Stock Valuation Professor Thomas Chemmanur. 2 Stock Valuation Common stock represents ownership of the firm: stockholders elect the board of directors

Stock Valuation

Professor Thomas Chemmanur

2

Stock Valuation

Common stock represents ownership of the firm: stockholders elect the board of directors who control the firm by overseeing management.

Stockholders are also entitled to the cash flows generated by the firm.

Stockholders Board of Directors Management

However, stock holders are residual claimants: they get only the cash flow left over after paying interest to debt holders and other claimants.

At this stage, we will assume that the firm is 100% equity financed, postponing a discussion of the impact of other claims till we talk about the firm's capital structure choice.

3

Stock valuation: constant dividend growth model

Consider a firm which pays dividends D1, D2, ...etc for the first H periods. Let PH be the price of the stock at t = H. Then, from our present value class, we know that the current market price of the stock, P0 is given by,

Where r is the discounting rate which corresponding to the riskiness of the stock. But we know that PH is given by,

31 20 2 3

(1) ... (1 ) (1 ) (1 ) (1 )

H HH

DD D D PP

r r r r

1 22

(2) ...(1 ) (1 )

H HH

D DP

r r

4


Using (2) in (1) and simplifying,

The problem in using (3) is that, to begin with, we don't know what the dividend stream is! Let us make the simplifying assumption that dividends grow at a constant rate g. Then, if D0

was the dividend in the last (current) period,

D1 = D0 (1 + g); D2 = D1 (1+g) = D0(1+g)2; D3 = D0(1+g)3, etc.

Using this assumption in (3),

31 2 1 20 2 3 1 2

(3) ... ...(1 ) (1 ) (1 ) (1 ) (1 ) (1 )

H H HH H H

DD D D D DP

r r r r r r

2 3

0 0 2 3

1 (1 ) (1 ) (1 ) (1 )(4) 1 ... , (1 ) .........

1 (1 ) (1 ) (1 ) (1 )

HH

H

g g g g gP D r

r r r r r

5


The term in the square brackets is an infinite geometric series of the form I talked about in the class on present values, with common ratio (1+g)/(1+r). Applying the formula for the sum of an infinite geometric series, (4) reduces to:

(5) is often referred to as the constant growth formula for stock valuation. The above formula can be used only when r > g, since otherwise the common ratio (1+g)/(1+r) of the geometric series is greater than 1, and P0 .

0 10

(1 )(5)

D g DP

r g r g

6


(5) is true only if the underlying assumption of a constant growth rate in dividends is true (which is often not the case).

We also assume that we know r and g. If we do not know g, we have to make additional assumptions.

Example:

If g = 10%, r = 14%, D0 = $2 / SH, what is the price?

10

D 2 (1.1) 2.2P = = = = $55 / SH

r - g 0.14 - 0.10 0.04

7

Approximating g from ROE the plow-back ratio

Remember that return on equity (ROE) is an accounting concept.

Unfortunately, accounting numbers represent the past, while in finance, we want to know about future cash flows to value securities. Accounting figures also have the problem that they can differ according to the choice of accounting method). However, they can act as a starting place.

Earnings per share (EPS) = Net Income (obtained from the income statement)/Number of shares

Book Value per share = Book value of equity (obtained from the balance sheet)/Number of shares.

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Approximating g from ROE the plow-back ratio

Return on equity (ROE) = EPS/Book Value Per Share Dividend payout ratio = Dividend per share/EPS Plow-back ratio = 1 - Dividend payout ratio. Then g ≈ Plow-back ratio x ROE, assuming that the firm's

earnings re-invested in the company earn the same return as in the past. (Again, it goes without saying that this assumption may not always be true).

9

Finding r given P0, D0 and g

From (5),

Again assuming constant dividend growth. We will now look at the situation when dividend growth is not

constant.

1

0

(6) D

r gP

1 10

0

- -

D DP r g

r g P

10

Stock Valuation with different growth rates in dividends

Often, the assumption of a constant growth rate in dividends is not at all realistic.

The dividend may grow at a certain rate for the first few years, and then settle down to a steady rate afterward.

To illustrate how to price equity in this case, consider the following problem.

Problem 1: A company's dividend grows at 10% rate for the first five years, and thereafter settles down to a steady growth rate of 6%. If stockholders expect a rate of return of 14% from investing in the equity, what is its share price? D0 = $2 per share.

11

Stock Valuation with different growth rates in dividends

First note we can rewrite the above stock valuation formula as:

Where g1 = high growth rate for the first “n” years

g2 = steady growth rate afterwards, in perpetuity

Here:

g1 = 10% = 0.1 g = 6% = 0.06

D0 = $2 / SHARE r = 14% = 0.14

n = 5 YEARS

0 10 1

(1 ) 1 1(7)

(1 ) - (1 )

tn n

t nt

D g DP

r r g r

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Problem 1: Solution

PV of dividends for the first 5 years:

= 2 (1.1) (0.87719)

+ 2 (1.1)² (0.76947) + 2 (1.1)³ (0.06749)

+ 2 (1.1)⁴ (0.59208) + 2 (1.1)⁵ (0.5193)

= 1.93 + 1.86 + 1.80 + 1.73 + 1.67

= $8.99 Dividends at the beginning of year 6:

= D6 = 2 (1.1)6 = 3.41

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Problem 1: Solution

PV of dividends starting year 6:

= $22.14 Total price of the stock = 22.14 + 8.99 = $31.13

Timeline:

t = 0 1 2 3 4 5 6 7

D0 D1 D2 D3 D4 D5 D6 D7

60 5 5

D 1 3.41 1P =

r - g (1 + r) 0.14 - 0.06 1.14

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Stock Valuation from the NPV of Growth Opportunities

Sometimes, it may be easier to value equity as the sum of two components: (a) The value of a share if the firm does not make any future investment, and therefore pays out all earnings as dividends (b) The NPV of future growth opportunities.

(a) If the firm does not make any investment, let us assume that it will earn EPS1 next year, and continue to earn the same

amount for ever. In this case, its value will be EPS1/r, since the

earnings stream will be a perpetuity.

(b) If the firm does make the investment, it will plow back a fraction of EPS1 into the firm. In that case, it will obtain

additional earnings from this new project. Denote the NPV of this new investment per share by PVGO.

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Stock Valuation from the NPV of Growth Opportunities

Then, price per share,

Re-arranging (8) we get,

The importance of (9) is that it gives a relationship between the earnings-to-price ratio of a company and its market capitalization rate r (rate of return expected by investors from the equity). When PVGO is zero (i.e., when the firm has no growth opportunities) both are the same.

10(8) ( ) ( )

EPSP a b PVGO

r

1

0 0

(9) 1EPS PVGO

rP P

1

0

(if PVGO = 0)EPS

rP

16

Problem 2

Consider a firm with existing assets that generate an EPS of $5. If the firm does not invest any further, EPS is expected to remain constant at this level.

However, starting next year, the firm has the chance to invest $3 per share a year in developing a newly discovered geothermal steam source for electricity generation. Each investment is expected generate a perpetual 20 % return. The geothermal source will be fully developed by the fifth year.

What will be the stock price and earnings-price ratio assuming investors require a 12% rate of return?

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Problem 2 - Solution

First compute PVGO The firm has five different projects, each requiring $3

investment, and earnings = 3*0.20 = $0.60 per year forever (in perpetuity).

NPV of each project, at r = 12%

Timeline:

t = 0 1 2 3 4 5

$2 $2 $2 $2 $2

$0.60 - 3 = $2/PROJECT

0.12

PVGO = $2 [PVIFA 12%, 5YRS]

18


PVGO = $7.21

10

EPS 5P = + PVGO = + 7.21 = $48.88

r 0.12

1

0

EPS 5 = = 0.102 or 10.2%

P 48.88

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Problem 3

Consider a firm with the following data: D1 = $10 per share; dividends are expected to grow by 5% a year; the company plows back 20% of earnings.

(i) What is the price of a share?

(ii) What is next year's expected earnings?

(iii) What is the return on book equity?

(iv) What is PVGO? (r = 15%).

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g = 0.05 D1 = $10/SH

PLOW-BACK RATIO = 0.2

= 1 – 0.2 = 0.8

Return on Book Equity: Book Equity increases as earnings are plowed-back (at the same

rate as dividends, assume)

0.2 (12.50) = 0.05 (BV/SH)

(Increase in BV)

1 11

1

D D 10 = 0.8 EPS = = = $12.50

EPS 0.8 0.8

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Problem 3 – Solution (Contd)

PVGO = 100 – 83.33 = $16.67 / SHARE

0.2 (12.50)BV = = $50 / SH

0.05

1EPS 12.50ROE = = = 0.25 OR 25%

BV 50

1D 10PRICE PER SHARE = = = $100 / SH

r - g 0.15 - 0.05

10

EPSPVGO = P -

r

1EPS 12.50 = = $83.33/SH

r 0.15

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Stock Price as the PV of Free Cash Flow / Share

We can also express the price per share as:

where Free cash flow = Revenue - Costs - Investment. However, free cash flow per share is the amount that the firm is free to pay out as dividends per share. Therefore (10) is essentially identical to (3).

0

Free cash flow per share at date t(10)

(1 )1

t

t

Pr

t

23

Price to Earnings Ratio

The P/E ratio is available in financial newspapers. It is the reciprocal of EPS/Price ratio

Used for valuation by comparables, i.e.

Value of equity

= (Earnings of firm to be valued) x (P/E of similar firms) From (9), we can see that the P/E ratio of a company will be

higher if (a) its r is low or (b) it has higher growth opportunities (PVGO is high).

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Stock Valuation Professor Thomas Chemmanur. 2 Stock Valuation Common stock represents ownership of the firm: stockholders elect the board of directors