Chapter-8
Bond Valuation
Bond Valuation
Bond is long term financial security which is used to raise long term fund.
Bond is long term promissory note, which is issued by corporation or government or specific
debt contract between two parties under which borrower of loan (issuer of loan) premise to pay
fixed period of interest upon maturity and repay the principle at the maturity to the lender (or
holder of bond).
Features of the bond:
a. Face value or Maturity Value(M)= Rs 1000
b. Market value of bond (Vo)= Determine by demand and supply of market
c. Actual value or Intrinsic value or Face value or Theoretical value or Calculated value (Vo)
= The present value which is calculated by using equation or formula.
d. Coupon(c)= The interest rate which is paid by the bond issuer to holder.
e. Interest (I)= c % of M
f. Market interest rate or going rate of interest or paid by others in a market or required on
bond or yield on bond or yield to maturity ( YTM or k)
g. Maturity or Remaining life (n)
h. Call features=An additional features provided to bond issuer to call the bond before the
maturity with call price.
i. Convertibility=An additional features provided to the bond holder to convert the bond
into fixed number of common stock at pre-determined conversion price within the
specified time period.
j. Default risk= Possibility of non-payment of interest and principal at that situation bond
holder sell the collateral and cover his claim.
k. Trustee= An institution or individual who plays the role of guarantee on behalf of holder.
l. Bond indenture= A legal document which specify the term and condition of debt
contract.
m. Sinking fund= A separate fund or reserve fund established by issuer to repay the bond.
Bond valuation:
It is the process of finding the present value or actual value of bond by
discounting all future cash flow to zero year.
Bond valuation answers how much should be paid for the bond.
Valuation of bond (Vo)= PV of future cash flow
= PV of interest + PV of maturity value
Vo = I x PVIFA@ k%, n year + Mx PVIF @ k%, n year
OR
I 1 M
Vo= [1- ]+
k ¿¿ ¿¿
Valuation of different types of bond:
a. Perpetual bond or irredeemable bond:
The bond which pays interest forever or if the life is not identified that is perpetual bond.
I
Vo=
k
b. Zero coupon bond:
The bond which never pays coupon or which only repays principal after certain period.
M
Vo=
¿¿
c. Normal bond or straight bond:
The bond which has the normal bond feature is normal bond.
I 1 M
Vo= [1- ]+
k ¿¿ ¿¿
d. Callable bond:
The bond which will be called before maturity period is called callable bond.
I 1 Call price
Vo= [1- ]+
k ¿¿ ¿¿
e.Convertible bond:
The bond which will be converted after certain period.
I 1 C.V
Vo= [1- ]+
k ¿¿ ¿¿
Eg
ABC company has issued 10 % coupon bond with 10 years maturity and current market interest
rate is 12 %.
i. How much should you pay for the bond?
ii. How much should you pay if it is callable in 4 years at 115%?
iii. What will be the price if it is zero coupon bond?
iv. How much will you pay if it is perpetual?
Note:
SN Condition Valuation Decision
1. Market value>Actual value Over valued Sell
2. Market value<Actual value Under valued Buy
3. Market value=Actual value Correctly valued No action
2. If interest is paid semiannually
I k
, ,nx2
2 2
3. If interest is paid quarterly
I k
, ,nx4
4 4
4.Effective YTM=( 1 + periodic rate)m-1
Some cases of the bond valuation:
1.There is inverse relationship between the interest rate and the value of the bond.
Eg
Consider the following information of the particular bond;
Bond A:coupon rate =8%, face value= Rs 1000, life of the bond = 10 years
i. Find the value of the bond if YTM is 12%
ii. Find the value of the bond if YTM is 10%
iii. Find the value of the bond if YTM is 8%
iv. What’s your conclusion?
2.i.If coupon rate>YTM then Vo>M, that bond is premium bond
ii. If coupon rate<YTM then Vo<M, that bond is discount bond
iii. If coupon rate=YTM then Vo=M, that bond is par bond
Consider the following bonds with different coupon rate.
Bond Face value Coupon YTM Nature of bond Life
A Rs1000 12% 10% Premium bond 10 years
B Rs1000 10 10 Par bond 10
C Rs1000 8 10 Discount bond 10
a. Find the value of each bond today or at 0 year. (Hints n=10 years)
b. Find the value of each bond 2 years from now or in 2 years. (Hints n=8 years)
c. Find the value of each bond 5 years from now or in 5 years. (Hints n=5 years)
d. Find the value of each bond 8 years from now or in 8 years. (Hints n=2 years)
e. Find the value of each bond 10 years from now or in 10 years. (Hints n=10 years)
f. What’s your conclusion from above calculation?
.Return on bond
Current yield=Yield on bond on market value
I
CY=
Vo
Coupon yield= Current yield
Ep−Bp+ I
HPR=
Bp
#Yield to maturity:
It is the actual yield if the bond is held up to maturity.
It is the discounted rate which equates & the market value and actual value.
It is IRR of bond because it makes zero NPV.
Assumption of YTM:
All cash flows are reinvested at same YTM.
Bond should hold up to maturity or should not be called.
Cash flow should be received without default.
Step i
Approx YTM
M −Vo
n
OR k=I+
M + 2Vo
3
Step ii
I 1 M
Vo= [1- ]+
k ¿¿ ¿¿
Step iii
By interpolation
Volr−Vo
YTM/2= LR + (HR-LR)
Volr−Vohr
Eg.
ABC Corporation issued Rs 1000 face value,10 % coupon bond paid semi annually and
has 10 years to maturity, current market price of Rs 1050.Calcualte the YTM.
Solution:
Face value(M) = Rs 1000
Maturity (n)= 10 years
Coupon rate(C)= 10 %
Interest (I)= 10 % of Rs 1000 = Rs 100
Market price (Vo)=Rs1050
YTM=?
As we know YTM equates the market value and present value.
I /2 1 M
Vo= [1- ]+
k /2 ¿ ¿ ¿ ¿
100/2 1 1000
1050= [1- ]+
k /2 ¿¿ ¿¿
Step 1
Approx YTM/2
M −Vo
nx 2
OR k/2=I/2+
M + 2Vo
3
1000−1050
10 x 2
= 100/2+
1000+2 x 1050
3
=4.6%
Step II
Let’s try at 5%
Vo=1000(YTM=C)
Vo=1000<1050,so let’s try at 4%
50 1 1000
Vo= [1- ]+
0.04 ¿¿ ¿¿
=1136>1050
Step III
Our required value, Vo=1050 lies between 1000 to 1136 so YTM/2 must lies between 4 to 5
%.
By interpolation
Volr−Vo
YTM/2= LR + (HR-LR)
Volr−Vohr
1136−1050
=4+ (5-4)
1136−1000
=4.6%
Nominal annual=YTM/2x2
=4.6x2
=9.2%
Effective annual rate=(1+ semi annual rate)m-1
=(1+0.046)2-1
=9.4%
#Yield to call:
Actual yield of the bond which is hold up to call period.
IRR of callable bond
It is the discount rate which equates market value and present value.
Assumption:
All cash flows are reinvested at YTC
Bond should be called
All cash flows should be received without default
Eg
If the above bond is called after 4 years at Rs 1100.What is YTC?
Hints;
YTM=YTC
M=Call price
n= c.p(call period)
# Duration:
It is the average length of time which is required to cover the investment on bond in
present value terms.
It is the discounted payback period of bond.
It is the recovery period of bond.
It is developed by Frederick R. Macaulay in 1938 so it is called Macaulay duration.
It shows the sensitivity of bond’s price with respect to YTM so it is also known as
Elasticity of bond price.so MD= EL
1+ y ( 1+ y )+ t(c− y )
M.D= -
y c¿¿
If coupon is paid semi annually
1+ y /2 ( 1+ y /2 )+ tx 2(c /2− y /2)
M.D=[ - ] ÷2
y /2 c /2¿ ¿
Eg
Consider a bond with following information
Face value=Rs 1000
Coupon rate=10%
Value of bond=Rs1000
Yield to maturity= 10%
Term to maturity=5 years
Duration=?
#Rules Regarding Duration:
Rule 1: Duration of a zero-coupon bond equals it’s time to maturity.
D = n (maturity period)
Rule 2: Holding maturity period constant, a bond’s duration is higher when the coupon rate is
low.
Rule 3: Holding the coupon rate constant, a bond’s duration generally increases with its time to
maturity.
Rule 4: Holding other factors constant, the duration of a coupon bond is higher when bond’s
yield to maturity is lower.
Rule 5: The duration for perpetual bond is:
1+ y
D=
y
Rule 6: Duration of a coupon bond equal is:
1+ y ( 1+ y )+ t(c− y )
D= -
y c¿¿
Rule 7: Duration of coupon bond selling at par value is:
1+ y
D= -¿]
y
Rule 8: Duration of bond portfolio (Dp)= WAxDA+ WBxDB+……+ WZxDZ
# Modified Duration:
It shows the bond’s percentage change in its yield. It is a better measure of the price risk of a
bond than the Macaulay’s duration because it shows how a bond’s duration changes in relation
to interest rate movements.
'
Macaula y s duration
MD=
(1+YTM )
# Bond immunization:
After purchasing the bond, bondholder has to bear two types of risk due to change in interest
rate. They are
1. Re investment risk
2. Price risk
If market interest rate change, you would not get the cash flow according to your expectation
due to reinvestment risk & price risk which is also known as systematic risk of bond so bond
immunization is the process of holding bond portfolio (combination of long term & short term)
to minimize the interest rate risk so that we would get the return according to our expectation.
If interest rate is expected increase, short term bond is suitable
If interest rate is expected decrease, long term bond is suitable
According to immunization holding period or investment horizon is always equal to the
weighted average duration of bond portfolio.
Eg.
Consider the following two bonds.
Bond A Bond B
Term 5 years 1 year
Coupon 12% 7%
Face value Rs 1000 Rs1000
YTM 10% 10%
a. Calculate the durations of both bonds.
b. Suppose a portfolio manager has an obligation to pay Rs 1,000,000 in two years. How
much the manger would need to invest now in the bonds so that he has cash to pay in
two years?
c. How much should he invest in Bond A and how much in Bond B to create fully
immunized portfolio?
Now
a.
Duration of A
1+ y ( 1+ y )+ t(c− y )
D= -
y c¿¿
1+ .1 ( 1+ .1 )+ 5(.12−.1)
= -
.1 .12¿ ¿
=4.07 years
Duration of B
1+ y ( 1+ y )+ t(c− y )
D= -
y c¿¿
1+ .1 ( 1+ .1 )+ 1(.12−.1)
= -
.1 .12¿ ¿
= 1 year
FV
b. PV=
¿¿
1,000,000
=
¿¿
=Rs 826,446.28
c.
According to immunization
Investment horizon = Weighted average duration of portfolio
Now
Holding period=WAxDA + WBxDB
Or 2 = WAx4.07 + (1-WA) x1
Or WA = 0.33
WB=1-0.33= 0.67
He should invest 33% of fund in Bond A and 67 % of fund in Bond B.
