Some Basic Formulas

1. Benefits of the company (B). We will suppose these are of 20 million dollars.
2. Equity or countable value (CV) that we use in our ratio q. We will suppose these are of 100 million dollars.
3. ROE or Return on Equity. This is the quotient between the benefits and the equity or countable value. ROE = B/CV. In our case the ROE is of 20 for 100 (20/100). Out of the previous equation we defer that B = CV x ROE, in other words, the benefits are equal to the ROE multiplied by the equity or countable value (CV) of the company.
4. Pay Out (p) is the percentage of benefits destined to dividends (DIV). We will suppose that it is of 50 for 100, therefore, XYZ Company pays 10 million dollars in dividends. In other words, DIV = B x p.
5. Growth rate of the dividends (g). We will suppose the dividends grow at 10 for 100 a year (g = 10%). In theory and supposing the company does not change its financial structure and does not become more indebted, this growth depends on the ROE and the pay out. In other words, if the company is profitable, generates a lot of income and can grow a lot; on the other hand if almost everything it obtains is distributed as dividends (high pay out) then it will not be able to grow that much. The formula that defines the sustainable growth is g = ROE x (1 – p). And clearing p we see the pay out depends on the ROE and growth of the benefits (g) according to the formula: p = 1 – g/ROE. In other words, we will pay more or less dividends in function of how much we want to grow. In our example the results would be:
   
   g = ROE x (1 – p) – 0,20 x (1 – 0,5) = 0,1 or 10% and also,
   p = 1 – g/ROE = 1 – 0,1/0,2 = 0,5 or 50%
   
   If XYZ Company wanted to grow more, without increasing their debt, they would have to increase their ROE or decrease their pay out or percentage of benefits distributed as dividends.
    
6. We suppose the earning power (K) the investors ask for from the XYZ Company is of 15 for 100. As we know, this K has two components, the risk free profit or market interest rate and the risk premium or profit above the risk free profit the investors ask the XYX Company for due to the risk that it entails. We will suppose that both are of 5 for 100 and of 10 for 100 respectively.
7. The market value (MV) of the company is given through the following formula:
   
   MV = DIV over (K – g)
   
   This is only in the assumption the company has been functioning for many years and that the financial structure of the company does not change. This is the known of formula of Gordon Shapiro of the Dividend Discount Model (DDM) that has a lot of restrictions but that also sheds a lot of light on things. In our example the XYZ Company would have a market value, MV = 10/ (0,15 – 0,1) = 200 million dollars.
8. In formula 4 we saw that the DIV = B x p and in formula 3 that B = CV x ROE, therefore, DIV = CV x ROE x p. We also saw that in formula 5 p = 1 – g/ROE. Therefore, DIV = CV x ROE x (1 – g/ROE). In the equation 7 we can substitute the dividends (DIV) for the example we have just obtained and will get:
   
   MV = CV x ROE x (1 – g/ROE) over (K – g)
    
9. 
   In order to obtain the ratio q (MV/CV) it is enough to divide the previous explanation the market value gives us, by the countable value (CV). Thereupon we will simplify it by eliminating the CV in the numerator and denominator, and will resolve the parenthesis of our numerator. Let’s take a look at this:

Q = MV over CV = 1 over CV x CV x ROE x (1 – g/ROE) over (K – g) = ROE – g over K – g

By using our example the result is:

q = ROE – g over K – g = 0,2 – 01 over 0,15 – 0,1 = 2

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