- Gerald Garvey
- Konark Saxena

*To order reprints of this article, please contact David Rowe at drowe{at}iijournals.com or 212-224-3045.*

## Abstract

How does the interest rate environment inform the allocation between risky and safe assets? The authors focus on the level rather than change in rates because rates are highly persistent; changes are infrequent and unpredictable while low versus high rate regimes are durable. U.S. history from 1926 through 2016 shows that equities have lower risk and higher excess returns when rates are low. Despite all the concern about low yield, a low rates environment is actually a good thing for more-aggressive investors.

The short rate plays a fundamental role in a market economy: It helps firms evaluate projects against their opportunity cost of capital, and it affects households’ ability to save for the future. It is less clear what rates mean for investors’ risk allocations. Recent experience and evidence suggests that rate reductions favor lower-risk equities such as dividend yield or minimum volatility strategies.^{1}

To act on this insight, however, one would need to forecast rate changes or at least move quickly in response. This is difficult because central bankers’ intentions are hard to divine and, even more fundamentally, because rates are highly persistent and changes are relatively rare. In the current U.S. context, the consensus forecast appears to be for a rate increase, but there is substantial uncertainty about the size and timing of any rate hikes. Nonetheless, essentially no one believes we will see short rate *levels* above 3% in the next five years. To put more systematic numbers on the issue, Exhibit 1 uses the Fama–French monthly risk-free rate (RF) field from 1926 to 2016 (expressed in annual terms for ease of interpretation).^{2} The horizontal axis is the current level of rates, and the data clouds represent next month’s RF level and change. The current month’s rate level is an excellent indicator of the future level and nearly useless as a forecast of the rate change.^{3}

We show that a standard mean–variance framework is extremely useful for understanding the implication of rate levels for future equity risk and return.^{4} The key intuition is that different combinations of risky assets are better suited for low- versus high-rate environments. High rates mean that investors desiring lower risk levels can invest in the riskless asset at more attractive terms. This implies that investors (1) allocate less to risky assets and more to the riskless asset, decreasing their overall portfolio risk, and (2) invest in a riskier market portfolio with higher returns to partially offset this risk reduction. A low-rate environment makes this strategy less attractive and instead tends to favor a larger allocation to a less risky mix of risk assets. Less risk-averse investors can then increase leverage to this less risky portfolio.

We first present the one situation in which this intuition does not hold and rate levels are uninteresting: if the entire risk–return frontier shifts one for one with rates. Investors are all better off (at least in nominal terms) in a high-rate environment, but there are no asset allocation implications. We strongly reject this view in the data. Our data support the more interesting and asset allocation–relevant case in which the risk–return frontier is *not* a direct function of rate levels. In that case, risk is lower and the equity risk premium is higher when rates are low. Low rates are bad for more risk-averse investors but a good thing for those who are less risk-averse. In addition, as documented in the existing literature, rate reductions are good for lower-risk equity portfolios because their relative share in the market needs to rise.

Our key contribution is to show in a standard mean–variance framework that investors should increase their allocation to equities when rates are low. It is widely accepted that equities carry a substantial long-term risk premium (see, e.g., Dimson, Marsh, and Staunton [2002] for comprehensive evidence using a century of data for 16 major markets). What the dynamics of the equity risk premiums imply for asset allocation is less well understood.^{5} ,Dimson, Marsh, and Staunton [2013] showed evidence from a global panel that long-term (five-year) real equity returns tend to be lower when real rates are lower. Our findings for the United States, using monthly returns, are consistent with their result. Furthermore, both findings are consistent with our key conclusion that investors should increase their allocation to equities when rates are low. This conclusion does not require expected equity returns to decrease with rates. Instead, we need to ask the following questions:

1. Does the equity market offer a higher return

*premium*when rates are lower? The real versus nominal distinction is not relevant here, because we are looking at the difference in returns over a common horizon. To underscore this point, Exhibit 2 reproduces Dimson, Marsh, and Staunton’s [2013] Figure 5, showing the positive relationship between equity returns and rates. In Exhibit 2, we also show that the relationship is eliminated or reversed if we simply shift to excess returns. Higher excess returns in low-interest-rate environments suggest that investors should increase their allocation to equities.2. Does the premium carry any additional risk? It is difficult to address this question with the annual data in Dimson, Marsh, and Staunton’s [2013] study, but our U.S. data are monthly, and we use a simple ARCH (autoregressive conditional heteroskedasticity)–GARCH (generalized autoregressive conditional heteroskedasticity) framework to simultaneously estimate the effect of rates on risk and expected return.

^{6}We find strong evidence that risk is lower in low-rate environments, and a less formal look at percentile returns suggests there is also less downside risk. This finding also suggests that investors should increase their allocation to equities in a low-interest-rate environment.

## FRAMEWORK

In a classical capital asset pricing model analysis, the expected return to the portfolio of all risky assets is the point at which the tangent from *R _{f,t}* intersects the efficient frontier of risky assets and is therefore uniquely determined by

*R*and the location of the frontier.

_{f,t}Points 1 and 2 in Exhibit 3 represent the case in which the level of rates has no implication for the risk–return trade-off or asset allocation, because neither the standard deviation nor the expected *excess* returns on the tangency portfolio change. This would be the case, for example, with pure and neutral inflation (i.e., when the expected returns of all assets change by the same amount, so that excess returns stay constant). Although this is an extreme case, Cochrane [2001] noted that it is actually the one most researchers implicitly accept:

Stocks are priced with stock factors, bonds with bond factors, and so on. More recently, stocks sorted on size, book/market, and past performance characteristics are priced by portfolios sorted on those characteristics.

In fact, there is extensive evidence that risk premiums are dynamic (for a survey, see Cochrane [2011]). If the dynamics of risk premiums are correlated with the dynamics of the riskless rate, then, by definition, the frontier will not move one-to-one with riskless rates. For example, this would be the case when riskless rate changes reflect changes in aggregate uncertainty (Saxena [2017]).

Now consider the case in which the frontier does not shift along with rates. In this case, changes in *R _{f,t}* will induce time variation in the expected returns and risk premium of the tangency portfolio (E

_{t}[

*R*

_{m,t+1}]). In Exhibit 3, the tangent line from the riskless rate hits the frontier at point 3, instead of point 2. The new tangency point has higher risk and a higher expected return. Because the efficient frontier is concave, the increase in risk is greater than the increase in return, so the Sharpe ratio of the tangency portfolio at point 3 is less than that of the tangency portfolio at point 1.

Under a reasonable choice of parameters, the excess return of the tangency portfolio decreases because the increase in expected return of the tangency portfolio is less than the increase in the riskless rate. This can be visualized by simply comparing the length of the vertical lines from points 2 and 3 to their respective riskless rate levels. This result holds so long as the slope of the efficient frontier at the tangency point is less than 1 (under a first order approximation), which is equivalent to a Sharpe ratio less than 1. This is a very mild condition. We provide a fuller derivation in the Appendix, but the key assumption is that the mean–variance frontier does not itself shift too much when rates change.

Exhibit 3, and the asset allocation decision more generally, highlights the simultaneous importance of both risk and return. This makes for a more powerful and interesting analysis than simply examining the effect on average returns. If there were a parallel upward shift in the entire frontier as rates increase (as in our baseline case), then neither the risk nor the excess return of the market would change with rates. The existing evidence that rate decreases are good for low-risk stocks is not consistent with this scenario; if the entire frontier moves with rates, then all risky assets are equally affected. By contrast, the immobile frontier case (in which the market portfolio moves from point 3 to point 1) is consistent with strong performance of low-risk stocks when rates fall. Intuitively, the market portfolio becomes more similar to the minimum variance portfolio as rates fall. This means that the weight of low-risk stocks must increase; hence, their returns must be temporarily elevated.

## EMPIRICAL ANALYSIS

For a formal test of whether the market risk premium and its volatility are related to the level of the riskless rate, we use an adapted ARCH model:^{7}^{,}^{8}

β_{f} measures the effect of the riskless rate on market expected returns and η on market volatility. The error variance (*h _{t}*) in this equation is likely to be time-varying, because market returns are known to exhibit periods of swings interspersed with periods of relative calm. The ARCH (ϕ) and GARCH (φ) terms capture such time variation and control for the effect of volatility clustering, improving estimates of time variation in both expected return (β

_{f}) and risk (η).

We use the risk-free and market return data from 1926 to 2016 from Ken French’s data website. The parameter estimates in Exhibit 4 strongly reject the rate-neutral view implied by points 1 and 2 in Exhibit 3. We find that β_{f} < 0, indicating that market excess returns increase when the riskless rate decreases. We also find that η > 0, indicating that market volatility increases when the riskless rate increases. This observation is robust to constraining β_{f} = 0 in an alternative (unreported) estimation. Based on these findings, we cannot reject the no shift in the frontier hypothesis (points 1 and 3) in Exhibit 3. The parallel shift case in Exhibit 3 implies β_{f} = η = 0, a hypothesis we can strongly reject. The risk of the market is higher and the expected risk premium is lower in high-rate environments.

Exhibit 5 adds the recent change in the short rate as an additional explanatory variable for both risk and return. We find that both effects are significant at the 10% level. The level effect is robust to the changes effect. As discussed earlier, this suggests that the information in rate levels can be used for strategic asset allocation.

Our key claim is that asset allocation needs to pay attention to rate levels in addition to more traditional metrics. However, it is possible that the relevant information could be gleaned by simply looking at equity valuations. At least directionally, low rates should increase valuations, which on its own conveys information about future returns (e.g., Campbell and Shiller [1988]; Berk [1997]).^{9} Exhibit 6 adds the marketwide P/E to the model in Exhibit 4. Data for the P/E are taken from Robert Shiller’s website (http://www.econ.yale.edu/~shiller/data.htm).

The expected-return results are intuitive. Higher valuation levels are associated with lower future average returns, and because valuations tend to be higher when rates are low, the independent effect of rates on expected equity returns is attenuated (but not eliminated).

The variance results underscore the unique information conveyed by the rate level. It is at least plausible that higher valuations could be associated with lower risk, but there is absolutely no evidence of this. Rate levels continue to be an informative guide to future risk.

## ASSET ALLOCATION IMPLICATIONS

Next, we analyze the asset allocation implications of different rate environments. Based on the evidence in the previous section, we analyze the allocation between the riskless rate and the tangency portfolio for risk-averse investors with mean–variance utility functions. In Exhibit 7, we present two cases in which the risk-aversion parameters are chosen such that investors are either (Panel A) net lenders at the riskless rate or (Panel B) net borrowers.

In Exhibit 7, Panel A, we see that in low-rate environments the indifference curves shift toward the lower right for more risk-averse investors who are net lenders in the riskless rate. This is associated with a decrease in the utility for such investors, who get less interest on their riskless rate investments. Notice also that as the riskless rate decreases, the asset allocation point 4 moves to point 3, which is higher up as a proportion of the tangency line. This shows that, in low-rate environments, these investors should invest *more* in the tangency portfolio.

In contrast to the previous exhibit, in Exhibit 7, Panel B, we see that the indifference curves shift higher for more risk-averse investors who are net borrowers at the riskless rate. This is associated with an *increase* in the utility for such investors, who pay less interest on their borrowings. As the riskless rate decreases, the asset allocation point 4 moves to point 3, which is still higher up as a proportion of the tangency line. This shows that, in low-rate environments, these investors should increase their leverage and invest *more* in the tangency portfolio.

To conclude, we calibrate these implications in the data. We divide our sample into three subsamples based on the 30th and 70th percentile cutoffs of the one-month Treasury bill rate from 1926 to 2016. In Exhibit 8, we plot the average market excess returns and Sharpe ratio for these three periods. We find evidence that both expected excess returns and the Sharpe ratio decrease in the riskless rate.

## CONCLUSION

We show that the interest rate environment should inform asset allocation decisions. Consistent with the theoretical framework that informs this decision, we find that a low-rate environment is associated with higher expected excess returns and Sharpe ratios for risky assets. We also show that, despite all the concern about low yield, a low-rate environment is actually a good thing for more aggressive investors. Our analysis uses a broad set of listed U.S. equity indexes as a proxy for risky assets, but the implications should carry over to risk assets more generally, including private equity, risky credit, and international equity markets. Lower short rates favor a higher allocation to these asset classes as well.

Although our main analysis focuses on U.S. data, Exhibit 2, derived from Dimson, Marsh, and Staunton [2013], shows some supporting international evidence. Across 20 countries with 113 years of investment history, equity markets with lower rates tend to offer a higher excess return in the subsequent five years. We leave the international study of interest rate levels with equity market valuations and volatility for future research.

In future research, it could also be valuable to better understand the leveraging implications. In particular, our article has focused exclusively on standard deviation and has neglected tail risk. At a high level, there does not appear to be a major issue; if we simply divide the sample at the median level of the risk-free rate (close to 3%), we continue to see a more attractive profile in the low-rate environment if we focus on the lower tails. The lower 5% monthly return is −7% in the high-rate months versus −5.99% in the low-rate months, and the lower 1% is almost −12% in the high-rate months versus just over −10% in the low-rate months. However, a deeper understanding of the drivers of tail risk and leveraging constraints in different rate environments does seem warranted.

## APPENDIX

The relations described in the theory section can be modeled using the results of MVE mathematics (see Roll [1977]). The return on a tangency portfolio (*R*_{m,t+1}) from a riskless rate of *R _{f,t}* is

where *a*, *b*, and *c* are mean–variance frontier parameters that depend on the expected returns and variance–covariance matrix of all conditionally risky assets. Specifically, *a* = **R′∑ ^{−1}R**,

*b*=

**R′∑**, and

^{−1}ι*c*=

**ι′∑**where

^{−1}ι**R**is the vector of expected returns of risky assets,

**∑**their covariance matrix, and

**ι**a vector of ones.

Although these parameters can be sensitive to changes in the riskless rate, we illustrate the relation by assuming them to be constant. Under this assumption, the sensitivity of E_{t}[*R*_{m,t+1}] to *R _{f,t}* can be approximated using the first derivative of Equation A-1:

This derivative is always positive (except when the denominator is zero, when it is not defined). For the risk premium to reduce with an increase in the riskless rate, this derivative should be less than one.

## ENDNOTES

^{1}Results supporting this view can be found in work by Fishwick, Muijsson, and Satchell [2014; 2016]; Driessen, Kuiper, and Beilo [2017]; and Saxena [2017].^{2}We use the short-term Treasury bill rate, because longer-term bonds are generally more subject to inflation risk and interest rate risk. The data were downloaded from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.^{3}The alert reader will also note heteroskedasticity; the variance increases with the level. This is a further challenge for analyzing rate changes; a 2-bp move could be quite different if the prevailing level is 15% versus 2%. We explicitly account for heteroskedasticity in our empirical analysis of levels.^{4}The intertemporal capital asset pricing model, starting with Merton [1973], suggests that long-term investors overweight (underweight) stocks that hedge unexpected changes to the investment opportunity set, such as to expected returns. It is well known that such investors deviate from the short-run mean–variance optimal portfolio. To the best of our knowledge, however, simple closed-form solutions for optimal portfolio choice in the general setting are not available.^{5}Notable academic work in this area used numerical methods and multiple variables such as short rates and the price-to-earnings ratio to predict equity risk premiums (see, e.g., Brennan, Schwartz, and Lagnado [1997] and Campbell, Chan, and Viceira [2003]). We focus on providing an intuitive framework to analyze the implications of a low-rate environment on asset allocation decisions.^{6}Dimson, Marsh, and Staunton [2016] analyzed trading strategies in the United States and the United Kingdom during easing and hiking cycles (identified using recent*changes*in rates). They found that the volatility and Sharpe ratios of these trading strategies were greater during periods of decreasing rates. We focus on the implications of the riskless rate*level*. Our empirical analysis is robust to control for the effect of recent changes in the riskless rate.^{7}Engle [2001] explained a key benefit of this approach: “The standard warning is that in the presence of heteroskedasticity, the regression coefficients for an ordinary least squares regression are still unbiased, but the standard errors and confidence intervals estimated by conventional procedures will be too narrow, giving a false sense of precision. Instead of considering this as a problem to be corrected, ARCH and GARCH models treat heteroskedasticity as a variance to be modelled. As a result, not only are the deficiencies of least squares corrected, but a prediction is computed for the variance of each error term.”^{8}Note that our test is not about whether an interest rate cut is good news for the stock market (see, e.g., Fishwick, Muijsson, and Satchell [2014; 2016]). It is about whether the*equity risk premium*changes along with changes in the riskless rate (see previous discussion and Exhibit 3 for details).^{9}Moreover, papers on strategic asset allocation, such as by Brennan, Schwartz, and Lagnado [1997] and Campbell, Chan, and Viceira [2003], included the price-to-equity ratio as one of their state variables.

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