# Term structure of Equity returns

What is the meaning of term structure of equity returns.

I know what term structure of interest rates means, but somehow i cant seem to relate them. Also, how would we measure them?

Also in this paper : https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3689947.

The author has mentioned term structure of dividend risk premia and discount rates.What do you mean by them?

• ok.but in case of interest rates you have live prices available.How will you measure for equity?forecasting dividends and expexted returns? – pppp_prs Apr 28 at 19:06
• "Dividend futures" although not very liquid do exist and they provide a measure of what the market expects in terms of dividends for various future periods. – noob2 Apr 28 at 19:14

## Intro: Duration-Based Asset Pricing

Similar to bonds, we can define the duration of stock $$i$$ as $$Dur_{i,t} = \sum_{s=1}^\infty s\cdot\frac{\mathbb{E}_t[CF_{i,t+s}]e^{-s r_{i,t}}}{P_{i,t}},$$ where $$P_{i,t}$$ is today's stock price, $$r_{i,t}$$ a discount rate and $$CF_{i,t}$$ are cash flows. The variable $$Dur_{i,t}$$ tells you the weighted average of when a firm receives its cash flows. In this sense, you can think of stocks as bonds whose coupons (= cash flows = dividends) occur at random time points and are of uncertain size.

As a good asset pricer, you now start sorting stocks into decile portfolios to see whether/how duration is priced. As it turns out, there's a short duration premium, i.e., the later firms receive their cash flows, the lower their average return. This observation leads to the term structure of equity, or equity yield curve: you plot maturities ($$x$$-axis) and their corresponding returns ($$y$$-axis). This is very similar to what we know from bonds.

One difficulty with this duration measure is that it is hard to compute. You need to find appropriate discount rates, predict future cash flows, determine when to truncate the sum etc. Many researchers follow the approach from Dechow et al. (2004, RAS), who essentially assume a mean-reverting process for book equity growth and return on equity (which determine cash flows). There are, however, many slight variations in methodology here if you compare different papers.

An alternative approach is to look at dividend strips (essentially, these assets pay you only the dividend of a stock at a particular time point; it's like stripping coupons from a bond). Futures on these dividends strips have been used by Binsbergen et al. (2013, JFE). Gormson and Lazarus (2021) also use dividend strips and describe the data as follows

The average annual volume is 11,692 contracts and the average open interest is 5,382 contracts. A contract is a claim to the dividends paid out on 1,000 shares and trades on average at around €2,000. The average notional outstanding is around €4 million. The total value of all the notional outstanding is around €4 billion at the end of the sample

## Some Results

To give you some numbers, Gonçalves (2021, JFE) finds that

The time series average of the median equity duration is 38.9 years, with the 10% and 90% quantiles being 17.1 and 94.7 years, respectively.

So, there quite some variation and large right tail. Binsbergen and Koijen (2017, JFE) summarise some stylised facts about the short duration premium.

Both risk premia and Sharpe ratios are higher for short-maturity claims than for the aggregate stock market. (...) The returns on short-term dividend claims are risky as measured by volatility, but safe as measured by market betas. (...) The volatility of equity yields is downward-sloping with maturity.

This already points at some problem. It's never nice when returns and betas don't align. So, the question is really what risk drives the high returns on stocks that receive their cash flows soonish?

Weber (2018, JFE) plots the term structure of equity, which nicely shows a negative slope.

[Due to methodological differences, Weber's duration is much shorter than Gonçalves'.]

## What's The Big Deal?

Binsbergen, Brandt and Koijen (2012, AER) report that

We find that both the long-run risks model and the external habit formation model predict that expected returns, volatilities, and Sharpe ratios of short-term dividend strips are lower than those of the aggregate market. Further, the risk premium on short-term dividend strips in those models is near zero. In the rare disasters model, the volatilities and Sharpe ratios of short-term dividend strips are lower than the aggregate market. Expected returns, on the other hand, are equal across all maturities of dividend strips and, therefore, also equal to those on the aggregate market. Our results suggest that risk premia on the short-term asset are higher than predicted by leading asset pricing models.

Essentially, the authors argue that many of the leading consumption-based asset pricing models don't line up with empirical facts. These models were developed to explain the equity premium puzzle from Mehra and Prescott (1985, JME) but they make “wrong” predictions about the term structure of equity.

Clearly, the guys who invented these models weren't too happy and that's where your paper comes in. Bansal and Yaron developed the long run risk model and in Bansal et al. (2019), they argue that estimating the slope of the equity yield curve accurately is not straight-forward and that it may be state-dependent. Cochrane (2017), who contributed to the habit models, is also critical of the original Binsbergen, Brandt and Koijen paper. Chen and Li (2020) suggest that the equity yield curve could be humped-shaped.

## Cross-Sectional Anomalies

While there are some open questions on how duration fits into the macro-finance literature, it has also found its way into the cross-sectional asset pricing literature. Lettau and Wachter (2007, JF) propose a duration-based model to explain the value premium. Gormson and Lazarus (2021) show how a duration factor can explain many other anomalies such as value, profitability and investment.

One question is obviously why do firms with short duration have higher returns? Weber (2018, JFE) points to mispricing arguments whereas Gonçalves (2021, JFE) offers a rational explanation using re-investment risk in an ICAPM, see also Gonçalves (2021, JF).

• Very interesting, +1. This might become the next puzzle, then? I was working with some consumption based models during my research time and I was always interested in whether and how newer models might reconcile some even newer, recently discovered, 'anomalies'. Thanks for the post. – Kermittfrog Apr 29 at 9:58
• @Kermittfrog It reminds me a bit of how many models that were built to explain the value premium cannot explain the (newer) profitability anomaly. So (even) newer models have to try to combine both. At least it keeps doing research interesting :) – Kevin Apr 29 at 10:17
• This is a much better and complete answer than mine. Upvioted – phdstudent Apr 29 at 11:22
• @phdstudent Thanks very much, I think both answers complement each other well. Yours is the much more to the point answer anyone would be looking for; I merely tried to add and give a broader literature review but I like your answer and upvoted it yesterday as I posted my answer. I very much enjoy the teamwork on this site. – Kevin Apr 29 at 12:44

The term structure of returns refers to returns on assets with the same underlying cash flows, where the return is measured over the same holding period, but for different maturities.

The price of a stock or equity index $$S_t$$ is given by the discounted value of its dividends $$D_t$$:

$$P_t = \sum^\infty_{n=1} E_t(M_{t:t+n}D_{t+n}) = \sum^T_{n=1} E_t(M_{t:t+n}D_{t+n})+\sum^\infty_{n=T+1} E_t(M_{t:t+n}D_{t+n})$$

Now define the price $$P_{t,n}=E_t(M_{t:t+n}D_{t+n})$$ as the price of $$nth$$-period dividend strip. The index is just the sum of all strips: $$P_t = \sum_{n=1}^\infty P_{t,n}$$

Some good references: