* 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.↵

There is a grain of truth in what the book says but I offer four caveats for the reader to consider.

1. Basing expected returns on the US market is an egregious case of selection bias. The US market is an outlier and an unexpected outlier at that. Suggesting in 1900 to anyone that they put their net worth into the US market would have been a brave move - a ruinous civil war, rampant corruption, booms and busts, etc - Are you Joking! "Triumph of the Optimists" has some figures for other markets but not for all - many markets went to zero and never recovered. Yes you can diversify globally but this cuts the returns almost in half compared to the US market's recent history.

People will often say they are bullish on America. This is an easier argument to dispute than it used to be - "So, nothing really serious can go wrong in a country that elected Donald Trump as president?". But more seriously one's feelings of confidence are a very poor prognostic indicator, as Japanese investors found post 1989.

2. Any mention of the normal distribution or the central limit theorem in relation to financial markets opens you to huge errors. This is the ludic fallacy - markets are not tame and do not comply with tractable probability distributions. The returns from one year to the next are not independent and identically distributed, and nor is the underlying distribution necessarily tame enough for the CLT to apply. I suggest to rework the numbers with more realistic distributions such as Student's t or a power law distribution. Results may be worse than you intuitively expect.

3. There is a more subtle problem... Books advocating leverage, stocks for the long run, index and forget etc, tend to appear after a run-up in the market (as in this case after a 50% surge after the GFC slump). People tend to invest in this way also. Last year, as the market was making new highs several people advised me that they had decided on stocks for the long run because stocks always outperform bonds (until they don't - consider the Japanese stock market, currently at 50% lower than its level in 1989). Many of these "long term holders" have since sold out, perhaps close to the bottom. My suggestion is that anyone feeling an urgent and pressing need to invest in the market for the long run may do well to trickle-feed their money into the market over a period of a few years.

4. Terrible market returns often coincide with hard times for the portfolio owner, such as unemployment, slumps in the value of other assets and other difficulties. Having a leveraged portfolio that went to zero or beyond ("losses can exceed your initial investment" as they say in the fine print) in 1932 would have been very unfortunate. Margin purchases of stocks were very popular in 1929, and in general high levels of margin lending seem to be an indicator of trouble ahead.

I do commend the study of markets to the LW community. There are so many interesting aspects to it - psychology (yours and others'), cognitive biases, subtle statistical issues and many lessons on the limitations of vanilla statistics, the subtleties of risk management in the real world, the difficulties of a system comprising intelligent adaptive agents, agency issues. And it is a way to put your insights into the nature of reality to the test.

FYI they redo the analysis for the FTSE and the Nikkei and they come to the same conclusion. Also, the theoretical analysis comes out the same even if returns are lower in the future than they have been in the past.

Lower expected return does mean putting a lower share into the risky asset, but expected returns would have to go very low indeed (w/o a corresponding drop in expected volatility) for the analysis not to suggest that those just starting out should use leverage. (2x leverage is way undershooting the target that the math suggests, but they suggest maxing out at 2x leverage for various practical reasons. If expected returns were a bit lower, then 2x would probably still be below the theoretical target for people at the beginning of their careers.)

I am curious about this. It's my impression that assets tend to become more correlated in a downturn. I'm not sure how much this, or the presence of fat tails, affects things, but their back test on at least three different countries' data mitigates my concern somewhat.

Happily, this was at least not the case here. The book was written in 2008/2009, and published in 2010, just after the financial crisis. And we're reading this review during the coronavirus pandemic when the S&P is still down 15% from the start of the year.

This is a fair point, which I think was not addressed well enough in the book. But which was addressed well in Jess's review! (See e.g. his discussion of disability insurance.)

(I don't know how it applies to this model, but...) price movements are not normally distributed, and any model that assumes they are carries a major risk of blowing up. For example: during the financial crisis Goldman Sachs chief financial officer David Viniar infamously told the Financial Times “we were seeing things that were 25-standard deviation moves, several days in a row."

What are the chances that a

25-sigma event strikes your investment portfolio?We should expect a 4σ event to happen twice in our lifetime. A 5σ event occurs about every 5000 years, or once since the beginning of recorded history. A 6σ event might have happened roughly twice in the millions of years since

homo sapiensbranched off from the other apes. A 7σ event comes along every billion years or so, or four times since our planet coalesced out of a cloud of interstellar dust. We pass the Big Bang somewhere around the 8σ mark. At 20σ, the number of years we’d have to wait is ~10x higher than the number of particles in the universe, etc.(which is to say, Goldman and friends' models were disastrously, absurdly, cosmologically wrong.)

AFAIK Benoit Mandelbrot was the first to start warning people about this, and his PhD student Eugene Fama wrote his thesis on it...back in 1965! Which gives you a sense of how crazy it is that people would still try to apply normal distributions to financial markets.

Mandelbrot's book The Misbehaviour of Markets is worth a read. I've also written a summary of his ideas here, in the context of stress-testing the assumptions of the 'early retirement' movement.

I endorse ESRogs' replies. I'll just add some minor points.

1. Nothing in this book or the lifecycle strategy rests on anything specific to the US stock market. As I said in my review

If you are bearish on stocks as a whole, this is incorporated by you choosing a lower equity premium and hence lower overall stock allocation. This choice is independent of the central theoretical idea of the book.

2. Yours is a criticism of all modeling and is not specific to the lifecycle strategy.

3. As ESRogs mentioned, neither this book nor my review has the timing you suggest, so the psychoanalysis of proponents of this strategy appears inconsistent.

4. I acknowledged this sort of argument in my review, and indeed argued that the best approaches hinges on such correlations. But consider: even in the extreme case where I believes my future income is highly correlated with the stock market and is just as volatile, the lifecycle strategy recommends that my equity exposure should start

lowwhen I'm young and thenincreasewith age, in opposition to conventional strategies! So even if you take a different set of starting assumptions from the authors, you still get a deep insight from their basic framework.Or, to put your comment more succinctly, the book talks about several variables as if though they are independent (e.g. market mid-term ROI, personal income, amount of leverage most brokers provide), but historically speaking these variables have always been heavily correlated.

FWIW, the book does discuss the correlation of one's income with stock market returns. They cite a study on this (from early in the 90s I believe) suggesting that most people's income correlations with the market are between 0 and 20% (with many retail workers even having negative correlation with the market!). I was surprised how low those numbers where when I read that. I'd be curious to look into this more.

Your DCA recommendation in 3 is found by several studies to be riskier and lower return than lump sum investing. Do you believe those are similarly flawed?

Young investors who shouldn’t do this. (from the book)

Other contraindications

Could there possibly be as much as 1% of the population for which this is an appropriate asset allocation?

It's always very frustrating to see people's critiques of these models rely on assumptions of high leverage ratios. Moderate leverage (in the realm of 1.5:1) is a mathematical no brainer given return to volatility ratios and borrowing rates. i.e. lever the 60/40 portfolio up to the same volatility as the sp500.

Practical note: NTSX is pretty cool.

I had not heard about this. This does seem cool. But, do you actually want exposure to treasuries right now?

The yields for the 2-30 year treasuries that NTSX invests in are between 0.2 and 1.4% right now. And while returns on treasuries have often been negatively correlated with stocks historically, it seems like that relationship might not hold going forward.

In particular, it's unclear to me how treasuries could appreciate in value. Unless the yields go negative, they can't appreciate in price from here by more than a fraction of a percent (or just over 1% for the 20 and 30 yr). And if rates increase, then both stock and treasury prices are expected to drop.

Is the value of holding treasuries just getting access to that extra ~0.8% return? Or is there something I'm missing?

Yeah, it isn't great to buy in to *now* now. But is a pretty good proof of concept for running a mutual fund like this.

Agreed. The optimal amount of leverage is of course going to be very dependent on one's model and assumptions, but the fact that a young investor with 100% equities does better *on the margin* by adding a bit of leverage is very robust.

Based on this approach, optimal allocation for equities for younger folks is probably well over 100% - and this isn't particularly complicated to do, contra the statements in the article. Long dated out-of-the-money stock index options are a viable retirement investment. I'd tell people to seriously consider buying out of the money calls. As an illustrative example, a 120% of future price once a year for 2-3 years away with, say, 5% of your portfolio.

BUT - warning to readers: If you don't know / understand the argument I'm making, please don't just go buy stock options. Certainly don't spend more than a small portion of your long-term savings on them!

Is this strategy just counting on bankruptcy as a personal bailout in case you're leveraged 3:1 and the stock market doesn't go back up at least a bit after the pandemic?

Thanks for the helpful summary. Does this imply that young people should invest in cryptocurrency?

Looking at the Samuelson share equation, I'm not sure that premia have been established for crypto? But they do seem more volatile, and perhaps you can invoke CAPM or something to therefore claim they have a premium?

My impression is that they are also easier to leverage, although I'm less sure of that.

One way to increase your market exposure would be to invest in high-beta assets, yes.

On the other hand, there's some evidence that high-beta stocks have returns lower than you'd expect, given their volatility. In other words, investors are not actually compensated for increased risk with increased reward. (See Betting Against Beta, or Eric Falkenstien's work.) The standard explanation is that many investors are unable to (or prefer not to) use leverage, so, in order to get their preferred market exposure they over-invest in high-beta stocks (rather than just diversifying and using leverage), driving up their prices and driving down their returns.

On the other other hand, cryptocurrency is pretty new and has had historical returns and volatility pretty far from the returns and volatility of typical of high-beta stocks (I believe; haven't checked the numbers), so it's not clear to me whether the same effect would be playing out.

(Personally, I have very much over-weighted cryptocurrency in my portfolio, but that's been based on an inside-view bet that it would do well. In the long run I would expect to treat it as an asset class like any other and diversify accordingly.)

EDIT: This or this is probably a better link for Eric Falkenstein.

Any idea why the strategy seems to work over the known period but the estimated period all three seem to converge?

I believe they're just graphing the current (as of 2010) value of the portfolio for hypothetical investors at each of the different target retirement years. So moving to the right along the horizontal axis from 2010 does not mean moving forward in time, It means moving to younger and younger cohorts.

The data point at 2010 is the value, in 2010, of the portfolio of someone retiring in 2010. The value at 2011 is the value, in 2010, of the portfolio of someone who will retire in 2011... And the value at 2054 is the value, in 2010, of the portfolio of someone who will retire in 2054. (That is, it's the value of the portfolio of someone who's just starting out on their assumed 44 year career. The number isn't 0 because they're taking into account the present value of future income.)

So the reason all the numbers converge as you go to the right is that you're looking at the values of (hypothetical) young people's portfolios, and they've only been following the strategy for a couple of years, so the performance of the different strategies hasn't had a chance to diverge yet.

Here's the relevant passage from the book:

Thanks. Convergence makes perfect sense now.

It seems the whole deal is dependent on margin interest rates, so I would appreciate more discussion of the available margin interest rates available to retail investors and what rates were used in the simulations. I would also like more evidence to the statement in comments that says "the fact that a young investor with 100% equities does better *on the margin* by adding a bit of leverage is very robust" to be able to take it as a fact, as it only seems true at certain rates that don't seem obviously available.

As one datapoint, my current broker retail margin rates are high, at 8.625% under $25k, 7.5% > $25k loans. Given that this is a period of relatively low interest rates, (and I would expect it to be higher in other periods), it would not seem like an indisputable fact to me that a young investor expect a better return by investing $1 on the margin after interest. But I have no idea how it compares to other brokers. (Unless you are considering leveraged funds, which I considered to be a different beast).

Indeed these margin rates are way too high and it would be madness to borrow at such rates.