Just reread the Ayres-Nalebuff paper. Some comments on it.

#1 They arbitrarily choose a long initial phase (as much as a decade or two, depending on how early returns pan out) of constant x2 leverage, relative to current financial assets, because it is the number used by a regulation.

This isn't rigorous. If the regulation said 4:1, apparently they'd use 4:1. If it said 1.5:1, they'd use 1.5:1. It's more reasonable to suppose that there is some non-regulatory reason for a bound or strategy on leverage to apply in the earliest phase. After all these regulations came about after the 1929 crash (first in the Securities Act of 1933). A reasonable investor (perhaps with math not yet developed...) would have had a way of determining the best margin to use other than calling their broker or lawmaker and asking their opinion on a good maximum.

"We propose a maximum leverage of 2:1."

"For most of the analysis, we assume that the maximum leverage on stocks is 2:1, pursuant to the Federal Reserve Regulation T."

Because this choice of how to leverage in the first phase is arbitrary and because the results of the first phase affect final savings, they haven't shown rigorously that their model of savings is optimal. Perhaps it should be lower, perhaps it should be higher and they should have used a technique for obtaining higher leverage. There's still a lot of value in the paper, but it does suggest more work could be done.

#2 They assume that it's only profitable to leverage stocks (implicitly or explicitly, since they don't investigate anything else), not profitable to leverage government bonds or another asset. Indeed their final asset allocation in retirement is divided between stocks and a risk-free rate of return, like T-bills. Since their model includes only stocks, margin rates, and risk-free rates, there's room for tinkering with other assets.

"the cost of borrowing on margin exceeds the bond rate"

This is true of brokerage margin rates, but with the borrowing costs implicit to the leverage implicit in treasury features, it's not true for longer duration government bonds, due to the low spread above LIBOR for borrowing and going out on the long end of the yield curve for investment. Also, government bonds have a non-interest component of their returns (price returns), which can play a part in constructing a balanced portfolio of stocks and bonds, which (under Modern Portfolio Theory) can increase risk-adjusted returns (Sharpe) and thus can increase returns over leveraging just stocks (whereas the author's framework is just leveraging up stocks).

Perhaps this chart may be illustrative of other possibilities for applying leverage to increase early returns (both positive and negative - thus also achieving temporal diversification by increasing the early volatility and returns if leverage is dialed down later), other than just leveraging stock:

https://www.portfoliovisualizer.com/bac ... ion3_3=-20
#3 They don't actually implement Samuelson-Merton's constant allocation.

They attempt to implement it by going as close as they can before bumping up into the 2:1 limit on current net assets. As Uncorrelated correctly points out, apart from the costs of the implementation (which leads the paper's authors to consider alternatives), the initial portfolio recommended by the paper is exactly the same as putting all of one's current financial wealth into a 2x stock LETF, such as SSO (Ultra S&P500) or the older mutual fund ULPIX (ProFunds UltraBull). This is constant leverage, not varying leverage, until the first phase is over.

This thread's approach is different and appears to attempt to vary leverage somewhat during the first phase, primarily in response to market movements and implicitly by holding options because this eliminates the path dependence (over a short term period). This difference isn't relevant to the point because it's still not implementing Samuelson-Merton's constant fraction of (lifetime) wealth in stocks.

The problem is that Samuelson-Merton's allocation is impossible to achieve for a very young saver with most wealth in the form of future savings, so an alternative must be considered, and the paper somewhat facilely slides into its arbitrary phase 1 recommendation as the alternative, when there are others as well. So not only is the idea of using the maximum leverage to buy stocks arbitrary, it doesn't satisfy the theory that is claimed to justify it, because it doesn't do what the theory asks for (only getting 'closer').

#4 The Samuelson-Merton recommendation

*assumes* that all your wealth is

*available to invest* (in stocks/t-bills) or consume.

When that assumption is contradicted, it's already unclear that Samuelson-Merton is the right framework. Instead of jerry-rigging its conclusions into a new theoretical framework where the assumptions are invalid, it would make more sense to develop the author's own justification regarding the optimal investment in stocks that is actually consistent with their assumptions. That is, to introduce the addition of income over time, as well as the various liquidity and leverage constraints, and derive the optimal investment based on utility with that model.

It's not easy, but it would seem that the right approach is to do it from scratch under the actual assumptions.

I've tried to work it out a bit on scratch paper. In the limit, when the additional income is a positive constant and it is the infinite case, the solution for the fraction to invest approaches the Merton solution (which is the Kelly limit modified by a utility function with a constant relative risk aversion) where wealth is just the same thing as current wealth. This suggests that the interesting work is to solve it for finite cases, especially in the early stages when the additional contributions are a considerable fraction of the initial investment, or in the late stages when taking money out. I'm not sure if this is an unsolved problem, or if someone has worked it out. It will take me some time to continue working on it or look up what's been done.

#5 They effectively defend the "leveraged glide path" as an approach during accumulation, empirically.

This much, taking more risk in early years, does seem justified by the empirical studies. They are persuasive in arguing that the basic problem with the old "glide path" is that it has too low of an average allocation to risky assets like stocks, as well as too small of a difference between starting risk and ending risk, to be effective. They are also persuasive in showing that a "glide path" incorporating leverage can overcome those two problems, provide enough risk up front and enough risk differential to matter and to reduce overall retirement risk through "temporal diversification."

However, the effect size isn't necessarily very large, when comparing to alternatives that are similarly exposed to stocks. Also, they've limited the solution space to those solutions that dial down to being unleveraged in retirement. This does represent a blind spot. If a portfolio is leveraged in retirement, there may be less room to create a large difference with the initial risk taken through leverage. They may have mostly just recreated an effect they criticize -- being able to achieve temporal diversification by artificially limiting later returns (as they point out, retirement accounts that are all or vastly bonds have no trouble being temporally diversified to stocks without leverage) -- in the new solution space where leverage is allowed.