What is the equity risk premium, abbreviated ERP? It’s the market’s best point estimate, today, of what “stocks in the aggregate” will return in the future — after subtracting a risk-free interest rate. By “stocks in the aggregate”, I am taking a US perspective, so thinking about an equity investment matching the S&P500 (after dividends are accounted for).

If the future is very distant, say the next 20 years, a widely held belief is that the ERP will average around 4-6% per year. The ERP has often been called “the most important number in finance”.

Speaking of interest rates, it’s well-known that rates vary in time and, at each time, have a term structure. For example, if r was a US Treasury rate, we would write r_{t,T} to denote the annual yield at time t for a Treasury bond maturing at time T.

Just like interest rates, the ERP is also time-varying and has a term structure — so we also write ERP_{t,T}. For example, we could ask, what is the ERP today for holding stocks over the next 6 months? Regardless of the horizon, which might be very close, the convention is to quote the ERP as an *annualized percentage rate*. This is the same convention as for interest rates: even if you are borrowing money for just a couple weeks, your borrowing cost will be quoted as an annualized percentage rate.

The fact that the ERP has a term structure is not widely appreciated. In contrast, the term structure of interest rates is well-known and readily visible. For example, just look in the Wall Street Journal for the current term structure of US Treasury rates — or find it in many places on the web. The ERP term structure is **not** directly visible and needs to be estimated. Exactly how? That’s the question I answer in a new research paper, recently posted at the arXiv, titled:

Option-based Equity Risk Premiums

As an example, I show below my estimate of the (US) ERP term structure for Feb 7, 2018, 2 days after the so-called `Volpocalypse’. That Feb 5, 2018 volatility event was the day of the Dow Jones Industrial Average’s largest point loss ever, although the percentage loss was less than 5%. The two charts show the same ERP, but the bottom uses a log time scale in order to better show the various, closely-spaced, option expirations. As you can, the ERP is declining from about 26% per year for the nearest dates (2 days away) toward the longer-term values mentioned above. The dark lines are central estimates and the red lines estimate the uncertainty. My assumptions and other details are found in the article.