somrh (90.42)

Valuation Risk

August 01, 2012 – Comments (1)

I associate valuation risk with the broad set of risks associated with correctly valuing an asset. As an example, one must choose a model (which model) and then put inputs into that model to get a result which is the estimated value. Any such process has limitations.

I'm going to look at one particular model and one aspect of valuation risk. That aspect is what people in finance called "sensitivity" and what physicists call "uncertainty". (As a side note, "uncertainty" doesn't seem to be too important in economics. Economists have borrowed many metaphors from physics but unfortunately have failed to integrate uncertainty into their models as far as I can tell. At the very least, its importance is much lower in economics than it is in physics.)

What this amounts to is how variations in inputs result in different outputs. The goal is to quantify these variations. 

In order to do this, we will need a specific model. I will start with a discounted cash flow (DCF) model. In particular I will make the following assumptions:

1) The asset is discounted at a rate R

2) The asset has annual cash flows for next year of C.

3) The cash flows grow perpetually at a constant rate G. 

4) The growth rate G is less than the discount rate R.

From these assumptions, the price of the asset is given by P:

P = C / (R - G)

As a quick example, suppose you have an asset that produces $1 in cash flows, discounted at a rate of 10% and growing at a rate of 4%. The resulting price is then:

P = $1 / (10% - 5%) =  $20

Obviously there is already risk in using this model. After all, no series of cash flows grow at a constant rate perpetually. But it's a half-way decent model for mature well established firms or even better for modeling stock indices. What we'll focus on is, given the model, how sensitive is the price (output of the model) to variations in the inputs. To do this we will need to take the total differential (dP, dC, etc, can be thought of as the "change in price" and the "change in cash flows", etc):

dP =  C / (R - G)^2 [dG - dR] + 1/(R-g) [dC]

To show how this works, let's suppose that in our previous example we were certain that the initial cash flows would be $1 (dC = 0) and we are certain that the discount rate should be 10% (dR = 0) but the growth rate is 5% +/- 1%. The above equation will estimate how much a change in the the growth rate effects the price:

dP = C / (R - G)^2 [dG] = $1 / (10% - 5%)^2 [+/- 1%] = +/- $4

In other words simply varying the growth rate by 1% will result in an estimated change in the price of $4. 

Now it's convenient to look at dP/P which tells you the sensitivity of the price with respect to the price in percentage terms. For the above example, dP/P = $4/$20 = 20%. 

We'll do this more generally to derive an interesting result. First we'll assume that the cash flows are known with certainty (dC = 0). This gives us:

dP =  C / (R - G)^2 [dG - dR]

Next, we observe that:

P^2 = C^2 / (R - G)^2.

or:

P^2 / C = C / (R - G)^2

We can then make the substitution:

dP = P^2 / C [dG - dR]

Next we divide both sides by the price P to give us the sensitivity in percentage terms (just as we did above in our example):

dP/P = P / C [dG - dR]

In other words, the senstivity of the price to the uncertainty of the discount rate and growth rate is proportional to the price multiple. Stocks with higher price multiples ("growth") are more sensitive to variations in the inputs than stocks with lower price multiples ("value"). 

It should be noted, of course, that this result was only derived for this particular model and this model might not be appropriate for most growth stocks. But I think it's an interesting result and I haven't seen it anywhere else.