# Price Earning Ratio

Most people know price earning ratio (P/E ratio or PE) and use it a way of value whether the stock is over-valued or not.

$PE=\frac{company\ market\ cap}{earning}=\frac{price\ per\ share}{earning\ per\ share}$

This number tells you how many years are required to earn your investment back (via earning) For example, usually $$PE\approx 15$$ is a standard for a fair value company. However, PE doesn’t consider the growth of company or the cost of your money.

The cost of money considers having x money now is more valuable than having the same amount of money in the future. The could result from the currency inflation, or you could invest this money in some almost safe way (e.g., US treasury anual return is about 1% now and was 4% before) and get more money back in the future.

# Price Earning Ratio for Growth Company

Let us consider this value for a growth company. Let assume the anual growth rate is x and cost of your money is y. We also starts with assumption that x and y is constant, which is unlikely to be true in real life.

To be able to profit from investment in T years, we should have:

$(1+y)^t\leq\sum_{t=1}^T{\frac{1}{PE}(1+x)^t}$

where PE is the PE of the invested company now. That means

$PE\leq\sum_{t=1}^T{(\frac{1+x}{1+y})^T}=\frac{1-(\frac{1+x}{1+y})^T}{1-\frac{1+x}{1+y}}$

Here we consider some examples here. Here I use US treasury bond long term anual return for y (4%) and gross profit year over year growth for x.

Company Current PE Growth x Years Required Value
Amazon 94 30% 14.3 Fair
Apple 35.67 10% 19.9 Slightly Over
Tesla 1334 50% 17 Fair

Again, I have assumed the company’s gross profit grows constantly in those years, which is unlikely to be true.

Here is the code

def year_of_return(PE, x, y=0.01):
ratio = (1 + x) / (1 + y)
if 1 + PE * (ratio - 1) <= 0.0:
return math.inf
return math.log(1 + PE * (ratio - 1)) / math.log(ratio)


# How Low Interest Rate Affects Valuation

People have observed the PE for the stocked market has increased significantly since 2020. To my understanding, this is because the low interest rate reduces the cost of your money (y). Let us check how it affects using the equation above with y=1%.

Company Current PE Growth x Years Required (y=4%) Years Required (y=1%)
Amazon 94 30% 14.3 13.2
Apple 35.67 10% 19.9 16.8
Tesla 1334 50% 17 16

Obviously, as the cost of your money reduced from 4.4% to 1%, the number of years required to get your investment back get reduced.

We could compute what will be the starting PE for fair value (assuming T=15).

Company Growth x Actual PE Fair Value PE for y=1% Fair Value PE fory=4%
Amazon 30% 94 150 109
Apple 10% 35.67 29 22
Tesla 50% 1334 775 547

It also indicates fair-value PE ratio is affected more by the intereste rate for the higher growth company than the lower growth company.

Here is the code

import math

def pe_from_return(x, t=15, y=0.01):
ratio = (1 + x) / (1 + y)
return (ratio ** t - 1) / (ratio - 1)


# How Growth Rate Affects Valuation

Similar we could compute the required years for return over different growth rate. Here we assume y=1%:

Growth x\Year Required for PE 15 50 100 1000
0% 16.2 68.7 463.8 Inf
10% 9.9 19.9 26.9 52.7
20% 7.8 13.6 17.3 30.4
30% 6.6 10.8 13.4 22.4
40% 5.9 9.2 11.3 18.2
50% 5.3 8.2 9.9 15.6
100% 4.0 5.7 6.7 10.1
150% 3.5 4.8 5.5 8.1
200% 3.1 4.2 4.9 7.0

Or y =4.4%.

Growth x\Year Required for PE 15 50 100 1000
0% 23.2 inf inf inf
10% 11.3 24.9 35.4 76.6
20% 8.4 15.3 19.9 36.0
30% 7.0 11.8 14.8 25.1
40% 6.2 9.9 12.1 19.9
50% 5.6 8.6 10.5 16.8
100% 4.1 5.9 7.0 10.5
150% 3.5 4.9 5.7 8.3
200% 3.2 4.3 5.0 7.1

This table shows how important the growth rate. Again constant growth rate is assumed hered and it is unlikely to be true.

# Reference

Written on January 30, 2021