week 1

(1) Like public companies, private companies can also use their share price as a measure of performance.

(2) Financing for public corporations must flow through financial markets.

Review Questions

Module 2

Valuation of Securities (Chapters 5, 6, 7)

1/ Future values and present values

2/ Compound interest

3/ Multiple cash flows, perpetuities and annuities

4/ Effective annual interest rates

5/ Inflation and time value

Chapter 5: Time Value of Money

Example 1

$1000

0 1

$900

$1000 $1000

$1000 $1030

$1000 $1100

(1)

(2)

(3)

(4)

?

Example 1 (cont.)

Compensation Expected inflation

Delayed consumption

$100 $30 $70 + =

r %CPI rreal + =

Inflation and Interest Rates

Approx.

r = %CPI + rreal

Nominal int. rate = Inflation rate + Real int. Rate

Exact

(1 + r) = (1 + %CPI) * (1 + rreal)

r = (1 + %CPI) * (1 + rreal) – 1

rreal = (1 + r) / (1 + %CPI) – 1

Example 1 (cont.)

0 1

$1000 $1100 (4)

10% = 3% + 7% At t = 0:

What happen if you find out that the price of a 32” LCD TV has increased to $1090 in stead of $1030 as expected?

How much are you rewarded for your delayed consumption? Or what is your actual rate of return?

At t = 1:

Actual inflation = 9% !!

Realized return = 10% - actual infl. = 1% !!

We are here now!!

We are here now!!

Example 2

0 1

$1000 $1000

0 1 Now Then This year Next year Present Future

X

Y

$1000 * (1 + r)

r = 10% $1100

$1000 / (1 + r)

r = 10% $909

Example 2 (cont.)

0 1

$1000 $1000

Present Future

$1000 $1100

$909 $1000

FV = PV * (1 + r)

PV = FV / (1 + r)

(1)

(2)

(3)

Example 2 (cont.)

$1000 $1100

0 1 2

$1210

$1100 * (1 + 10%) $1210

FV2 = PV1 * (1 + r) $1000 * (1 + 10%)

$1100 FV1 = PV0 * (1 + r)

$1100 * (1 + 10%) $1210

FV2 = FV1 * (1 + r) FV2 = PV0 * (1 + r) * (1 + r)

FV2 = PV0 * (1 + r)2

Future Value:

FV = PV * (1 + r)t

Present value:

PV = FV / (1 + r)t

Where • r is interest rate per period • t is the number of periods

Future & Present Values

Example 2 (cont.)

$1000

0 1 2

$1000 * (1 + 10%) $1100

$1000 * (1 + 10%) $1100 $1000

$100 $100 * (1 + 10%)

$110

= +

$1210

+

At t= 2: $1210 = $1000 + $200 + $10

Original CF Simple interest Compounding

Compounding Effect

Year FV Principal Simple interest

Compounding %Compound

0 100.00 100 0 0.00 0.00

5 133.82 100 30 3.82 0.03

10 179.08 100 60 19.08 0.11

15 239.66 100 90 49.66 0.21

20 320.71 100 120 100.71 0.31

25 429.19 100 150 179.19 0.42

30 574.35 100 180 294.35 0.51

Future value and its components (r = 6% per year)

Compounding Effect

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

FV (

$)

Year

Future Value Components

Principal Simple Compound

r = 6%

Compounding Effect

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

% o

f Fu

ture

Val

ue

Year

Compounding Effect r = 3% r = 6% r = 12%

Compounding Effect

0

500

1000

1500

2000

2500

3000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Futu

re V

alu

e

Year

Total Future Value

r = 3% r = 6% r = 12%

Compounding Effect

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Pre

sen

t V

alu

e

Year

Present Value r = 3% r = 6% r = 12%

r = %CPI + rreal

Nominal int. rate = Inflation rate + Real int. Rate

FV = original CF + simple interest + compounding

When n compounding effect

If r1 > r2 then

(comp. eff.)1 > (comp. eff.)2 and

%(comp. eff.)1 > %(comp. eff.)2

For the same FV, if r1 > r2 then PV1 < PV2

What did we learn so far?

Example 3

$1000

Mar 12 Sep 12 Mar 13

National Bank

(1)

3% 3% (2)

6% $1060.0

$1000 * (1 + 6%)

$1030 $1060.9 $1000 * (1 + 3%) $1030 * (1 + 3%)

$1000 * (1 + 3%) * (1 + 3%) (1 + 6.09%) $1000 * =

(1 + 3%)2 = (1 + 6.09%)

Effective Annual Interest Rate

(1 + 6.09%) = (1 + 3%)2

6% 2

(1 + 6.09%) 1 + 2

=

EAR

Effective Annual Rate (EAR) Annual interest

Rate (r)

Number of compounding periods in a year (m)

+ ) r 1 m

+ m

= - 1 (

Effective Annual Interest Rate

• m = 1 (annual): EAR = (1 + 6%)1 – 1 = 6%

• m = 2 (semi-annual): EAR = (1 + 6%/2)2 – 1 = 6.09%

• m = 4 (quarterly): EAR = (1 + 6%/4)4 – 1 = 6.14%

• m = 12 (monthly): EAR = (1 + 6%/12)12 – 1 = 6.17%

• m = 365 (daily): EAR = (1 + 6%/365)365 – 1 = 6.18%

• m (continuous): EAR = e0.06 – 1 = 6.184%

EAR r 1 m

+ m

= - 1

Future & Present Values Revisited

… CF0

1 2 3 0 n

FV – single cash flow

FV1 = CF0(1 + r)1

FV2 = CF0(1 + r)2

FV3 = CF0(1 + r)3

FVn = CF0(1 + r)n

FV1 < FV2 < FV3 < … < FVn

Future – Present Values Revisited

CF0 CF2 CF3

FV – multiple cash flows

CF1

CFn-1 (1 + r)1

CF3 (1 + r)n-3

CF2 (1 + r)n-2

CF1 (1 + r)n-1

CFn-1

… 1 2 3 0 n n-1

CF0 (1 + r)n

= FV +

FV = CFt n

t

(1 + r)n-t


FV – A general case

CF3 / (1 + r)3

CF1 /(1 + r)1


… 1 2 3 0 n

PV – A general case

CF2 / (1 + r)2

CFn (1 + r)n

CF1 CF2 CF3 CFn

PV = +

PV = CFt n

t (1 + r)t _____



PV = CFt n

t (1 + r)t _____



• CFt = cash flow at the period t

• n = total number of periods

• r = interest rate per period

4. An annual percentage rate will be equal to

an effective annual rate if:

(a) simple interest is used.

(b) compounding occurs continuously.

(c) compounding occurs annually.

(d) an error has occurred; these terms cannot

be equal.

Review Questions

How much more would you be willing to pay today for an investment offering $10,000 in four years rather than the normally advertised five-year period? Your discount rate is 8%.

Review Questions

• Cash price: $3,200 • 20% deposit : $3,200 * 20% = $640 • Financed amount = $3,200 - $640 = $2,560 • Weekly payment = $66.46 • Number of weeks = 52

Annuities

An annuity is an equally spaced level stream of cash flows with a finite maturity.

…

c c c c c

1 2 8 9 10 0

= = = = ...

is known

PV = ?

c =

What is the annuity value, i.e. the combined value of all cash flows, today? What is the annuity value at maturity?

FV10 = ?

n …

Annuities

PV =

Ordinary annuity

C C

… C C C C

1 2 8 9 0 n …

End of year End of year End of year End of year End of year

Annuity due

C C

… C C C C

1 2 8 9 0 n …

Begin of year Begin of year Begin of year Begin of year Begin of year

+

+ PV =

Annuities

PV of an ordinary annuity

C

… C C C C

1 2 3 n-1 0 n

C

(1+r)1 PVann. =

C

(1+r)2

C

(1+r)3

C

(1+r)n-1

C

(1+r)n + + +…+ +

C r PVann. =

1 (1 + r)n

1 -

C = cash flow per period r = interest rate per period n = number of periods till maturity

1 – [1/(1+r)]n

1 – [1/(1+r)]

Annuities

PV of an annuity C

(1+r)1 PVann. =

C

(1+r)2

C

(1+r)3

C

(1+r)n-1

C

(1+r)n + + +…+ +

C

(1+r) PVann. =

1

(1+r)1

1

(1+r)2

1

(1+r)n-2

1

(1+r)n-1 + +…+ + 1 +

Because: (1 – vn) = (1 – v) * (1 + v + v2 +…+ vn-1)

C

(1+r) PVann. =

C

(1+r) PVann. =

1

(1+r)n

(1 + r)

r 1 –

Annuities

PV of an annuity due

C

… C C C C

1 2 3 n-1 0 n

C PVann. = C

(1+r)1

C

(1+r)2

C

(1+r)3

C

(1+r)n-1 + + +…+ +

C = cash flow per period r = interest rate per period n = number of periods till maturity

C r PVann. due =

1 (1 + r)n

1 - (1 + r)

Annuities

FV of an ordinary annuity

C (1+r)n-1 FVann. = C (1+r)n-2 C (1+r)2 C (1+r)1 C + + +…+ +

C r FVann. = [(1 + r)n – 1]

FV of an annuity due

C r FVann. = [(1 + r)n – 1] (1 + r)

Perpetuities

PV of a perpetuity

A perpetuity is an annuity with infinite maturity.

C r PVper. =

1 (1 + r)n

1 -

When n (1 + r)n and 1 / (1 + r)n 0

C r

PVper. =

1

Example - Annuity

You are purchasing a car from Continental Car Service. You are scheduled to take 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the effective price you are paying for the car (i.e. what is the PV)?

4000 4000 4000

1 2 3 0

4,000

(1 + 0.10)1 Pricecar = + +

PVann. =

4,000

(1 + 0.10)2

4,000

(1 + 0.10)3

4,000

0.10

1

(1 + 0.10)3 1 –

Pricecar = $9,947.41

=

Example - Annuity

What if Continental Car Service allows you to start your first installment payment 2 years from now?

4000 4000 4000

2 3 4 0

4,000

(1 + 0.10)2 Pricecar = + +

4,000

(1 + 0.10)3

4,000

(1 + 0.10)4

Pricecar = $9,043.10

0

1

0

(1 + 0.10)1 +

9,947.41 (1 + 0.10)1

Pricecar = = $9,043.10

1/ Interest rates and bond prices

2/ Bond yields

3/ The yield curve

Chapter 6: Valuing Bonds

Review

Balance Sheet

Assets

Use of funds Source of funds

Liabilities Bonds

Equity Stocks

Source: Asset Allocation Advisor

Source: NZX.com

Bond investors

=

A bond is a security that obligates the issuer to make specified payments to the bondholder

Bond Pricing

90 90 90 90

2 3 4 0 1

Issuer

$1000 bond

1000

+

Principal value: paid at maturity

Coupon: fixed, % of principal

Maturity

Coupon: fixed, % of principal Coupon: fixed, % of principal Coupon: fixed, % of principal ?

Bond Pricing

P = n

t=1 (1 + r)t

Ct

(1 + r)n

PP +

P = 2n

t=1 (1 + r/2)t

Ct/2

(1 + r/2)2n

PP +

Annual coupon payment:

Semiannual coupon payment:

PV = CFt n

t (1 + r)t _____

PV = CFt n

t (1 + r)t _____

Bond Pricing

Maturity: n = 4

Cash flows: CF1 = CF2 = CF3 = 90,

CF4 = 90 + 1,000 = 1,090

90

(1 + 0.09)2 PVbond = + +

90

(1 + 0.09)3

1,090

(1 + 0.09)4

90

(1 + 0.09)1 +

PVbond = $1000

Interest rate: r = 9% Interest rate: r = 10%

90

(1 + 0.10)2 PVbond = + +

90

(1 + 0.10)3

1,090

(1 + 0.10)4

90

(1 + 0.10)1 +

PVbond = $968.30

Interest rate: r = 8%

90

(1 + 0.08)2 PVbond = + +

90

(1 + 0.08)3

1,090

(1 + 0.08)4

90

(1 + 0.08)1 +

PVbond = $1,033.12

PV = CFt n

t (1 + r)t _____

Bond Pricing

Bond prices are inversely related to interest rates, i.e bonds exhibit interest rate risk.

Interest Rate Risk

0

500

1,000

1,500

2,000

2,500

3,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

P

R

I

C

E

Interest rates (%)

Par value = $1,000, maturity = 15 years,

coupon rate = 10% paid annually

10

Premium r < coupon

Discount r > coupon

Par = Price

r = coupon

Interest Rate Risk

0

500

1,000

1,500

2,000

2,500

3,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

P

R

I

C

E

Interest rate (%)

2YRS

15YRS

6YRS

Par value = $1,000, coupon rate = 10% paid annually

10

Interest Rate Risk

The inverse relationship between bond prices and interest rates are more pronounced for longer-term bonds than shorter-term bonds.

At r < coupon: premiumlong > premiumshort

At r > coupon: discountlong > discountshort

At r = coupon: pricelong = priceshort

For the same percentage change in interest rate, |%∆r| |%∆P|long > |%∆P|short

Interest Rate Risk

Maturity 0 Bond price Par value

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

P

R

I

C

E

YEARS TO MATURITY

Premium prices

Discount prices

0

Bond Yields

Yield-to-maturity (YTM):

Interest rate at which the present value of a bond’ CFs is equal to its price.

YTM at time of purchase (t=0) is equal to the realized rate of return (at t=T) if:

(1) All coupons are reinvested at the same rate as YTM0; and (2) The investor holds the bond to maturity;

Bond Yields

2 3 4 0 1 r = 9% r = 9% r = 9% r = 9%

90 90 90 90

1,000

98.10 90*1.09

106.93 90*1.092

116.55 90*1.093

$1,411.58 $1,000

Investment Total income

CASE 1: No change in interest rates

Bond Yields

2 3 4 0 1 r = 9% r = 9% r = 9% r = 9%

$1,411.58 $1,000

HPR = Total income

Investment

1/ What is the total rate of return over the 4-year period, i.e. holding period return?

Coupon income + price change

Investment =

HPR = $1,411.58 - $1,000

$1,000 = 41.16%

This is a 4-year return

Bond Yields

2 3 4 0 1 r = 9% r = 9% r = 9% r = 9%

$1,411.58 $1,000

$1000 = $1,411.58

2/ What is the annual rate of return from investment?

(1 + r)4 = $1,411.58

$1,000

r = 1.411581/4 – 1 = 9% per annum

$1,000

* (1 + r)4 This is also the promised YTM when the bond was purchased 4 years ago.

Bond Yields

r

Bond Yields

2 3 4 0 1 r = 9% r = 9% r = 9% r = 5%

90 90 90 90

1,000

$1,000

CASE 2: A change in interest rates

1/ What is the HPR from the 4-year investment?

HPR =

90*(1.05)

103.01 90*(1.09)*(1.05)

$1,399.79

$1,399.79 - $1,000

$1,000 = 39.98% This is a 4-year

actual return

94.5

112.28 90 * (1.09)2 * (1.05)

Bond Yields

CASE 2: A change in interest rates


HPR = $1,399.79 - $1,000

$1,000 = 39.98%

$1000 = $1,399.79


(1 + r)4 = $1,399.79

$1,000

r = 1.39981/4 – 1 = 8.77% per annum

* (1 + r)4

This is lower than the promised YTM

Bond Yields

2 3 4 0 1 r = 9% r = 9% r = 9% r = 5%

90 90 90 90

1,000

$1,000

CASE 3: A rate change and early divestment

Sell P3 = ?


HPR = Coupon income + price change

Investment

$1,038.10

90*(1.09) 98.10 90*(1.09)2 106.93

1090/1.05

$1,333.13

$1,333.13 - $1,000

$1,000 = 33.31%

This is a 3-year return

Bond Yields

CASE 3: A rate change and early divestment


HPR = $1,333.13 - $1,000

$1,000 = 33.31%

$1000 = $1,333.13


(1 + r)3 = $1,333.13

$1,000

r = 1.33311/3 – 1 = 10.06% per annum

* (1 + r)3

why is it > YTM??

Bond Yields

Try this at home:

CASE 4: rate change to 5% at year 2 and sell bond at year 3.

What is the annual rate of return?

r = %CPI + rreal

Assume: %CPI = 3% and rrealNZ = 4% r = 3% + 4% = 7%

Example 4

$1000

$1070

0 1 10

$2000

…

$1070 / (1 + 0.07)1 $2000 / (1 + 0.07)10

$2000 / [(1 + 0.07) * (1 + 0.07) * … * (1 + 0.07)]

1000 = $2,000

(1 + r2010 ) * (1 + r2011) * … * (1 + r2119)

= 7%

Our concerns between short-term and long-term bond investments include:

1/ The stability of the components in the risk-free interest rate, i.e. how we expect them to change in the future.

2/ Liquidity preference

LP indicates that long term rates have to pay a premium over short term rates because:

– Investors need a premium to compensate for the added price risk associated with long-term bonds.

– Borrowers are willing to pay higher rates on long-term debt to avoid refinancing risk.

Term Structure of Interest Rates

Yields

Time to maturity

7%

Expectations of higher short-term rates

Liquidity premium


r = %CPI + rreal + liquidity



Default (Credit) risk: The risk that a bond issuer may default on its bonds.

Default Risk

r = %CPI + rreal + Liquidity + Default

Review

Balance Sheet

Assets

Use of funds Source of funds

Bonds

CommonStocks

1/ Stock markets

2/ Valuing common stocks

3/ Growth versus income stocks

Chapter 7: Valuing Stocks

Common stocks represent ownership shares in a publicly held corporations.

Holders of common stocks

- Are owners of the firm;

- Are able to be involved in some of the firm’s management;

- Can vote to select the BOD; and

- Have residual claims against the firm’s assets.

Common Stocks

Stock Market

Primary markets – New issues are sold.

• Seasoned issues: new shares offered by an already listed firm.

• Initial public offerings (IPOs): new shares offered to the public for the first time.

Secondary markets – Outstanding issues, i.e. stocks already sold to the

public, are traded.

Stock Market

Why secondary stock markets are important?

Provide liquidity to investors who acquire securities in the primary market

• Helps issuers raise needed funds in the primary market since investors want liquidity

Help determine market pricing for new issues

Stock Valuation

Equity Valuation

Discounted Cash Flow Techniques

Relative Valuation Techniques

Present value of • dividends (DDM) • free CF to equity • operating free CF

• Price/Earnings ratio (P/E) • Price/Cash flow ratio (P/CF) • Price/Book value ratio (P/BV) • Price/Sales ratio (P/S)

Relative Valuation

Relative valuation focuses on how the market is currently valuing financial assets.

– This does not necessarily imply that current valuations are appropriate.

– The overall market or a particular industry can become seriously overvalued or undervalued for a period of time.

Relative Valuation

Appropriate to use under two conditions:

– You have a good set of comparable entities.

• Similar size, risk, etc.

– The aggregate market or the relevant industry is not at a valuation extreme.

• It is fairly valued.

Should be used together with the DCF models to determine equity value.

Price/Earnings Ratio

Measure investors’ attitude toward the firm’s earnings power

How many dollars investors are willing to pay for a dollar of expected earnings.

Average P/E ratio for IT industry

0.00

50.00

100.00

150.00

200.00

250.00

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Source: Cooper et al. (JF, 2001)

Dividend Discount Model (DDM)

PV = n

t=1 (1 + r)t CFt PV =

n

t=1 (1 + r)t Dt


PV = n

t=1 (1 + r)t Dt

PV =

(1 + r)

D1 + (1 + r)2

D2 + (1 + r)3

D3 + … + (1 + r)n

Dn

where

PV = Intrinsic value of common stock at t = 0 r = Required rate of return for common stock Dt = Expected dividend for period t


Constant growth (g)

D1 = D0(1 + g) D2 = D1(1 + g)

PV = (1 + r)

D0(1+g) +

(1 + r)2 +

(1 + r)3 + … +

(1 + r)n

D0(1+g)n D0(1+g)2 D0(1+g)3

Dt+1 = Dt(1 + g)

PV = (1 + r)

D1 + (1 + r)2

D2 + (1 + r)3

D3 + … + (1 + r)n

Dn


Constant growth (g)

D1 = D0(1 + g) D2 = D1(1 + g)

PV = (1 + r)

D0(1+g) +

(1 + r)2 +

(1 + r)3 + … +

(1 + r)n

D0(1+g)n D0(1+g)2 D0(1+g)3

Dt+1 = Dt(1 + g)

PV = r - g D1


Constant growth (g)

Example:

Stock A, D0 = $1, r = 0.10, g = 0.06

D1 = D0(1 + g) = $1*(1 + 0.06) = $1.06

r – g = 0.10 – 0.06 = 0.04

PVA = $1.06/0.04 = $26.50


Constant growth (g)

Special case: g = 0 D0 = D1 = … = Dn

PV = r

D0 PV =

r E0

Assuming all earnings are paid out as dividends

V(D4-D∞) = (r – g2)

D4 (1 + r)3

1


2-stage growth

PV = (1 + r)

D0(1+g1) +

(1 + r)2 +

(1 + r)3 +

D0(1+g1)2 D0(1+g1)3 (r – g2)(1 + r)3

D0(1+g1)3(1+g2)

0 1 2 3 4 ∞

D1 D2 D3 D4 D∞

g1 g2

V(D1-D3) = 3

t=1 (1 + r)t Dt


2-stage growth

PV = (1 + r)

D0(1+g1) + … +

(1 + r)m +

D0(1+g1)m (r – g2)(1 + r)m

D0(1+g1)m(1+g2)

0 1 ... m m+1 ∞

D1 D… Dm Dm+1 D∞ g1 g2

V(1) = m

t=1 (1 + r)t Dt V(2) =

(r – g2) Dm+1

(1 + r)m 1

(1) (2)


2-stage growth

Example : Stock A, D0 = $1, r = 0.10, Growth rate in the next 3 years (g1) = 0.12, Growth rate in year 4 onward (g2) = 0.06.

0 1 2 3 4 ∞

D1=1.12

g1=0.12

D0=1

g2 =0.06

D1 = D0(1 + g1) = $1*(1 + 0.12) = $1.12 D2 = D0(1 + g1)2 = $1*(1 + 0.12)2 = $1.2544 D3 = D0(1 + g1)3 = $1*(1 + 0.12)3 = $1.4049 D4 = D3(1 + g2)= $1.4049*(1 + 0.06) = $1.4892

D2=1.25 D3=1.40 D4=1.49


2-stage growth

Example 2: Stock A, D0 = $1, r = 0.10,

g1=0.12 g2 =0.06

0 1 2 3 4 ∞

D1=1.12 D2=1.25 D3=1.40 D4=1.49

P = $4.57

+

Use D4 and the constant growth model to estimate the present value of all dividends from year 4 onwards. Then bring (discount) that value to time 0.


2-stage growth

Example : Stock A, D0 = $1, r = 0.10,

g1=0.12 g2 =0.06

0 1 2 3 4 ∞

D1=1.12 D2=1.25 D3=1.40 D4=1.49

P = $31.08

+ P3 = 1.49/(0.10 – 0.06)


2-stage growth

Example 2: Stock A, D0 = $1, r = 0.10, Growth rate in the next 3 years (g1) = 0.12, Growth rate in year 4 onward (g2) = 0.06.

D1 = D0(1 + g1) = $1*(1 + 0.12) = $1.12 D2 = D0(1 + g1)2 = $1*(1 + 0.12)2 = $1.2544 D3 = D0(1 + g1)3 = $1*(1 + 0.12)3 = $1.4049 D4 = D3(1 + g2)= $1.4049*(1 + 0.06) = $1.4892

P = (1 + .10)

1.12 +

(1 + .10)2 +

(1 + .10)3 +

1.2544 1.4049 (.10 – .06)(1 + .10)3

1.4892

P = 31.08 VD1-D3 VD4-D∞


Multi-stage growth

PV = (1 + r)5

D4(1+g1) +

(1 + r)6 +

(1 + r)7 +

D4(1+g1)2 D4(1+g1)3 (r – g2)(1 + r)7

D4(1+g1)3(1+g2)

(1 + r)4

D4 +

0 5 6 7 8 ∞

D5 D6 D7 D8 D∞

g1 g2

4

D4

No dividend

V(D4-D7) = 7

t=4 (1 + r)t Dt V(D8-D∞) =

(r – g2) D8

(1 + r)7 1


Multi-stage growth

PV = (1 + r)m+1

Dm(1+g1) + … +

(1 + r)n +

Dm(1+g1)n-m (r – g2)(1 + r)n

Dm(1+g1)n-m(1+g2)

(1 + r)m

Dm +

0 m+1 … n n+1 ∞

Dm+1 D… Dn Dn+1 D∞ g1

(2)

m

Dm

No dividend

V(1) = n

t=m (1 + r)t Dt V(2) =

(r – g2) Dn+1

(1 + r)n 1

g2

(1)


Multi-stage growth

Example : Stock A, D4 = $1, r = 0.10, Growth rate in the following 3 years (g1) after year 4 = 0.12, Growth rate in year 8 onward (g2) = 0.06.

D5 = D4(1 + g1) = $1*(1 + 0.12) = $1.12 D6 = D4(1 + g1)2 = $1*(1 + 0.12)2 = $1.2544 D7 = D4(1 + g1)3 = $1*(1 + 0.12)3 = $1.4049 D8 = D7(1 + g2)= $1.4049*(1 + 0.06) = $1.4892

VA = (1 + .10)5

1.12 +

(1 + .10)6 +

(1 + .10)7 + 1.2544 1.4049

(.10 – .06)(1 + .10)7

1.4892

(1 + .10)4 1.00

+

VA = 21.91 VD4-D7 VD8-D∞

What if the analyzed company is not paying dividend, i.e. Dt = 0? PV = 0?

Alternative model: PV = n

t=1 (1 + r)t FCFEt

PV = n

t=1 (1 + r)t Dt


Return on equity (ROE) = EPS / Book equity per share

Payout ratio: % of earnings paid out as dividends

Plowback ratio: % of earnings retained by the firm

Sustainable growth rate (g): Steady rate at which a firm can growth; Growth rate that can be sustained without additional equity issues, i.e. all increases in equity come from retained earnings.

g = ROE * plowback ratio

Growth Stocks

Example: Blue Skies E1 = $5, r = 12%, ROE = 20% What is stock price of Blue Skies?

Case 1: payout = 100% D1 = 100% * E1 = 1 * $5 = $5 g = ROE * (1 – payout) = 0.20 * (1 – 1) = 0 P0 = $5 / (0.12 – 0) = $5 / 0.12 = $41.67

Case 2: payout = 60% D1 = 60% * E1 = 0.60 * $5 = $3 g = ROE * (1 – payout) = 0.20 * (1 – 0.60) = 0.08 P0 = $3 / (0.12 – 0.08) = $3 / 0.04 = $75

P0 = D1 / (r – g)

Growth Stocks

PV of growth opportunities = Pgrowth – Pno growth

= $75 - $41.67 = $33.33

P0 = D1 / (r – g)

Growth Stocks

PV of growth opportunities = Pgrowth – Pno growth

= $0 ?????

Example: Blue Skies E1 = $5, r = 12%, ROE = 12% What is stock price of Blue Skies?

Case 1: payout = 100% D1 = 100% * E1 = 1 * $5 = $5 g = ROE * (1 – payout) = 0.12 * (1 – 1) = 0 P0 = $5 / (0.12 – 0) = $5 / 0.12 = $41.67

Case 2: payout = 60% D1 = 60% * E1 = 0.60 * $5 = $3 g = ROE * (1 – payout) = 0.12 * (1 – 0.60) = 0.048 P0 = $3 / (0.12 – 0.048) = $3 / 0.072 = $41.67

g = ROE * (1 - dividend payout ratio) g = (1 – D1/E1) * ROE

(1) D1/E1 = 1, i.e. all earnings are paid out ROE is irrelevant because g = 0 and P0 = E1 / r

(2) ROE = r , i.e. P0 = D1/[r – (1 – D1/E1)*ROE] P0 = D1/[r * (1 – 1 + D1/E1)] P0 = E1 / r How much the firm pays out is irrelevant because it does not affect the stock price.

Growth Stocks

Growth stocks must have (1) positive retained earnings, and (2) the retained earnings are reinvested to earn a higher rate of return

(ROE) than required by investors (r).

Growth Stocks

STOCK VALUATION

Example:

A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. Investors require 12% rate of return. The stock has an P/E ratio of 9; and its expected EPS is $3. What is the stock worth?

Recommended end-of chapter questions/problems:

• Chapter 5: – Questions: 4, 5, 7, 11, 12, 18

– Problems: 22, 25, 30, 32, 34, 40, 48, 51, 60-64

• Chapter 6: – Questions: 1, 2

– Problems: 11, 12, 17, 18, 23, 25

• Chapter 7: – Questions: 1, 6

– Problems: 13, 16, 18, 31, 32

week 1

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