How to determine duration of a common stock whose dividends grow in perpetuity?

Question

What is the duration of a common stock whose dividends are expected to grow at a constant rate in perpetuity? Specifically, how can I compute this, since the dividends will grow forever?

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Update: Ok, to make this more realistic, assume that the discount rate r is higher than the growth rate g, and both r and g are positive.

For reference, the formula for the duration is = summation of t x PV(period t) / (Total PV)

The only thing I don't get is to sum up the t x PV(period t). I have a feeling it would sum to infinity though.

Answer 1

The Dividend Discount Model is based on the concept that the present value of a stock is the sum of all future dividends, discounted back to the present. Since you said:

dividends are expected to grow at a constant rate in perpetuity

... the Gordon Growth Model is a simple variant of the DDM, tailored for a firm in "steady state" mode, with dividends growing at a rate that can be sustained forever.

Present Stock Value = DPS / (r - g)

Consider McCormick (MKC), who's last dividend was 31 cents, or $1.24 annualized. The dividend has been growing just a little over 7% annually. Let's use a discount, or hurdle rate of 10%.

1.24 / (.10 - .073) = $45.93

MKC closed today at $50.32, for what it's worth.

The model is extremely sensitive to inputs. As g approaches r, the stock price rises to infinity. If g > r, stock goes negative. Be conservative with 'g' -- it must be sustainable forever.


The next step up in complexity is the two-stage DDM, where the company is expected to grow at a higher, unsustainable rate in the early years (stage 1), and then settling down to the terminal rate for stage 2. Stage 1 is the present value of dividends during the high growth period. Stage 2 is the Gordon Model, starting at the end of stage 1, and discounting back to the present.

Consider Abbott Labs (ABT). The current annual dividend is $1.92, the current dividend growth rate is 12%, and let's say that continues for ten years (n), after which point the growth rate is 5% in perpetuity. Again, the discount rate is 10%. Stage 1 is calculated as follows:

( DPS  * (1+g) * [1 - ( (1+g)^n / (1+r)^n )] ) / (r - g)
( 1.92 * 1.12  * [1 - ( 3.1058  /  2.5937 )] ) / (.10 - .12)
( 1.92 * 1.12  * -0.1974                     ) / -0.02       = 21.22

Stage 2 is GGM, using not today's dividend, but the 11th year's dividend, since stage 1 covered the first ten years. 'gn' is the terminal growth, 5% in our case.

DPS_11 = DPS * ((1+g)^(n+1)) = 6.68

then...

DPS_11 / ( (r - gn)   * (1 + r)  ^(n+1) )
6.68   / ( .10 - .05) * (1 + .10)^11    ) = 51.50

The value of the stock today is 21.22 + 51.50 = 72.72

ABT closed today at $56.72, for what it's worth.

Answer 2

The fact that dividends grow in perpetuity does not prevent one from calculating duration. In fact, many academic papers look at exactly this problem, such as Lewin and Satchell. This Wilmott thread discusses some of the pros and cons of the concept in some detail.

PS: Although I was already broadly familiar with the literature and I use the duration of equities in some of my every-day work as a professional working in finance, I found the links above doing a simple google search for "equity duration."