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 Bond
Current 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 pa

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 KShs
i
i i
i i
 
  
 
  
   
   
  
   
   
5
1
1
1 1 1 1
. 1,000*
1 1 1 1
1 1 1 1 1
Pr . 1,000*
1 1 1 1 1
1
. 1,000*
1
i
Amount receivable after one year hence KShs
i i i i
esent Value KShs
i i i i i
KShs
i
 
 
 
    
     
       
     
      
         
 
  
 
5
1
1,000 ,
1 i
 
  
 

SIR – CONT’D
 Thus:
5
5
5
1
600 1, 000 *
1
1 600
1 1, 000
1
0.6
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
1
t 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
ond
Cost 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
min
t
n
t
n
Annual Interest receivable Ter al Value of a Bond
Cost of Bond
i i
 
  

  


           
           
1 2 3 4 5 5
1 2 3 4 5 5
Pr ., .700
80 80 80 80 80 1,000
700
1 1 1 1 1 1
1 1 1 1 1 1,000
80
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 1,000
700 80
1 1 0.18
1 0.18
1,000
80 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 17
17% 712.4 700
712.4 687.27
1
17% 12.04
247.70
17% 0.05%
17.05%
YTM
YTM
 
      
 
   
 
 


SHORTCUT METHOD FOR ARRIVING AT YTM
 
 
/
/ 2
I FV C n
YTM
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 60
0.1647
850
16.47%
I FV C n
YTM
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 0.12
min
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
n
t
t
t
t
n
t
t
t
t C
D
C
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
 
 1
t
tC
itC
 
 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 S
S
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
02 %
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 S
S
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 n
o n
D D SD D
S
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 n
o n
D D SD D
S
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 g
S
k k k k
   
    
   
 
 
0 1
–
o
D
g
S
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 D
P V
k k k k
    
   
 
 
  
1
.
1–
. N
N
II
P
g
k
V
k
D
g



   . . . .I II
PV PV 
 
 
  1
. .
1 1
1
–
NtN
o t Nt
i e
D gD
k
S
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 PV
Intrinsic 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
D
Intrinsic 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
D
Intrinsic 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 01 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 g
S
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 g
Intrinsic value of the equity share S
k g
H
KSh
g
s
k

 

 


 
 
H model

THANK YOU!!!

Bond and share valuation

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