 # Equity Duration Calculation

I know that duration mainly makes sense in fixed income world, but I am currenlty trying to “estimate” the duration of S&P 500. Does anyone know of a straightforward way of doing this?

John Hussman points out that - for dividend paying stocks, it’s roughly 1/div yield, and there’s a nice derivation of it based on differentiating the GGM with respect to the discount rate. Since the S&P 500’s div yield is about 2%, that suggests a duration of 50, which is an important figure to keep in mind when thinking about stocks.

Thanks BChadwick, I found Hussman’s piece (an excerpt from it is pasted below, and the full article can be found here: http://www.hussman.net/wmc/wmc040223.htm).

[Geek’s note: to see this, consider the dividend discount model P = D/(k-g). Differentiate with respect to k to get dP/dk = -D/(k-g)^2. Divide through by price , which is D/(k-g), and then substitute P/D for 1/(k-g). Notice that this result is independent of g. For stocks that don’t pay a predictable stream of dividends, you have to calculate duration explicitly from the stream of expected free cash flows, but for blue-chip indices, the price/dividend ratio is an excellent proxy for modified duration.]

I have 2 questions - one conceptual and one technical:

1) Can the duration be approximated by taking a partial derivative with respect to the discount rate, which includes the risk premium? Isn’t this more like an equivalent of the spread duration vs. interest rate duration? I think it is intuitive that the price of the stock will have much higher spread duration (i.e. ~ sensitivity to a change in the risk premium) than to interest rates. Would you agree?

2) This may be a naive question, but I do not see how the partial differential of price (dP/dk) divided by price reprsents the sensitivity of price to interest rates (or even to the discount rate for that matter).

Thanks!

1. you can think of the risk premium as a spread, since, that is effectively what it is (in both equity and fixed income contexts). And in reality, when things go haywire, the risk premium goes up, and that is often a bigger punch (but it’s also a bigger opportunity, since it may mean revert faster).

But since k = rfr + rp for small interest rates, then if rp is held constant, dk = d(rfr)

For answering the question “how do equity prices react, if rfr goes up,” it’s not necessarily a bad idea to hold the rp constant, though, more likely, you will want to assume that people panic in this situation and that rp goes up a bit too (and that would ideally be modeled explicitly).

1. a little algebra can help with this point:

(dp/dk) / p = (dp / p) / dk = (% change in price) / dk = equity duration

1. I agree that IF the risk premium (i.e. spread) is constant, then dk ~ d(rfr), but in reality the risk premium is very volatile, so is that a reasonable assumption to make?

2. OK, maybe I am alittle rusty on my math (although I have taken upper division calculus courses at UCLA), but isn’t dp/dk already the price change rate? So then, why would you divide it by the price to get the % change? Isn’t dP/dK already the limit of [(change in P)/(change in k)]? And in general, can you divide a rate by an absolute value, i.e. wouldn’t you divide change in price (in dollars) by the price to get the % change in price? Again, my apologies if I am missing anything that’s obvious.

1. Yes, in reality, you are going to see both the interest rate and the risk premium change, however, for long term analysis, it is not unreasonable to assume that the RP stays relatively constant over the long term. When you feel that the market is overvalued as a whole, for example, you are really saying that the risk premium is too small for the true risk of the market, and that you are expecting the risk premium to mean revert.

But, when you’re doing all these calculations, and you want to know what the effect of an interest rate change is on prices, it is ok to go ahead and hold the RP constant. Just, when the crapola hits the fan, you have to realize that in all probability, you’re going to be estimating only a lower bound or an upper bound by holding the RP constant.

Another point is that you get the same duration equation for risk premia changes. So if the interest rate stays constant, but the risk premium increases by 1%, you’re going to get a huge change in the index price. That’s interesting, because a 1% increase in the risk premium would supposedly shave 40-50% off of the S&P 500 valuation. If you believe the math, what it really tells you is that all the changes in the long term risk premium are actually pretty tiny.

There’s also an issue with the fact that some companies distribute “dividends” through share repurchases, which are qualitatively the same thing. Probably we would want to adjust the model to assume that (adjusted dividend yield) = (div yield) - (net change in shares outstanding). So the effect would not necessarily be quite so strong, but it would be a pain in the butt to calculate.

1. dp/dk is the change in price (dp) per change in interest rate. Duration is the percentage change in price (dp/P) per change in the interest rate. So that’s where the divide-by-P part comes in. It’s not super obvioius until you already know the answer.