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Trading Strategies

A Quantitative Method for Asset Allocation

The prevailing valuation of stocks and the relative valuation of bonds can be used to adjust an allocation above or below its target.

by Joseph Harding | June 2018

Article Highlights

  • Conventional rebalancing readjusts the portfolio back to the predetermined target allocation for stocks.
  • The quantitative method considers not only the target allocation but also the prevailing valuation of stocks and the relative valuation of bonds.
  • Returns are enhanced for portfolios holding allocations of 50%, 60% and 70% to stocks; volatility is reduced for equity allocations up to 90%.

One of the most important aspects of portfolio management is asset allocation.

An individual investor must decide how to allocate portions of their portfolio among different asset classes, such as stocks, bonds or cash. A diversified portfolio allocated among different asset classes has been proven to be an excellent strategy for obtaining good returns over the long term, with the added benefit of reduced volatility (i.e., avoiding large changes in portfolio value) over the short term.

Indeed, AAII conducts an ongoing Asset Allocation Survey, with results updated every month  (www.aaii.com/assetallocationsurvey). The survey measures the percentage holdings of members in five asset categories: stock funds, stocks, bond funds, bonds and cash.

Conventional Method for Asset Allocation

So how does an investor go about determining the appropriate asset allocation for their portfolio? There are two generally accepted guidelines for making this determination.

The first guideline is that an investor must determine their tolerance for risk. In other words, how willing are you to endure large swings in portfolio value over the short term in exchange for potentially higher returns over the long term? If you determine that you are willing to endure large swings in portfolio value over the short term, then a higher percentage of stocks compared to bonds or cash in your portfolio would be appropriate. If, on the other hand, you determine that you are not as willing to endure large swings in portfolio value over the short term, then a lower percentage of stocks compared to bonds or cash in your portfolio would be appropriate.

The second guideline is based on the expected retirement date of the investor. An investor who is many years away from retirement should have a higher allocation to stocks compared to bonds or cash. However, an investor who is either retired or is only a few years away from retirement should have a lower allocation to stocks compared to bonds or cash. Indeed, this is the idea behind the various target date funds that are available today.

After determining the appropriate asset allocation for your portfolio, it is not wise to simply “set it and forget it.” Over time, the value of one asset class will likely change compared to other asset classes. This means that over time, the asset allocation of your portfolio will also change compared to your original target allocation. Therefore, it is recommended that an investor should rebalance their portfolio at least annually. During the rebalancing process, portions of portfolio assets that have increased in relative value are sold, with the proceeds invested in assets that have decreased in relative value. The result is that the portfolio is readjusted back to the original target allocation after the rebalancing is completed.

The Quantitative Method


Figure 1. Stock Allocations Based on the
Conventional Method
Figure 1. Stock Allocations Based on the Conventional Method

The conventional method for asset allocation described so far has been proven over time to be a good model for individual investors to follow. But what if there is a better way?

When an investor rebalances their portfolio, what if the investor adjusts their actual target allocation after rebalancing based on not only their target allocation but also the current prevailing relative value of the stock and bond market?

In the quantitative method described herein, the allocation to stocks would be equal to the investor’s target allocation if the current prevailing valuation of the stock market is historically average compared to the current prevailing 10-year Treasury note yield. However, the allocation to stocks would be greater than the investor’s target allocation if the current prevailing valuation of the stock market is historically below average compared to the current prevailing 10-year Treasury note yield. The allocation to stocks would be less than the investor’s target allocation if the prevailing current valuation of the stock market is historically above average compared to the current prevailing 10-year Treasury note yield.


Figure 2. Stock Allocations Based on the
Quantitative Method
Figure 2. Stock Allocations Based on the Quantitative Method

Figure 1 shows the conventional method for rebalancing. When an investor rebalances, their portfolio is always readjusted back to the “target” allocation for stocks, regardless of the valuation of the stock market compared to the current 10-year Treasury note yield.

Figure 2 shows the quantitative method for rebalancing. When an investor rebalances, their portfolio is adjusted to an allocation of stocks that is based on a combination of the investor’s target allocation for stocks as well as the valuation of the stock market compared to the current 10-year Treasury note yield.

 

Stock Market Valuation

A well-known method for determining the valuation of the U.S. stock market (specifically the S&P 500 index) is the cyclically adjusted price-earnings ratio, or CAPE ratio. This ratio, invented by Robert Shiller of Yale University, compares the current level of the S&P 500 to its average earnings over the last 10 years, adjusted for inflation. A high CAPE value indicates that the stock market is overvalued, while a low CAPE ratio indicates that the stock market is undervalued. A spreadsheet that includes all of the relevant data for calculating CAPE is publicly available at www.econ.yale.edu/~shiller/data.htm.

The quantitative method for asset allocation described here is based on the data contained in this spreadsheet but does not use the calculated CAPE Shiller from the spreadsheet directly. Instead, the quantitative method uses a 20-year period of earnings for calculating CAPE (denoted as CAPE20) and also takes into account the current S&P 500 dividend yield compared to the current 10-year Treasury note yield.

Taking the S&P 500 dividend yield and the 10-year Treasury note yield into account makes logical sense. Low bond yields compared to their historical average (as has been the case for the last several years) should support a stock market with a higher than average CAPE. The opposite should be true if bond yields are high compared to their historical average.

Quantitative Method Formula

The following formula is used in the quantitative method for determining the actual allocation to stocks during rebalancing:

Stock allocation = [target stock allocation + (2 × CAPE avg) – (2 × CAPE20)] × K

Where:

  • Stock allocation is the actual stock allocation used by the investor during rebalancing, in percent;
  • Target stock allocation is the investor’s target allocation to stocks;
  • CAPE avg is the average value of CAPE20 over the last 600 months (50 years);
  • CAPE20 is the cyclically adjusted price-earnings ratio over the last 20 years; and
  • K is 1.5 × (S&P 500 dividend yield ÷ 10-year Treasury note yield).

The CAPE20 uses 20 years of earnings versus the 10 years of earnings used in Shiller’s CAPE ratio. The S&P 500 yield can be calculated using data from Shiller’s spreadsheet; simply divide the dividend by the S&P 500 composite value. The 10-year Treasury yield can be found at www.multpl.com/10-year-treasury-rate/table/by-month.

Here’s an example of how the formula works using a target allocation for stocks of 50%. As of early April 2018, the CAPE avg was 22.04, the CAPE20 was 34.29, the S&P 500 dividend yield was an estimated 1.95% and the 10-year Treasury note yield was 2.86%. Using whole numbers to represent the target allocation, dividend yield and bond yield, the stock allocation formula would be:

= [50 + (2 × 22.04) – (2 × 34.29)] × (1.5 × (1.95% ÷ 2.86%)
= [50 + (44.08 – 68.58)] × (1.5 × 0.682)
= 25.5 × 1.023
= 26.09

This reflects a 26% target allocation to stocks.

The formula can result in values above 100% and below 0% (meaning negative values). If the result is greater than 100%, then 100% is used. If the result is negative, then 0% is used.

Comparison of Results

In order to compare the results of the conventional method to the quantitative method, we need to first establish a model portfolio. The rules governing this model portfolio are as follows:

  1. The portfolio is rebalanced once per year.
  2. The stock portion of the portfolio achieves the same return as the S&P 500, including dividends. Dividends are not reinvested in stocks but are deposited into the bonds/cash portion of the portfolio.
  3. The bond and cash portion of the portfolio collectively achieves a return equal to 80% of the 10-year Treasury note yield. This is considered to be a conservative estimate.

Using the above model portfolio as a basis, over the time period of 1950 through 2017 the quantitative method not only increases the average return of a portfolio compared to the conventional method in almost every case, but the quantitative method also reduces the volatility of the portfolio in every case studied (as measured in the number of years with a negative portfolio return) when compared to the conventional method.

The benefits of increased returns to an investor are obvious. The benefits of reduced volatility, in this case a significant reduction in the number of years with a negative portfolio return, are more psychological. Even in years where both models have a negative return, the losses are reduced with the quantitative method. As an example, in 2008, an investor with a target allocation of 70% stocks and 30% bonds/cash would have experienced a loss of 23.4% with the conventional method. That same investor would have experienced a loss of 16.3% using the quantitative method.

Figure 3. Average Annual Portfolio Return
(1950–2017)
Figure 3. Average Annual Portfolio Return (1950–2017)

Figure 3 shows a comparison of average annual returns between the two methods over the time period of 1950 through 2017, depending on the investor’s target allocation to stocks and bonds/cash.

Figure 4 shows a comparison of the number of years with a negative return between the two methods over the same time period, once again depending on the investor’s target allocation to stocks and bonds/cash. Both charts are based on calendar-year holding periods.

As can be seen in Figure 3, the quantitative method provides a distinct advantage in returns (approximately 1% annually) for target allocations of 50%/50%, 60%/40% and 70%/30% stocks to bonds/cash. The advantage in returns is reduced for an 80%/20% allocation. The return advantage disappears for a 90%/10% target allocation.

Figure 4. Number of Years With Negative
Portfolio Return (1950–2017)
Figure 4. Number of Years With Negative Portfolio Return (1950–2017)

Figure 4, on the other hand, shows that the quantitative method reduces the volatility of the portfolio irrespective of the target allocation, although the effect is reduced somewhat for higher stock target allocations.

It should be pointed out that both Figures 3 and 4 are based on an investor rebalancing on the first trading day of October each year. This seems to be the ideal day to rebalance compared to the first trading day of any other month. On the other hand, the first trading day of May or June seems to be the least favorable. However, the quantitative method still provides an advantage in returns for lower stock allocations (i.e., 50%/50%, 60%/40% and 70%/30% stocks to bonds/cash) no matter when rebalancing takes place. The advantage in returns for higher stock allocations (80%/20% and 90%/10% stocks to bonds/cash) can disappear depending on the time of rebalancing. However, the advantage of the quantitative method in reducing the volatility of the portfolio is not significantly affected by the rebalancing date for any target allocation.

Let’s look at another scenario. What if an investor were able to achieve higher returns on the bonds/cash portion of their portfolio (e.g., by holding higher-yielding bonds), for example 120% of the 10-year Treasury note yield? As expected, the average annual returns are increased for both the conventional and quantitative methods, but the annual returns are increased by a wider margin for the quantitative method. The difference in annual returns between the two methods increases from 0.3% to 0.6%, depending on the investor’s target allocation. In this scenario, the quantitative method continues to provide significantly reduced volatility compared to the conventional method.

The Test of Time

Figure 3 shows average annual returns for the quantitative method compared to the conventional method over a time period of 68 years (1950 to 2017). Is it possible that the quantitative method was only better than the conventional method for a brief period of time during that period? For example, did it outperform in the 1950s but hasn’t offered a significant advantage since then?

My analysis shows that irrespective of the target allocation to stocks, the quantitative method provided superior returns compared to the conventional method in the 1950s, 1960s, 1970s, 2000s and so far in the 2010s (i.e., 2010 to 2017).

The quantitative method slightly underperformed the conventional method in the 1980s, and significantly underperformed in the 1990s. This result makes logical sense.

The 1990s were famously dubbed as the era of “irrational exuberance” by Alan Greenspan, who was chairman of the Federal Reserve Board during that period. Dot-com stocks had extremely high valuations during this period, with a resulting high value of CAPE20 for the S&P 500. Therefore, the quantitative model was suggesting a lower allocation to stocks during that period. However, stocks kept going up anyway, until they didn’t!

The next decade of 2000 to 2009 was a dismal period for stock returns, with two major stock market declines. In this decade, an investor with a target allocation of 90%/10% stocks to bonds/cash, using the conventional method, would have experienced an overall portfolio loss of 1% over the entire 10-year period. That same investor would have experienced a gain of 44% if the quantitative method was used. Although 44% is not a great portfolio return for a 10-year period, it sure beats a loss!

Table 1. Allocations Suggested by the Quantitative Model

If an investor rebalanced at the time that these calculations were run in early April 2018, the model would have suggested a below-target allocation to stocks and an above-target allocation to bonds and cash.
Target Allocation Suggested Allocation
50% Stocks/50% Bonds and Cash 26% Stocks/74% Bonds and Cash
60% Stocks/40% Bonds and Cash 36% Stocks/64% Bonds and Cash
70% Stocks/30% Bonds and Cash 47% Stocks/53% Bonds and Cash
80% Stocks/20% Bonds and Cash 57% Stocks/43% Bonds and Cash
90% Stocks/10% Bonds and Cash 67% Stocks/33% Bonds and Cash

What the Model Currently Says

As of early April 2018, the quantitative model calculates a current CAPE20 of 34.3, with an average CAPE20 over the last 50 years of 22.0. Therefore, the model is suggesting a below-target allocation to stocks at the time the calculations were run. The suggested allocations can be seen in Table 1.

Looking to the Future

Although the quantitative method of determining asset allocation that I have described here has been shown to provide advantages over the conventional method, it is likely that further improvements could be made to the formula presented in this article. Such improvements will be the aim of future research in this area.

Discussion

D Wilson from FL posted over 8 years ago:

How exactly did you determine the values in the formula from Robert Shiller's spreadsheet? The article was very thought provoking and appears to be based on a logical foundation.


J Harding from OH posted over 8 years ago:

I created my own spreadsheet that uses the data from Shiller's spreadsheet. The formula uses a 20-year CAPE (as opposed to the 10-year CAPE in Shiller's spreadsheet. I simply went through a process of assuming different structures for the formula (e.g. linear, exponential, etc.) and then played around with different parameters such as slope, exponents, number of years for CAPE, etc. The formula in the article is the best I have come up with so far for the time period 1950-2017. There may be an even better formula out there but I haven't found it yet. I have not given up on trying!


Ken Smith from AB posted over 8 years ago:

Why is October 1 the best day to rebalance your portfolio compared to other days of the year?


Joe Harding from OH posted over 8 years ago:

In my analysis, I looked at rebalancing on the first trading day of every month, i.e. 12 different scenarios. For the time period of 1950 to 2017, rebalancing on the first trading day of October would have resulted in the best returns using the quantitative model presented. As far as why, I am not completely sure except to say that there are well-known seasonal influences in the stock market, i.e. some months have historically provided better returns than others.


Harry Rich from OH posted over 8 years ago:

Thanks for a thoughtful article. What led you to use a 20 year CAPE rather than a 10 year CAPE? What now leads you to believe that the 20 year CAPE is superior?


Joe Harding from OH posted over 8 years ago:

During the development of the model, one of the variables in the spreadsheet I used was the number of years to use in the CAPE. I have always thought that 10 years was a bit arbitrary. What I found is that for the time period of 1950 - 2017, using a 20 year CAPE gave the best results. Of course, past results are no guarantee of future results, but 68 years is a pretty good track record in my opinion.


Bruce Bodner from MA posted over 8 years ago:

The danger of your approach of course, is that you have discovered correlations which are not causative. For example you could have uncovered an association with which conference won the super bowl and whether it was sunny on Groundhog day, or if the stock market went up or down in the first 9 days in March, etc. If you "mine" enough data you will find strong associations which then evaporate going forward. But your approach is intriguing, investing less in the stock market when it is overvalued. It is a variety of 'market timing'. Caveat circumsedere.


Bruce Bodner from MA posted over 8 years ago:

The last column in the data set from Schiller's website is labeled Cyclically Adjusted Price Earnings Ratio P/E10 or CAPE for your Cape20 for April 2018 I assume you average the last 20 years of this column or 240 rows(months) ending on the April 2018 row. For the CAPEavg you average 50 years, or 600 rows(months. Is that correct ? also for the SP500 dividend yield do you use the "Real Dividend" or the "Dividend" ? do you use the S&P Comp P for price or the "Real Price" column ? thanks


Joe Harding from OH posted over 8 years ago:

Here are answers to Mr. Bodner's three questions: 1. Yes CAPE20 and CAPEavg are exactly as you describe. 2. I use "Dividend" in the model. 3. I use "S&P Comp P" in the model.


Joe Harding from OH posted over 8 years ago:

Regarding Mr. Bodner's comment on correlations which are not causative, of course you may be right. Past performance is no guarantee of future performance. Having said that, the idea of reducing allocations to stocks when they are expensive and increasing them when they are cheap seems intuitive to me. The fact that the idea seems to work very well over over the time period studied (1950 - 2017), plus the fact that it also had the effect of reducing volatility in one's portfolio, is intriguing. As an engineer, finding something that can be expressed as a mathematical formula and also makes logical sense (at least to me) is attractive.


Mark Kelly from PA posted over 8 years ago:

Can you provide a copy of your spreadsheet with all of the underlying calculations?


Rao Bhamidipati from IN posted over 8 years ago:

Thanks for a very thoughtful article. Did your model indicate overweighting stocks at different points? It would help if the time periods, as well as the frequency of under vs. oeverweighting are indicated. In your analysis, did you separately simulate the number of years for the avg vs. the 'current cape'? I'm wondering for e.g. how your metrics would look if you used something like the (2xCAPEAVG20 - 2xCAPE5) or (2xCAPEAVG20 - 2xCAPE10) etc. I would expect the performance metrics to improve with this, but just curious if you had already done this.. The basic idea is that the CAPEAVG20 remains as the long term average, but the decision/reaction time can be shorter than 20 yrs.. Thanks


Joe Harding from OH posted over 8 years ago:

Dear Mr. Bhamidpati, Thank you for your comment. In my analysis, I never mixed the number of years of earnings in the current CAPE vs. the average CAPE, although that is an intriguing idea that I will look at down the road. Thanks for suggesting it. What I have looked at so far is the number of years of earnings to use for the current CAPE as well as the number of months to use for the average CAPE. I found that 20 years of earnings for the current CAPE and a 600 month (50 year) average CAPE works the best for 1950 - 2017.


Joe Harding from OH posted over 8 years ago:

Dear Mr. Bhamidpati, Regarding your first question, stocks would be overweighted after rebalancing if the formula yielded a result that was greater than the investor's target allocation to stocks. If that sounds simplistic, it's because it is! Again, the model is based on rebalancing once per year. Also, I found the first trading day of October was the best time to rebalance (compared to the first trading day of any other month) for the time period 1950 - 2017.


V Rao Bhamidipati from IN posted over 8 years ago:

Thanks for your response.. Could you please indicate how often or how many times (and perhaps indicate the actual time periods) during which the model suggested stock overweight? Thanks in advance.


Joe Harding from OH posted over 8 years ago:

Dear Mr. Bhamidipati, The answer below assumes rebalancing on the first trading day of October each year: The model suggested being overweight in stocks in 1950-1958, 1960, 1962, 1974, 1977, 2008, 2010-2012 and 2014-2016.


Perry Susskind from CT posted over 8 years ago:

Dear Mr. Harding, Thank you for your article! An interesting (set of) graph(s) would be to fix a time period, e.g. 5 years, (10 years, etc.) and then plot the difference in percent gain between your quantitative method and the conventional fixed allocation method (annualized), beginning in 1950, 1951, 1952, ... , 2013. This, it seems to me would provide a strong indication of whether the method is sound, especially for shorter windows of time such as 5 years and arbitrary (say, by year) start times. I wonder if you have looked at that? (And is your data in a form that would make generating this information reasonably efficient?) Thank you again!


David Phillips from AL posted over 8 years ago:

Mr. Harding, Several people have asked this question: "Could you please indicate how often or how many times (and perhaps indicate the actual time periods) during which the model suggested stock overweight?" Would you please answer? Thanks very much.


Joe Harding from OH posted over 8 years ago:

Mr. Phillips, I answered the question 7 days ago (please see above). I will copy my answer for you: The answer below assumes rebalancing on the first trading day of October each year: The model suggested being overweight in stocks in 1950-1958, 1960, 1962, 1974, 1977, 2008, 2010-2012 and 2014-2016.


Peter Polcari from TX posted over 8 years ago:

Mr Harding, Thank you for a very informative article. A possible idea for future research would be to provide a method that allows the investor to pick the upper and lower bounds for stock allocation in the portfolio that they are comfortable with and have a metric to determine where in that range they should be invested during a given period.


Matt Schott from NC posted over 7 years ago:

Thanks for a provocative article. I have a couple of questions. Calculation When I downloaded the Shiller spreadsheet this past week, it looked like CAPE10 was calculated by Real price (Col H for the current month) divided by the average for real earnings (Col J) for the 120 months prior to the current month. So for CAPE20 I did the same, only going back 240 months for real earnings. Is that right? For April 2018 CAPE20, I calculated a value of 34.44 (compared to your 34.29 from the article) and for CAPE avg I got 22.08 (compared to your 22.04). I figure the difference is in the fact that you did yours in early April and the April values in the spreadsheet I downloaded this past weekend were for the end of April. Spreadsheet The spreadsheet I downloaded did not have Real Earnings for most recent 5 months April-Aug or for Real Dividend for July-Aug. Is this typical? If you do rebalancing on October 1, how do you calc the values through Sept 30 without that data? Decumulation Have you considered how this approach would work during decumulation? Do you rebalance then take the withdrawal proportionally from the asset classes based on the new allocation? My current thinking is to take withdrawals from bonds/cash after a down year for stocks and from stock after an up year. However, I'm not sure how that would work when the actual asset mix can change so dramatically from the target. Thanks again. The article really has me thinking with the market so overvalued at present.


Joe Harding from OH posted over 7 years ago:

Hello Mr. Schott, My apologies for not looking at these comments for awhile. You ask some good questions. I will try to answer them as follows: For CAPE20, going back 240 months is correct. My April 2018 CAPE20 is calculated on the first trading day of April, which in 2018 is April 2. You are right, some of the data is missing in Shiller's spreadsheet that you need to do the necessary calculations. I put in estimates in order to get my final numbers. Even if I am a little off, it should not make a big difference due to the long time period involved. My article did not really cover how to do withdrawals. However, my current thinking is along the line of James Cloonan - i.e. allocate maybe 3 years of withdrawals in a separate bucket that is held in low risk investments. Withdraw from that fund, and refresh that fund with profits (in years with profits) from the "investment" portion (i.e. the portion NOT in the separate bucket).


D Akerhielm from IN posted over 7 years ago:

If I look at the current CAPE, the September result is 33.18 and the Calculated CAPE20 (Sum of CAPE rows 1542 to 1781, divided by 240) is 26.84. The net result is a suggested allocation of 66.14% to equities which can't be "right". What am I doing wrong?


D Akerhielm from IN posted over 7 years ago:

Clarifying my question, my "baseline" equity exposure is 60% and my "K" was (1.75/2.89) x 1.5. I end up with an answer of 66.14%. Ps - in the June article you state the CAPE in early April was 22.04. I did not find a 22.04 on Schiller's data sheet. Furthermore, an A[ril CA{E of 22.04 makes no sense in light of the 33.18 end-of-September CAPE. Obvioulsy I am doing something wrong or reading something wrong. Thanks for the help.


Bruce Bodner from MA posted over 7 years ago:

Joe ! I actually tried to use your formulas but got different results than you. I downloaded the spreadsheet from the shiller site and tried to derive this variable: CAPE20 is the cyclically adjusted price-earnings ratio over the last 20 years I did this by averaging the 240 entries in the last column, the K column, Cyclically Adjusted Price Earnings Ratio P/E10 or CAPE column, for 240 months ending on the 2018.4 month I made a cell with the formula : =AVERAGE(K1536:K1776) as you can see its 240 months and my result was 26.96, not your value of 34.29. So I looked back at this K column, the last column and found very few values above 30. So, how can the average of the past 240 months be 34.29 ? And you mention in your article, that "As of early April 2018, the CAPE avg was 22.04," But its not ! it's 30.97 ! Please explain


D Akerhielm from IN posted over 7 years ago:

A few of us need help! Thank you


Joe Harding from OH posted over 7 years ago:

Dear Mr.Kuiken,


Joe Harding from OH posted over 7 years ago:

Dear Mr, Bodner,


Joe Harding from OH posted over 7 years ago:

Dear Mr, Bodner,


Joe Harding from OH posted over 7 years ago:

Dear Mr, Bodner,


Joe Harding from OH posted over 7 years ago:

Dear Mr. Akerheim,


Bruce Bodner from MA posted over 7 years ago:

Mr Harding appears to have abandoned us. No wonder it is often said, "do not attempt to time the market."


C Myers from FL posted over 7 years ago:

Has anyone ran this model with rebalancing in December to see how the results vary?


Maurice from ME posted over 6 years ago:

I'm getting crazy results this month, negative equities based on 60% goal. Is anyone else having the same problem?


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