Do equities (stocks) have duration? Of course, they do and in most dividend paying stocks the duration is very high, thereby signifying . But it may not be apparent at the first point of analysis. In a recent training session with a group of equity analysts we had an interesting discussion on this topic.

With stock markets worldwide hitting record highs in the last many months and with such uncertainty about the short term interest rates and the US Dollar, it is an interesting problem to study: how exactly does the equity of a firm behave with respect to the change in interest rate.

Duration is the measure of interest rate risk and equities are affected by the interest rates, or more appropriately the short rate. Generally one values and analyzes equities, quantitatively at least, within a factor framework and interest rates are an important factor. There is a direct deterministic link - via a mathematical formula - between the value of a stock and the short term (risk free) interest rate and hence theoretically equities should have a first order variation with the interest rate.

In a very simple, but elegant dividend discounting model, the change in value of a stock is completely determined by the growth in the dividends that the stock would pay in the future. If you approximate the growth in dividend yield,^{ D },by an exponential formula whereby ^{ D } grows by a rate of ^{ g }, then the value of the equity, ^{P } is given by:

The above formula simply says that the value of equity today is simply the discounted value of the growth in the value of the dividends that the stock pays in future. The limits of the integration simply tell us that starting today, time t = 0, the stream of dividends goes on infinitely in the future. Hence we see that the interest rate ^{ r }enters the valuation model via the process of discounting.

The above reduces to:

(Remember that D to be the dividend yield and not the dollar dividend.) From the above if we were to calculate the interest rate sensitivity, i.e. duration of the value of the stock, by taking the first order shift (first mathematical derivative of the stock value with respect to the interest rate, ^{r }) and putting it in the duration formula (rate of change of the stock value with respect to the rate) we will get:

Therefore, the duration of a stock is inversely related to its dividend yield. Higher the dividend yield the higher is the duration of the stock. Normally, the dividend yield of stocks in the G7 economies vary from 1% to 5% thereby implying that the duration of stocks in these economies vary from 20 years to 100 years (a fairly high degree of first order risks).

**Questions: **

- How we interpret such high duration numbers for equities?
- How would conceptually and intuitively explain the linkage of duration, which is an interest rate risk measure and the dividend yield of a stock?

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