Bond and share valuation

BOND AND SHARE VALUATION

Richard Wamalwa Wanzala

HFIN 4104: Corporate Finance Theory

OUTLINE

INTRODUCTION

BOND VALUATION

Bonds

Return from Bonds

Coupon rate

Current yield

Spot interest rate

Yield to maturity or redemption yield:

Shortcut Method for Arriving at YTM

Yield to call

Bond Duration

Macaulay's Duration (D)


OUTLINE – CONT’D

EQUITY MARKET

Equity Instruments

SHARE VALUATION

Share Valuation Models

Discount Rate

Dividend Discount Models

One year holding period model

Multiple Year Holding Period Model

Constant Growth Model (CGM)

Constant Growth Model (CGM)

Zero Growth Models (ZGM)

H-Model


INTRODUCTION

Debt markets are used by both firms and

governments to raise funds for long-term

purposes, though most investment by firms is

financed by retained profits. Bonds are long-

term borrowing instruments for the issuer.

Major issuers of bonds are governments

(Treasury bonds in US, gilts in the UK, Bunds in

Germany) and firms, which issue corporate

bonds.


BOND VALUATION


BONDS

Bonds/ debentures are fixed income securities

containing acknowledgement of indebtedness

by the issuing company and are similar in

nature except for the issuing company.

The bond/ debentures will contain a promise to

pay interest for a specified period and then to

repay the principal at the end of specified

period, on a given date of maturity.


TYPES OF BONDS

• Callable and putable bonds

• Convertible bonds

• Eurobonds

• Floating rate notes (FRNs)

• Foreign bonds

• Index-linked bonds.

• Junk bonds

• Covered bonds


RETURN FROM BONDS

Bonds returns are expressed in the following five

forms:

Coupon rate

Current yield

Spot interest rate

Yield to maturity or redemption yield:

Yield to call


COUPON RATE

It is the rate of interest fixed for a bond and

printed on the bond certificate. Interest payable

is calculated at the coupon rate on the face

value of the bond. If the face value of the bond

is, say Shs. 1000 and if the coupon rate is fixed

at, say 9% p.a. the annual interest payable by

the company to the bond holder is Shs. 90 per

bond.


CURRENT YIELD

• It is the rate of return available from a bond on its

current price in the secondary market.

• It fluctuates depending on the bond price.

• It only measures the annual rate of return accruing to an investor who purchase the bond from the secondary market.

Illustration

• A bond has a face value of KShs. 1,000 and a coupon rate of 9 per annum. The bond is currently selling in the secondary market at a price of KShs. 800. Calculate the current yield.


CURRENT YIELD – CONT’D: SOLUTION

Annual interest payable (calculated at the rate

on the face value)

Annual interest payable (calculated at the rate

on the face value)


*100

Pr

Annual Interest Payable on the BondCurrent Yield

Current ice

9*100

100

.90

90*100

800

11.25%

KShs

Current Yield

SPOT INTEREST RATE (SIR)

It is the annual rate of return on a bond that offers

only one cash inflow to the investor. A zero coupon

bond offers only one cash inflow to the investor on

its maturity and hence the rate of return offered by

a zero coupon bond calculated on an annualized

basis is as “SIR”.

Mathematically, SIR of a zero coupon bond is the

discount rate that equates the present value of the

single cash inflow available to the investor on

maturity of the bond to the price of the bond.


SIR – CONT’D: ILLUSTRATION

• A zero coupon bond of face value KShs. 1,000 is

issued at discounted price KShs. 600. The bond has a maturity period of 5 years. Find the Spot Interest Rate of the bond.

Solution: • Let be the spot interest rate • The face value of KShs. 1,000 will be received

after a period of 5 years • Amount receivable after five years hence = KShs.

1,000


% . .i p a

SIR – CONT’D

Thus if the present value is at the end of 5

years a sum KShs. 1,000 would be available.


1 4 . 1,000*

1

1 1 3 . 1,000*

1 1

1 1 2 . 1,000*

1 1

Amount receivable after years hence KShsi

Amount receivable after years hence KShsi i

Amount receivable after years hence KShsi i

5

1

1

1 1 1 1 . 1,000*

1 1 1 1

1 1 1 1 1Pr . 1,000*

1 1 1 1 1

1. 1,000*

1

i

Amount receivable after one year hence KShsi i i i

esent Value KShsi i i i i

KShsi

51

1,000 ,1 i

SIR – CONT’D

Thus:


5

5

5

1600 1,000*

1

1 600

1 1,000

10.6

1

10.76%

Present value Discounted price of bond

i

i

i

YIELD TO MATURITY

It is the internal rate of return earned from a

bond, holding the bond till its maturity.

It is calculated by equating bond the cost of

bond to the present value of cash inflows from

the bond held till maturity. Hence, the YTM is

the discount rate makes the cost of bond equal

to the present value of cash inflows from the

bond held till maturity.


YTM – CONT’D

Illustration Bond of face value KShs. 1,000 with a coupon rate of 8% p.a.

and present value of KShs. 700 has a maturity period of 5

years. Calculate the (YTM) on the bond.


1

mi

.

1

n

1t n

n

t

Annual Interest rece

Cost of b

ivable Ter al Value of a B

ond annual interest receivable plus terminal value of bo

ondCost of Bond

where

i yield to maturity

t time peri

nd

od

n no of year

i i

s to maturity

YTM – CONT’D: SOLUTION

Annual interest rate at the coupon rate of 8% = KShs.

80 (i.e. on the face value of KShs. 1,000)


1 1 1

mint

n

tn

Annual Interest receivable Ter al Value of a BondCost of Bond

i i

1 2 3 4 5 5

1 2 3 4 5 5

Pr ., .700

80 80 80 80 80 1,000700

1 1 1 1 1 1

1 1 1 1 1 1,00080

1 1 1 1 1 1

Cost of Bond esent Value viz KShs

i i i i i i

i i i i i i


The value of the YTM can be arrived at any trial and error

method assuming a certain yield to maturity are reworking with

a different value until the value of the RHS matches with the

value on the LHS of the equation. (Try YTM= 9.5%; 10%; 17%;

and 18%).


1 2 3 4

5

5

1 1 1 1

1 0.18 1 0.18 1 0.18 1 0.18 1,000700 80

1 1 0.18

1 0.18

1,00080 0.8474 0.7182 0.6086 0.5158 0.4371

2.2878

250.17 437.10

687.27


The value of RHS is now less than the value of

LHS. The YTM lies in between 17% and 18%

which can be arrived at by interpolating

between 17% and 18%.


18 1717% 712.4 700

712.4 687.27

117% 12.04

247.70

17% 0.05%

17.05%

YTM

YTM

SHORTCUT METHOD FOR ARRIVING AT YTM


/

/ 2

I FV C nYTM

FV C

where I annual interest

FV face value of bond

C current price of bond

n bond period

SHORTCUT … CONT’D

Illustration

Bond of face value KShs. 1,000 with a coupon rate of 8%

p.a. and present value of KShs. 700 has a maturity period

of 5 years. Calculate the Yield To Maturity (YTM) on the

bond using shortcut method.

Solution


/

/ 2

80 1,000 700 / 5

1,000 200 / 2

80 600.1647

850

16.47%

I FV C nYTM

FV C

Appprimate value of YTM

YIELD TO CALL

YTC is the discount rate that makes the present

values of cash inflows to call (i.e. the annual

interests till the call date and the specified call

price of the bond at which it is redeemed)

equal to the cost of the bond.

YTC can be calculated on a similar lines as a

YTM as calculated


BOND DURATION

• It is defined as the time period at which the price risk and the investment risk of the bond are equal in magnitude but opposite in direction (Nagarajan and Jayabal, 2011) .

Illustration • A 5 years bond with a face value of KShs.

100 has a coupon rate of 10%. What is the terminal value that will be available to the investor in this bond, if the market interest rate is 12%?


BOND DURATION – CONT’D

At the end

of year 1

At the end

of year 2

At the end

of year 3

At the end

of year 4

At the end

of year 5

cashflow to

the bond

owner

10.00 10.00 10.00 10.00 10.00

100.00

(redeemed

value of the

bond)

110.00


4 3 2

1

10 1 0.12 10 1 0.12 10 1 0.12min

10 1 0.12 110

.163.53

Ter al Value

KShs

MACAULAY'S DURATION (D)

It is “the weighted average of time periods to

maturity” and is given by the following formula:


1

1

.

:

int

1

/ or " "

in

1

t

nt

t

t

t

nt

t

t

t C

DC

where t time period of each cashflow

C erest and principal payment that occurs in period t

i market erest rate

n maturity per

i

i

iod of the bond

MACAULAY'S … CONT’D

Illustration

A new bond is issued by a company with a

maturity period of 5 years and a coupon rate of

12%. The face value of the bond is KShs. 100

and the bond is redeemable at par after its

maturity period of 5 years. The market interest

rate is 12%. Prove that the duration of the bond

is less than its period of maturity



t

1 12.00 10.714 10.714

2 12.00 9.566 19.132

3 12.00 8.541 25.623

4 12.00 7.626 30.504

5 112.00 63.552 317.769


1t

tC

itC

1

.t

tt C

i

99.999 403.733


Thus:


403.733

99.999

4.03733 ( . . 4.04 )

Duration

i e years

EQUITY MARKET


EQUITY MARKET

• Equity markets are markets which organize trading

nationally and internationally in such instruments, as common equity, preferred shares, as well as derivatives on equity instruments.

• The purpose of equity is the following:

a. A new issue of equity shares is an important source of external corporate financing;

b. Equity shares perform a financing role from internally generated funds (retained earnings);

c. Equity shares perform an institutional role as a means of ownership.


Equity Instruments

a. Common or ordinary share (stock) is an equity share

that does not have a fixed dividend yield. It represent partial ownership of the company and provide their holders claims to future streams of income, paid out of company profits and commonly referred to as dividends.

b. Preferred share is an equity security, which carries a predetermined constant dividend payment. It is a financial instrument, which represents an equity interest in a firm and which usually does not allow for voting rights of its owners.


EQUITY INSTRUMENTS – CONT’D

The decision to issue equity against debt is based on several factors:

• Tax incentives. In many countries interest payments are tax deductible,

however dividends are taxed. Thus the tax shield of debt forms incentives to

finance company by debt.

• Cost of distress. Increase of company leverage, increases the risk of

financial insolvency and may cause distress as well as lead to bankruptcy.

Thus companies tend to minimize their credit risk and increase the portion

of equity in the capital structure.

• Agency conflicts. When a company is financed by debt, an inherent conflict

arises between debt holders and equity holders. Shareholders have

incentives to undertake a riskier operating and investment decisions,

hoping for higher profits in case of optimistic outcomes. Their incentives are

mainly based by limited liability of their investments. In case of worst

outcome debt holders may suffer more, in spite of their priority claims

towards company assets.

• Signaling effect. The companies, which issue equity to finance operations,

provide signals to the market, that current share selling price is high and

company is overvalued.


EQUITY INSTRUMENTS – CONT’D

Types of Preferred shares

• Cumulative preference shares

• Non-cumulative preferred shares

• A redeemable preferred share

• Convertible preferred shares

• Participating preferred shares

• Stepped preferred shares

• Specific adjustable rate preferred shares

• Auction rate preferred shares (ARPS) or Single point adjustable rate shares (ARPS)

• Preferred equity redemption cumulative stocks (PERCS)


SHARE VALUATION


SHARE VALUATION MODELS

Share valuation means arriving at the intrinsic

value of shares. A share possess intrinsic value

because it offers returns to the shareholder in

the form of dividends and capital appreciation.


DISCOUNT RATE

The discount rate is the rate of return required

by the investor on his investment in the share.

This rate of return is arrived at by taking into

account the risk involved in the investment. The

discount rate consist of two components: risk

free interest rate and risk premium for the

share concerned


DIVIDEND DISCOUNT MODELS

• Based on the holding period and the dividend

expected to be received, share valuation

models can be grouped into two:

a. Holding period model

• One year holding period model

• Multiple year holding period model

b. Dividend growth models

• Constant growth model

• Multiple growth model


ONE YEAR HOLDING PERIOD MODEL

Assumptions

The investor purchases the share now

The investor intends to hold the share for a period of one

year

The investor intends to dispose off the share at the end of

one year

The present value (or intrinsic value) of the share is given by

the following relationship:


1 1

1 1

0

1

1

,

1

1

o

where

S present value of the share

D dividend expected to be received at the end of the first year

S expected selling price of the share at the end of the fir

D SS

k k

k

st year

rate of retu

‘ ’rn required by the investor also called capitalization rate

ONE YEAR… CONT’D: ILLUSTRATION 4.2

An investor desires to purchase the equity share of a

company from the secondary market. The investor

prefers to hold share for one year and dispose off the

share after on year. The investor expects to get a

dividend of KShs.5.00 per share next year and he is

hopeful of selling the share in the secondary market

at a price of KShs.70 after one year. He expects a

return of 20% on his investment, considering the level

of risk associated with it. Calculate the present value

of the share to the investor.


ONE YEAR… CONT’D: SOLUTION

• Hence, present value of the share for the investor is given the

following relationship

Therefore, intrinsic value of the share for the investor KShs. 62.50


1

1

. 5.00

. 70.00

0 2 %

Dividend expected after one year D KShs

Expected sales realization after one year S KShs

Return required by the investor k

1 1

1 1

1 1

1 1

5.00 70

1 0.20 1 0.20

4.17 58.93

.62.50

o

D SS

k k

KShs

MULTIPLE YEAR HOLDING PERIOD MODEL

Assumptions

The investor purchases the share now

The investor intends to hold the share for a

certain number of years

The investor will dispose off the share at the

end of the holding period


MULTIPLE YEAR…CONT’D

Accordingly, the present value of the share as

per multiple years holding period model is

given by the following relationship


31 2

1 2 3...

1 1 1 1

n no n

D D SD DS

k k k k

0

1 2 3, , ,.

:

.

.

, n

n

where

S present value of the share

D D D D annual dividend that will be received in the respective years

S expected sales price of the share at the end of the holding period

k rate of retu

rn required for the investor

n holding period in years

MULTIPLE YEAR…CONT’D: ILLUSTRATION

An investor desires to purchase the equity share of a

company from the secondary market. The investor prefers

to hold the share for a period of four years and dispose off

the share after four years. He expected to get a dividend of

KShs. 6.00, KShs. 6.50, KShs. 7.50 and KShs. 9.00 per

share in the next four years respectively. He is hopeful of

selling the share in the secondary market at a price of

KShs. 120 after the end of four years. He expects a return

of 22% on his investment considering the level of risk

associated with it. Calculate the present value of the share

to the investor.


MULTIPLE YEAR…CONT’D

Solution


31 2

1 2 3

1 2 3 4

...1 1 1 1

6.00 6.50 7.50 9.00 120

1 0.22 1 0.22 1 0.22 1 0.22

4.92 4.38 4.13 58.23

Pr .71.66

n no n

D D SD DS

k k k k

esent value of the share KShs

CONSTANT GROWTH MODEL (CGM)

Assumptions

The investor buys and holds the share forever

The dividends from the share grow at a

constant rate.

The discount rate (used for arriving at the

present value of the share) is greater than the

dividend growth rate.


CGM – CONT’D

the present value of the share is the sum of

present value of all future dividends.

When “n” approaches infinity the above formula

can be simplified as:

Since the present value can also be written

as


1 2 3

1 2 3 3

1 2 3

1 1 1 1...

1 1 1 1

n

o n

D g D g D g D gS

k k k k

0 1

–o

D

gS

g

k

–oS

g

D

k

1 0 , 1D D g

CGM – CONT’D

LIMITATIONS OF GORDON DIVIDEND

MODEL The use of this model is restricted to firms that have

been growing and are expected to grow at a stable

growth rate forever and are expected to offer

dividends at the stable growth rate forever.

If the dividend growth rate in more than the required

rate of return, the model cannot be used since the

value of the stock will become negative

If the dividend growth rate is equal to the required rate

of return, the model cannot be used since the value of

the stock will become infinity.


CGM – CONT’D: ILLUSTRATION

• M/s. Good tread company ltd, has declared a dividend of KShs. 4.50 per equity share for the current year. As a policy, the company has been enhancing its dividend by 12% every year. The company is expected to follow its dividend policy in the future also. An investor is interested to invest in the equity shares of the company as a long term investment. The investor expects a return of 18% on his investment considering the level of risk associated with the share of the company. Estimate the intrinsic value of the share for the investor.


CGM – CONT’D: SOLUTION

The intrinsic value of the share is given by:


1 ’ . 4.50

18% . .

12% . .

Current year s dividend D KShs per share

Return required for the investor k p a

Growth rate of dividend g p a

0

4.50 1 0.1

1

–

0.18 012

. 84

2

o

D g

k g

KShs

S

CONSTANT GROWTH MODEL (CGM)

Two stage growth model

Assumptions

The time period (in which the investor will be getting

streams of dividends) is divisible into two different

growth stages.

During the initial growth stage (stage -1) the growth

rate of dividend is variable from year to year.

During the latter growth stage (stage II) the dividend

growth rate will remain constant from year to year.

This stage will have indefinite time duration.


MGM – CONT’D

The intrinsic value of the share is the sum of

the present values of the dividend flows from

the two stages; Stage 1

Stage II

Intrinsic value


1 1 1

1 2 3. . ...

1 1 1 1

N

NI

DD D DPV

k k k k

1 .

1–.

N

N

IIP

g

kV

k

D

g

. . . .I II

PV PV

1. .

1 1

1

–

NtN

o t Nti e

D gD

kS

k kg

MGM – CONT’D: ILLUSTRATION

• A Company has paid a dividend of KShs. 150 per share during the current year. The company is expected to pay a dividend of KShs. 2.00 per share during the next year. The company is expected to pay a dividend of Analysts forecasts a dividend of KShs. 3.00 and KShs. 3.50 per share during the subsequent two years. After three years, the company is expected to pay dividends that are expected to grow at the rate of 10% every year. The investor expects a return of 20% on his investment. Calculate the intrinsic value of the share.


MGM – CONT’D: SOLUTION

I. Present value of stage growth model

II. Present value of dividends from the fourth

year to infinity


1 2 3

1 0.20 1 0.2

2.00 3.00 3.

0 1 0.20

50

. 5.78KShs

–

0.20 – 0

3.

.1

50 1 0

0

.10

. 38.50

k

D

KShs

g

MGM – CONT’D: SOLUTION

Present value of KShs. 38.50 receivable after

three years is arrived at as below:


3

3

38.50

38.50

. 22.80

1

1 0.20

PV

5.78 22.28

. 28.06

o

k

of dividends from Stage I PVIntrinsic value of the share S

of dividends from Stage II

KShs

KShs

ZERO GROWTH MODELS (ZGM)

The dividend per share will remain constant

every year, forever.

The above assumption means that the dividend

stream is a long-term annuity

Symbolically it means:

The intrinsic value of the share as per zero

growth models is given by:


o

DIntrinsic value S

k

1 2 3 4...D D D D D D

ZGM – CONT’D: ILLUSTRATION

A company pays a dividend of KShs. 20 per equity

share. The dividend is expected to remain the same.

An investor requires a return of 16% for investment in

equity shares of the company. Calculate the value of

equity share.

Solution


20

0.16

. 125

o

DIntrinsic value of the equity share S

k

KShs

H-MODEL (TWO STAGE DIVIDEND MODEL)

Assumptions of H-Model

Current above-normal growth rate is

The dividend growth rate falls down gradually

over a period of years

After a period of years, the dividend growth

rate falls down to .


0 0

1

f c f

o

f f

The intrinsic value of the share under the above conditions is given by

the following relationship :

D g D g gS

k – g k – g

.H

cg

2H

2H

fg

H-MODEL – CONT’D: ILLUSTRATION

The equity share of a company offers dividend

of KShs. 4.00 at present. The present dividend

growth rate is 40%. Analysts predict that the

dividend growth rate will decline linearly over a

period of 12 years after which it will stabilize at

15%. An investor requires a return of 18% for

his investment in the equity share of the

company. What is the intrinsic value of the

equity share?


H-MODEL – CONT’D: SOLUTION

The dividend pattern resembles


4.00

1

– –

1 0.15 4.00 6 0.40 0.15

0.18 – 0.15 0.18 – 0.

153.33 200

. 353

1

.33

5

o c fo

o

f f

D g gD gIntrinsic value of the equity share S

k g

H

KSh

g

s

k

H model

THANK YOU!!!


Economy & Finance

Bond and share valuation