Tuesday, February 25, 2014

Stock/Bond Asset Allocation

A friend asked me about stock & bond allocations, and I decided to review the data.  Of course you're all doing it wrong.  I explain below.  Make sure you read to the end....You're welcome.  :-)


For bonds I have used the Moody's AAA bond rate series, from Fred, not adjusted for defaults, and adjust the value of the bond portfolio monthly, based on the monthly rate, assuming average bond duration of 30 years.  For stocks, I use Robert Shiller's historical S&P500 data.

Here is a graph of cumulative gains (rebalanced monthly) for different allocations:

The following two graphs are comparisons of the returns to these allocations over time, for a 10 year holding period and for a 20 year holding period.

There is a dispute in finance about the size and persistence of the equity risk premium (the additional return that an investor gets by taking on the extra risk of owning equities instead of bonds).  One issue is that these relationships just aren't stationary enough for a century's worth of data to give us an answer.

As can be seen in these graphs, almost all of the excess gains to equities came in the period from 1940 to 1965.  At shorter holding periods, there is a significant amount of variance in equity returns.  But, with a longer holding period, returns revert to long-term trends, so that with very long holding periods, there are two distinct eras:

(1) holding periods beginning from 1940 to 1965 where equities earned a tremendous premium with much less risk than bonds.

(2) all other periods, where equities earned a slightly higher return with slightly more volatility and the occasional crisis.

So the allocation decision boils down to two questions.

1)   Are we about to see another 1940-1965?  This is impossible to answer.  We are currently in a period of very low interest rates.  My intuition wants to say that in a low interest rate environment, bond allocations should be very low, because (a) in a low rate environment it would be very difficult for bond yields to match the combined growth and income value of stocks and (b) the risk profile would skew very negatively for bonds since bonds have defined income potential and a limit on valuation gains due to the zero lower bound on interest rates.

The one era of high interest rates that we have seen in the past century was associated with relatively high returns with relatively low variance for long holding periods, along with a modest equity premium.

Eras of low interest rates have had a more bifurcated set of outcomes, with periods of either very high equity premiums or periods of very negative returns on equities compared to bonds.  This is largely because the most damaging economic crises have come during deflationary liquidity crises associated with monetary policy.  These episodes have been associated with low interest rates.  (The graph at right shows the difference between stock and bond returns <blue line> overlaid on top of the 10 year treasury yield <red>.)

So, the current level of interest rates isn't as useful of an indicator as I would hope.  This is again due to the fact that the serial correlations we see in these broad market behaviors mean that over a century there have only been a few separate regimes of return behavior, with no clear forward indication of when the regimes might switch again.

2) What is your holding period?  This is a question that we can answer.  And, considering the lack of a definitive answer to number 1, the answer to this question should generally guide our decisions.  The decisions here are going to be imprecise, but the good news is that we can get a pretty good idea of the appropriate allocation within 10% to 20%, and over the holding period, the differences between the outcomes of allocations within that range are going to be pretty marginal.  As the two line charts to the right demonstrate, as the holding period is extended, the number of periods where stocks underperform bonds declines.  At a 20 year holding period, there are just a few times where stocks slightly underperformed bonds during the last 94 years.

This is the classic chart of returns for portfolios with different allocations.  These three return/risk curves assume annual rebalancing and reflect the average returns and variance of holding periods with start dates after 1919 and end dates prior to 2013.

Equity returns can have large annual variance, but they tend to revert to long term trends.  The effect that we can see from this is that, as the holding period is extended, the standard deviation of holding period returns of equity heavy portfolios is mitigated, and these return/risk curves rotate counterclockwise with longer holding periods.  The extra returns available from equities can be captured during these long holding periods without taking on so much extra risk.

In any holding period, there are very high expected costs from being too highly-weighted in bonds.  Even at short holding periods, holding more than 70% bonds is suboptimal.  But, once a portfolio reaches the efficient frontier where there is a reasonable trade-off between higher risk and higher return, there isn't that much difference between portfolios.  Even at 20 years, the difference in expected returns between a 70% equity allocation and a 100% equity allocation is about 1% annually.

So, here's where I would say that as long as you pick an allocation to stocks in the 40% to 90% range (lower for shorter holding periods and higher for longer holding periods), depending on your risk aversion, you'll be fine.  But, I would be wrong.

These models are based on normal distributions and linear correlations with homoscedastic errors.  Both of these assumptions are surprisingly wrong.

Buy Equities, but not for the reason you think you should.

Looking again at the cumulative holding period returns for different portfolios, you can see that bonds have a bifurcated return distribution.  There are long periods with low returns and long periods with high returns, and not much in the middle.  I thought maybe all of this would disappear with inflation adjustments, but the nominal and real data are surprisingly similar.

Here is a histogram of real bond returns over a 20 year time horizon.  Instead of providing a large basket of moderate return outcomes, bonds provide a bunch of relatively poor outcomes and a bunch of relatively good outcomes, without much in the middle.  The relatively low average returns from bonds are supposed to be the cost we incur for their moderating influence.  I would say that for the past century, bonds have not been doing that job well.

Next is the histogram of real stock returns over a 20 year time horizon.  There is a similar bifurcation here, but less pronounced than with bonds.  Out of 74 holding periods, a full 32 bond return outcomes are worse than any stock outcomes.  Bonds aren't just failing at mitigating failures at the left end of the outcome distribution - they are creating them by the bushel.

And, it's even worse than that.  Over 20 year holding periods, there is little correlation between stocks and bonds, so you would think that at least bonds would provide diversification benefits.  But this is not the case either.  Holding periods with low bond returns have no correlation with stock returns.  But, as bond returns increase, they tend to correlate relatively strongly with stocks.  In other words, bonds only perform well when you don't need them to perform well, and when you do need them to perform well, they never perform well.

On these scatterplots, the diagonal line represents equivalent outcomes between the two portfolios.  Outcomes above the line are outcomes where 100% stock allocations outperformed the other portfolios.

I have included two additional graphs comparing the fully invested stock portfolio with a 80/20 mix and a 60/40 mix.  In both cases, we can see again that the allocation to bonds has not provided any systematic protection against relative losses over the 20 year time frame.  As the bond allocation is increased, the average return decreases substantially, with no improvement in the number of poor outcomes.  There is one holding period that begins in 1930 where bonds provide a 70% return compared to 25% for stocks.  One holding period out of 74.

Even for a 10 year holding period, the past 94 years (84 holding periods) provides little reason to utilize bonds for stability related to the holding period.

For 10 year holding periods, I show the comparison between stocks & bonds, and between stocks and a 60/40 and 40/60 portfolio.  There are two holding periods (the two starting in 2000 and 2001) where bonds provide significantly positive returns while stocks provide significantly negative returns.  But, outside of these two holding periods, for every period where a bond allocation would have provided a boost, there is at least one other holding period where a bond allocation created a similarly negative outcome.

The final graph below shows the cumulative distribution of returns for a portfolio of 100% stocks and a portfolio of 60/40 stocks and bonds.  There is very little gain at the bottom of the outcome distribution from the 60/40 portfolio with which to justify so many holding periods in the heart of the distribution where significant losses are taken, relative to a 100% stock portfolio.

Some analysis of portfolio allocations might suggest a relatively large bond allocation, because on an annual basis, bonds and stocks do provide significant diversification.  Portfolios with a higher bond allocation will produce a higher annual Sharpe Ratio.  So, the paths these 10 and 20 year stock portfolios take en route to their final values will probably tend to be more volatile than the diversified portfolios.  And that is probably an important factor to consider if you have to deal with agency issues that come along with managing other people's money.  But, the non-normal distribution of outcomes, the heteroskedastic behavior of the correlations, and the mean reverting nature of these asset classes are strong enough influences on the performance of long term portfolio behaviors, that if you can commit to these longer holding periods without risking capital flight during potentially volatile return paths, the past century's experience gives absolutely no reason to allocate to bonds on a 20 year time scale, and scant reason to do it even at a 10 year horizon. (follow up post).

community animated gif on Giphy


  1. This is a really, really good piece. There's just one thing: bonds aren't the only alternative to stocks.

    Obviously you can look at precious metals, and you should; I believe they don't do well, though.

    The other major omitted alternative is real estate, with profits from rent. There are long (pre-industrial) historical periods over which real estate has massively outperformed other investments while having lower risk. It makes sense to properly analyze the real estate alternative.

    1. Thanks for all of your comments. On this one, if you go on to the follow up posts, I ended up doing 5 parts, and I did come to a similar conclusion to yours.

      Because of the peculiarities of the real estate market, which forces middle class savers to buy real estate in large chunks with leverage, real estate has tended to provide excess returns. But, it leaves households with a large short bond position, because of the mortgage. By the end of my series of posts, I determined that a home with a mortgage, stocks, and bonds, might be the best allocation. But, the bonds are there to hedge the mortgage, not to hedge the stocks.

      The model at the end probably needs to be taken a little deeper to fully account for all the cash flows of home ownership, and the peculiar risks of a mortgage in situations where the home is sold non-financial reasons. But, I think homes probably come out even a little better than my basic model suggested if all those factors are accounted for.

      As home prices increase again, though, and make home values more bond-like in their behavior, which I think is justified, I'm not sure if homes still are as obvious an addition for all investment portfolios. They probably start to only make sense for investors with a strong need for fixed income. The decline in home ownership is probably fitting, considering this issue.

  2. Great piece.

    I wish there were a convenient way to invest in farmland. I can invest in timberlands, commercial industrial or residential real estate through REITs - but it's farmland I want.

    The excess returns in the stock market for the past hundred years are partly the result of a transition to more and more of the economy being handled by large publicly traded companies. How many mom and pop retailers and restaurants have disappeared under the relentless march of Walmart, McDonalds and Starbucks? That transition seems to be complete.