Mitigating risk with a variable stock-to-bond ratio

Would you consider checking your assumptions about a fixed stock-to-bond ratio if your retirement depended on it? Do the projected rates of return for stocks and bonds matter? I am presenting nothing novel here; in fact, my point is that the observations from the past are a good complement to advice that is presently offered. This is an invitation to think critically; please do not construe it as a statement that "you're doing it wrong".

Lucile Tomlinson (Wessman) started her career at Barron's before becoming managing editor of Investment Companies [1, 2]. Her 1953 publication, Practical Formulas for Successful Investing [2], presented an "insider's view" of formulas that endowment managers and fund managers were using at the time. She analyzed each formula in detail, including its strengths and weaknesses; as a good journalist, she would not proclaim any one as better than the others. Although she and her book may be almost forgotten, they merit our attention. She devoted three chapters to explore some "variable [stock-to-bond] ratio formulas" that were in use at the time. Some might consider these "market timing", although that phrase means different things to different people.

Investment advisors warn that "the research" has shown that [active fund managers] can't beat the [stock] market [consistently over the long term]. This has been uncritically generalized into the axiom, "Market timing doesn't work". In my experience, the first (if not only) risk-mitigation strategy offered by conservative investing advisors is to maintain constant percentages of stocks and bonds in an investment portfolio. We are expected to assume that this is the only way "not to time the market". In my opinion, that is merely saying that stocks are always poised to give substantial returns in the long run, which is itself a judgement about when is a good time to buy stocks. What this ignores is the finiteness of corporations' ability to generate earnings and (irresponsibly in my view) encourages investors to ignore any "margin of safety" [3]. Investors are told by an asset manager to have faith that short-term losses incurred in lump-sum reallocation of their portfolios will be more than rewarded by eventual gain in the long term; the trouble is that the net gain may be only slight.

I find far more convincing Benjamin Graham's recommendation to consider average earnings over the previous ten years ("E10") as the best (albeit weak) predictor of future earnings. (Similarly, I consider ten-year interest rates the best [albeit weak] predictor of ten-year interest rates for the next decade.) By contrast, I see no rational basis to predict movement of stock prices, but this does not preclude assigning a discount rate on the basis of the E10 divided by the current price and projecting the that rate relative to the current price for the next ten years will experience little more than cyclical variation. Ms. Tomlinson presented an early version of Graham's "Central Value" computation ([4], pp. 192-195 of [2]) and analyzed how this valuation metric would be applied by one endowment's plan to reduce volatility while matching returns from the Dow-Jones Industrial Average over the preceding 30 year period ("A Compromise Solution", pp. 232-238 of [2]). One may wonder how similar performance of such a formula might have been over the past seventy years. I did, and I took the bait....

I agree with the implication of Graham's "Central Value" that it is important to consider the projected yields of bonds and stocks when deciding the ratio of stocks and bonds in a portfolio. I wondered whether formulating this decision in terms of getting good long-term return on investments (rather than in terms of getting the best short-term gain in price) would have outperformed a fixed-percentage allocation in the US market since 1911. (I chose this year because it is a few years before 1916, which would have been a very difficult year to retire, and because it is, to my view, early in the "modern era"; the previous awful year to retire, 1899, just didn't seem to be at all representative of the past 120 years.) The data that I used are (as is customary in such investigations) the historical spliced S&P 500 data and ten-year Treasury bonds (GS10) from Robert Shiller's web site [5].

I projected the rate of return for stocks at any given date (i.e., dividend yield plus retained earnings) as E10/P (average earnings over the past ten years divided by current price, from Shiller); for bonds, the projected yield is the interest rate for GS10 (from Shiller). Total returns for the S&P 500 and GS10 are Shiller's calculations; I added a few more tabs to his spreadsheet. My modification of Dr. Shiller's spreadsheet with the calculations and graphs presented below may be found at [7]. In my simulation, I applied the rule not to exchange stocks and bonds except (1) to sell bonds and buy stocks when stocks are below the minimum required and (2) to sell stocks and buy bonds when stocks are above the maximum allowed; the minima and maxima are computed by the equations shown in the figures.

The first figure shows the cumulative returns for a variable asset ratio (VAR; orange in the figure), a 66% stock allocation (which is the average for the VAR from 1911-1922; light blue), a 60% stock portfolio (pale violet), a 100% stock portfolio (olive), and a 100% bond portfolio (tan); below the returns is indicated the allocation ratio (purple dot-dash line), which is limited at the high end to 85% (because few of us are likely to have the stomach for 100% stock allocation even if projected returns are very high) but which is allowed to descend to zero when bond yields swamp stock yields (although, as it turns out, keeping a 10% stock allocation doesn't hurt performance very much); when visible, red and blue dotted lines respectively indicate the upper and lower stock allocation limits. My inspiration to incorporate bond returns into the variable-allocation maximum equation was in part from the Graham's "revised formula" [8] and in part because I experienced the ridiculous interest rates of the 70s and 80s before the Federal Reserve decided to take measures to (try to) prevent them from happening again for a protracted period.

To reduce the possibly favorable effect on returns of short-term fluctuations in the VAR, I used a weighted 3 month moving average for the VAR. I chose the subtrahend 3.03% as the "point of diminishing returns" for stocks since my analysis [6] suggests that this corresponds to the 95th percentile for the P/E10. I chose the addend 3.64% in the numerator of the VAR maximum specifically to raise the return for the 20-year periods beginning in August, 1929 and January, 1962 (which may not be at all representative of future market conditions!), although the performance of the model is not very sensitive to small adjustments of this value.

The VAR portfolio would have outperformed the 66% stock portfolio overall, with no more (and, at times, much less) volatility. I'm not thrilled by the pole-to-pole swings in stock allocation, but they are relatively infrequent, with a few exceptional periods (1965-1987 and 2002-2009; see below). The VAR would have produced a stock-like gain overall and would have showed much lower transient losses than any of the non-zero fixed stock allocations after the bursting of the 2000 and 2008 bubbles.

The next figure shows theoretical 20-year returns for the data in the preceding figure. The volatility of these returns for the VAR portfolio is no greater than the volatility for the 66% stock portfolio, yet the gains are usually higher. Most importantly, the VAR portfolio's subsequent 20-year return was the only one not to fall below the subsequent 20-year return observed for August 1929. (Hopefully, we won't ever again see 20-year returns as bad as for January, 1962!)

Notably, each of the periods of frequent reallocation (1965-1987 and 2002-2009, as seen in the first figure) spans one of the "local minima" for returns over 20-year periods (those starting in January 1962 and December 1989).

Overall, the VAR shows modestly reduced variation in 20-year returns (about a six-fold difference between the minimum and maximum, vs. an eight-fold difference for the fixed 66% allocation). Only for 20-year periods beginning from 1977 through 1994 did the fixed-percentage allocation outperform the VAR, although not by much, and during that period, returns were quite favorable, so the impact of this disadvantage is (in my mind) negligible.

I fully recognize the limitations of parameters chosen on the basis of past market behavior. I consider statistics to be the science of interpreting observed data, so I don't think that this is avoidable; the only things that we can be sure could happen are things that did happen, even though we can bank on them never happening exactly that way again. (Of course, reliance on historical data is a factor for any modeling exercise; even the inputs to Monte Carlo simulation are based on historical performance metrics.) Conversely, I would consider as pure speculation any formula to be considered for application in the future that would not have been effective in the past over some multi-decade periods.

With those caveats in mind, I think that one would do well to consider this counter-example to the assertion that "it cannot work in the long term to take projected returns of stocks and bonds into account when deciding to allocate between stocks and bonds". I do not assert that this exact model would be a good candidate as an allocation strategy in the future; my specific claim is that the effectiveness of this approach with data since 1911 suggests that a fixed-percentage stock-to-bond allocation has not been the best way to handle fluctuation in the yield of stocks relative to the yields of bonds (especially not from 1962 to 1981). In other words, I think that the fixed-percentage allocation is oversold regarding the benefits it may bring; this is not a claim that it has no benefit, but rather that it is not a panacea to accept without engaging critical thinking.

Not shown: These calculations are on the basis of GS10 interest rates. For example, VWEAX, which includes B-grade bonds (and which has a performance correlation with stocks of about 0.3), now yields 200+ basis points more than GS10, which may skew the choice somewhat in favor of bonds. (There is a parameter in the spreadsheet to add a hypothetical risk premium to the interest rates, but it was set to zero for this demonstration).

Although the demonstration here includes consideration of GS10 yield, this is not necessary to mitigate risk more effectively than a constant percentage stock allocation. Work by Wade Pfau in 2011 [9, 10] a "25-50-75 decision rule" based on the P/E10 without consideration of the GS10 yield, concluding, "Valuation-based market timing demonstrates strong potential to improve risk-adjusted returns for conservative long-term investors." Since that time, Dr. Pfau seems to have lost his enthusiasm for the P/E10 as a stock valuation metric [11].

Regarding a practical formula for variable allocation, I have not yet incorporated the GS10 into my formula [6], but I might need to if inflation were to remain very high to such a point that the Federal Reserve would need to raise interest rates substantially for a many years.

I don't expect to convert anyone to my point of view; in fact, I don't want anyone to change their investment strategy (or even become uncomfortable with it) because of this. Rather, I hope that this investigation makes it clear that the fixed stock-to-bond ratio may occasionally incur (unnecessarily) greater losses than we have been led to believe and should therefore be viewed with a healthy (i.e., small) bit of skepticism.

• I am indebted to my father, who taught me the principle of tying stock allocation to risk (and so much more).
• Robert Shiller's data page and book Irrational Exuberance, which taught me history's lessons that prices reflect an aggregation of real and imaginary bases.
• Wade Pfau's postings on Bogleheads and papers (referenced above), which confirmed my suspicion that the correlation that Shiller showed between Benjamin Graham's cyclically-adjusted price-to-earnings ratio can theoretically be applied to portfolio allocation. He also brought Luclie Tomlinson's 1953 book to my attention.
• Lucile Tomlinson's 1953 book Practical Formulas for Successful Investing, which provided "A Compromise Solution" that inspired this study.

[1] "A graduate of Mount Holyoke College, Miss Tomlinson started working for Barron's, the national business and financial weekly, first as secretary to the editor and later as member of the editorial staff and associate editor. She originated Barron's investment company gauge and has done considerable writing about investment companies."
The Westfield Leader, March 20, 1952, available from
https://web.archive.org/web/20231026231946/http://www.digifind-it.com/westfield/leader/1952/1952-03-20.pdf
[2] Tomlinson, Lucile. Practical Formulas for Successful Investing. 1953, New York, Wilfred Funk.
https://books.google.com/books?id=iBVyxv5Y2UcC
This book is out of print but may be obtained by inter-library loan or (occasionally) in the used-book market. I'd love to make it available online if only I could find the copyright holder to ask permission. You can find a nice bio and review of the book at
https://memphreinvestments.com/book-blog/f/8-lucile-tomlinson-practical-formulas-for-successful-investing
[3] https://en.wikipedia.org/wiki/Margin_of_safety_(financial)
[4] Tomlinson quoted Graham: "A reasonably sound central value, or intrinsic value, of the Dow-Jones list can be ascertained by capitalizing the average earnings for the past ten years on a basis equivalent to twice the yield of high-grade bonds."
[5] Robert Shiller's data page lists the sources for the data in the spreadsheet
http://www.econ.yale.edu/~shiller/data.htm
[6] https://eschenlauer.com/investing/risk_based_allocation/
[7] ie_data_VariableAssetRatio.xlsx; unfortunately, I implemented some of the functionality with LAMBDA functions, which would not translate to LibreOffice Calc.
[8] https://en.wikipedia.org/wiki/Benjamin_Graham_formula#Revised_formula
[9] Pfau, Wade D., Long-Term Investors and Valuation-Based Asset Allocation (November 1, 2011).
http://dx.doi.org/10.2139/ssrn.2544636
[10] Pfau, Wade D., Withdrawal Rates, Savings Rates, and Valuation-Based Asset Allocation (December 1, 2011).
http://dx.doi.org/10.2139/ssrn.2544635
[11] "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 glean a whole lot of information from something like the CAPE ratio."
– "Retire with Style" podcast episode 57, at 16:41
https://risaprofile.com/wp-content/uploads/2023/03/Episode-57-Market-Timing-Still-Doesnt-Work-1.pdf