S 2-2
Debt Financing and Valuation
Concepts
B.B.Chakrabarti
Professor of Finance
How to Value Bonds?
• Primary Principle:
– Value of financial securities = PV of expected
future cash flows
• Bond value is, therefore, determined by
the present value of the coupon payments
and par value.
• Interest rates are inversely related to
present (i.e., bond) values.
Price of Bonds
• Bond prices are quoted in rupees or dollars and thirty-
seconds (1/32) of a dollar in US.
• The quoted price is for a bond with a face value of 100.
• Thus, a quote of 90-05 in US indicates that the quoted price
for a bond with a face value of $100,000 is $90,156.25
• The quoted price is not the same as the cash price that is paid
by the purchaser.
• Cash price = Quoted price + Accrued Interest since last
coupon date
• Quoted price is referred to as Clean Price and Cash price as
Dirty Price by the traders
Price of Bonds - Example
Suppose that it is March 5, 2022, and the bond
under consideration is an 11% coupon bond with
semi-annual coupon maturing on July 10, 2024, with
a quoted price of 95-16 or $95.50.
- The most recent coupon date is January 10, 2022,
and the next coupon date is July 10, 2022.
- The number of days between January 10, 2022,
and March 5, 2022, is 54, and 181 between January
10, 2022, and July 10, 2022.
Price of Bonds - Example
- On a bond with $100 face value, the coupon
payment is $5.50 on January 10 and July 10.
- The accrued interest on March 5, 2022, is the
share of the July 10 coupon accruing to the
bondholder on March 5, 2022.
- Because actual/actual in period is used for
Treasury bonds, this is
(54/181)*$5.5 = $1.64
- The cash price per $100 face value for the July
10, 2022, bond is therefore
$95.5+ $1.64 = $97.14
Valuation of a Bond
N= no. of coupons P = price today
C= annual coupon y = yield
N
C/2 100
P=å +
j =1 (1 + y / 2) (1 + y / 2)
j N
Price of Bonds Between Coupon
Dates
Settlement date
0 1 2 N
z
x
Last Next
N= no. of remaining coupons
Coupon Coupon
Z= no. of days between SD and NCD
date date
X= no. of days between LCD and NCD
C= annual coupon
Price of Bonds Between Coupon
Dates
N -1
C/2 100
P=å j+z / x
+ N -1+ z / x
j = 0 (1 + y / 2) (1 + y / 2)
Valuation of a Bond using Spot
Rates
N= no. of coupons P = price today
C= annual coupon rj = jth. period spot rate
N
C/2 100
P=å +
j =1 (1 + r j / 2) j
(1 + rN / 2) N
Pure Discount Bond: Example
Find the value of a 30-year zero-coupon
bond with a $1,000 par value and a YTM
of 6%.
$0 $0 $0 $ 1,000
0 1 2 29 30
FV $1,000
PV = = = $174 .11
(1 + y ) T
(1.06 ) 30
Level Coupon Bonds
• Make periodic coupon payments in
addition to the maturity value
• The payments are equal each period.
Therefore, the bond is just a combination
of an annuity and a terminal (maturity)
value.
• Coupon payments are typically
semiannual.
• Effective annual rate (EAR) =
(1 + R/m)m – 1
Level Coupon Bond: Example
• Consider a U.S. government bond with a 6
3/8% coupon that expires in December 2010.
– The Par Value of the bond is $1,000.
– Coupon payments are made semi-annually (June
30 and December 31 for this particular bond).
– Since the coupon rate is 6 3/8%, the payment is
$31.875.
– On January 1, 2006 the size and timing of cash
flows are:
$31.875 $31.875 $31.875 $ 1,031 .875
1 / 1 / 06 6 / 30 / 06 12 / 31 / 06 6 / 30 / 10 12 / 31 / 10
Level Coupon Bond: Example
• On January 1, 2010, the required annual
yield is 5%.
$31.875 é 1 ù $1,000
PV = ê1- 10 ú
+ 10
= $1,060.17
.05 2 ë (1.025) û (1.025)
Bond Pricing with a
Spreadsheet
• There are specific formulas for finding
bond prices and yields on a spreadsheet.
– PRICE(Settlement,Maturity,Rate,Yld,Redemption,
Frequency,Basis)
– YIELD(Settlement,Maturity,Rate,Pr,Redemption,
Frequency,Basis)
– Settlement and maturity need to be actual dates
– The redemption and Pr need to be given as % of par
value
• Click on the Excel icon for an example.
Bond Concepts
q Bond prices and market interest rates
move in opposite directions.
q When coupon rate = YTM, price = par
value
q When coupon rate > YTM, price > par
value (premium bond)
q When coupon rate < YTM, price < par
value (discount bond)
YTM and Bond Value
When the YTM < coupon, the bond
1300 trades at a premium.
Bond Value
1200
1100 When the YTM = coupon, the
bond trades at par.
1000
800
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
6 3/8 Discount Rate
When the YTM > coupon, the bond trades at a discount.
Bond Example Revisited
• Using our previous example, now assume
that the required yield is 11%.
• How does this change the bond’s price?
$31.875 $31.875 $31.875 $ 1,031 .875
1 / 1 / 06 6 / 30 / 06 12 / 31 / 06 6 / 30 / 10 12 / 31 / 10
$31.875 é 1 ù $1,000
PV = ê1- 10 ú
+ 10
= $825.69
.11 2 ë (1.055) û (1.055)
