% equities felt vaguely dangerous. Now that there is a good argument that the optimal allocation is *greater* than 100% equities — though that is by no means assured — this no longer feels so extreme. Crossing the 100% barrier by acquiring leverage involves many real-world complications, but in the platonic realm there is nothing special about the divide.

### Footnotes

(↵ returns to text)

- I have edited the broken link for “fallacy of time diversification” to point to an archived version of the web page.↵
- The approximation is valid for .↵
- I thank Will Riedel for this compelling phrasing.↵
- Although the authors don’t quantitatively explore this enough. See criticisms below.↵
- Looks like Merton’s version of the problem is the most well known. Here are the references taken directly from the book: “Paul A.. Samuelson, “Lifetime Portfolio Selection by Dynamic Stochastic Programming,” Review of Economics and Statistics 51 (1969): 239-246; Robert Merton, “Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Case,” Review of Economics and Statistics 51 (1969): 247-257; and Robert Merton, “Optimum Consumption and Portfolio Rules in a Continuous Time Model,” Journal of Economic Theory 3 (1971): 373-413.”↵
- Ayers replies here.↵
- Indeed, I suspect that many of the valid criticisms of their strategy apply almost as well to conventional investment strategies. For example, many of us should probably have more disability insurance; if you lost the ability to work when you were 30, would things work out OK? The lack of leverage in a conventional portfolio, combined with the fact that the stock market is quite unlikely to lose more than, say, 60%, means that the conventional portfolio naturally includes some weak coverage of bad scenarios. But this is essentially accidental, and very unlikely to be optimal.↵
- The author mention in passing that their friend Moshe Milevsy has written an entire book on the question: “Are You a Stock or a Bond?”. But as their entire strategy hinges on this question, they should have addressed it much more deeply themselves.↵