# How to evaluate companies with different rate of growth rate?

I'm trying to do a value analysis within a group of companies with very different growth rate, here are some method I've explored:

1. P/E ratio. By this measurement most top value companies are those with declining revenue (which make sense why they have a low PE). Example is WDR in 2017, which has a low P/E of 10 but revenue is declining for three consecutive years.

2. PEG ratio. This works for high growth companies but for very stable business like Walmart the PEG ratio doesn't work.

Are there any recommendations, preferably with a link to a research paper, about the consistent method to evaluate these companies?

Thank you!

## 1 Answer

The annuity formula. It's the best and easiest way to approximate a discounted cash flow analysis (DCF) using minimal data. PE is a decent "first" approximation that compares cash flow to price, but totally ignores the growth component. The PEG is a convenient tool for standardizing PEs to growth rates. PEG, however, has no mathematical basis in the time value of money principle (and/or no-arbitrage theory of pricing model) and it goes totally asymptotic (i.e., meaningless) for a large range of values for P, E, and G.

Although imperfect, the "gold standard" of valuation methods is still the DCF analysis. Involved DCF analyses can become very complex and ornate, and still don't solve the "forever" problem -- it is common therefore to just assume an exit multiple of EBIT or EBITDA. Moreover, with increasing complexity, the risk of utilizing garbage data increases as does the likelihood of getting a garbage result (i.e., garbage-in, garbage-out). Fortunately, you can approximate a DCF pretty closely and with far less complexity if you utilize the annuity formula given only an initial cash flow, a growth rate, and a discount factor. Moreover, you can compare the resultant expected present value of an annuity to its market value, much like the PE and PEG ratios.

Working with continuous time is easiest -- it is also usually sufficient for modeling purposes. If we assume that an investor receives continuously accruing annualized cash flows, $C_t$, which grow at a continuous annualized rate, $g_t$, then the present value of those cash flows is given by:

$V_t = \int_t^T C_t e^{g*t} \frac{1}{e^{r*t}} = \int_t^T C_t e^{(g-r)*t}$

where:

$r$ is the force of interest

For a finite $T$, this becomes:

$V_t = C * (e^{T(g-r)}-e^{t(g-r)})\frac{1}{g-r}$

[EDIT: intial formula had error; apologies]

As $T$ goes to $\infty$, this becomes the perpetuity formula:

$V_t = C * (e^{-r*t})\frac{1}{r-g}$

Note that $t$ is usually set to $0$ -- a thing to the $0^{th}$ power equals $1$.

Also note that the perpetuity will go asymptotic to $\infty$ if the growth rate exceeds the force of interest rate. Fortunately, you mix and match any combination of finite and infinite annuities which achieves your desired end-state. For a growing annuity, it is often reasonable to sum the values of an annuity (initial growth period) plus a discounted perpetuity (terminal value), i.e.,:

$$V_t = (C * (e^{T(g-r)}-e^{t(g-r)})\frac{1}{g-r}) + (C*e^{(g-r)*T}\frac{1}{r+d})$$

where $d$ now equals the terminal exponential decline rate of $C$ starting at time, $T$.

If future cash flows are known and discrete, more accurate results can be had by using discrete time models. There are plenty of discrete annuity models to choose from... just be careful on the difference between annuities and annuities due.

As far as references, I'll just point you to Aswath Damodoran: http://seekingalpha.com/article/4027440-myth-5_1-believe-forever-dcf. He says basically the same things.

• Thank you David for taking the time providing so much details! It's very helpful, appreciate it! – Hao Mar 8 '17 at 0:59
• You're welcome. I have to apologize, though. I made an error in my initial answer. I gave the formula for the future value of an annuity. You usually want the present value for relative valuation purposes. – David Addison Mar 8 '17 at 2:33