Bond Valuation
by Binam Ghimire
1
Learning Objectives
Bond valuation
Understand the relationship between bond and interest rate
Compute Yield to Maturity
Work out bond valuation in Excel
The alternative bond yields that are important to investors
Spot rates and forward rates and how do you calculate these rates
from a yield to maturity curve
 What is the spot rate yield curve and forward rate curve
 How and why use the spot rate curve to determine the value of a
bond
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2
The Fundamentals of Bond Valuation
The present-value model
Pp
Ct 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Where:
Pm=the current market price of the bond
n = the number of years to maturity
Ci = the annual coupon payment for bond i
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond
The Present Value Model
 For example. 8% coupon, 30-year maturity bond with
par value of $1,000 paying 60 semi-annual coupon
payments of $40 each. Suppose interest rate is 8% p.a
or 4% per-6month. What is the price of the bond?
4
The Present Value Model
 Then bond price is =
60
𝑡=1
$40
1.04
𝑡
+
$1,000
1.04 60
= $810.71
5
The Price Yield Curve
 When we increase the required rate of return, the
market price falls down
6
The Price Yield Curve
 If yield < coupon rate, bond will be priced at a
premium to its par value
 If yield > coupon rate, bond will be priced at a
discount to its par value
 Price-yield relationship is convex (not a straight line)
The Price Yield Curve
 X Company has just issued 8 percent, 10-years, £ 1,000
par bond. Current market interest rate is 8 percent.
What is the price of the bond?
 What will be the bond price if interest rate were to a)
rise to 10% b) fell to 6%
8
The Price yield curve
 The results:
Required rate of return
Bond value
Status
10%
£ 877.06
Discount
8
1,000.00
Par value
6
1,147.21
Premium
9
The Yield Model
The expected yield on the bond may be computed from
the market price
Pp
Ci 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Where:
i = the discount rate that will discount the cash flows to equal the
current market price of the bond
YTM
 Discussed before but see again
 The Yield to Maturity (YTM) of a bond represents the
rate of return investors earn if they buy the bond at a
specific price and hold it until maturity
 YTM is the interest rate that makes the present value of
a bond’s payments equal to its price
 So when price of bond = face value of bond then YTM =
Coupon Interest Rate
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YTM
 In October 2007 Tesco raised $2bn (£990m) of debt in its first
dollar-denominated bond issue.
 The bond issue includes 10-year notes paying 5.5 per cent interest
(US$ 850m) and 30-year notes paying 6.15 per cent interest (US
1150m).
 The proceeds of the debt raising, which was jointly arranged by
Citigroup and JP Morgan Cazenove, would be used for "general
corporate purposes“.
 What does this bond offer?
 the first one pays 5.5/2 = 2.75% every six months until
November 2017 then it pays the coupon and the par value of
$100.
 The observed price of the first bond in Datastream was $ 96.28.
Find the YTM for the first bond
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YTM
24/07/2008 Year (24/07/08)
15/11/2008
0.31
15/05/2009
0.81
15/11/2009
1.31
15/05/2010
1.81
15/11/2010
2.31
15/05/2011
2.81
15/11/2011
3.31
15/05/2012
3.81
15/11/2012
4.31
15/05/2013
4.81
15/11/2013
5.31
15/05/2014
5.81
15/11/2014
6.31
15/05/2015
6.81
15/11/2015
7.31
15/05/2016
7.81
15/11/2016
8.31
15/05/2017
8.81
15/11/2017
9.31
YTM:
CF
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
102.75
Sum=
6.28%
PV
2.70
2.62
2.54
2.46
2.39
2.32
2.25
2.18
2.12
2.05
1.99
1.93
1.87
1.82
1.76
1.71
1.66
1.61
58.30
96.28
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Computing Bond Yields
Yield Measure
Purpose
Nominal Yield
Measures the coupon rate
Current yield
Measures current income rate
Promised yield to
maturity
Promised yield to call
Measures expected rate of return for bond
held to maturity
Measures expected rate of return for bond
held to first call date
Realized (horizon) yield Measures expected rate of return for a bond
likely to be sold prior to maturity. It
considers specified reinvestment
assumptions and an estimated sales price. It
can also measure the actual rate of return on
a bond during some past period of time.
Nominal Yield
 Measures the coupon rate that a bond investor receives
as a percent of the bond’s par value
Current Yield
 Similar to dividend yield for stocks
 Important to income oriented investors
CY = Ci/Pm
where:
CY = the current yield on a bond
Ci = the annual coupon payment of bond i
Pm = the current market price of the bond
Promised Yield to Maturity
 Widely used bond yield figure
 Assumes
Investor holds bond to maturity
All the bond’s cash flow is reinvested at the
computed yield to maturity
Pp
Ci 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Solve for i that will equate
the current price to all
cash flows from the bond
to maturity, similar to IRR
Computing the
Promised Yield to Maturity
Two methods
 Approximate promised yield
Easy, less accurate
 Present-value model
More involved, more accurate
Approximate Promised Yield
APY 
Ci 
Pp  Pm
n
Pp  Pm
2
=
Coupon + Annual Straight-Line Amortization of Capital Gain or Loss
Average Investment
Computing the
Promised Yield to Maturity
Example
 8%, 20 Year bond, is priced at $900, what is the
promised yield to maturity?
 Answer: 4.45%
Promised Yield to Call
Approximation
 May be less than yield to maturity
 Reflects return to investor if bond is called and
cannot be held to maturity
Pc  Pm
AYC = approximate yield to call (YTC)
Ct 
nc
P = call price of the bond
AYC 
P = market price of the bond
Pc  Pm
C = annual coupon payment
2
nc = the number of years to first call date
Where:
c
m
t
Promised Yield to Call
Present-Value Method
2 nc
Ci / 2
Pc
Pm  

t
2 nc
(1  i )
t 1 (1  i )
Where:
Pm = market price of the bond
Ci = annual coupon payment
nc = number of years to first call
Pc = call price of the bond
Realized Yield Approximation
ARY 
Ci 
Pf  P
hp
Pf  P
Where:
2
ARY = approximate realized yield to call (YTC)
Pf = estimated future selling price of the bond
Ci = annual coupon payment
hp = the number of years in holding period of the
bond
Realized Yield
Present-Value Method
2 hp
Pf
Ct / 2
Pm  

t
2 hp
(1  i 2)
t 1 (1  i 2)
Calculating Future Bond prices
 We need to compute a future price (Pf) when estimating
the expected realised (horizon) yield performance of
bonds
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Calculating Future Bond Prices
Pf 
2 n  2 hp

t 1
Pp
Ci / 2

t
2 n  2 hp
(1  i 2) (1  i 2)
Where:
Pf = estimated future price of the bond
Ci = annual coupon payment
n = number of years to maturity
hp = holding period of the bond in years
i = expected semiannual rate at the end of the holding
period
Calculating Future Bond prices
 Assume you bought 10% 25 year bond at $842 giving it
a promised YTM of 12%. Based on the analysis of the
economy and the capital market, you expect this bond’s
market YTM to decline to 8% in 5 years. Therefore you
want to compute its future price at the end of year 5 to
estimate the expected rate of return, assuming you are
correct in your assessment of the decline in overall
market interest rate.
 Above you estimated the holding period of 5 years
which means the remaining life is 20 years and
estimated future market YTM is 8%
 Find out the future selling price of the bond
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Calculating Future Bond Prices
1
1
P  50
 1,000
(1.04)
(1.04)
40
f
t 1
t
40
Calculating Future Bond Prices
1
1
P  50
 1,000
(1.04)
(1.04)
40
f
t 1
t
 50 (19.7928) + 1,000 (0.2083)
 $1,197.94
40
Bond Valuation using spot rates
 We said we discount all CFs by one common yield but
having one YTM is not realistic
 Investors at any point in time require a different rate of
return for flows at different times
 For example in a zero coupon bond, investors will expect
different rates for bond maturing at 2, 5 or 10 years
 The rates that is used to discount a CF at a certain point
are called spot rates.
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Bond Valuation using spot rates
 See the excel file
 It shows the desire for different rates.
 Analysts recognise that it is inappropriate to discount all
the flows for a bond at one single rate
 For example see page 615, Brown and Reilly (2012)
Analysis Investments And Management Of Portfolios,
10th Ed., Cengage
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Zero Coupon Bonds and Treasury
Strips
 Brown and Reilly (2012) Analysis Investments And
Management Of Portfolios, 10th Ed., Cengage, Page: 443
 Determinants of Bond Safety
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Further Exercise
 You can find more exercise on different bond yields in
Bodie, Kane and Marcus (2008) Investments,
International Edition. Pages: 468-479,
33
Thank you