## Summary

- A new
*Journal of Banking and Finance*paper uses Stochastic Dominance to show stocks yield dominant returns over bonds in the long run. - The caveat is that a portfolio of equities or risky bonds (or both) must include a risk-free asset, such as TIPS, for the result to hold.
- Over an investment horizon of 15 years or more, the dominant asset allocation strategy is to have a stocks-bonds ratio of at least 60:40 in favour of equities, as long as a risk-free asset is included.

## Introduction

Do stocks outperform bonds in the long run? Yes, according to a recently published paper in the highly rated *Journal of Banking and Finance*. It states that ‘any investor would be better off investing in stocks rather than in risky bonds, as long as the portfolio included a riskless asset – a Treasury Inflation-Protected Security (TIPS)’.

This result gives weight to a famous idiom, ‘stocks for the long run’, which papers have previously struggled to justify empirically. Why? They have been measuring risk inappropriately. They tend to use the variance of returns, or of log-returns, which only works if returns are normally distributed.

In the long run, returns are not normally distributed, especially annual returns. They do not even follow any specific parametric form. So, the authors use a distribution-free Stochastic Dominance approach to capture the possibility that investors may prefer an investment with a higher variance over one with a lower variance. They find:

- Neither stocks nor bonds empirically dominate one another for any investment horizon because stocks have a higher mean, but also a more negative minimal return (fatter tail).
- However, when the riskless asset is available, any combination of stocks plus TIPS dominates the expected utility of any combination of bonds plus TIPS.
- By expected utility, it means anyone looking for higher returns in expectation would do better with a portfolio of risky stocks plus a riskless asset. It does not imply returns are always higher ex-post.

## Theory

Bear with me – some simple distribution theory will help us through this paper. Let *B *and *S *denote the cumulative distributions of the returns of a risky bond-only (*B*) and stock-only *(S*) strategy. Then, a stocks strategy is preferred to a bonds strategy by all investors if *S *is below *B. *In other words, loosely speaking, stocks provide a better chance of higher returns at any point along the distribution. And so, any rational investor would choose a portfolio of stocks over bonds. If the distributions cross, neither dominates (Chart 1).

The paper argues that when they add US TIPS to a portfolio of stocks (say 40% stocks, 60% TIPS), the three-year+ return distribution is always lower than the one with bonds. This dominating portfolio has a higher Sharpe ratio, and it remains higher for any combination of stocks and TIPS.

## Data and Methodology

To get the results, the authors must generate the cumulative distributions. First, they create empirical return distributions from the annual real returns on the S&P500 (including dividends) and US 10-year bonds between 1928 and 2019. For the riskless asset, they obtain the returns on TIPS with one- to 30-year maturities.

With the empirical return distribution to hand, they randomly draw annual returns 100,000 times to build the cumulative return distributions for investment horizons of between one and 30 years. Intuitively, this simulation gives the *ex-ante* annual returns an investor could have expected to receive at any point over the last 90 years when investing in stocks, bonds, or any combination of stocks and bonds with a riskless asset.

Using this information, and eyeballing Chart 1, we could say the following. An individual investing only in stocks could have expected to achieve annual S&P500 real returns of over 50%, roughly 40% of the time. A bonds-only investor could have expected to achieve the same annual returns less than 5% of the time. Equally, a bonds-only investor was less likely to have experienced large negative annual returns.

## The Main Result

First, the authors examine the cumulative return distributions of stock- and bond-only investments over increasing time horizons. They find, as in Chart 1, the distributions cross in all horizons. Therefore, neither strategy dominates the other. This is mostly because the minimal return on stocks is always lower than the minimal return on bonds. Notably, however, the mean return on bonds is always smaller than on stocks.

Under what circumstances, then, could a portfolio of stocks always beat a portfolio of bonds? At shorter-term investment horizons, like one year, the authors find that a portfolio with equities requires 70% in TIPS to ensure that the chance of low returns in the worst-case scenario is lower than in a portfolio with bonds. However, by removing downside risk, the upside probability is actually higher in a risky bonds portfolio, which is hardly appealing.

So, while there is no combination of stocks with a riskless asset that dominates at a short-term horizon (less than three years), there is when the investment horizon increases. At a five-year horizon, the authors find a portfolio with 40% stocks and 60% TIPS dominates any combination of bonds with a riskless asset. As the horizon increases from there, any combination of stocks and TIPS (e.g., 90% stocks, 10% TIPS) outperforms one with bonds and TIPS.

In reality, however, portfolios usually contain a mix of risky stocks and bonds. In this scenario, as the investment horizon increases, the most dominant strategy is one with a higher percentage of stocks in the portfolio (Chart 2). For example, if an investor’s horizon is 15 years or more, the investor should invest at least 60% in stocks and 40% in bonds, irrespective of the proportion of TIPS included – so, say 10% TIPS, 54% stocks, and 36% bonds.

In essence, the paper says that the longer the investment horizon, the more an investor should include stocks into the portfolio, *as long as a riskless asset is included*. This is because, in expectation, they will achieve higher returns with a higher probability than one with a higher allocation of bonds.

## Different Samples, Expected Returns and Disaster Risk

Crucially, the result is robust even when the authors look at the most recent 30-year period. Indeed, when dividing the sample into three sets of 30 years (1928-1957, 1958-1987 and 1988-2018), stocks are still preferable to bonds in the long run when a riskless asset is included.

The result is also robust for periods where the annual expected returns on stocks are lower. ‘*If the annual expected return on stocks is 2.30-2.86% lower than the average real return of 8.1% in the 1928–2018 period, stocks still dominate bonds.’ *

Interestingly, the result does not just hold when expected returns are lower, but also in rare disaster events, like the GFC. For example, for a disaster return of -30%, stocks continue to dominate bonds for horizons of three or more years. So, by adding a reasonable probability of large crashes during the investment horizon, the authors still find that a portfolio of stocks with a riskless asset dominates bonds with a riskless asset.

## Bottom Line

For investors deciding on an asset allocation framework, the paper helps confirm one thing – holding a larger share in equity indices is more fruitful over time, even when adjusting for risk. By doing so, you can trump the alternative of a bonds-heavy portfolio in periods of both market tranquillity and chaos. The key is to ensure you include a risk-free asset, otherwise, there is no guarantee holding more stocks than bonds yields higher returns in the long run.

**Citation**

Levy H., Levy M. (2021), Stocks versus bonds for the long run when a riskless asset is available, *Journal of Banking & Finance*, (Volume 133), https://www.sciencedirect.com/science/article/abs/pii/S0378426621002314

*Sam van de Schootbrugge is a Macro Research Analyst at Macro Hive, currently completing his PhD in international finance. He has a master’s degree in economic research from the University of Cambridge and has worked in research roles for over 3 years in both the public and private sector.*