Finance 3310

Lecture Notes

Lecture 8

I. Preferred stock

III. Common stock

a) definitions

b) valuation - no growth

c) valuation - growth

PREFERRED STOCK:

• A hybrid between common stock and a bond
• Fixed dividend payment
• Must be paid before common stock dividends.
  
  

Preferred stock involves a fixed or defined dividend payment, forever. Therefore, the cash flow stream to be valued is simply a perpetuity

Dp = dividend

Rp = required return on preferred

Pp = price of preferred

Thus: Pp = Dp/Rp

Note that if we know price and the dividend, then we can solve the above equation for the investor's expected return. The investor's expected return is: Rp = Dp/Pp, which is just the 'dividend yield.'

COMMON STOCK:

Features:

1. Has residual claim

2. Dividends are not promised or guaranteed

3. Returns = dividends, capital gains


(Note: For puposes of these notes, because I cannot create appropriate symbols, assume the following:

Re (Normal font) = required return

Re (Bold) = expected return

Re (Italic) = realized return

Terms: Dt = dividend paid at t

P0 = market price of stock today

Pt = expected price at time t

g = expected growth rate in dividends

Re = required return (comes from where?)

Re = realized return (i.e., actual return)

When we buy a common stock, we usually expect to get dividends and appreciation in price. More specifically, the expected return from holding a stock consists of:

1. Expected capital gain (yield) ==> (P1 - P0) / P0

2. Expected dividend yield ==> D1/P0

Total expected return: (P1 + D1 - P0) / P0

Realized return: substitute actual P1 and D1 for expected

How do we value a common stock?

A common stock is also a stream of cash flows! What are the cash flows? Based on the above equations, the future cash flows consist of D1 and P1, in other words dividends plus a future price. So we could simply write:

P0 =

But where do we get P1? Well, we could just substitute for P1 in the above equation to see that P1 is the present value of D2 and P2. And, if we continually substitute for the P's we see that the value of stock is the present value of all the future dividends. So, conceptually we can just find the present value of all future dividends.

The problem is, dividends continue forever and could follow any path. So we consider three special cases that we can solve for analytically, and then attempt to approximate the true, but unknown, path of dividends with one of these three assumptions. We always assume that the price is 'ex-dividend,' or without the current dividend. So P0 does not include D0.

1. Zero Growth Stock:

==> dividends expected to remain constant forever ==> same as perpetuity

P0 = D / Re

Also, if we know D and P0, we can get calculate our expected return:

Re = D/P0

 

2. Constant Growth Stock:

A second assumption we can make about dividends is that they grow at a constant rate forever. That is,

D_t+1 = D_t * (1 + g)

where g represents the long-run growth rate of dividends. That is, each dividend is g% higher than the previous one. For example, suppose that:

D0 = $1.50

g = 6%

Then: D1 = $1.50 x 1.06 = $1.59.

The interesting question is, what is g, and how do we go about estimating it. Using the same algebra we used previously for a perpetuity we can solve for the present value of this infinite stream of growing dividends. Sparing you the algebra, and assuming g < Re, the result is:

P0 = D1/(Re - g) or, more generally: P_t = D_t+1/(Re - g)

(Why does g have to be less than Re?)

The above model is called the Gordon model. In words, the model says the present value of a constant growth stock at any time, t, is the next dividend divided by (Re - g).

Example:

Suppose D0 = 2.00

g = 6%

Re = 15%

What is P0? -------------> P0 = D1/(Re-g)

D1 = 2.00 x 1.06 = $2.12

P0 = 2.12/(.15 - .06)

P0 = $23.56

Growth:

1) inflation

2) reinvested earnings

3) return on equity (earnings)

For mature companies, sustainable growth rate may be a good estimate.

If we know the current price, dividend and expected growth rate in dividends, then we can solve for the expected return as follows:

Re = D1/P0 + g

= expected dividend yield + expected growth rate (capital gains yield)

Example:

Suppose P0 = $42; D0 = 4.00; g = 5%

What is Re? -----------------> Re = D1/P0 + g

We need D1; we have D0; thus D1 = 1.05*4.00= 4.20

Thus: ----------> Re = 4.20/42 + 5% = 15%

= dividend yield + capital gain

Suppose it is one year later: What is P1 if Re = 15%?

4.00 ----------4.20 ---------- 4.41----------etc

|____________|___________|___________|___________|
0----------------1----------------2---------------3

------------------?

P1 = D2/(Re-g)

P1 = 4.41/(15-5) = $44.10

What is expected capital gain (yield)? (Recall P0 = 42)

(44.10 - 42.00)/ 42.00 = 2.10/42.00 = 5% (growth rate)

What is expected dividend yield?

4.20/42.00 = 10% (constant)

Thus: for a constant growth stock:

1. Stock price grows at the constant rate g
   
2. Dividend grows at the constant rate g
   
3. The dividend yield is constant
   
4. The capital gains yield is a constant g
   
5. The total return = div yield + cap gains yield
   

Note that for a zero growth stock you can substitute 0 for g in the above statements.

3. Supernormal Growth:

Some firms in their early stages go through periods of 'supernormal' growth. (ex, Wal Mart, Apple, Intel, Microsoft, etc.), and then settle down to a constant growth rate. The time path of dividends might appear as follows:

(GRAPH TO BE SUPPLIED IN CLASS. Remember, the notes are not a substitute for class)

How do we find value? Visually we just break the dividend stream into two pieces. The piece to the right of the juncture is a 'constant growth' dividend stream. To the left is a set of individual dividends that we find the present value of one at a time. The trick is knowing which dividend to plug into the constant growth model. Whichever dividend you choose to plug in the model, along with every dividend after it, must be growing at the rate g.

Example: Suppose the following:

Dividend expected to grow at 30% for 5 years

D0 = $3.00 (last dividend paid by company)

Dividend expected to grow at 6% constant rate thereafter

Re = 15%

3.00----3.90-----5.07-----6.59-----8.57-----11.14----11.81-----12.52

|_______|_______|_______|_______|_______|_______|_______|

0 --------1 ---------2 ---------3 --------4 ---------5 ---------6 --------7

The first question would involve where to apply the 'constant growth' model. We cannot start with D4 (8.57) because it grows at the rate of 30% to become D5. We can use D5 (11.14) however, because it grows by 6% to become D6, and so on. We could also start with any dividend after D5, but doing so leaves more individual dividends to discount.

1. Find value at t=4; That is: P4 = 11.14/(15% -6%)= $123.78

2. Now discount this and dividends to t=0.

3. P0 = PV(P4 + 8.57 + 6.59 + 5.07 + 3.90)

4. P0 = $123.78/(1+15%)⁴ + 8.57/(1+15%)⁴ + 6.59/(1+15%)³ + 5.07/(1+15%)² + 3.90/(1+15%)

P0 = 70.77 + 4.90 + 4.33 + 3.83 + 3.39

P0 = $87.22

Consider 4 companies, all with same expected return:

1. Declining growth: negative growth, large dividend yield

2. 0 growth; dividend yield = expected return

3. Normal growth; div yield + growth = expected ret.

4. Supernormal growth; smaller div; larger cap gain

P/E ratio = price / earnings

How much investors will pay for a dollar of earnings. If earnings are growing, they will pay more.

==> high growth co's have higher p/e ratios, on average; higher risk companies should have lower P/E ratios.

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