Valuation of Stocks

Stock ownership produces cash flows from viz dividends and capital gains. The valuation of stock can be done by discounting present value of all future dividends. The different stocks have different growth potential and so there are three type of situations:

– Zero growth of dividends
– Constant growth of dividends
– Differential growth

Zero Growth Stocks

Let us make following assumption

1. Going concern - company will remain forever

2. Investor will remain invested forever

3. Dividend pay-out will remain at the same level forever

4. Perpetual growth remain same

R = discount rate, expected return from the market perpetually (in the asset class of same risk level)

Constant dividend D0 = D1 = D2 = ...

Present Value, PV0 = D1 / (1 + R) + D2 / (1 + R)^2  +...

PV0 = D0 / R

Constant Growth Stocks

Let us now assume that dividend growth at the constant rate, g
D1 =  D0 (1 + g)
D2 =  D1 (1 + g) = D0 (1+g)^2
....

Present Value, PV0 = D1 / (1 + R) + D2 / (1 + R)^2  +...
PV0 = D0(1+g) / (1 + R) + D0 (1 + g)^2 / (1 + R)^2  +...

PV0 = D0 (1+g) / (R - g)
PV0 = D1 / (R-g)

When company do not give the dividend this model of prediction cannot be applied.

Differential Growth Stocks

Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter. To value a differential growth stock:
– Estimate future dividends in the foreseeable future
– Estimate the future stock price when the stock becomes a constant growth stock
– Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.

As an example, let us say in initial n years dividend growth rate are g1, g2,.. gn. And after nth years, it remains constant for perpetuity with value g.

D1 =  D0 (1 + g1)
D2 =  D1 (1 + g2)
....
Dn = Dn-1 (1+gn)
Dn+1 = Dn (1+g)
Dn+2 = Dn (1+g)^2
....

Present Value, PV0 = D1 / (1+R)+ D2 (1+R)^2 + ... Dn / (1 + R)^n + Dn (1 + g) / (1 + R)^(n+1)  +...

Now let us assume that for initial n years, there is a constant rapid growth rate of dividend, say gr, then

g1 = g2 = g3 ... = gn = gr

And, after n years, there is continuous slower growth rate of dividend, say g = gc

then,

Present Value at after n years till infinity, 

PVn = Dn (1+gc) / (R - gc)

PVn = D0 (1+gr)^n (1+gc) / (R - gc)

Hence, present Value at 0 till infinity,

 PV0 = D0 (1+gr) / (1 + R) + D0 (1+gr)^2 / (1+R)^2 + .... D0 (1+gr)^n/(1+R)^n + PVn / (1+R)^n

This model is commonly referred as Two State Growth Model.

Estimation of Parameters

The value of a firm depends upon its growth rate, g, and its discount rate, R. 

Where does g come from?

g = Retention ratio × Return on retained earnings

Where does R come from?

R = D1 / P0 + g = D0 (1+g) / P0 + g

i.e. the discount rate can be broken into two parts viz  dividend yield and growth rate on dividends. In practice, there is a great deal of estimation error involved in estimating R.

Price-Earnings Ratio

One should not discount earnings to obtain price per share because part of earnings must be reinvested. Only dividends reach the stockholders, and only they should be discounted to obtain share price. 

Price per share = EPS / R + NPVGO

Price to earning ratio is a functions of three factors:
1. Per-share amount of the firm's valuable growth opportunities
2. the risk of the stock
3. the type of accounting method used by the firm.

Capital Asset pricing model is only applicable for the perfectly "diversified portfolio" i.e. no non-systematic risk and only systematic risk

Beta captures the systematic risk or return

Alpha captures the unsystematic risk  factor