PDF Summary:The Missing Billionaires, by Victor Haghani and James White

Most people struggle with key financial questions: How much should I invest in stocks? How much can I safely spend each year without running out of money? What's the right balance between present enjoyment and future security? In The Missing Billionaires, Victor Haghani and James White apply mathematical frameworks to answer these questions and help you make better decisions about managing wealth over a lifetime.

The authors explain how to use expected utility theory and the Merton-Samuelson framework to optimize your investment and spending choices. You'll learn how to calculate optimal portfolio allocation based on your risk tolerance, why dynamic asset allocation can outperform static strategies, and how to adjust your spending in response to portfolio performance. The guide also covers practical considerations like tax implications, the role of annuities in retirement planning, and how to manage concentrated stock positions.

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Practical Considerations: Sizing, Unpredictability, and Allocating Assets

Uncertainty in anticipated returns can significantly impact the building of a portfolio and choices related to risk. Haghani and White state that misjudging anticipated returns is ten times as influential as misjudging variability and correlations. When anticipated returns are uncertain, they heighten actual volatility and boost the asset's projected price by the end of the time frame. For average risk-aversion levels, the heightened volatility outweighs the increased expected price, leading to decreased allocations as uncertainty and time rise. Uncertainty about parameters affects the risk-taking decisions of more risk-averse investors.

Parameter Uncertainty in Dynamic Portfolio Models

The authors’ discussion of uncertainty in anticipated returns and its impact on portfolio choices is rooted in the tradition of modern portfolio theory, which has evolved to incorporate dynamic, multi-period models that account for parameter uncertainty. In Strategic Asset Allocation, John Y. Campbell and Luis M. Viceira explain that in a dynamic setting, the parameters that describe the distribution of asset returns—such as expected returns, variances, and covariances—are themselves uncertain and must be estimated from historical data. This uncertainty about the parameters themselves introduces an additional layer of risk that investors must consider when making portfolio decisions. The authors highlight that investors can use Bayesian methods to update their beliefs about these parameters as new information becomes available, allowing for a more adaptive approach to portfolio management.

Practical Applications: Investing and Spending Rules

Haghani and White suggest adjusting spending according to your portfolio's performance to avoid depleting your funds. If you spend a fixed amount annually, you risk depleting your funds if your investments perform poorly. Spending a fixed percentage of your portfolio each year will cause your spending to diminish as time goes on due to volatility drag. Instead, spending the expected compound growth from your investments maintains the consistency of your spending and portfolio value over time. Therefore, you should adjust your spending based on how your portfolio performs—spending less when its value decreases and more when it grows.

The Constant-Percentage Withdrawal Rule

In Safety-First Retirement Planning, Wade D. Pfau discusses the pros and cons of different withdrawal strategies, including the constant-percentage withdrawal rule. He notes that this rule doesn’t necessarily mean your spending will decrease over time. Instead, your spending will fluctuate based on market performance. If your investments do well, you can spend more; if they perform poorly, you’ll need to cut back. Pfau explains that this approach ensures you never run out of money, as you’re always spending a fixed percentage of your portfolio. However, it requires flexibility and the ability to adjust your lifestyle based on market conditions.

We’ll look at how income yield and anticipated real returns affect portfolio construction and risk management. We’ll also cover sustainable wealth withdrawal and lifecycle planning.

Portfolio Construction and Risk Management

When determining the amount to invest in the stock market, Haghani and White advise considering the earnings yield and anticipated returns adjusted for inflation. The earnings yield equals earnings divided by the price, while the anticipated inflation-adjusted return is the expected inflation-adjusted return.

The authors note that the earnings yield can reliably forecast the stock market's inflation-adjusted real return over extended periods. What the stock market is anticipated to return changes over time based on supply and demand. The supply of investment opportunities and the demand from savers shift over time, leading to changes in the market's anticipated returns. The stock market's anticipated returns are higher if the yield from earnings is greater and lower if it's lesser. It is similarly reduced with greater uncertainty and increased with less uncertainty.

Earnings Yield and Long-Run Real Returns

The authors’ claim that the earnings yield is informative about the stock market’s inflation-adjusted real return over long periods is supported by research. In a 1988 paper, economists John Campbell and Robert Shiller used a present-value framework to show that when the earnings yield is high, subsequent decade-long real stock returns are higher on average. This suggests that the earnings yield contains information about the long-run inflation-adjusted performance of the stock market. Their analysis used long-horizon regressions to link the earnings yield to future real returns, providing a statistical basis for the authors’ argument.

We'll go over the core strategy and allocation, implementation details, and risk mitigation.

Core Investment Strategy & Allocation

Haghani and White explain that the Merton share formula helps determine optimal investment size based on expected return, risk, and personal risk tolerance. The formula is k̂ = μ / γσ², where k̂ is the optimal fraction of wealth to invest in the risky asset, μ is the anticipated additional return of the risky investment, σ is the riskiness of that investment expressed as standard deviation of returns, and γ is your personal degree of risk-aversion.

The formula calculates the best amount to invest in a single high-risk asset, on the assumption that it has a short-term return with a normal distribution and the investor has CRRA utility. The ideal bet size is directly tied to the gamble’s expected return, and inversely related to both its variance and your personal risk aversion.

The Merton Share Rule and Drawdown Constraints

The Merton share rule is optimal only as long as you don’t impose a drawdown constraint. In a 1993 paper, Sanford J. Grossman and Zhijian Zhou showed that if you require your wealth to never fall below a fixed percentage of its previous peak, the optimal investment strategy becomes path-dependent. The optimal fraction of wealth to invest in the risky asset becomes a decreasing function of the ratio of current wealth to its historical maximum. When your wealth is at its peak, the optimal fraction matches the unconstrained Merton share. But as your wealth approaches the drawdown boundary, the optimal fraction declines toward zero. This means you should progressively de-leverage your portfolio as your wealth nears the drawdown limit, to avoid breaching the constraint. This path-dependent strategy differs fundamentally from the constant-share rule prescribed by the Merton formula. The drawdown constraint introduces a dynamic element to the optimal policy, requiring you to adjust your risk exposure based on your wealth trajectory.

Implementation Details & Risk Mitigation

For managing risk, Haghani and White recommend considering tax implications when deciding how to invest. Utilizing tax losses can enhance your investment returns after taxes. The tax code incentivizes transferring wealth through philanthropy or to future generations earlier rather than later. The government taxes gains once they're realized, and losses when realized produce a credit for carrying forward a tax loss, which can be applied solely to offset gains realized in the future. The authors note that the regulations around taxation are complicated, so it’s typical to concentrate on improving post-tax financial assets in one main scenario. For the majority of choices regarding taxes, risk is a key component, so analyzing it statically by comparing likely scenarios of after-tax wealth might not yield the right conclusion.

(Shortform note: The authors don’t explain why static analysis of after-tax wealth can be misleading. George M. Constantinides, a finance professor at the University of Chicago, explains that the tax code’s rules for capital gains and losses make the after-tax value of an investment depend on the entire sequence of gains and losses over time, not just the final outcome. This means that two investment strategies that look similar in a few static scenarios can have very different expected after-tax outcomes. For example, a strategy that realizes losses early and defers gains can be much more valuable than one that does the opposite, even if they have the same expected pre-tax returns.)

A superior analysis involves estimating your Expected Utility for varying decreases in equity holdings, and then selecting the level of decrease, and resulting realization, that yields the best outcome. The optimal decision results in 3% more Certainty-equivalent Wealth at the horizon than not realizing any gains at all. The ideal Merton allocation to stocks under identical assumptions but without considering taxes is 37.5%, much less than the 59% in this scenario with a valuable asset.

(Shortform note: In Investment Science, David G. Luenberger explains that Certainty-equivalent Wealth is the amount of wealth that would make you indifferent between receiving that amount for sure and keeping your risky portfolio. In other words, it’s the amount of wealth that gives you the same utility as the expected utility of your risky portfolio. The difference between the expected value of your risky portfolio and your Certainty-equivalent Wealth is called the risk premium.)

Certain investors are drawn to the potential of their appreciated asset's basis being raised to the present market price upon death. As long as this tax rule stays unchanged, your capital gains might not be taxed at all if you hold them until you die. However, applying the previous analysis with the assumption of a 0% tax on capital gains, you should lower your equity holdings from 75% to 62%. This yields a welfare boost of approximately 2%, similar to an annualized 0.1% rise in the risk-adjusted portfolio return over two decades. The framework also helps determine whether it's sensible to cash in gains now and reinvest if you anticipate future tax increases.

(Shortform note: In Death by a Thousand Cuts, Michael J. Graetz and Ian Shapiro explain that the rule allowing an appreciated asset's basis to be raised to the present market price upon death is a product of early twentieth-century compromises over how to coordinate the income tax with estate and inheritance taxes. This rule has survived repeated attempts at repeal because it quietly allows large amounts of wealth to pass between generations without ever being subject to capital-gains taxation. Proposals to tax accrued gains at death or to replace step-up with carryover basis have surfaced repeatedly in Congress and in expert tax-reform plans, yet they have been defeated time and again under pressure from well-organized interests that benefit from shielding inherited appreciation—an arrangement that, as they emphasize, is largely invisible to most voters even though it is central to the politics of the estate tax.)

This type of analysis can also be applied to cases involving a single, significantly appreciated stock. Your anticipated return for the stock will be a key factor, but in most scenarios we've evaluated using a realistic expectation, the Expected Utility analysis consistently suggests reducing highly appreciated, concentrated holdings. That's because the reduction in risk often compensates for the diminished expected wealth due to paying taxes on gains earlier.

(Shortform note: The authors’ recommendation to reduce highly appreciated, concentrated holdings is not universally accepted. In Concentrated Investing, Allen C. Benello, Michael van Biema, and Tobias E. Carlisle argue that concentrated value investors should be prepared to hold a small number of deeply researched positions for very long periods, even after substantial appreciation. They contend that the informational and control advantages of large positions can outweigh the diversification and tax arguments for selling.)

Sustainable Wealth Withdrawal and Lifecycle Planning

Responsible expenditure strategies are crucial for managing lifelong wealth. Haghani and White explain that the ideal way to spend is by determining a percentage of your current wealth to use annually, explicitly acknowledging that spending is connected to investment performance. The higher your time preference, the higher your current spending rate, as you want to spend more of your wealth sooner. If the risk-adjusted return and your time preference rate are almost equal, the best spending rate is just the risk-adjusted portfolio return.

(Shortform note: While Haghani and White’s rule of spending a percentage of current wealth is a good general guideline, it can break down in certain situations. For example, economist Christopher D. Carroll argues that when households face significant labor income risk and borrowing constraints, optimal spending is driven by the need to maintain a buffer of safe assets rather than by a simple fraction of current wealth. This suggests that the ideal spending strategy may need to be adjusted when income is unpredictable and borrowing is limited.)

The optimal spending rate to a very long horizon is: rra - (rra - rtp)/γ, where rra is the portfolio's return adjusted for risk, rtp is the investor’s time preference rate, and γ is the investor’s level of risk aversion relative to the constant. As your risk-adjusted yield rises, your optimal spending can increase, but if that yield increases by 1%, it doesn't boost your current spending by 1%. This happens since the wish for a consistent, smooth spending rate must be balanced with the potential to spend more in the future by saving today to invest at the higher rate.

(Shortform note: In Strategic Asset Allocation, John Y. Campbell and Luis M. Viceira explain that this formula is the only one that allows the ratio of wealth to desired spending to settle down to a steady value in the long run. If you spend more than this, your wealth will eventually run out, and if you spend less, your wealth will grow without bound. This is because the formula balances the desire to spend now with the potential to spend more in the future by saving and investing at a higher rate.)

For any limited time period, the best spending approach at time t is to convert your wealth into annuities over your remaining period, T, applying the long-term ideal spending rate to determine the annuity amount. The equation is: ct equals c infinity divided by 1 minus the quantity 1 plus c infinity to the negative T power.

(Shortform note: In 1965, the economist Menahem E. Yaari published a paper on optimal consumption with annuities. He argued that, in the absence of a bequest motive, the optimal policy for a utility-maximizing individual is to hold their entire wealth in the form of life annuities, thereby eliminating all uncertainty about their future consumption stream by transforming their given stock of resources into a certain, constant flow of income for the remainder of their lifetime.)

We’ll go over withdrawal strategies, guidelines for spending, lifecycle considerations, and risk management.

Withdrawal Strategies and Plans for Use

According to Haghani and White, ideal strategies for spending from endowments consider risk-adjusted return, time preference, and risk-aversion. The ideal spending rate depends on three factors: the portfolio's risk-adjusted return at the best risk level, the time preference rate, and the endowment's risk-aversion level.

Endowment Spending and Intergenerational Equity

In Pioneering Portfolio Management, David F. Swensen, former Chief Investment Officer of Yale University, argues that a fourth factor is crucial in determining an endowment’s ideal spending rate: intergenerational equity. Swensen contends that the fundamental objective of a permanent endowment is to generate a sustainable stream of distributions for current needs while simultaneously preserving, and if possible enhancing, the inflation-adjusted value of the portfolio so that future generations receive at least the same level of support as the present generation.

While sustainable strategies for spending are simple, they may result in less optimal choices compared to ideal expenditure guidelines. A sustainable spending approach doesn't directly consider the endowment’s risk aversion or time preference, which can lead to less effective spending decisions.

(Shortform note: In Pioneering Portfolio Management, David F. Swensen suggests that endowments should evaluate spending rules by modeling how they affect the probability of having to cut core programs in bad times. He argues that endowments should favor conservative, well-tested policies that provide stable, predictable support for essential activities while minimizing the likelihood of disruptive spending reductions.)

Lifecycle Considerations and Risk Management

Haghani and White suggest that annuities can help manage the risk of unexpected longevity and optimize spending in retirement. Annuities are financial products generally available through insurers that supply a guaranteed monthly payment for the duration of the buyer's life. They allow you to pool your longevity risk with others, significantly lowering the chances that you'll outlive your savings. This lets you increase your retirement expenditures without having to worry about exhausting your funds.

However, in the US, annuities represent under 8% of all privately held retirement funds. This is partly because insurers design most annuity contracts with excessive complexity and fees, reducing their value. Additionally, many individuals are unaware of how annuities can be advantageous.

(Shortform note: The study of how individuals should manage their finances over their lifetimes, including the use of annuities, is a key focus of life-cycle finance. This field of economics examines how people can optimize their consumption and investment decisions throughout their lives, taking into account factors like income, expenses, and risk preferences. Brown, a leading researcher in this area, explains that life-cycle finance uses mathematical models to help individuals make decisions about saving, investing, and purchasing insurance products like annuities.)