What is the value of a stock (price)? It comes from the earnings it generates! Let us first consider constant earnings and then growing earnings.
Constant Earnings
P = E/r , where P is the price of the stock determined by the earnings potential E.
This formula follows from the definition of the expected rate of return ‘r’, as the capital used to purchase the share is the market price of the stock. For our purposes, we can take this as an annual rate of return.
But to be clear, let us work out the math in a different way.
The earnings E in a future year has a present value that is determined by how much you have to invest now to have a total (capital +profit) of E in the future. This amount works out to be E/(1+r), as if you invested this amount, you would have the original investment and a return of r, a year later. So, if a stock generated E every year, its value would be:
P = E ( present value of year1) + E (pv year 2) +E (pv year 3)……
P = E/(1+r) + E/(1+r)2+E/(1+r)3+….
This can be rewritten to include year 0 as,
P +E= E+ E/(1+r) + E/(1+r)2+E/(1+r)3+….
Using the formula for goemetric progressions with a factor of 1/(1+r),
P+E = E * (1-xn)/(1-x), where x = 1/(1+r)
Since x <1 (as r>0), as n becomes larger, xn approaches zero.
Hence, P = E/(1-x) – E, which simplfies to P = E/r
Growing Earnings
Let us consider earnings E at present and a continuous growth of E by a set amount of E* every year… forever
Then, P = E/r + E*/r2
Let’s discuss this formula and then derive it. Firstly no company grows forever, so we need a terminal valuation. But that consideration aside, we see how in low growth – low interest rate economies like the US at present, growth stocks are very attractive. As r <<1, 1/r becomes large and 1/r2 becomes really large, making the value/price increase greatly. For example, if the rates drop in half, the earnings value becomes double but the earnings growth value becomes four times!
Now to derive, we have to consider the new price with the increased earning:
P = E ( present value of year1) + E (pv year 2) +E (pv year 3)…
P = E/(1+r) + (E+E*)/(1+r)2+(E+E*+E*)/(1+r)3+….
Let’s separate the E terms, then keep the E* terms separate by the year in which the growth of E* was achieved, set x = 1/(1+r):
P = Ex+ Ex2+Ex3+…. (constant earning value)
+ E*x2+E*x3+… (year 1 growth increases earnings in all future years)
+E*x3+… (year 2 growth increases earnings of all future years in present value)
Let’s name these terms in each line above for clarity,
P = Earnings value + Present value of Growth in year 1 + P.V.G in year 2+ P.V.G in year 3…
P.V.G (year 1) = E*x2+E*x3+… = E*x [x +x2+…] = E*x [1/r]
P.V.G (year 2) = E*x2 [1/r]
Adding them all together, Total PVG = (E*/r) [x +x2 +…]
Total PVG = E*/r [1/r] = E*/r2
P = Earnings value + Total PVG
P = E/r + E*/r2
In recent decades, it is such a well established truth, that I won’t even bother to cite references, that value investing has underperformed growth stock investing. This is not necessarily always the case but recent US market history is so skewed in this direction that it is a truth universally accepted that a growing company with a high share price is in need of a higher share price. (Jane Austen reference)
I had the fair opportunity of reading on another blog (and I agree with it) that when growth is prevalent in the market place, value is at a premium and when growth is scarce (like in low GDP growth economies), growth stocks are at a premium. I must also qualify the word growth as “realized growth rate”, since market conditions are dynamic. Else, the price would not appreciate over time (instead the market would price in the future value of expected growth from day 1). But as is the case, the performance of the company has to make a case for a growth trajectory with some proof from its history in past quarters/years. This accompanied by suitable market conditions and a vision from management, is usually necessary for increased share price. In recent decades, the US has seen relatively low GDP growth. This macro condition would explain the great performance of many growth fund managers in the past two decades to a degree.
Cars and Physics 