Financial Mathematics Text

Monday, March 24, 2014

Uncertainty of the Constant Growth Assumption

In Uncertainty and Margin of Safety, I discussed the role uncertainty plays in valuation. Today I want to take a look at one of the assumptions built into many discounted cash flow (DCF) models. That assumption is the constant growth rate assumption.

Real World Growth

Growth in the real world is quite lumpy. As an example here is real and nominal dividend growth for the S&P 500 (data courtesy of Robert Shiller).

Here are some summary statistics:

As you can see, actual dividend growth is quite choppy with a standard deviation of almost 12%. 

Growth Volatility and Uncertainty

Typically models assume constant growth (iow, there is no volatility in the growth rate.) For example, the Gordon Growth Model assumes that growth occurs at a constant rate from here to eternity:
$$P = \frac{d}{i- g}$$
where $P$ is the price, $d$ is next year's dividend payment, $i$ is the discount rate (required rate of return) and $g$ is the constant annualized growth rate.

But what if, even if we precisely know CAGR, our estimate can still be off? How much uncertainty does that growth volatility contribute to value estimates?

For that we'll need a model!

Modeling Growth Uncertainty

To model growth uncertainty we'll need to make some assumptions. The assumptions will actually be false and therefore we will not be able to draw precise conclusions from them. However, I do believe they will be very valuable for illustration purposes.

  1. I'm going to assume that year over year growth will follow a normal distribution with a mean $\mu$ and standard deviation $\sigma$.
  2. I will assume that cash flows are discount at a constant rate $i$. 
  3. I will assume the cash flows will be paid once annually and grow for a period of $t$ beginning with an initial cash flow of $d$. 
The assumptions are likely problematic. For instance, actual dividend growth for the S&P 500 failed the Shapiro-Wilk normality test with p<0.001. The model assumes that growth rates are constant across time but there may be periods in which average growth is higher or lower than others.

We'll begin by assuming 4% mean growth with 11% standard deviation (roughly in line with the S&P 500 nominal data.) We'll discount the dividends at a rate of 10%. Trailing dividend will be $1.00 and the dividends will be paid for 100 years.

We shall then calculate the actual present value of the dividends versus the model which assumes constant growth at the same rate growth actually occurred. Here are graphical representations of 20 trials:

Price of the stock

Difference Between Actual and Constant Growth

So as you can see, even if we precisely know CAGR, there can be a significant amount of deviation between the actual value versus the model value.

Now I also ran this simulation with 1000 trials. Here are the summary statistics of the difference between the actual growth versus the constant growth models:

While the difference between the two averages out to about 0%, it varies quite a bit trial to trial. The constant growth assumption in one case predicted a 114% higher value than what would actually be obtained. The overall standard deviation was 25%.

Do you want to play with this model yourself?

If you're bored and want to play with this on your own (or just get a better sense of what this model is doing), I've created a variation of this in Google Docs:

Constant Growth Assumption Model

Feel free to save a copy to your drive (goto File and then click Make a copy) or you can download it as an Excel file.

You can alter the mean and standard deviation of the growth rate and also the discount rate. To recalculate, CTRL + R acts as the equivalent of Excel's F9.

Concluding Remarks

While there are obvious problems with this model, I think it offers some valuable insight into the role the constant growth rate assumptions introduces uncertainty into valuation models. This again illustrates why following Benjamin Graham's advice of having an adequate margin of safety when purchasing a financial asset is so important. Even if we precisely know the long term growth rate of a stock, we may still under or overestimate the value of that stock.

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