VALUING A STOCK

·        Stock price = PV of all future dividends

P₀ = D₁/(1+r) + D₂/(1+r)² + D₃/(1+r)³ + D₄/(1+r)⁴ + . . . (1)

r is the required return or market capitalization rate

·        Even though it is expressed as the PV of dividends alone, it includes both dividends and capital gains

·        Consider a stockholder who plans to hold the stock for three years.

Her cashflows will be D₁, D₂, D₃ & P₃ (P₃ reflects her expected capital gains)

She could compute stock price as the PV of her cashflows:

P₀ = D₁/(1+r) + D₂/(1+r)² + D₃/(1+r)³ + P₃/(1+r)³          (2)

But P₃ itself is the PV at time 3 of the future dividends (time 4 onwards)

P₃ = D₄/(1+r) + D₅/(1+r)² + D₆/(1+r)³ + . . .                   (3)

If you substitute equation 3 in equation 2, this will just give equation 1

=>

·        even a s/h who plans to hold the stock only 3 years can compute the stock price as the PV of all future dividends

·        D₁ through D₃ are the dividends she receives; D₄ onwards determine her capital gains

CONSTANT GROWTH STOCK

Assume dividends grow at a constant rate g forever

·    Dividends are a growing perpetuity

·    P₀ = D₁/(r-g)

·    r = (D₁/ P₀) + g   

First term is dividend yield; second term is capital gains yield, i.e. rate at which the stock price grows => when dividends grow at g, price also grows at g.

Also, this equation can be used to estimate required return without using CAPM

Need only:

·     expected dividend yield (based on next dividend, nor previous one)

·     estimated long-term growth rate: analyst use analyst forecast or compute as

g = plowback ratio * ROE

(if you earn 20% on equity, and reinvest 40% of that, then equity grows by .2 * .4 = 8%.   Earnings and dividends will grow at same rate.)

·        Example:  Ptennisnet Inc.’s EPS last year was $2.50.  They pay out 60% of their earnings, and their return on equity is 16%.  If the required return is 14%, 

      a) what is the stock price today? 

      b) verify that dividend yield + growth rate equals r

NON-CONSTANT GROWTH STOCK

Not always reasonable to assume dividends will grow at a constant rate forever.  Even then we assume that dividends will grow at a constant rate beyond some future date.

·        All divs = first-stage divs (not a growing perpetuity) + second-stage divs (growing perpetuity)

·        Use growing perpetuity formula to compute value of second stage divs one year before they start

·        Discount this value, along with first stage dividends, back to time zero

·        Example: Company will pay no dividends next 2 years.  At time 3, they will pay a dividend of 50c.  For two years dividends will increase at 12%; and then they will settle down to a constant growth rate of 7%.  If the required return on the stock is 10%, what’s the stock price today?

o       D₃ = 0.50

o       D₄ = 0.50*1.12 = 0.56

o       D₅ = 0.56*1.12 = 0.6272

o       D₆ = 0.6272*1.07 = 0.671104

o       PV₅(D₆ onwards) = 0.671104/(.1 - .07) = 22.37

o       P₀ = 0.50/1.1³ + 0.56/1.1⁴  + (0.6272 + 22.37)/1.1⁵ = 15.04

·        Note that PV₅(D₆ onwards) is just P₅.  This is no different from what is called “horizon value” in section 4.5.  We’re using a 5 year horizon.  Stock price today is PV₀ of dividends over the 5-year horizon + PV₀ of the horizon value at the end of the 5 years.

·        There’s no simple formula for estimating required return like we have for a constant growth stock. Given expected future dividends and today's stock price, we can still estimate required return, though, by doing an IRR type trial-and-error computation.

VALUING A BUSINESS BY DISCOUNTING FREE CASHFLOW

o       The approach we use to value a non-constant growth stock can also be used to value the firm as a whole

o       When we use it for stock valuation, we compute PV of all future dividends

o       When we use it for valuing a firm, we compute PV of free cashflows

o       Not a per share computation; numbers used are total earnings etc.

o       Free cashflow = net cashflow after investment

o       negative FCF on a given date => new shares issued on that date to raise capital to finance investment (we assume for now all new capital is equity)

o       positive FCF on a given date => dividend paid out on that date (if new shares were issued in the past, part of these dividends go to those new s/h)

o       growing firm will typically have negative FCF to start with and positive FCF later

o       The nature of the computation is exactly the same as what we did for non-constant growth stock:

       Value of firm = PV₀ of first stage FCF (taken individually) + PV₀ of second stage FCF (growing perp. formula)

o       Computing the value of the firm and dividing by the number of shares outstanding today gives the current stock price

STOCK PRICE, PVGO AND P/E RATIO (FOR CONSTANT GROWTH STOCK)

·        Firm A generates $10 a year forever from its existing assets.  Has no investment opportunities.  Pays out all its earnings as dividends each year.  Required return is 10%.

o       Dividends are $10 forever

o       P_o = PV of dividends = 10/.1 = $100

o       No growth in dividends or stock price

·        Firm B is the same firm (still has no positive NPV investments) but it decides to achieve growth by paying out only half its earnings, and reinvesting the other half in zero NPV projects

o       Will earn 10% on the re-invested earnings (zero NPV => E(R) on investment = required return)

o       D₁ = $5

o       g = plowback ratio * ROE = 0.5 * .1 = 5%

o       P_o = 5/(.1 – 0.05) = $100

=> Stock price still consists of value of existing assets.  Investing in zero NPV projects gives you growth in dividends/earnings.  But this growth is meaningless.  Does not increase stockholder wealth.

o       Form C is the same firm, but now it has managed to come up with positive NPV investments.  Existing assets still generate $10; each year it still pays out half its earnings and reinvests half.  But now it earns 15% on re-invested earnings

o       D₁ = $5

o       g = plowback ratio * ROE = 0.5 * .15 = 7.5%

o       P_o = 5/(.1 – 0.075) = $200

o       Existing assets are the same; where does the extra value come from?

o       Stock price has two components:  P_o = PV of existing assets + PVGO

o       PVGO is the PV today of the positive NPV investments the firm is expected to make in the future

With no investment or zero NPV investment, the stock price just equals the PV of existing assets.  It is only when you invest in positive NPV projects that PVGO turns positive and increases stock price 

=> growth is meaningful only when it comes from positive NPV investment.  Only then does growth increase the wealth of stockholders. 

·        We can confirm that PVGO is just the PV today of the NPV expected to be earned in the future:

NPVs in the last column are growing at 7.5%.  

Thus firm C will earn a stream of future NPVs which constitute a growing perpetuity, starting with $2.50 and growing at 7.5%.  PV today of this stream of future NPVs = 2.50/(.1 - .075) = $100

·        Growth and P/E ratios:

     Assume that firm does not make zero NPV investments, and that without new investments EPS generated by existing assets would be a perpetuity

o       PV of existing assets = EPS/r

o       P₀ = EPS/r + PVGO

o       EPS/ P₀ = r(1 – PVGO/P₀)

o       P/E ratio is high when earnings-price ratio is low; that happens when required return is low or firm has valuable growth opportunities (PVGO is positive)

o       Thus a firm’s P/E ratio reflects a) its systematic risk, and b) its growth opportunities

o       If the market believes you have better growth opportunities in the future, then your PVGO goes up and your P/E ratio goes down.  Hence your P/E ratio reflects what PVGO the market believes in for you.

o       If you want to value a stock by taking its EPS and multiplying by the P/E ratio of a “similar” firm, you have to come up with a firm that has the same systematic risk (easy) and the same growth opportunities (not so easy)

STOCK PRICE DECOMPOSITION FOR NON-CONSTANT GROWTH STOCK