Financial Mathematics Text

Tuesday, September 2, 2014

More on the Constant Growth Assumption

So earlier, in Uncertainty of the Constant Growth Assumption, I discussed how even if we know with certainty (hah!) what rate of growth cash flows (in our example, dividends) will grow at, we may still end up getting the value incorrect. This is due to the fact that actual cash flows are lumpy and don't grow at a nice constant rate but vary quite a bit. The result is that the present value of the asset can be a good deal more or less than where we estimated it to be.

Today I want to look at this from a different perspective. I also wanted to change some of the inputs of the model I used (which, you could do on your own if you downloaded or made a copy of the original spreadsheet.) Instead of focusing on how the actual present value of the cash flows differs from what the model predicts, I want to look at it from the perspective of what returns you'll actually get.

Regarding the Inputs

I began by using Robert Shiller's data and showed the year over year dividend growth of the S&P 500 with a chart that looks somewhat like this:

I also gave some summary statistics for the above graphs:

What should be striking of course is how much different the last, say, 50 years or so have been. Year over year dividend growth was actually much more volatile in the beginning of the time series compared with the end of the time series. What should we expect going forward?

If I had to guess, I'm guessing it will look more like the end of the time series than the beginning. Companies now payout a much lower portion of their dividends than they have in the past. As a result, it makes it easier for them to continue paying the same dividend and increasing it at regular intervals compared to if they were paying a higher payout. This is partly due to companies shifting to using share buybacks instead of dividends.

So here's what the summary statistics look like from 1960-2013:

Average (both arithmetic and CAGR) real growth rates were about the same for both periods. However, the 1960-2013 period is less volatile. We can also note that nominal growth rates were much higher due to higher inflation during this period.

That of course raises the question of whether or not we ought to use a "real" growth rate instead of a "nominal" growth rate which is an important question but one I won't tackle here.1 I'll stick with the nominal discount rate of 10% that I previously used so I'll go with a nominal growth rate.

Now, we might ask whether it's appropriate to use the 5.5% growth rate during this period (which included a very high inflationary period during the late 1970's.) I'll basically use the real dividend growth rate and add to that the current market implied expected inflation rate:

The 30 year expected inflation rate is about 2.3%. I'll round it up to an even 2.5% and add that to the real growth rate of about 1.5% to get a nominal growth rate of 4%.

The standard deviation I'll round off to a nice even 6%.

Uncertainty of Returns with the Constant Growth Assumption

As we had seen before, the constant growth assumption is a simplification of actual returns. The result is that you can get a vastly different value depending the exact manner of how those cash flows actually growth lumps and all (as opposed to the mathematically pretty constant growth rate.)

So what happens if you pay "fair value" as your model predicts assuming constant growth rates. What sort of returns can you expect?2

Well, if you discount at a rate of 10%, you can expect about 10% returns. However, your annualized returns can vary quite a bit from there.

Running 1000 trials, here's what the distribution of annualized returns look like if you pay fair value:

I overlaid it with a normal distribution for context. For what it's worth, while it appears to be a close fit to a normal distribution, it actually fails tests for normalcy.

Here are the summary statistics:

So what this shows is that if you pay "fair value" (using the constant growth assumption and a 10% discount rate), on average you'll earn about a 10% rate of return.

But that annualized rate of return may be higher or lower depending on how the cash flows arrive. In this simulation (1000 trials), it was possible to get returns as low as 7.14%.

Playing with the Model

If you want to play with the model, you can download or make a copy to your own google drive account.

Constant Growth Assumption Returns

It's only set up for 20 trials but you can extend that if you wish.

Does It Really Matter?

So how much of a difference is that, roughly, 1% standard deviation? Is it really a big deal if I only get 9% instead of 10% returns?

Here's a chart (using continuous compounding interest) that shows the difference sub par returns can make over time.

For example, underperforming by 1% over 20 years results in missing out on 22.14% more at the end of the period. That would mean missing out on over 20% more income in retirement, for example.

Concluding Remarks

As before, since actual cash flows deviate from the constant growth assumption present in many models, the actual present value of those cash flows will vary (or the subsequent returns assuming you pay "fair value" will vary.) This illustrates another source of uncertainty present in the valuation process.

As I mentioned in Uncertainty and Margin of Safety, I believe that uncertainty present in valuation models is intricately related with Benjamin Graham's concept of margin of safety. By building in a margin of safety into your purchases you have some buffer room in case reality differs from your assumptions. It's important to be aware of the various sources of uncertainty to guard against them as best as possible.

1 I think the decision will depend a good deal on what liabilities you have. If your goal is to save for retirement, then what you care about is how much money you'll have to pay future expenses (which will go up with inflation over time.) in that case a real discount rate would be appropriate. On the other hand, if your concern is using those cash flows to pay fixed payments (say on a debt that was used to finance those assets) then a nominal discount rate would be more appropriate.

Suffice to say, whichever one you choose, the growth rate and the discount rate ought to be the same (both real or both nominal.)

2 Keep in mind here that we are assuming that we know, with absolute precision, the exact rate at which dividends will grow (CAGR) over the next 100 years. Our goal is to only illustrate that the constant growth assumption introduces some uncertainty into our estimates of value and future returns. The fact that we don't precisely know this figure means that's an entirely separate but equally important (if not moreso!) layer of uncertainty.

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