Stock Valuation

Stock Valuation

Stocks are infinitely lived assets

They pay dividends

The residual claim on the firm i.e. stockholders get dividends after bondholders get coupons.

Dividends are discretionary

To value the stocks, we estimate the present value of expected dividends for the life of the firm.

A closely related concept is the free cash flow to equity valuation model.

Free cash flow is defined as the cash flow generated by the firm that could have paid out as a dividend after funding required capital expenditures and making required payments to other claimants.

A group of models exists that are collectively referred to as DCF models. Theoretically, all of these models should provide an identical estimate for the stock price if properly estimated.

Present value computations

Consider an equity of $d per period. We know how to discount each period back by calculating the price of a zero coupon bond. So, the present value of an n period annuity can be calculated by summing the discounted cash flows. Recall that earlier we developed a formula for computing the value of an n period annuity.

A = a * ( z * ( 1 - zⁿ ) / ( 1 – z ) )

For a stock we need to make three modifications.

Infinite life

Discounting a dividend cash flow which is a decision variable, not a coupon cash flow.

Stocks are more risky than bonds- default and residual claimant.

So a general formula for pricing the stock will be

P = d₁z¹ + d₂z² + d₃z³…

Z1 = 1 / ( 1 + k ) , z² = 1 / ( 1 + k ) ² and so on…

We must be careful using k instead of r.

K is market capitalization rate.

Simplifying assumption

Dividends grow at a constant rate for ever

P = do ( 1 + g ) / ( 1 + k ) + do ( 1 + g )² / ( 1 + k )² .

Take x = ( 1 + g ) / ( 1 + k )

We get P = do * x / ( 1 – x )

Substitute and get P = do( 1 + g ) / ( k – g ) = d1 / ( k – g )

K > g otherwise this stock has greater returns than the required rate of return.

If not, this would drive the price of the stock up and the price of the other stocks down.

Price earning ratios and constant growth

Assumptions

Dividend payout ratio is constant

dt = a * E

Where E is earning in period t and a is the payout ratio. So the firm pays out a constant proportion of their earnings as dividends.

Earnings grow at a constant rate g forever

Starting with the constant growth model

P = d1 / ( k – g )

Substituting in for d1 = a * E

P = a * E / ( k – g )

Now we can write the price earning ratio of the form as a function of the dividend payout ratio, growth rate in earnings and the required rate or return.

Divide both sides by E1

P / E1 = a / ( k – g ) = ( d/ E ) / ( k – g )

Limitations of constant dividend growth model

*Unlikely that the firm dividends grow at a constant rate. Most firms have rapid growth at beginning followed by normal growth

*Some firms pay no dividends followed by a policy of rapid dividends.

*K may be time dependent. That is a firm may change their business mix.

According to standard dividend discount model, a theoretical stock price is computed by discounting the stream of all future dividends at the market rates.

P = D1/( 1 + k ) + D2/ ( 1 + k)² + D3/(1 + k )³…

Where D1 = dividend at time 1

If dividends grow at constant rates g

P = D1 / ( k – g )

E1 = r B_i-1

Beta = ( 1 – alpha )

Bi = beta ( 1 + beta r ) ^I Bo

Ei = r( 1 + beta r ) ^i-1 Bo

Since Di = alpha r ( 1 + beta r ) ^i-1 Bo

Di = ( 1 + beta r ) ^i-1 D1

.g = Beta r = ( 1 – payout ratio ) ROE

P = alpha E1 / ( k – g )

Theoretical P / E = alpha / ( k – g )

P / E = ( 1 – beta ) / ( k – r beta )

P / E = ( 1 / k ) * ( 1 + beta ( r – k ) / ( k – r beta ) )

P = E / k + (r – k ) * ( g ) * ( B o ) / ( (k) * ( k – g ) )

Present value of growth opportunities

Pso = E1/ k + PVGO

An,k = E1 ( 1 – ( 1 + k ) ^–n ) / ( k )

In erms of formula

Pso = E1/ k + PVGO

E1/ P = k ( 1 – PVGO / P)

PVGO = ( gP – ( 1 – a ) E1 ) / k

Issues in valuation

Dividend discount in real terms

Vo = d1 / ( k – g )

We can separate real parts and inflation

Vo* = d1* / ( k* - g* )

Dividend forecasting and macroeconomic information

A structured way of pricing the dividends is as follows

First project the real GNP of a particular country for a number of years.

Second, determine the historical relationship between a company’s real assets and real GNP.

Project earnings given your GNP forecasts.

Third make an assumption about payment policies of the firms that you are examining. You can determine what the expected dividends will look like. Discount these back to the present – using the real required return for the firm.

Forecasting holding period returns.

It is quite difficult to forecast holding period returns for any investment. However these are some ways to get approximation to the holding period return. We will examine some simple ways to forecast the holding period return for the stocks and the bonds.

Stocks

A simplified approach will invariably make simple assumptions.

V1 = d2/ ( k – g ) = d1*( 1 + g ) / ( k – g ) = d1 * ( 1 + g ) / ( k – g )

V1 = Vo * ( 1 + g )

HPRcg = ( V1 – Vo ) / Vo = ( Vo * ( 1 + g ) – Vo ) / Vo = g

HPR = HPR^cg + d1/ Vo = g + d1 / Vo

Bonds

Forecast of holding period return on bond will be linked to our estimate of what interest rates looked like in one year.

First the modified duration of the bond must be estimated. Second, make a forecast of interest rates.

HPR^cg = - Modified Duration * Predicted Rate changes.

Finally, we need to allow for the expected coupon payment over the year that we are holding the bond.

HPR = HPR^cg + c / B

So the holding period return is the sum of holding period capital gain plus the coupon yield.