Free reference data (Damodaran, Shiller), company filings, official pages on margin loans, leveraged products and retirement income, and the Bogleheads wiki on withdrawal methods and leverage. Listed, not scored. We score videos (VettedVids score), books (Book Rating) and channels (Channel Rating).
Damodaran Online — Equity risk premium and cost of capital data
Aswath Damodaran, NYU Stern School of Business
Best for: Free, regularly updated datasets used in valuation: the implied equity risk premium for the S&P 500 back to 1960, cost of capital by industry, and country default spreads and risk premiums.
Who writes it: Teaches corporate finance, valuation and investment philosophies at New York University's Stern School of Business, per his channel description.
Caution: Raw tables with short notes, mostly US-centred, though some datasets are global. The implied premium and country premiums are one academic's estimates, built on his own modelling choices.
Data: Current · Historical Implied Equity Risk Premiums for the US · Cost of Capital by Sector (US) · Country Default Spreads and Risk Premiums
Robert Shiller — Online Data (US stock market and CAPE since 1871)
Robert J. Shiller, Yale University
Best for: Monthly US stock prices, dividends, earnings and consumer prices from January 1871, with the cyclically adjusted P/E (CAPE) ratio. It is the dataset behind Irrational Exuberance and is used in many CAPE studies, including the AQR paper in Research.
Who writes it: Sterling Professor of Economics at Yale; awarded the 2013 Nobel Prize in Economic Sciences jointly with Eugene Fama and Lars Peter Hansen, per his Yale biography.
Caution: A spreadsheet with little explanation; the download link on this page points to a separate site (shillerdata.com), which couldn't be checked for this review. US market only, and the early decades are reconstructed from older sources. The data show valuations, not when to act on them.
Online Data - Robert Shiller
Investor.gov — Margin, leveraged ETFs and annuities
U.S. Securities and Exchange Commission (SEC)
Best for: Plain-English US regulator bulletins on how margin accounts and margin calls work, why daily-reset leveraged and inverse ETFs drift from their multiple over time (with worked examples), and what annuities are, including surrender charges and the risk that the insurer can't pay.
Who writes it: The SEC's Office of Investor Education and Advocacy, part of the US securities regulator.
Caution: US-specific rules and products. Bulletins describe risks and mechanics; they don't compare the costs of particular products.
Investor Bulletin: Understanding Margin Accounts · Investor Tips: Margin - Borrowing Money To Pay for Stocks (SEC.gov) · Updated Investor Bulletin: Leveraged and Inverse ETFs · Annuities
FINRA — Margin accounts and margin calls
Financial Industry Regulatory Authority (FINRA)
Best for: How a margin call is triggered, what a brokerage firm can do when equity falls below the maintenance requirement, and the margin rules brokers must follow, including the standard margin disclosure statement.
Who writes it: FINRA, the US broker-dealer self-regulatory organisation.
Caution: US brokerage rules; margin terms vary by firm and can be stricter than the regulatory minimum.
Know What Triggers a Margin Call · Margin Regulation · Margin Disclosure Statement (PDF)
Moneysmart — Borrowing to invest, annuities and lifetime income
Australian Securities and Investments Commission (ASIC)
Best for: How margin loans and gearing work in Australia: lenders usually cap the loan-to-value ratio at around 70%, a margin call generally gives about 24 hours to restore it, and the lender can sell investments if it isn't restored. Separate pages explain fixed-term and lifetime annuities and lifetime income streams, with their trade-offs.
Who writes it: ASIC's Moneysmart, the Australian corporate and financial-services regulator's consumer education service.
Caution: Australia-specific (super, Age Pension, Australian products). General information, not product comparisons.
Borrowing to invest · Annuities · Lifetime income streams
MoneyHelper — Annuities and pension drawdown
Money and Pensions Service (UK government-backed)
Best for: UK guides comparing the two main ways to turn a pension pot into income: buying an annuity for a guaranteed income for life or a fixed term, or leaving the pot invested and drawing from it (drawdown), where income isn't guaranteed.
Who writes it: MoneyHelper is run by the Money and Pensions Service, a UK government-backed body.
Caution: UK-specific tax and pension rules (for example, the tax-free lump sum). General guidance rather than personal advice.
Take your pension as a guaranteed income: annuities explained · Take money from your pension when you need it: pension drawdown explained · Compare annuities: get a guaranteed retirement income
FCA — Retirement income data and leverage limits on CFDs
Financial Conduct Authority (FCA)
Best for: Official UK statistics on how people take their pensions (annuity, drawdown or cashing in), and the regulator's rules for contracts for difference, which cap retail leverage at between 30:1 and 2:1 and require positions to be closed out when funds fall to 50% of the margin needed.
Who writes it: The UK's financial services regulator.
Caution: UK market data and rules written for firms. The statistics show what people chose, not whether it suited them.
Retirement income market data 2025/26 · PS19/18: Restricting contract for difference products sold to retail clients
SEC EDGAR — Company filings search
U.S. Securities and Exchange Commission (SEC)
Best for: Free access to every filing by US-listed companies: annual reports (10-K), quarterly reports (10-Q) and proxy statements, the raw material for working out owner earnings, reinvestment, buybacks and acquisitions over a decade.
Who writes it: The US securities regulator's official filing system.
Caution: Primary documents with no analysis. US-listed companies only.
EDGAR Full Text Search · Search Filings
ASX — Company announcements
ASX (Australian Securities Exchange)
Best for: Today's and historical announcements by ASX-listed companies, including annual reports, results, capital raisings and buyback notices.
Who writes it: The operator of Australia's main stock exchange, where listed companies lodge their market announcements.
Caution: Primary documents with no analysis. Australian-listed companies only.
Announcements · Historical announcements
Companies House — Find and update company information
Companies House (UK government)
Best for: Free search of the UK company register: filed accounts, officers and filing history for UK companies, including private ones.
Who writes it: The UK government's official registrar of companies.
Caution: Filings only, with no analysis. Small companies may file abridged accounts with little detail.
Search the register
Bogleheads wiki — Withdrawal methods and leverage
Bogleheads (volunteer investor community)
Best for: Community-written explanations of retirement withdrawal methods (constant-dollar, constant-percentage, 1/N and variable percentage withdrawal, which sets each year's withdrawal as an age-based percentage of the current balance), and of the forms and risks of leverage.
Who writes it: Maintained by members of the Bogleheads community, an investor forum founded around the ideas of Vanguard founder John C. Bogle.
Caution: Volunteer-edited and US-centric. The leverage page takes an editorial stance: it strongly discourages leverage, especially for retirement savings.
Withdrawal methods · Variable percentage withdrawal · Leverage
YouTube channels with credentialed presenters on valuation, portfolio choice, leverage and retirement spending, with their stated credentials and cautions. Each channel is rated from the videos of theirs we have scored; channels with fewer than 2 scored videos are marked not yet rated.
Not yet rated: none of their videos has been through our scoring yet.
Best for: University lectures and webcasts on valuing a business: intrinsic value from cash flows, estimating the equity risk premium implied by market prices, and how companies return cash through dividends and buybacks.
Claim: Teaches corporate finance, valuation and investment philosophies at New York University's Stern School of Business, per the channel description.
Caution: Long, technical classroom recordings with a US focus. His annual data updates reflect his own estimates and views.
Session 2: Intrinsic Value - Foundation · Valuation Tools Webcast #3: Implied Equity Risk Premiums · Data Update 8 for 2026: Dividends and Buybacks - The Investing Harvest!
Not yet rated: none of their videos has been through our scoring yet.
Best for: Research-led explainers on how much risk to take, what leverage does to returns and drawdowns, and safe retirement spending rates.
Claim: Portfolio Manager and Chief Investment Officer at PWL Capital; CFA charterholder and CFP professional; securities-licensed in Canada by CIRO, per the channel description.
Caution: Canada-based; content is produced and owned by PWL Capital, the presenter's employer. Leans heavily on academic papers and can be dense.
How Much Risk Should You Take? · How Leverage Can Boost Returns (And What Can Go Wrong) · The 2.7% Rule for Retirement Spending
Not yet rated: none of their videos has been through our scoring yet.
Best for: Long interviews with the people behind the ideas on this page: Robert C. Merton on portfolio choice and retirement, Victor Haghani and James White on bet sizing, and Michael Mauboussin on market expectations, intangible assets and valuation.
Claim: Hosts Cameron Passmore and Benjamin Felix are portfolio managers with PWL Capital, securities-licensed by CIRO, per the channel description.
Caution: Hosted by PWL Capital portfolio managers. Episodes run over an hour, and guests (Haghani and Mauboussin among them) work at investment firms.
Prof. Robert C. Merton: ICAPM, Retirement, and Models in Finance | Rational Reminder 234 · Victor Haghani and James White: The Missing Billionaires | Rational Reminder 270 · Michael Mauboussin: The One Job of an Equity Investor | Rational Reminder 344
Not yet rated: none of their videos has been through our scoring yet.
Best for: Full lectures from MIT's Finance Theory I course, taught by Andrew Lo: present value (the basis of discounted cash flow) and portfolio theory.
Claim: MIT's official open courseware channel; the 15.401 lectures are taught by Andrew Lo, per the video descriptions.
Caution: Recorded in 2008; long classroom lectures that assume some maths. A small part of a very large general channel.
Ses 2: Present Value Relations I · Ses 14: Portfolio Theory II
Not yet rated: none of their videos has been through our scoring yet.
Best for: Robert Shiller's Financial Markets course: diversification and mean-variance analysis, and why he concludes that the efficient markets hypothesis is a 'half-truth'.
Claim: Yale University's official course channel; lectures by Robert Shiller (ECON 252), per the video descriptions.
Caution: Recorded in 2011; long lectures, US-focused. A small part of a broad university channel.
4. Portfolio Diversification and Supporting Financial Institutions · 7. Efficient Markets
Not yet rated: none of their videos has been through our scoring yet.
Best for: Conference talks by the researchers behind the main spending rules: William Bengen on the history of safe withdrawal rates, Jonathan Guyton on retirement spending, and Wade Pfau on retirement income planning.
Claim: Official channel of the Bogleheads investor community; speakers vary by session.
Caution: Conference recordings: long, with variable audio and slides. US-centric. Speakers' views are their own, and several work in the financial industry.
Bogleheads® Conference 2024 The History of Safe Withdrawal Rates with William Bengen · Bogleheads University 501 2024 Retirement Spending Deep Dive with Jonathan Guyton · Bogleheads® Conference 2023 - Wade Pfau Discusses Retirement Income Planning with Jim Dahle
Not yet rated: none of their videos has been through our scoring yet.
Best for: Data-heavy explainers on borrowing to invest when young ('time diversification'), why leveraged ETFs can disappoint, and how to reduce the chance of running out of money in retirement.
Claim: Ramin Nakisa, PhD (Imperial College London); worked in investment banking as a strategist before founding PensionCraft in 2016.
Caution: UK-based. Sells memberships, courses and coaching. The borrowing-to-invest video is sponsored by WisdomTree, which sells short and leveraged products. Some videos discuss market outlooks and the presenter's own portfolio choices.
Should Younger Investors Borrow To Invest? · Leveraged ETF Problems · What Is The Safe Withdrawal Rate In Retirement?
Not yet rated: none of their videos has been through our scoring yet.
Best for: Clear introductions to building a discounted cash flow model, the case for and against share buybacks, and how stretched market valuations look.
Claim: Richard Coffin, CFA charterholder and CFP professional, working as an analyst at an investment management company in Ottawa, per the channel description.
Caution: Takes sponsored segments (the channel lists a sponsorship contact). Some explainers are several years old.
The DCF Model Explained - How The Pros Value Stocks/Businesses · Stock Buybacks - The Good And The Bad Explained · How BAD Are Stock Valuations Really?
Not yet rated: none of their videos has been through our scoring yet.
Best for: A walk-through of the Kelly criterion and how leverage affects trading returns, and a case study of margin calls hitting more than 1.2 million South Korean retail accounts during a sharp market fall.
Claim: Hedge fund manager, university professor and former investment banker, per the channel description.
Caution: Mostly market-news commentary in a dry, sceptical style; only a few videos fit this topic. Videos carry sponsored segments.
How Does Leverage Affect Trading Returns? The Kelly Criterion | Coffeezilla Follow-up · How to Be Right and Lose Everything
What the research found on expected returns and valuation, risk-taking and the Merton share, Kelly sizing and concentration, leverage, buybacks and acquisitions, capital and credit cycles, and withdrawal rates (42 sources). Several findings come from fund managers, advisers or their founders (AQR, Elm Partners, PIMCO, E.O. Thorp and Associates, Cornerstone Wealth Advisors, Finance Concepts); we say where. Contested findings are shown both ways. General information, not financial advice.
Estimate expected returnsEvidence: Cohort or observational study
Past equity returns ran well above what dividend and earnings growth implied
▸ Key points▾ Key points
- In practice
- Past average stock returns can overstate what to expect: in US data for 1951–2000, part of them came from valuations rising, which need not repeat.
- The number
- US, 1951–2000: the equity premium estimated at 2.55% a year from dividend growth and 4.32% from earnings growth, against 7.43% from the average realised stock return.
- The limit
- US data for one half-century; the estimate depends on which growth measure is used.
Fama and French estimate the US equity premium from dividend and earnings growth (the expected capital gain) rather than from realised returns. For 1951 to 2000 their estimates are 2.55% a year (dividend growth) and 4.32% (earnings growth), against 7.43% from the average realised stock return. They attribute the gap to a fall in discount rates that produced a large unexpected capital gain, and conclude that the average return of that half-century 'is a lot higher than expected'.
Limits: US data, 1951–2000. The gap between the growth-based and realised figures is the 'change in valuation' part of a return: a re-rating that need not repeat. The estimates depend on which growth measure is used.
Fama & French 2002, Journal of Finance
Estimate expected returnsEvidence: Cohort or observational study
Starting valuation explained about a quarter of 10-year returns, but timing on it barely helped
▸ Key points▾ Key points
- In practice
- In US data, starting valuation partly explained next-decade returns in sample, but varying the share held in stocks with it barely changed results in this test.
- The number
- US, 1900–2015: starting valuation explained about a quarter of the variation in next-10-year returns in sample (R-squared 0.24); Sharpe ratio 0.37 for a valuation-based strategy, 0.38 for buy-and-hold.
- The limit
- Contested: the authors' firm runs timing strategies, results are hypothetical and before costs and fees, and in-sample figures defined 'cheap' with hindsight.
For US equities from 1900 to 2015, the Shiller earnings yield (the inverse of CAPE) explained about a quarter of the variation in next-10-year returns in sample (R-squared 0.24), 0.05 for next-year returns and 0.00 for next-month returns. A rules-based strategy that held between 50% and 150% in equities depending on valuation had a Sharpe ratio of 0.37, against 0.38 for buy-and-hold, over 1900–2015. Over 1958–2015 it earned 5.4% a year above cash, against 5.5% for buy-and-hold. Neither difference was statistically significant. The authors argue that combining valuation with momentum signals, with only modest tilts, worked better than valuation alone, though this improvement was also not statistically significant.
Limits: Contested. The in-sample link between valuation and later returns is often read as strong support for valuation-based shifts; this strategy test found little benefit, and Welch & Goyal (Level 2) found predictors weak out of sample. All three authors work at AQR Capital Management, a fund manager that runs timing and factor strategies. Results are hypothetical and before costs and fees. In-sample figures use the full history to define 'cheap' and 'expensive', which investors at the time couldn't do.
Asness, Ilmanen & Maloney 2017, Journal of Investment Management, Market Timing: Sin a Little
Estimate expected returnsEvidence: Cohort or observational study
Most published return predictors failed out of sample, in an update to 2021
▸ Key points▾ Key points
- In practice
- In US data, many published signals for forecasting stock market returns weakened or failed when re-tested with later data, so original results may not carry forward.
- The number
- US data to the end of 2021: more than one-third of 29 newer predictors were no longer significant even in sample, and half of the rest performed poorly out of sample.
- The limit
- US data; statistical performance alone doesn't show that a timing rule would have paid after costs.
Goyal, Welch and Zafirov re-tested 29 variables from 26 papers published after Goyal and Welch (2008), plus the original 17, with data to the end of 2021. More than one-third of the new variables were no longer significant even in sample. Of those that were, half performed poorly out of sample. A small number still performed reasonably well both in and out of sample.
Limits: US equity premium data. Valuation ratios are among the variables tested. Updates the 2008 study cited at Levels 2 and 4 rather than repeating it. Statistical performance alone doesn't show that a timing rule would have paid after costs.
Goyal, Welch & Zafirov 2024, Review of Financial Studies
Estimate expected returnsEvidence: Cohort or observational study
The world equity premium since 1900, and lower forward estimates
▸ Key points▾ Key points
- In practice
- Expected-return assumptions based on long-run history may run high: removing past windfalls lowered the World premium, and the authors' 2010 forward estimate was lower still.
- The number
- 19 countries, 1900–2010: World equity premium over Treasury bills 4.5% a year (geometric), 4.0% with windfalls removed; the authors' forward estimate was 3–3.5% geometric.
- The limit
- Data end in 2010; the forward figures are the authors' year-end 2010 estimates and depend on the growth assumption; all 19 markets survived the period.
Across 19 countries from 1900 to 2010, the annualised (geometric) equity premium over Treasury bills was 5.3% for the US, 4.3% for the UK, 4.6% averaged across the countries and 4.5% for the World index. Over long government bonds it averaged 3.8%. Removing windfalls from currency moves and rising valuations lowers the World figure from 4.5% to 4.0%, and the average country's US-dollar premium from 4.0% to 3.4%. From a 2.5% dividend yield and an assumed 2% real dividend growth, the authors put the prospective World premium at 3–3.5% geometric, or around 4.5–5% arithmetic. They warn that investors relying on figures as high as 7% geometric or 9% arithmetic 'are likely to be disappointed'.
Limits: Data end in 2010. The 2026 Yearbook extends the series to 2025, but its full premium tables aren't public; Level 1 cites the summary edition for long-run returns. The forward figures are the authors' estimates as of year-end 2010, and depend on the growth assumption. All 19 markets survived the period: see the survivorship finding.
Dimson, Marsh & Staunton 2011, Equity Premiums around the World, in Rethinking the Equity Risk Premium (CFA Institute Research Foundation)
Estimate expected returnsEvidence: Cohort or observational study
Survivorship: is the US equity record the exception?
▸ Key points▾ Key points
- In practice
- US-only history may be a narrow base for estimating future returns, since the US was one of the period's strongest markets; how much this matters is contested.
- The number
- US real return 4.3% a year in 1921–1996, median elsewhere 0.8%; even if missing markets lost everything, the world premium would be overstated by at most about 0.1 percentage point a year.
- The limit
- Contested: the two studies measure different things, one using price indexes without dividends against the median country, the other total returns weighted by market size.
Jorion and Goetzmann collected capital-appreciation indexes for 39 markets back to the 1920s. From 1921 to 1996, US equities had the highest real return, 4.3% a year, against a median of 0.8% for the other countries. They conclude that the high US equity premium 'appears to be the exception rather than the rule'. Dimson, Marsh and Staunton reach a different conclusion for the world as a whole. Their 19 surviving markets made up an estimated 89% of world equity value in 1900. Even if the missing 11% had all become worthless (investors in Russia in 1917 and China in 1949 lost everything), the world equity premium would be overstated by at most about 0.1 percentage point a year.
Limits: Contested, and the two studies measure different things. Jorion and Goetzmann use price indexes without dividends and compare the US with the median country. Dimson, Marsh and Staunton use total returns weighted by market size. Both treat the US as one of the strongest markets of the period, which makes US-only history a narrow base for forward estimates.
Jorion & Goetzmann 1999, Journal of Finance · Dimson, Marsh & Staunton 2011, Equity Premiums around the World, in Rethinking the Equity Risk Premium (CFA Institute Research Foundation)
Decide how much risk to takeEvidence: Theory or logical argument
The average return overstates what a typical investor ends up with
▸ Key points▾ Key points
- In practice
- Plans built on the expected cumulative return can overstate the typical long-horizon outcome; to forecast the median, the authors compound the average log (geometric) return.
- The number
- In the authors' simulations of a large-cap stock index, there was only about a 30% probability of meeting or exceeding the expected 30-year cumulative return.
- The limit
- A statistical argument with simulations, not a study of investors; the 30% figure depends on the assumed volatility and horizon.
Long-horizon cumulative returns are skewed, so the median outcome sits below the mathematical expectation. The chance of reaching the expected cumulative return falls toward zero as the horizon lengthens. In the authors' simulations of a large-cap stock index, there was only about a 30% probability of meeting or exceeding the expected 30-year cumulative return. To forecast the median outcome, they compound the average log (geometric) return instead. Even with 54 years of data, typical stock volatility of 15% leaves historical averages uncertain by about ±400 basis points.
Limits: A statistical argument with simulations, not a study of investors. The 30% figure depends on the assumed volatility and horizon.
Hughson, Stutzer & Yung 2006, Financial Analysts Journal
Decide how much risk to takeEvidence: Theory or logical argument
The Merton share: a constant risky share set by excess return, volatility and risk aversion
▸ Key points▾ Key points
- In practice
- In this model, the optimal share in the risky asset depends on its expected excess return, its variance and risk aversion, not on wealth or time.
- The number
- Two-asset case with constant relative risk aversion: the optimal fraction in the risky asset is its expected excess return divided by relative risk aversion times its return variance.
- The limit
- Theoretical model assuming a known, constant mean and volatility, no labour income and unchanging risk aversion; very sensitive to the expected excess return, which is hard to estimate.
Merton solved the combined problem of lifetime consumption and portfolio choice in continuous time. In the two-asset case with constant relative risk aversion, the optimal fraction of wealth in the risky asset is its expected excess return over the riskless asset, divided by relative risk aversion times the asset's return variance. That fraction stays constant over time and doesn't depend on wealth.
Limits: A theoretical model. It assumes returns follow a random walk with known, constant mean and volatility, no labour income and unchanging risk aversion; changing these changes the answer. The result is very sensitive to the expected excess return, which is itself hard to estimate (see the expected-returns findings). The formula is the paper's two-asset, constant-relative-risk-aversion solution.
Merton 1969, Review of Economics and Statistics
Decide how much risk to takeEvidence: Cohort or observational study
Given a coin weighted 60/40 in their favour, 28% of trained players went bust
▸ Key points▾ Key points
- In practice
- Favourable odds didn't stop players in this game going bust; for this coin, betting a constant 20% of the bankroll gives a probability of ruin close to zero.
- The number
- Sixty-one quantitatively trained players given $25 and a coin said to land heads 60% of the time: 28% went bust, 21% finished with at least $200 (cap $250), average payout $91.
- The limit
- Both authors are practitioners (Haghani founded Elm Partners, a fund manager; Dewey worked at PIMCO); a small, non-random sample in a one-off capped game, not investing.
Sixty-one young, quantitatively trained participants, mostly economics and finance students and young finance professionals, were each given $25 to bet for 30 minutes on a coin they were told came up heads 60% of the time. Winnings were capped at $250. 28% went bust and only 21% reached the cap (finishing with at least $200); the average payout was $91. Eighteen players bet their whole bankroll on a single flip. Betting a constant 20% of the bankroll on heads (the Kelly fraction, 2 × 0.6 − 1) gives a probability of ruin close to zero. The authors also show why a positive average isn't enough: betting 20%, a win followed by a loss leaves 1.2 × 0.8 = 0.96 of the starting amount.
Limits: Both authors are practitioners: Haghani is founder and CEO of Elm Partners, a fund manager, and Dewey worked at PIMCO. A small, non-random sample playing a one-off game with a payout cap; not a study of investing. The abstract says 'about 30%' went bust; the body gives 28%, defined as finishing with under $2.
Haghani & Dewey 2016, Working paper (arXiv 1701.01427; SSRN 2016)
Size and concentrate positionsEvidence: Theory or logical argument
Kelly betting maximises long-run growth, but practitioners bet a fraction of it
▸ Key points▾ Key points
- In practice
- Sizing bets for maximum long-run growth (full Kelly) 'can be very risky in the short term'; when inputs are uncertain, the authors advise smaller bets.
- The number
- In one example of 700 bets with a 14% edge, full Kelly turned $1,000 into at least $100,000 16.6% of the time, half Kelly 0.1%; they could leave $18 and $145 respectively.
- The limit
- Mathematical results and worked examples, not a study of investors; in investing, the true edge and odds are unknown.
Kelly showed that a gambler with an edge can make wealth grow exponentially, at a maximum rate equal to the information rate of the channel. Betting to maximise the expected logarithm of wealth (the Kelly criterion) gives the highest long-run growth rate; for a simple two-outcome bet, the Kelly wager is the edge divided by the odds. MacLean, Thorp and Ziemba list the drawbacks: the wagers can be very large, so Kelly 'can be very risky in the short term', and in continuous time, betting twice the Kelly amount cuts the growth rate to the risk-free rate. In one example of 700 bets each with a 14% edge, full Kelly turned $1,000 into at least $100,000 16.6% of the time but could also leave $18. Half Kelly could still leave $145, and reached $100,000 or more only 0.1% of the time. Citing Chopra and Ziemba, they note that errors in estimated means mattered roughly 10 times as much as errors in variances and 20 times as much as errors in covariances, and they advise smaller bets when the inputs are uncertain.
Limits: Mathematical results and worked examples, not a study of investors. Kelly worked at Bell Labs; Thorp is listed with E.O. Thorp and Associates. In investing the true edge and odds are unknown, which is the main argument given for betting less than full Kelly.
Kelly 1956, Bell System Technical Journal · MacLean, Thorp & Ziemba 2010, Quantitative Finance
Size and concentrate positionsEvidence: Cohort or observational study
Shortfall risk kept falling beyond 100 stocks
▸ Key points▾ Key points
- In practice
- On a wealth-shortfall measure, holding more stocks kept lowering risk in this US test, and did more than spreading a few holdings across industries.
- The number
- Portfolios simulated from 1,000 large US stocks over a 20-year holding period: the chance of ending below a wealth target kept falling even above 100 stocks.
- The limit
- US large-cap stocks only; Statman (Level 2) found 30–40 stocks enough under a volatility-based definition, so the answer depends on the measure.
Simulating portfolios drawn from 1,000 large US stocks over a 20-year holding period, the authors measured shortfall risk: the chance of ending below a wealth target. It kept falling as portfolio size grew, even above 100 stocks. Spreading holdings across industries lowered risk slightly in small portfolios, but adding more stocks lowered it more.
Limits: US large-cap stocks only; the abstract doesn't give the sample years. Uses ending wealth rather than volatility. Statman (Level 2) found 30–40 stocks enough under a volatility-based definition, so the answer depends on the measure.
Domian, Louton & Racine 2007, Financial Review
Size and concentrate positionsEvidence: Cohort or observational study
Concentrated portfolios: signs of skill, more risk, and costly on average
▸ Key points▾ Key points
- In practice
- In these US studies, concentration was costly for most under-diversified individuals and for funds on average, though concentrated households' stock picks outperformed diversified households'.
- The number
- US discount-broker accounts, 1991–1996: concentrated households' trades outperformed diversified households', but their portfolios had more total risk and lower Sharpe ratios. Focused mutual funds did significantly worse than diversified funds after expenses.
- The limit
- Contested: the household result shows better stock selection, not better risk-adjusted portfolios; all are US observational studies where investors and funds chose their own concentration.
Using US discount-broker accounts from 1991 to 1996, Ivković, Sialm and Weisbenner found that stock trades by households with concentrated portfolios outperformed those of diversified households, especially in larger accounts and in local or non-S&P 500 stocks. But the concentrated portfolios carried more total risk and had lower Sharpe ratios. Goetzmann and Kumar found most US individual investors under-diversified, in patterns linked to overconfidence, trend-following and local bias, and that under-diversification was costly for most of them; only a small subset appeared to concentrate because of superior information. Among mutual funds, Sapp and Yan found that focused funds (few holdings) did no better than diversified funds before expenses, significantly worse after them, and closed more often.
Limits: Contested. The concentrated-household result shows better stock selection on average, not better risk-adjusted portfolios; the other two studies find concentration costly on average. All are US observational studies in which investors and funds chose their own level of concentration.
Ivković, Sialm & Weisbenner 2008, Journal of Financial and Quantitative Analysis · Goetzmann & Kumar 2008, Review of Finance · Sapp & Yan 2008, Financial Review
Size and concentrate positionsEvidence: Cohort or observational study
Professional managers bought well but sold worse than random
▸ Key points▾ Key points
- In practice
- Institutional managers in this study bought well but sold worse than at random, which the authors link to a low-attention rule of thumb used when selling.
- The number
- 783 institutional portfolios: added positions earned over 100 basis points a year more than the benchmark and random buying; sales gave up 80 basis points a year versus random selling (preferred specification).
- The limit
- Institutional managers, not individuals; the shortfall was largest for fundamentals-oriented managers with concentrated portfolios, and other selling rules weren't tested.
In data on 783 institutional portfolios averaging $573 million, positions the managers added earned over 100 basis points a year more, per dollar bought, than both the benchmark and a strategy of randomly adding to existing holdings. Their sales, by contrast, underperformed a random-selling strategy: in the preferred specification they gave up 80 basis points a year relative to selling at random. The authors link this to a costly, low-attention heuristic used when selling but not when buying.
Limits: Institutional portfolio managers, not individuals. Figures are from the 2021 NBER working paper; the study was published in 2023. The shortfall was largest among fundamentals-oriented managers with concentrated, high-tracking-error portfolios. The study doesn't test alternative selling rules.
Akepanidtaworn, Di Mascio, Imas & Schmidt 2023, Journal of Finance · Akepanidtaworn et al. 2021, NBER Working Paper 29076
Use (or avoid) leverageEvidence: Cohort or observational study
Daily-reset leveraged funds drift from their stated multiple over longer periods
▸ Key points▾ Key points
- In practice
- Daily-reset leveraged funds can end well away from their stated multiple of an index's return over longer periods; one study concludes they 'may be unsuitable for buy-and-hold investors'.
- The number
- SEC example over four months: an index gained 2% while a fund seeking twice its daily return fell 6%; a triple fund fell 53% while its index gained about 8%.
- The limit
- SEC examples are selected real cases plus a hypothetical; in periods of large returns and low volatility, leveraged funds can beat the static multiple.
In an SEC example, over four months an index gained 2% while a fund seeking twice its daily return fell 6%, and a fund seeking twice the inverse fell 25%. Over the same period a fund seeking three times another index's daily return fell 53% while that index gained about 8%. The SEC's two-day illustration: an index falls 10% then rises 10%, ending down 1%, while a 2x fund falls 20% then rises 20%, ending down 4%. Avellaneda and Zhang derive an exact formula linking a leveraged fund's return to the multiple of the index's return and the index's realised variance. It closely matched the quarterly returns of 56 leveraged funds (44 double, 12 triple) from January 2008, and they conclude such funds 'may be unsuitable for buy-and-hold investors'.
Limits: The regulator examples are selected real cases plus a hypothetical. In periods of large returns and low volatility, leveraged funds can beat the static multiple; Avellaneda and Zhang note a few such periods. Their study tested the formula over quarterly horizons from 2008. The authors are listed with NYU's Courant Institute and Finance Concepts, a firm.
SEC Office of Investor Education and Advocacy 2023, Updated Investor Bulletin: Leveraged and Inverse ETFs · Avellaneda & Zhang 2010, SIAM Journal on Financial Mathematics
Use (or avoid) leverageEvidence: Cohort or observational study
Leverage when young: a case for it, and the margin-call risk
▸ Key points▾ Key points
- In practice
- In this contested US test, leverage early in working life with less stock exposure later narrowed the spread of retirement wealth, but some cohorts suffered severe monthly losses.
- The number
- US data back to 1871: same average retirement wealth as a constant 75% stock allocation with a 21% smaller standard deviation; a fully leveraged young investor's portfolio fell 53% in October 1929.
- The limit
- Contested: depends on cheap borrowing, monthly rebalancing and continuing to invest through crashes; regulators warn that margin calls can force sales at low prices.
Using US stock data back to 1871, Ayres and Nalebuff find that holding stocks at up to 2:1 leverage early in working life, then reducing equity exposure later, produced the same average retirement wealth as a constant 75% stock allocation with a 21% smaller standard deviation. For an investor with relative risk aversion of 4, they estimate a 37% gain in certainty-equivalent wealth. The worst month was October 1929, when a fully leveraged young investor's portfolio fell 53% in that month. About one cohort in eight (11 of 96) lost more than 40% in at least one month. At 2:1 leverage with a 25% maintenance margin, forced selling starts after a market fall of about one-third; in their monthly-rebalanced data there were no margin calls under a 25% maintenance requirement, and six months with calls under 35%.
Limits: Contested and dependent on assumptions: cheap borrowing, monthly rebalancing and continuing to invest through crashes. Figures are from the authors' 2010 working paper; the 2013 Journal of Portfolio Management version may differ. Regulators (the SEC, FINRA and ASIC's Moneysmart) warn that margin calls can force sales at low prices; see Websites.
Ayres & Nalebuff 2013, Journal of Portfolio Management · Ayres & Nalebuff 2010, Diversification Across Time, Yale Law School working paper
Value a businessEvidence: Cohort or observational study
Long-run earnings growth barely persisted beyond chance
▸ Key points▾ Key points
- In practice
- In US data, persistently high earnings growth was rare, so valuations that assume it 'rest on shaky foundations', in the authors' words.
- The number
- US firms, 1951–1997: about 3% of surviving firms beat the median earnings growth rate five years in a row, close to the 3.1% expected by chance.
- The limit
- US firms; sales growth showed some persistence, earnings growth essentially none. Figures are from the 2001 working paper.
Across US firms from 1951 to 1997, the median growth rate of operating performance roughly matched GDP growth, and only about 10% of firms grew faster than 18% a year over ten years. For income before extraordinary items, about 3% of surviving firms beat the median growth rate five years in a row, close to the 3.1% expected by chance. Analysts' long-term (IBES) growth forecasts were too optimistic and added little predictive power. The authors conclude that valuations assuming persistently high growth 'rest on shaky foundations'.
Limits: US firms. Figures are from the 2001 NBER working paper; the study was published in 2003. Sales growth showed some persistence; earnings growth showed essentially none. Relevant to reading what a price implies: high long-run growth has been rare.
Chan, Karceski & Lakonishok 2003, Journal of Finance · Chan, Karceski & Lakonishok 2001, NBER Working Paper 8282
Value a businessEvidence: Cohort or observational study
Profitability reverts toward the mean, at about 38% a year
▸ Key points▾ Key points
- In practice
- In US firms, profitability on average moved back toward the mean, making current profitability an imperfect guide to the future and earnings changes partly predictable.
- The number
- US firms, simple model: profitability is estimated to move back toward the mean at about 38% a year, faster when below the mean or further from it.
- The limit
- US firms, sample period not given in the abstract; an average across firms, as some keep high returns on capital for long periods.
In a simple partial-adjustment model, Fama and French estimate that firms' profitability moves back toward the mean at about 38% a year. Reversion is faster when profitability is below its mean and when it is further from the mean in either direction. This mean reversion makes changes in earnings partly predictable.
Limits: US firms; the abstract gives no sample period. An average across firms: some keep high returns on capital for long periods, which is what judging competitive advantage tries to identify.
Fama & French 2000, Journal of Business
Value a businessEvidence: Cohort or observational study
Companies use hurdle rates about twice their cost of capital
▸ Key points▾ Key points
- In practice
- A company's discount rate for projects can reflect limited management capacity and company-specific risk, not just its cost of capital, in this US survey.
- The number
- In a survey of US chief financial officers matched to company data, firms used discount rates averaging twice their cost of financial capital.
- The limit
- Self-reported hurdle rates from surveyed US firms; describes how companies set them, not what rate an investor needs to require.
Matching a survey of chief financial officers to company data, the authors find that firms use discount rates averaging twice their cost of financial capital. Firms with plenty of capital but limited management or staff capacity appear to pass up profitable projects to keep room for better ones later. Firms using high discount rates had strong balance sheets, low leverage and large cash holdings, and firms also raised discount rates to allow for company-specific risk.
Limits: US firms responding to a survey; hurdle rates are self-reported. Describes how companies set hurdle rates, not what rate an investor needs to require.
Jagannathan, Matsa, Meier & Tarhan 2016, Journal of Financial Economics
Value a businessEvidence: Cohort or observational study
Returns after buyback announcements: strong in the 1980s, gone by 2003–2012
▸ Key points▾ Key points
- In practice
- Above-benchmark returns after US buyback announcements in 1980–1990 had disappeared for 2003–2012 events, so a pattern found in one era may not persist.
- The number
- US open-market buybacks announced 1980–1990: average abnormal four-year return 12.1%, and 45.3% for 'value' stocks; for 2003–2012 events, the long-run abnormal returns disappeared.
- The limit
- Contested: results differ between eras, long-run abnormal returns depend heavily on the benchmark, and neither US study judges whether a buyback used cash well.
For open-market buyback announcements from 1980 to 1990, the average abnormal four-year buy-and-hold return was 12.1%, and 45.3% for 'value' stocks, with no positive drift for 'glamour' stocks. Fu and Huang find that the long-run abnormal returns after buybacks (and after share issues) disappeared for events in 2003–2012. They link the change to more institutional investment, lower trading costs, better liquidity and tighter governance and disclosure rules.
Limits: Contested over time. Both are US studies, and long-run abnormal returns depend heavily on the benchmark used. Neither judges whether a particular company's buyback was a good use of its cash.
Ikenberry, Lakonishok & Vermaelen 1995, Journal of Financial Economics · Fu & Huang 2016, Management Science
Value a businessEvidence: Cohort or observational study
Acquirers' shareholders lost $240 billion in 1998–2001
▸ Key points▾ Key points
- In practice
- Most of the 1998–2001 losses to US acquirers' shareholders came from a small number of deals by very highly valued firms, and large-loss acquirers performed poorly afterwards.
- The number
- Around announcements, US acquirers' shareholders lost 12 cents per dollar spent in 1998–2001, $240 billion in total, against $7 billion (1.6 cents per dollar) in the 1980s.
- The limit
- US public acquirers; announcement returns are the market's immediate verdict, not the deal's eventual value, and averages hide wide variation between deals.
Acquiring-firm shareholders lost 12 cents per dollar spent around acquisition announcements from 1998 through 2001, $240 billion in total, against $7 billion (1.6 cents per dollar) in all of the 1980s. Most of the 1998–2001 loss came from a small number of deals by firms with very high valuations; without those deals, acquirers' shareholders would have gained. Firms that made large-loss acquisitions performed poorly afterwards.
Limits: US public acquirers. Announcement returns are the market's immediate verdict, not the deal's eventual value, and averages hide wide variation between deals.
Moeller, Schlingemann & Stulz 2005, Journal of Finance
Read capital and market cyclesEvidence: Cohort or observational study
Shipping booms: high earnings and heavy investment came before low returns
▸ Key points▾ Key points
- In practice
- In dry bulk shipping, high earnings and heavy investment came before low returns; the authors' model explains this as firms over-extrapolating demand and partly neglecting rivals' investment.
- The number
- Dry bulk shipping, 1976–2011: annual returns to owning a ship ranged from −76% to +86%; new-ship orders rose from under 10% of the active fleet (December 2001) to over 75% (August 2008).
- The limit
- One industry, where new ships take 18–36 months to deliver: an illustration of the capital cycle, not evidence that every industry behaves this way.
In dry bulk shipping from 1976 to 2011, high current ship earnings went with high secondhand ship prices and heavy investment in new ships, but forecast low future returns. Annual returns to owning a ship ranged from −76% (December 2007 to December 2008) to +86% (June 1978 to June 1979). The baseline regressions put expected one-year excess returns between −43% and +16%. Orders for new ships rose from under 10% of the active fleet in December 2001 to over 75% in August 2008. The authors' model explains this as firms over-extrapolating demand and partly neglecting their competitors' investment.
Limits: One industry, where new ships take 18–36 months to deliver. Figures are from the 2013 NBER working paper; the published 2015 version may differ. An illustration of the capital cycle, not evidence that every industry behaves this way.
Greenwood & Hanson 2015, Quarterly Journal of Economics · Greenwood & Hanson 2013, NBER Working Paper 19246
Read capital and market cyclesEvidence: Cohort or observational study
Fast-growing balance sheets predicted lower stock returns
▸ Key points▾ Key points
- In practice
- In US stocks, faster growth in a firm's total assets came before lower benchmark-adjusted returns, including among large companies, in this observational study.
- The number
- Across US stocks, compared with book-to-market, size, past returns, accruals and other growth measures, annual asset growth stood out as an economically and statistically significant predictor.
- The limit
- Observational, with no magnitudes or sample period in the abstract; published return patterns often weaken after publication.
Across US stocks, a firm's annual growth in total assets strongly predicted lower future abnormal returns, including among large companies. Compared with book-to-market, size, past returns, accruals and other growth measures, asset growth stood out as an economically and statistically significant predictor.
Limits: Observational, and the abstract gives no magnitudes or sample period. A related 'investment' factor appears in the Fama-French five-factor model (Level 4), and published return patterns often weaken after publication (Level 4).
Cooper, Gulen & Schill 2008, Journal of Finance
Read capital and market cyclesEvidence: Cohort or observational study
Frothy credit markets came before wider spreads and weaker growth
▸ Key points▾ Key points
- In practice
- In US data, signs of froth in credit markets tended to come before weaker economic activity and low excess returns to corporate bondholders.
- The number
- Elevated credit-market sentiment was followed two to three years later by a decline in economic activity (US, 1929–2015); falling issuer quality signalled overheating more reliably than rapid overall credit growth.
- The limit
- US data; these are average forecasting relationships, not precise timing signals, and neither study says when spreads will widen.
Using US data from 1929 to 2015, López-Salido, Stein and Zakrajšek find that elevated credit-market sentiment in one year was followed two to three years later by a decline in economic activity. When credit risk was priced aggressively, spreads later widened, and the timing of the widening was closely tied to the start of contractions. Greenwood and Hanson find that the credit quality of corporate bond issuers falls during credit booms, that this fall forecasts low excess returns to corporate bondholders, and that it was a more reliable sign of overheating than rapid overall credit growth.
Limits: US data. Average forecasting relationships, not precise timing signals; neither study says when spreads will widen. See the Goyal, Welch and Zafirov finding on how forecasting variables have held up out of sample.
López-Salido, Stein & Zakrajšek 2017, Quarterly Journal of Economics · Greenwood & Hanson 2013, Review of Financial Studies
Read capital and market cyclesEvidence: Cohort or observational study
Credit growth has been a strong predictor of financial crises
▸ Key points▾ Key points
- In practice
- Rapid credit growth may signal rising crisis risk in developed economies, and crises stayed costly for output even as monetary policy responses became more aggressive.
- The number
- 14 countries, 1870–2008: credit growth was a powerful predictor of financial crises; total credit rose strongly relative to output and money in the second half of the 20th century.
- The limit
- Country-level data for developed economies; predicts crises, not market returns, and crises are rare, so estimates rest on relatively few events.
Across 14 countries from 1870 to 2008, total credit rose strongly relative to output and money in the second half of the 20th century. Credit growth was a powerful predictor of financial crises. Monetary policy responses to crises became more aggressive over time, but the output costs of crises stayed large.
Limits: Country-level data for developed economies. Predicts crises, not market returns, and crises are rare, so the estimates rest on relatively few events.
Schularick & Taylor 2012, American Economic Review
Set a spending ruleEvidence: Cohort or observational study
Where the 4% rule came from, and how much inflation adjustment matters
▸ Key points▾ Key points
- In practice
- In US history, raising withdrawals with inflation made portfolios fail far more often in one test, and nearly all failures in another spanned the high-inflation late 1970s.
- The number
- US, 1926–2009: a 75/25 stock/bond portfolio supported a 7% withdrawal for 30 years in 91% of past periods if fixed in dollars, but only 45% if raised with inflation.
- The limit
- US history only, before fees and taxes; overlapping periods give far fewer independent observations than the counts suggest. Bengen was a practising financial planner.
Bengen tested a 50/50 mix of US stocks and intermediate-term Treasuries from 1926. A 4% first-year withdrawal, raised with inflation each year, never exhausted the portfolio in under 33 years; 4.25% could run out in as little as 28 years. In the Trinity study authors' 2011 update (US large-company stocks and high-grade corporate bonds, 1926–2009), a 75/25 portfolio supported a 7% withdrawal for 30 years in 91% of past periods when withdrawals were fixed in dollars, but in only 45% when they rose with inflation. All but one of the 18 failed 30-year periods for 5% inflation-adjusted withdrawals from a 50/50 portfolio spanned the high-inflation late 1970s.
Limits: US history only, before fees and taxes. Overlapping periods give far fewer independent observations than the counts suggest. Bengen was a practising financial planner. Level 1 also cites Bengen, together with Pfau's 17-country test; this entry adds the inflation-adjustment and failure-period detail.
Bengen 1994, Journal of Financial Planning (reprinted 2004) · Cooley, Hubbard & Walz 2011, Journal of Financial Planning, Portfolio Success Rates: Where to Draw the Line
Set a spending ruleEvidence: Cohort or observational study
A 38-country sample points to a much lower sustainable withdrawal rate
▸ Key points▾ Key points
- In practice
- Withdrawal rates drawn from US history may look high against broader data: this contested study calls its 38-country estimate 'materially lower' than the 4% rule.
- The number
- Asset-class returns from 38 developed countries: the authors estimate a 65-year-old couple accepting a 5% chance of financial ruin could withdraw just 2.31% a year.
- The limit
- Contested: far below US-history results like Bengen's 4%, with a sample including poor-return countries and a different failure definition, portfolio and data.
Using asset-class returns from 38 developed countries, the authors estimate that a 65-year-old couple willing to accept a 5% chance of financial ruin could withdraw just 2.31% a year. They call this 'materially lower' than conventional advice such as the 4% rule.
Limits: Contested: far below US-history results such as Bengen's 4%. The sample includes countries with poor real returns, and the paper's keywords point to survivor bias and 'easy data' bias in narrower samples. As with the 4% rule, withdrawals are a fixed inflation-adjusted amount set as a percentage of the balance at retirement, and 'ruin' means running out of money (confirmed from the working-paper version; the published text is paywalled). The comparison with Bengen still differs in how failure is defined (a 5% lifetime chance of ruin for a couple, versus no failure over 30+ years), the portfolio and the data. Level 2 cites a related paper by three of the same authors on 30-year stock returns.
Anarkulova, Cederburg, O'Doherty & Sias 2025, Journal of Pension Economics and Finance
Set a spending ruleEvidence: Cohort or observational study
Guardrails: rules that cut or raise spending let higher starting rates last
▸ Key points▾ Key points
- In practice
- In these simulations, rules that cut withdrawals after poor markets and raise them after good ones allowed higher sustainable starting rates, especially with more in equities.
- The number
- Simulated 40-year retirements, 1928–2004 data: with at least 65% in equities, starting rates of 5.2–5.6% were sustainable at a 99% confidence standard; with 50% equities, the maximum was as low as 4.6%.
- The limit
- Higher starting rates are paid for with spending cuts after poor markets, so real income can fall; results depend on simulation assumptions and applying the rules yearly.
Guyton and Klinger ran Monte Carlo tests of 40-year retirements, using return data from 1973–2004 and from 1928–2004. They added two 'guardrail' rules to two rules kept from Guyton's earlier work: no inflation increase after a year of negative portfolio returns (when the withdrawal rate is above its starting level), and a rule on which assets to sell. If the current withdrawal rate rises more than 20% above the initial rate, that year's withdrawal is cut by 10%; this rule stops 15 years before the end of the planning horizon. If the rate falls more than 20% below the initial rate, the withdrawal is raised by 10%. With at least 65% in equities, initial withdrawal rates of 5.2–5.6% were sustainable at a 99% confidence standard, which also required median purchasing power to be kept; with 50% equities, the maximum fell to as low as 4.6%. The two data periods gave virtually identical results.
Limits: The higher starting rate is paid for with spending cuts after poor markets, so income can fall in real terms. Guyton was principal of a wealth-advisory firm (Cornerstone Wealth Advisors); Klinger's background is in software development. Results depend on the simulation assumptions and on applying the rules every year.
Guyton & Klinger 2006, Journal of Financial Planning, Decision Rules and Maximum Initial Withdrawal Rates
Set a spending ruleEvidence: Theory or logical argument
In theory, retirees gain from annuities, and pricing is close to fair for typical buyers
▸ Key points▾ Key points
- In practice
- In theory, annuitising some or all savings is often optimal for retirees; May 2024 US annuity prices were estimated to be close to fair for typical annuity buyers.
- The number
- US immediate annuities on sale in May 2024, bought at 65: about 87 cents per premium dollar at average population mortality, about one dollar for a typical annuity buyer.
- The limit
- Theory plus US pricing estimates that depend on mortality and discount-rate assumptions; buying an annuity gives up access to the lump sum and any bequest from it.
Yaari (1965) showed that a consumer with an uncertain lifespan and no bequest motive is best off fully annuitising. Davidoff, Brown and Diamond show that full annuitisation remains optimal under much weaker conditions when markets are complete, and that some annuitisation remains optimal in many cases when they are not, a considerable share in their calculations. They suggest that limited annuity purchases are plausibly due to psychological or behavioural biases. On pricing, Poterba and Solomon (2021, revised 2025) estimate that US immediate annuities on sale in May 2024, bought at 65, paid out about 87 cents per premium dollar for a buyer with average US population mortality, and about one dollar for a typical annuity buyer. For a deferred annuity bought at 65 with payments from 75, the figures were 73 cents and close to one dollar.
Limits: Theory plus US market pricing. Yaari's 1965 paper was not read directly: its result is given here as Davidoff, Brown and Diamond summarise it. Money's-worth estimates depend on the mortality and discount-rate assumptions. Buying an annuity gives up access to the lump sum and any bequest from it.
Yaari 1965, Review of Economic Studies · Davidoff, Brown & Diamond 2005, American Economic Review · Poterba & Solomon 2021, rev. 2025, NBER Working Paper 28557 (revised October 2025; prices from May 2024)
Set a spending ruleEvidence: Cohort or observational study
Few retirees buy annuities, and framing may explain part of why
▸ Key points▾ Key points
- In practice
- In this survey, framing alone shifted stated preference for a life annuity sharply, so how options are presented may affect annuity choices.
- The number
- Survey of 1,342 people over 50: 72% preferred a life annuity under consumption framing, 21% under investment framing; UK, year to 31 March 2026: 100,144 annuities bought, 401,137 plans into drawdown.
- The limit
- Hypothetical survey choices between actuarially equivalent options, not real purchases; the UK data show what people chose, not why, and many pots are small.
In a four-arm internet survey of 1,342 people over 50, 72% preferred a life annuity to a savings account when the choice was framed in terms of consumption, but only 21% when it was framed in terms of investment features. In the UK, in the year to 31 March 2026, 100,144 annuities were bought and 401,137 plans entered drawdown, out of 1,047,008 pension plans accessed for the first time.
Limits: The survey asked about hypothetical choices between options designed to be actuarially equivalent, not real purchases. The UK figures are regulator data on what people chose, not on why; 45.8% of pots accessed were cashed in entirely, and many pots are small.
Brown, Kling, Mullainathan & Wrobel 2008, American Economic Review (Papers and Proceedings) · Financial Conduct Authority 2026, Retirement income market data 2025/26
