Stocks&bonds2214 1

Valuation of Bonds and Equities

The value of any business asset depends on its expected future cash flows.

If you buy a bond you are effectively buying a stream of cash flows.

Bond with face (nominal) value of €100: coupon rate of 8% with 5 years to maturity.

If the yield to redemption on this type of bond is 10% at the moment how much should you pay for it?

0 1 2 3 4 5

? 8 8 8 8 108

Get the PV of the cash flows of the bond @ 10%

Annuity of €8 p.a. for 5 yearsPV = 8 X 3.791 = 30.33

€100 at the end of 5 yearsPV = 100 X 0.621 = 62.10

The value of the bond is €92.43

Topics Covered

How To Value Common Stock Capitalization Rates Stock Prices and EPS

Stocks & Stock Market

Common Stock - Ownership shares in a publicly held corporation.

Secondary Market - market in which already issued securities are traded by investors.

Dividend - Periodic cash distribution from the firm to the shareholders.

P/E Ratio - Price per share divided by earnings per share.

Stocks & Stock Market

Book Value - Net worth of the firm according to the balance sheet.

Liquidation Value - Net proceeds that would be realized by selling the firm’s assets and paying off its creditors.

Market Value Balance Sheet - Financial statement that uses market value of assets and liabilities.

Valuing Common Stocks

Expected Return - The percentage yield that an investor forecasts from a specific investment over a set period of time. Sometimes called the market capitalization rate or the cost of capital.

0

011 )()(Return ExpectedP

PPEDivEr


Example: If Fledgling Electronics is selling for $100 per share today and is expected to sell for $110 one year from now, what is the expected return if the dividend one year from now is forecasted to be $5.00?

15.100

1001105Return Expected


The formula can be broken into two parts.

Dividend Yield + Capital Appreciation

Expected Return r Div

PP P

P1

0

1 0

0

What is the value of a share?

)1(11

0 rDP

P

If an investor buys a share it is worth the PV of the future cash flows it gives her. If she plans to hold the share for one year (period). (Note I have dropped E() for convenience.)


ExampleCurrent forecasts are for XYZ Company to pay dividends of $3, $3.24, and $3.50 over the next three years, respectively. At the end of three years you anticipate selling your stock at a market price of $94.48. What is the price of the stock given a 12% expected return?


ExampleCurrent forecasts are for XYZ Company to pay dividends of $3, $3.24, and $3.50 over the next three years, respectively. At the end of three years you anticipate selling your stock at a market price of $94.48. What is the price of the stock given a 12% expected return?

PV

PV

3 001 12

3241 12

350 94 481 12

00

1 2 3

.( . )

.( . )

. .( . )

$75.

Value of Shareholders Funds

222

11

0 )1()1( rPDiv

rDivP

If an investor buys a share it is worth the PV of the future cash flows it gives her. If she plans to hold the share for 2 years the following formula applies.

If the investor she sells to at t2 plans to hold the share for H years its value is :

2HHH

24

13

2 r)(1PDiv...r)(1

Divr)(1

Div

P

Substituting this into the previous equation gives:

Substituting this into the previous equation gives

P Divr

Divr

Div Pr

H HH0

11

221 1 1

( ) ( )...

( )

This logic can be applied to the investor who buys the sharein year H and so on terminal value PH is so far in the future that it can be until the ignored. Thus, the value of a share is theoretically equal to the PV of all the futuredividends discounted at the cost of capital

E

VdV

110

EE

VdV

110

E

VdV

221

22

221

EEE

Vdd

E

VdV

32

3

33

33

221

EEEE

Vddd

44

4

10

Et

tE

t TVdV

...........VdVE

443

1t

tE

td...

The Dividend Discount Model:

Let (1+r) = ρE

Assume an investor holds a share for one year and sellsTo another investor who also holds the share for 1 year..

The Basic Dividend Valuation Model is:

(PVED) R

]d[E = PF

+tt

=1t

~

The value of a share is the present value of all the dividends that It pays to infinity.


Dividend Discount Model - Computation of today’s stock price which states that share value equals the present value of all expected future dividends.

Making the Basic DDM practical

If we assume all dividends are the same forever this implies we forecast no growth and will then value the stock as a PERPETUITY.

rDP 1

0


rEPSor

rDPPerpetuity 11

0

Assumes all earnings are paid to shareholders.

This essentially assumes that the company does not grow.

No earnings are retained so all earnings are paid out as dividends.

This is essentially the P/E ratio method of valuing a stock

Re-arranging the above equation we get

EP

EPSP

ror

PEPSr

rEPSP

1

0

0

1

10

1

Constant Dividend Model

This is obviously unrealistic since it assumes that no earnings are retained and there is no growth.

Accordingly, we need to adjust this formula for the value of growth.

Constant Growth Model

Constant Growth DDM - A version of the dividend growth model in which dividends grow at a constant rate (Gordon Growth Model).

grDivP

10

Dividends Growth at a constant rate g

d1 = d0(1+g)

If the most recent dividend paid was 100 and the growth rate is 8%.

The next dividend is d1 = 100(1.08) = 108

In two years time the dividend d2 is

100(1+g)2 = 100(1.08)(1.08)=116.6

Where does g come from?

It come from retained earnings which are reinvested at the cost of capital.

This increases subsequent earnings and dividends.


If a firm elects to pay a lower dividend, and reinvest the funds, the stock price may increase because future dividends may be higher.

Payout Ratio - Fraction of earnings paid out as dividends

Plowback (Ploughback) Ratio - Fraction of earnings retained by the firm.


Growth can be derived from applying the return on equity to the percentage of earnings plowed (ploughed) back into operations.

g = return on equity X plough back ratio

The ploughback ratio is 1 – payout ratio

g is a sustainable growth level

Sustainable Growth Rate - Steady rate at which a firm can grow: plowback ratio X return on equity

Notation

BVP: Book value per sharePayout Ratio: The proportion of earnings paid

out. DPS=EPS X Payout Ratio REPS: Retained Earnings Per Share: that part

of earnings per share not paid in dividends and ploughed back into the business = EPS X Ploughback Ratio

ROE: Return on Equity = EPS/BVP

Example

An all equity company has 1,000,000 shares and a book value of €10m

The BVP is €10If we assume the ROE is 10% the EPS is

0.1 X 10 = €1 or 100 centIf we assume the payout ratio is 40% the

DPS is 40 cent.

Number of Shares

Net Income (NI) 1000000 1000000 EPS 1

Dividend 400000 1000000 DPS 0.4

Retained Earnings 600000 1000000 REPS 0.6

Book Value (BV) 10000000 1000000 BVP 10

ROE = NI/BV 0.1

Summary of Accounts: year 0

T BVP EPS(cents)

PayoutRatio

DPS REPS ROE g

0 €10 100 0.4 40 60 10%

1 €10.6 106 0.4 42.4 63.6 10% 6%

2 €11.23 112.36 0.4 44.94 67.42 10% 6%

3 €11.91 119.1 0.4 47.64 71.46 10% 6%

How Growth affects earnings and dividends

Number of Shares

Net Income (NI) 1191016 1000000 EPS 1.191

Dividend 476406.4 1000000 DPS 0.476

Retained Earnings 714609.6 1000000 REPS 0.715

Book Value (BV) 11910160 1000000 BVP 11.91

ROE = NI/BV 0.1

Summary of Accounts: Year 3


ExampleOur company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to plough back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?



P0512

67 .

$41.

No Growth With Growth



P0512

67 .

$41.

No GrowthWith Growth

g

P

. . .

. .$75.

20 40 083

12 08000


Example - continuedIf the company did not plowback some earnings, the stock price would remain at $41.67. With the plowback, the price rose to $75.00.

The difference between these two numbers (75.00-41.67=33.33) is called the Net Present Value of Growth Opportunities (PVGO).


Net Present Value of Growth Opportunities (PVGO) - Net present value of a firm’s future investments.

Value of a share with growth

1

10

1EPS

PVGOrE

PAND

PVGOr

EPSP

Examples – using a Dividend Discount model to Value shares

How much is a share worth if it yield DPS of 100 cent forever. The cost of capital or expected rate of return is 10%

Answer: 100/0.1 = 1000 cent or €10

Suppose the company did not pay a Div in year 3

It reinvests the 100 cent per share at 10%.

What happens to the value of the share? First need to consider what happens to

dividends Assume that dividends are reinvested at

the cost of capital i.e. 10%

Reinvestment of Profits in year 3

Dividends in year three are zero Dividends from year 4 onwards

increase to 110 cent per annum. (The 100 cent yields are return of 10%)

Accordingly we have dividends of 100 cent for years one and two. Zero divs for year 3 and a perpetuity of 110 cent from year 4 onwards.

3210 r)r(1DiV4...r)(1

Div2r)(1

DIV1

P

33210 r)r(1110

r)(10

r)(1100

r)(1100

P

1tt

t0 R)(1

dP

.....

43210 r)(1DiV4

r)(1DIV3 r)(1

Div2r)(1

DIV1P

Divs 100 100 110PV of Divs from year 4 at year 3. 1100

Value Discount 1.1 1.21 1.331

1000.00 PV 90.91 82.64 826.45

Why is there no change in value?

Because the investment of retained earnings only yields the same rate of return as the cost of equity.

We could use the formula

10001.0

1000

10

P

PVGOr

EPSP

Real Growth Opportunity

What if the company did not pay any Divs in year three but invested in a positive NPV project

For example a project yielding 20 cent per share per annum forever beginning in year 4

NPV of the project is 100

1.020100 NPV

But this NPV is at year 3

NPV now is 100 x 0.7513 = 75.13

centP 13.107513.751.0

1000

3210 r)r(1DiV4...r)(1

Div2r)(1

DIV1

P

3210 r)r(1120...r)(1

100r)(1

100

P

1tt

t0 R)(1

dP

.....

43210 r)(1DiV4

r)(1DIV3 r)(1

Div2r)(1

DIV1P

1 2 34 to infinity

Divs 100 100 0 120

PV of Divs from year 4 at year 3. 1200

ValueDiscount Factor 1.1 1.21 1.331

1075.13 PV 90.91 82.64 901.58

Stocks&bonds2214 1

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