# Basic Investment Formulas

In order to understand the explanation we will first have to start from some basic equations that are used in company valuation. These equations are always only models of the reality that are only true if certain hypothesis’ are accomplished – which is sometimes very restrictive -. In other words, these models cannot be taken as a given fact, but they certainly do give us valuable information in order to see what factors and variables influence on the price of a share. To make the explanation a little simpler we will start from the following data of a hypothetical company XYZ.

- Company benefit (B). Let’s suppose this is of 20 million dollars.
- Equity (E). Let us suppose this is of 100 million dollars.
- Earning power over Return on Equity (ROE). This is the quotient between the benefits and return on equity. ROE = B/E. In our case the ROE is of 20 for 100 (20/100).
- Number of shares in circulation. Let us suppose there are 10 million shares.
- Basic earnings per share (EPS) will be of 2 dollars per share.
- Pay out (P) is the percentage of benefits destined to dividends. Let us suppose it is of 50 for 100, therefore, XYZ Company pays 10 million dollars in dividends, equivalent to 1 dollar of dividend per share (DPS). In other words, DPS = EPS x P. in our case, DPS = 2 dollars x 0,5 = 1.
- Growth rate of the dividends (g). We will imagine the dividends grow at 10 for 100 annually (g = 10 for 100). In theory and supposing the company does not change their financial structure (or does not become more indebted), this growth depends on the Return on Equity (ROE) and the pay out. In other words, if the company is very profitable, it generates a lot of expenses and can grow a lot; on the other hand, if almost everything it gains is distributed as dividends (high pay out) then it will not be able to grow a lot. The definition that defines sustainable growth is: g = ROE x (1 – P). And if we clear P we will see that the pay out depends on the ROE and of g since p = 1 – g/ROE. In our example, the results would look like this:

G = ROE x (1 – P) = 0,20 x (1 – 0,5) = 0,1 and also,

P = 1 – g/ROE = 1 – 0,1/0,2 = 0,5

If XYZ Company had the desire to grow more, without increasing their debt, they would have to increase their ROE or diminish their pay out of percentage of undivided profit as dividends. - We are supposing the profit (K) the investors ask from a share of XYZ is of 15 for 100. Since we know this K has two components, the risk free profit or market interest rate and the risk premium or extra yield above the risk free profit that the investors ask from the XYZ share due to the risk it has. We will suppose that both are of 5 for 100 and 10 for 100 respectively.
- The theoretical price of the share is given by the following formula:

P = DPS

(K –g)

This would only be in the case that we hold on to the share for many years and that the financial structure of the company does not change. This is the known about formula of Gordon Shapiro of the dividend discount that has a lot of restrictions but it can also be used to shed some light. In our case we will say the XYZ share has a price of, P = 1/ (0,15 – 0,1) = 20 dollars. - In point 6 we have seen that the DPS = EPS x P and in point 7, that P = 1 – g/ROE. Therefore, DPS = EPS x (1 – g/ROE). We can substitute in the equation of point 9, the DPS by the expression we have just obtained and will have:

P = EPS x (1 – g/ROE) over (K – g)

- If in the example of point 10 we pass the EPS to the left side of the equation we will obtain the PER of the share of the company since the P/EPS is equal to the PER, by definition.

P over EPS = 1 – g/ROE over (K – g) = PER

- Before tiring your eyes we will try to get to the conclusion. For this we will observe that the denominator of the previous explanation 11 can be transformed in the following way: (K – g) = K (1 – g/k). Substituting this last example in the formula we are left with:

PER = 1 over K x 1 – g/ROE over 1 – g/K = 1 over K x F

Therefore we see that the PER depends in the first place on the earning power the shareholders (K) ask for and in second place of a factor F that contains the growth (g), the ROE and again the K. Let’s see now what interpretation this complicated formula has.