Graphical Summary

“Buy as the bears appear; hold while the air is clear; sell as stocks get too dear.”

Summary

This is a practical guide to a risk-assessment formula that, historically (when applied to portfolios whose stock component tracks the S&P 500 stock index), would have produced almost 50% better “reward for the risk taken” than would have either a 100% stock portfolio or a strictly balanced 60% stock portfolio. The objective of the formula is to seek an efficient limit to the damage caused by weak market returns, not to beat the market.

The formula presented here is approached as follows:

This is a “very aggressive” formula in the sense that it can call for 100% of the portfolio to be allocated to stocks for prolonged periods. It is imperative that one understand that other assets be planned, maintained, and coordinated with the portfolio to meet expenses, particularly during periods of reduced market returns, to avoid depleting the portfolio while its value is in decline.

(Note the disclaimer below.)

(Also note that “risk” has different meanings in different domains, and this document uses the term more in the generally understood sense of “possibility of realizing hazard” rather than in the financial sense of “uncertainty” which, confusingly to the layman, includes possible gains as well as possible losses.)

Introduction (or “Why Bother with This?”)

One aims to make a retirement investment portfolio last a lifetime. To forestall its depletion, substantial returns are required during the withdrawal phase. However, a portfolio with high price volatility runs the risk of excessive depletion if withdrawals are necessary when the portfolio value is low.

Bonds experience less price volatility than stocks, but often with starkly reduced returns; for 1911–2022, the S&P 500 stock index has produced “real” (i.e., inflation adjusted) “total returns” (dividends plus price change) at a rate that is over four times the rate of real total returns for 10-year US Treasury bonds. It would be helpful to know the best “allocation ratio” (i.e., the value of stocks in one’s retirement portfolio divided by the sum of the values of stocks and bonds) to balance returns against risk.

“The Sharpe ratio compares the return of an investment with its risk.” Nobel Laureate William Sharpe more descriptively called this the “reward-to-variability ratio”, i.e., the standard deviation divided into the difference between the return for a portfolio and the so-called “risk-free rate of return” that would be expected with government bonds. Thus, higher values represent more reward for the risk taken. (The total returns for 10-year US Treasury bonds is the relatively “risk-free-rate” for the purpose of this introductory section, and the standard deviation herein is robustly approximated by the \(25^\text{th}\) through \(50^\text{th}\) percentiles of annual real total return.) For 1911–2022, the S&P 500 index’s Sharpe ratio = 0.44 (computed in simulations using data from Robert Shiller’s data page). (In fact, total returns from 10-year US Treasury bonds are highly susceptible to changes in interest rates, but the “financial risk”, i.e., uncertainty, of their total returns is considerably less than the uncertainty of total returns from stocks.)

“Periodic rebalancing” is commonly presented as a formula to get fair returns with reduced price volatility. The procedure is to adjust the balance of stocks and bonds in a portfolio periodically to maintain a constant ratio of stocks to bonds. For 1911–2022, with monthly rebalancing to a 60:40 stock:bond ratio, this “periodic rebalancing” formula (applied to the S&P 500 index and 10-year US Treasury bonds) would have produced only about 72% of the returns of the S&P 500 with about 66% of the downward volatility (Sharpe ratio = 0.42). Thus, although intuition suggests that this formula might be helpful, the 60:40 portfolio nevertheless merely reduces risk without effectively improving risk’s relationship with reward. (Other fixed-percentage strategies share this weakness.)

Assessing the probability and magnitude of a potential loss in stock value may play a role in the search for a more rewarding way to address downward stock price volatility. An intuitive view of the possible magnitude of an “intrinsic value” of stocks (explored below) is that stocks have returned almost 6.5% per year for the last 110 years (adjusted for inflation), meaning that the dividend-adjusted price has approached fifteen times earnings (very roughly, and with much variation). However, stock prices fluctuate wildly, perhaps because of pessimism or alternative investments that might have a higher yield in the near term; several years may pass until stocks return to such a relative valuation. How might one estimate when stocks are not grossly overpriced?

Benjamin Graham developed a formula for estimating the intrinsic value of a company’s stock that included the “reasonably expected 7 to 10 Year Growth Rate of earnings per share” that came with a warning:

Let the reader not be mislead into thinking that such projections have any high degree of reliability, or, conversely, that future prices can be counted on to behave accordingly as the prophecies are realized, surpassed, or disappointed.

It seems obvious to the writer of this document that past earnings arose during the environment of the past, that the environment in the future cannot be exactly the same as the environment of the past, and that therefore earnings must differ. Yet there is work that is suggestive that this “smoothed” earnings, while not being a good predictor of earnings, may be one of the only objective predictors of earnings for the stock market as a whole.

Extending the observation of Benjamin Graham (Graham & Dodd, 1934), the book Irrational Exuberance (by Nobel Laureate Robert Shiller) illustrated that a ratio commonly referred to as “the P/E 10 Ratio” (or “CAPE” or “PE10”, i.e., the price of the S&P 500 stock index divided by the average earnings over the previous ten years) is inversely related to total returns over the subsequent 20 years. Thus, when the P/E10 has been extremely high, there has been a greater likelihood of downward volatility. (The applicability of such a ratio as a limited predictor has been shown for stock markets in other countries as well.)

According to Lucile Tomlinson (1953), “A Variable Ratio formula provides for smaller percentages of stocks in high [market-price] areas, where the risk of owning stocks is greatest, and for larger percentages in low [market-price] areas, where the risk of loss is bound to be considerably less” (page 167). (I am indebted to Wade Pfau for bringing this excellent book to my attention by quoting from it in his 2011–2012 research papers. See the “More questions & answers” section below for discussion of Pfau’s observations.)

This document presents a simple “risk-bounded buy-and-hold” stock investment formula to limit the proportion of stock in a portfolio when the P/E10 is very high. As shown in the document linked below, this formula would have given 109% of the returns of the S&P 500 index during 1911–2022, with about 75% of the downward volatility (Sharpe ratio = 0.65). Thus, the “risk-bounded buy-and-hold” formula would have rewarded the risk taken with about 48% more returns than either the “periodic rebalancing” formula or the “100% stock” formula. Much of the increase in returns would have come as the market passed through the “price bubbles” of 1929, 2000, and 2008.

Note that the results above were obtained by simulating portfolios composed of US stocks and US government bonds. However, the returns of international bonds and stocks are fairly correlated respectively with those of domestic bonds and stocks and constitute only about half as much global market capitalization as the US counterparts do, so this formula ought to be representative of the results that might be expected even when one takes advantage of international securities to diversify a portfolio. (Reasons to employ this practice are discusssed at https://www.investopedia.com/ask/answers/061515/what-are-advantages-foreign-portfolio-investment.asp.)

“Risk-Bounded Buy-and-Hold” Formula

See “Decision-making examples” and “Example funds that could be employed” below.

(A reader of Tomlinson’s book might recognize this formula as a particular variant of “A Compromise Solution” that she presented on pp. 232–238.)

Allocation table for 2023

The columns of this table are described immediately after it. A graphical summary of this table is presented at the beginning of this document.

SP500 price P/E10 E10/P Relative PE10 Percentile Buy boundary Sell boundary Moderated hazard
6371 46.3 2.16% 2.16 100.0 0% 0% 0%
6031 43.8 2.28% 2.05 99.4 0% 0% 0%
5710 41.5 2.41% 1.94 98.8 0% 0% 0%
5406 39.3 2.55% 1.84 98.5 0% 0% 0%
5118 37.2 2.69% 1.74 97.7 0% 0% 0%
4846 35.2 2.84% 1.65 97.1 0% 0% 0%
4588 33.3 3.00% 1.56 95.8 0% 0% 0%
4343 31.5 3.17% 1.48 94.6 8% 17% 5%
4112 29.9 3.35% 1.40 92.2 15% 34% 9%
3893 28.3 3.54% 1.32 89.2 22% 52% 12%
3686 26.8 3.74% 1.25 83.6 29% 71% 14%
3490 25.3 3.95% 1.19 75.3 34% 92% 15%
3304 24.0 4.17% 1.12 66.5 39% 100% 12%
3128 22.7 4.40% 1.06 57.8 45% 100% 8%
2961 21.5 4.65% 1.01 50.9 50% 100% 3%
2804 20.4 4.91% 0.95 43.6 56% 100% 0%
2654 19.3 5.19% 0.90 37.8 61% 100% 0%
2513 18.2 5.48% 0.85 33.8 66% 100% 0%
2379 17.3 5.79% 0.81 29.0 70% 100% 0%
2253 16.4 6.11% 0.77 24.2 75% 100% 0%
2133 15.5 6.46% 0.72 18.8 79% 100% 0%
2019 14.7 6.82% 0.69 16.0 83% 100% 0%
1912 13.9 7.20% 0.65 12.9 87% 100% 0%
1810 13.1 7.61% 0.61 10.4 91% 100% 0%
1713 12.4 8.04% 0.58 7.8 94% 100% 0%
1622 11.8 8.49% 0.55 5.3 97% 100% 0%
1536 11.2 8.97% 0.52 2.4 100% 100% 0%
1454 10.6 9.47% 0.49 1.0 100% 100% 0%
1377 10.0 10.00% 0.47 0.7 100% 100% 0%
1303 9.5 10.57% 0.44 0.3 100% 100% 0%
1234 9.0 11.16% 0.42 0.1 100% 100% 0%

The S&P prices here are for 2023; they vary with inflation and the “relative PE10” (which is derived in the reference cited below).

  • SP500 price – price of the S&P 500 index, during 2023, in 2023 US Dollars
  • P/E10 – The “cyclically adjusted price-to-earnings ratio” (CAPE), given by \[\frac{\text{S&P 500 price}}{average(\text{S&P 500 earnings per year for preceding 10 years})}\]
    • This is computed with inflation-adjusted dollar values. It is conveniently available at
      https://www.multpl.com/shiller-pe
    • The expected median of the P/E10 increases gradually with time, as discussed in the the document linked below; a regression model predicts the median value of P/E10 to be: \[(0.0578777)(\text{Year}) - 95.694931\]
  • E10/P – The reciprocal of the CAPE, i.e., the statistically expected rate of return for the S&P 500, given by \[\frac{average(\text{S&P 500 earnings per year for preceding 10 years})}{\text{S&P 500 price}}\].
  • Relative PE10 – Expresses P/E10 as a multiple of the expected median value for 2023 (\(21.3\))
  • Percentile – percentile rank of Relative PE10
    • This reflects the statistical expectation (based on past historical performance) that the Relative PE10 will fall to a lower value.
  • Buy boundary – proportion of stock in portfolio to which one may increase as the S&P 500 index falls below SP500 price
  • Sell boundary – proportion of stock in portfolio to which one should decrease as the S&P 500 index rises above SP500 price
  • Moderated hazard – statistically expected loss (if any) expected for remaining investments in stock, after application of the sell boundary, when the S&P 500 index is at the level of SP500 price

Note that there is nothing “magical” about the boundaries in this table; they have been chosen to limit risk to reasonable levels during “stock market bubbles” and, in their aftermath, to take good advantage of the the eventual rebound in stock prices. However, if one wishes to assume a bit less risk in exchange for a modest reduction in returns, it would be acceptable to limit risk further by decreasing the proportions assigned to the “buy boundary”, by choosing a maximum proportion of stock less than 100%, or even by moving the “buy boundary” lower, as desired. Little advantage can be gained by moving the “buy boundary” higher or by moving the “sell boundary” either way. If you want to experiment, please consider the “development and testing” document linked below.

Graphical summary of allocation boundaries vs.\(\frac{\text{E10}}{\text{P}}\)

One may ask for whether there is an explanation for why the stock-trading boundaries presented here might have been effective in the past. Perhaps one may get an intuitive feel for this by inspecting a plot of the boundaries versus \(\frac{\text{E10}}{\text{P}}\) (i.e., the reciprocal of the P/E10, which is not the reciprocal of the “Relative PE10”).

As the P/E10 assigns a value to stocks based on their present price and past earnings, \(\frac{\text{E10}}{\text{P}}\) reflects the index’s yield on the same basis. This is consistent with the valuation approach of Graham, Buffet, and Shiller. I will call this the “statistically expected rate of return”, which in no way implies that it predicts a specific return at a specific time in the future.

With this rate in hand, we may relate it to the historical CAGR (compound annual growth rate) for S&P 500 stocks (about 6.5% real rate = \(\frac{1}{15.46}\)), high-yield bonds (about 4.75% real rate = \(\frac{1}{21.05}\)), and the “risk-free rate”. The “risk-free rate” is a theoretical rate of return expected for taking no risk with an investment. As a proxy, I am using the 30 day SEC yield for the Vanguard Short-Term Inflation-Protected Securities ETF (VTIP), currently (in late 2023) about 2.7% (real rate = \(\frac{1}{37.45}\)). The spread between TIPS and short-term Treasury bonds suggests inflation expected by the bond market to be about 2.5% (in late 2023).

Here is a plot of the boundaries versus \(\frac{\text{E10}}{\text{P}}\), with vertical lines indicating the above rates as points of reference.

In summary:

  • As \(\frac{\text{E10}}{\text{P}}\) decreases toward the risk-free rate, incrementally decrease stock proportion.
    • Stocks are reduced to 50% as their expected return descends to the risk-free rate.
  • As \(\frac{\text{E10}}{\text{P}}\) increases toward the high-yield bonds’ real rate, incrementally increase stock proportion.
    • Stocks are increased to 50% as their expected return ascends to the high-yield bonds’ real rate.
  • As \(\frac{\text{E10}}{\text{P}}\) increases toward the stocks’ real rate, incrementally increase stock proportion.
    • Stocks are increased to 75% as their expected return ascends to the expected real rate for the S&P 500.
    • Stocks are increased to 100% as their expected return ascends to the rate of inflation plus the expected real rate for the S&P 500.
  • When stock prices are descending, stock are not purchased until \(\frac{\text{E10}}{\text{P}}\) falls below the “buy boundary”.
  • When stock prices are ascending, stock are not sold until \(\frac{\text{E10}}{\text{P}}\) rises above the “sell boundary”.

“Excess yield of stocks” is weakly predictive of “excess return of stocks”

If the \(\frac{\text{E10}}{\text{P}}\) is considered as the “effective yield of stocks”, then the (sometimes negative) difference between twelve-month moving averages of the S&P 500 yield and the yield of ten-year US Treasury bonds is weakly predictive of the difference between the moving averages of the respective ten-year total returns (again, using Shiller’s monthly data), consistent with Graham’s formula specifying consideration of performance over the preceding 7-10 years when estimating the “intrinsic value” of a stock. This bolsters the case that relative yield may (in part) explain why the stock-trading boundaries presented here might have been effective in the past.

Example funds that could be employed

Potential guidelines

  • Presume that short-term expenses are managed outside of the portfolio in a short-term fund.
    • The short term fund must be sized to sustain spending during market downturns lasting several years.
    • This allows the portfolio to be both more aggressive and quicker to recover from adverse outcomes.
  • Employ exchange-traded-funds (ETFs) when possible to minimize expense ratios and taxable events.
  • Employ stock index funds weighted to represent the total world markets (roughly \(\frac{2}{3}\) US and \(\frac{1}{3}\) “ex-US”).
    • Exposure to international securities maximizes diversification to “achieve a higher risk-adjusted return” (Investopedia).
    • World markets are loosely correlated with US markets, so models considering US markets represent more than \(\frac{2}{3}\) of a total-market portfolio’s stock performance.
  • Intermediate-term expenses can be funded from more volatile assets having more earning potential, e.g., the portfolio may include high-yield corporate bonds.
  • Compose the portfolio simply to ease adjustment of stock proportion, including:
    • One broadly-diversified domestic stock fund.
    • One broadly-diversified international stock fund.
    • One high-yield bond fund, accepting the volatility and correlation with stock prices that comes with such funds.

Potential alternatives

One could choose to minimize the short-term expense management fund within the portfolio and replace a high-yield bond fund with a blend of a shorter term bond fund, more stock, and other uncorrelated asset classes. That approach seems to this author to provide little advantage and to complicate the math immensely.

Specific example funds

Decision-making examples

Consider a few examples (for 2023):

  • The S&P 500 index is at \(4850\) and \(\frac{\text{value(Stock fund)}}{\text{value(Stock fund) + value(Bond fund)}} = 0.74\)
    • Sell Stock fund and buy Bond fund to bring ratio to \(0.65\)
  • The S&P 500 index is at \(4850\) and \(\frac{\text{value(Stock fund)}}{\text{value(Stock fund) + value(Bond fund)}} = 0.45\)
    • Do nothing.
  • The S&P 500 index is at \(4000\) and \(\frac{\text{value(Stock fund)}}{\text{value(Stock fund) + value(Bond fund)}} = 0.22\)
    • Do nothing.
  • The S&P 500 index is at \(4000\) and \(\frac{\text{value(Stock fund)}}{\text{value(Stock fund) + value(Bond fund)}} = 0.08\)
    • Sell Bond fund and buy Stock fund to bring ratio to \(0.15\)

Demonstration of effect of formula on medium-to-long term returns

Comparison of formula to fixed allocation with same average stock percentage

For a detailed discussion of the effect of the formula presented here, see the document linked in the next section. Here is a brief demonstration of the effect of the risk-bounded buy-and-hold formula for 1911–2022 (during which time its stock allocation averaged 89.7%) relative to a rebalanced, fixed 89.7% portfolio. These plots show a ratio between the returns of these two allocations over the return periods 5, 10, 20, and 30 years. Typically, but not always, the risk-bounded allocation would have substantially outperformed the fixed allocation; when it would have fallen short, it would only have done so by less than 20%.

(A fixed 89.7% portfolio was used above because the 60% portfolio does not have sufficient earning power to compete; see below for comparison to a 60% stock portfolio.)

Comparison of formula to fixed 60% stock allocation

Would a fixed 60% stock allocation provide more protection agaist downside volatility of the formula described here? Compared to the fixed 60% allocation, when the formula would have fallen short, it would only have done so by less than 20%, and then only for time frames less than 20 years. (As for the previous figures, the average stock allocation for the formula for 1911–2022 was 89.7%.) Thus, the fixed 60% stock allocation would have provided little if any more protection against downside volatility of the formula than the fixed 89.7% stock allocation.

Lucile Tomlinson’s report of a hazard-limiting “Compromise Solution”

This chart is from Chapter 17 of Lucile Tomlinson (1953) Practical Formulas for Successful Investing, p. 237. It is based on the “Compromise Solution” that she presented, pp. 232–238, which takes a (very) similar approach to the formula presented here (but with a narrower band between buy- and sell-boundaries), producing analogous reduced downward volatility for 1924–1952. Tomlinson points out:

The problem of selecting a median or range, while not entirely eliminated by this formula, is reduced to relative unimportance.

Generally, return would have been higher than the Dow-Jones Industrial Average except during the bubble in the run-up to the Great Depression, notwithstanding the fact that the stock allocation would have remained at or below 60% stocks.

Chart from Lucile Tomlinson (1953) p. 237

Development and testing of this formula

A report of the derivation of the “relative PE10” and of the “Risk-Bounded Buy-and-Hold” formula, along with a test of how it would have performed historically, may be reviewed at https://eschenlauer.com/investing/risk_based_allocation/HistoricalReturnByAllocationStrategy.pdf; the abstract is presented here:

Since 1871, US stocks have produced greater “total returns” (dividends plus price change) than bonds in “real” (i.e., inflation-adjusted) dollars but have demonstrated much greater price volatility. One commonly recommended formula to reduce the down-side volatility of a portfolio is to perform “periodic reallocation” (e.g., reallocating monthly, quarterly, or annually) to maintain a fixed percentage of stocks. An alternative to periodic reallocation (due to Benjamin Graham) is to base the choice to purchase stock on to the price of divided by an estimate of the “intrinsic annual earning power”. Robert Shiller used the inflation adjusted earnings averaged over the preceding ten years to estimate the earning power of the S&P 500 stock market index; Shiller investigated the history of the ratio commonly named “the P/E10” (i.e., the current price of the S&P 500 index divided by this estimate) to show that its increase is generally predictive of reduced total returns over the subsequent 20 years. Using Shiller’s inflation-adjusted data for S&P 500 prices, earnings, and dividends and for US Treasury 10-year bonds, this study introduces a date-insensitive “relative PE10” and simulates the returns resulting from application of three allocation strategies: 60% stock “periodic reallocation”, P/E10-driven “risk-bounded buy-and-hold”, and 100% stock buy-and-hold. From 1911 through 2022, over time spans of \(\ge\) 5 years, the “risk-bounded buy-and-hold” formula generally would have produced greater annualized real returns than the 100% stock allocation with less down-side volatility, while the 60% stock “periodic reallocation” formula would have produced significantly lower returns with only moderately less down-side risk.

Afterword

What if everyone did this?

If this formula were widely applied, stock market price bubbles might become much more rare, and the portfolio might typically remain nearly 100% allocated into stocks, so the reward-to-variability ratio might diminish toward that of a 100% portfolio. Perhaps bond prices might remain lower (so that their yield could compete with that of stocks), which might make deficit spending, leveraged business activity, and consumer indebtedness less attractive.

This scenario seems extremely unlikely, however, because many investors and investment managers “seek inefficiencies in the market” or choose to respond to their estimate of “the current phase of the business cycle”, etc., and they would likely be seeking a considerably greater reward by taking greater risk (which seems like gambling to this author).

Can the P/E10 predict the future?

The P/E10 cannot predict the future. That being said, any formula that is to be effective for conditions that might happen should indeed have been effective for conditions that did happen (because it is indisputable that those past conditions were possible). Any proposed formula for portfolio management should pass testing against what has happened historically; if it does not, is it not “magical thinking”?

Is this formula “just market timing”?

Any decision regarding stock allocation is actually market timing, including even “buy and hold 100% stocks in a portfolio”. Any (wise) investment decision should take into account expected market returns. However flawed it might be, a formula exists to make this decision objective:

A formula is simply a method of forcing oneself to do something, whether it is popular at the time or not, … setting in advance the conditions under which securities are bought and sold. … The term “formula investing” [covers] any method which determines ahead of time action that is to be taken in the future.
– Tomlinson (1953), p. 6.

The formula presented here is “tuned” to maintain a very high stock allocation (typically 100%) at most times and to reduce it only slightly at most other times. When high-yield bonds are the alternative investment to stocks, choosing such an alternative is hardly “getting out of the market” because the latter are highly correlated with stocks (albeit with much less volatility than stocks but much greater volatility than investment grade bonds).

Ultimately, it is up to the reader to decide whether this formula relies upon market timing more or less than other approaches.

More questions & answers

  • Q: Is this formula a “new discovery”? Who was Lucile Tomlinson?
    • A: This is not new. Lucile Tomlinson’s 1953 book described a very similar formula applied by “a large endowment fund” as of that year. This exceptionally crafted book does not advocate for any one investment formula over another; Tomlinson was a journalist who, rather, described in detail the formulas with which she was familiar and presented the advantages and challenges that each might experience. Although the market has experienced many changes since the book was published, there are many fundamentals applicable to today that are explained in plain language. Without grossly overstating the case, Lucile Tomlinson might considered “the mother of mutual fund”; see https://memphreinvestments.com/book-blog/f/8-lucile-tomlinson-practical-formulas-for-successful-investing for a biography and summary of the 1953 book. She also she wrote and edited publications about how to establish and run mutual funds.
  • Q: If the main benefit of the formula is because other strategies perform poorly during bubbles, how does the performance compare if you don’t analyze through bubble periods? Does this require a bubble to be competitive?
    • A: During non-bubble periods, the formula here allocates up to 100% into stocks and, hence, performs almost identically to the latter. There is advantage only when risk is excessive, which the formula handles (advantageously) by decreasing stock allocation temporarily and incrementally.
  • Q: Why is the standard deviation for real total return approximated by the \(25^\text{th}\)–\(50^\text{th}\) percentile, and not something that is balanced around the \(50^\text{th}\) percentile? Why not use the standard deviation?
    • A: Down-side volatility is the risk mitigated here, so only downward deviation is considered. The standard deviation is not used because it assumes “a Gaussian distribution” (a.k.a. “the bell-shaped curve”) of independent observations; observations of real returns are not normally distributed; and returns are not independent because they are autocorrelated.
  • Q: Is the cost of changing percentage stock allocation taken into account here?
    • A: There are indeed costs associated with trading. This is true of trades to “rebalance” a portfolio periodically as well as to change allocation incrementally (e.g., “dollar cost averaging”). The approach presented here only trades when necessary, which might be expected to be less often than periodic allocation strategies require. To decrease trading frequency, one could increase the size of incremental changes in allocation.
  • Q: The outcomes are compared to a 100% stock and a 60% stock periodic reallocation. Are these fair comparisons? What do people actually do with their investment on average, and could typical strategies be compared?
    • A: A 60% stock allocation is so typical that there exist funds (such as VBIAX, the Vanguard Balanced Index fund) that specifically target this ratio. 100% stock represents the ultimate aggressive portfolio. Results shown here are representative of what is seen for other (fixed) percentages of stock in periodically rebalanced stock portfolios (lower or higher percentage stock portfolios generate correspondingly lower or higher returns over the long term).
  • Q: Why average earnings over ten years rather than some other number of years?
    • A: The P/E10 is widely discussed and easily found, e.g., https://www.multpl.com/shiller-pe, making it practical and relatively easy to use as a metric. Benjamin Graham (1934) originally suggested at least seven to ten years as an estimator of the earnings rate for a company. Ten years (and other periods) were shown effective in Campbell and Shiller (1988). (Five years seems a bit too short, and more than ten years seems past the point of diminishing advantage.)
  • Q: Is the effect of using the Relative PE10 attributable to knowing the expected value during the entire period tested?
    • A: No, results for 2003–2022 would have been comparable even if the the trend in the P/E10 had been computed using years only through February 2003.
  • Q: Does Wade Pfau recommend using a Variable Ratio formula for stock allocation in an investment portfolio?

    • A: Definitely not. He wrote research papers (2011, 2012) regarding a “safe withdrawal rate from a retirement portfolio” and, in the context of that research, demonstrated (using Shiller’s data) that, “most every permutation of valuation-based allocation strategies based on P/E10 demonstrate strong potential to improve risk-adjusted returns for conservative long-term investors. Such valuation-based strategies provide comparable returns as 100% stocks, but with substantially less risk according to a wide variety of risk measures.” However, he made this evaluation only in that context and recently commented:

      Shiller wrote the his article on the CAPE ratio, the cyclically adjusted price earnings ratio, I think the first one came out in 1998. And when you looked historically, before that point, it provided pretty good predictive power about just that’s fitting the line through the data. When the CAPE ratio was high, the subsequent stock market returns tended to be lower, and vice versa. But it’s kind of like what happens with a lot of this, once people saw that point, the relationship broke down. And if you were following that sort of guidance, you would have been spending most of your time since the mid 1990s, with the lower stock allocation, because you were worried that markets were overvalued, and you would be experiencing a lower return from those stocks. Now, we’ve seen there’s been a lot of years that have passed since the mid 1990s. And for the most part, it probably wouldn’t have been a great idea to not to have a lower stock allocation that whole time, because you were worried that because the CAPE ratio is predicting lower subsequent stock market returns. And that’s where I think ultimately planning conservative assumptions for retirement. You might not want to assume historical stock market returns when the CAPE ratio is higher. But beyond that, I don’t think we can really gleam a whole lot of information from something like the CAPE ratio.
      – “Retire with Style” podcast episode 57, at 16:41

      Pfau considers fixed rates of withdrawal impractical for many people. Rather, he urges individuals to discover an overall financial plan for retirement that is tailored to their “retirement income style”. (See https://risaprofile.com/for-individuals-risa/ to learn more, and also his excellent and extensive, consumer-oriented books on the subject at https://retirementresearcher.com/books/).

  • Q: Do you agree with Wade Pfau on this point?
    • A: Not entirely. Wade does excellent work for which I have immense respect. However, while I agree that the CAPE ratio is not something on which to base decisions to make radical changes in stock allocation in response to rapid change in the S&P index’s price, I similarly do not think that Graham’s theory cannot be ignored when the P/E10 is in the 90th percentile and above. I also agree with Graham that one would do better not completely divest themselves of stock at the extremes, but it’s worth considering the earning potential relative to the cost of any investment, as I developed above in the section “Graphical summary of allocation boundaries vs.\(\frac{\text{E10}}{\text{P}}\)”.

Acknowledgements

  • This formula was inspired by thoughts and procedures received from my father, Arthur Charles Eschenlauer, who drew inspiration from Benjamin Graham’s and Warren Buffet’s writings.
  • I have little doubt that this formula owes much to Lucile Tomlinson’s “Compromise Solution” (pp. 232-233 of her 1953 book, chapter 17), although I only learned of that source after I had tested the procedure and established the boundaries described here.
  • Jeremy Chacón provided helpful review and questions.

References

Campbell, J., & Shiller, R. Stock Prices, Earnings, and Expected Dividends. (1988). The Journal of Finance (New York), 43(3), 661-676. Available at https://scholar.harvard.edu/files/campbell/files/campbellshiller_jf1988.pdf

Graham, Benjamin, & Dodd, David L. (1934). Security analysis. New York: Whittlesey House, McGraw-Hill book company.

Pfau, Wade D., Long-Term Investors and Valuation-Based Asset Allocation (2011). Available at SSRN: https://ssrn.com/abstract=2544636 or http://dx.doi.org/10.2139/ssrn.2544636

Pfau, Wade D., Withdrawal Rates, Savings Rates, and Valuation-Based Asset Allocation (2012). Available at SSRN: https://ssrn.com/abstract=2544635 or http://dx.doi.org/10.2139/ssrn.2544635

Shiller R. J. (2016). Irrational exuberance (Revised and expanded third edition). Princeton University Press.

Robert Shiller’s data page http://www.econ.yale.edu/~shiller/data.htm.

Tomlinson, Lucille (1953). Practical Formulas for Successful Investing. New York: Wilfred Funk.

Disclaimer and License

Disclaimer

All investment involves risk. I am not a financial advisor; I am merely presenting the results of my investigations. This document is not financial advice; please treat it with a commensurate degree of skepticism. If you cannot serve responsibly as your own financial advisor, please hire one. Thank you.

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