Chapter Three

Interest Rates and

Security Valuation

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 Various Interest Rate Measures

•• Coupon
Coupon rate
rate
•• Required
Required Rate
Rate of
of Return
Return
•• Expected
Expected rate
rate of
of return
return
•• Realized
Realized Rate
Rate of
of Return
Return

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 COUPON RATE

• is the annual (or periodic) cash flow that the

bond issuer contractually promises to pay
the bond holder.

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 Required Rate of Return

• The interest rate used to find the fair present

value of a financial security
• (Effective interest rate)

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 Required Rate of Return

~~ ~~ ~~ ~~
FPV
FPV == CFCF11 ++ CF CF22 ++ CF CF33 ++… …++ CF CFnn
(1 + rrr) 11
(1 + rrr)
(1 + rrr) (1 + rrr)
22
(1 + rrr)
(1 + rrr)
33
(1 + rrr)
(1 + rrr)
nn

Where:
Where: rrr
rrr == Required
Requiredraterateof
ofreturn
return
CF
CF11 == Cash
Cashflow
flowprojected
projectedininperiod
periodtt(t(t==1,1,…,
…,n)n)
~~ == Indicates
Indicatesthat
thatprojected
projectedcash
cashflow
flowisisuncertain
uncertain
(due
(dueto
todefault
defaultand
andother
otherrisks)
risks)
nn == Number
Numberof ofperiods
periodsininthe
theinvestment
investmenthorizon
horizon
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 Required Rate of Return

• PV: Current Market Price

• Market Price < PV= Undervalued : Buy!
• Market Price > PV= Overvalued: X
• MP = PV: Fairly priced

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 Undervalued or overvalued?

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 Expected Rate of Return

• interest rate a market participant expects to

earn by buying the security at its current
market price , receiving all projected cash
flow payments ( CF s) on the security, and
selling the security at the end of the
participant’s investment horizon

• Basis: Current Price

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 Expected Rate of Return

~~ ~~ ~~ ~~
PP == CFCF11 ++ CF CF22 ++ CF CF33 ++… …++ CF CFnn
(1
(1++Err)
Err)1 (1
1
(1++Err)
Err)2 (1
2
(1++Err)
Err)3
3
(1
(1++Err)
Err)n
n

Where:
Where: Err
Err == Expected
Expectedrate
rateof
ofreturn
return
CF
CF11 == Cash
Cashflow
flowprojected
projectedininperiod
periodtt(t(t==1,1,…,
…,n)n)
~~ == Indicates
Indicatesthat
thatprojected
projectedcash
cashflow
flowisisuncertain
uncertain
(due
(dueto
todefault
defaultand
andother
otherrisks)
risks)
nn == Number
Numberof ofperiods
periodsin
inthe
theinvestment
investmenthorizon
horizon
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 current market price ≠ FPV

 Efficiency

• undervalued
• Overvalued

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 MARKET EFFICIENCY

• The process by which financial security

prices move to a new equilibrium when
interest rates or a security-specific
characteristic changes.

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 Realized Rate of Return

• is the interest rate actually earned on an

investment in a financial security

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 Realized Rate of Return

The
The actual
actual interest
interest rate
rate earned
earned on
on anan
investment
investment inin aa financial
financial security
security

PP == RCF
RCF11 ++ RCFRCF22 ++ …… ++ RCF
RCFnn
(1 + rr)
(1 + rr)
11
(1 + rr)
(1 + rr)
22
(1 + rr)
(1 + rr)
nn

Where:
Where: RCF
RCF == Realized
Realizedcash
cashflow
flowininperiod
periodtt(t(t==1,1,…,
…,n)
n)
rrrr == Realized
Realizedrate
rateof
ofreturn
returnon
onaasecurity
security
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 • realized rate of return > required rate of
return : actually earned more than was
needed for the risks
• If the realized rate of return < the required
rate of return : actually earned less than the
interest rate required for the risks
•

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 Bond Valuation

•• The
The valuation
valuation of
of aa bond
bond instrument
instrument employs
employs
time
time value
value of
of money
money concepts
concepts
–– Reflects
Reflects present
present value
value of
of all
all cash
cash flowsflows promised
promised or or
projected,
projected, discounted
discounted at at the
the required
required rate rate of
of return
return
(rrr)
(rrr)
–– Expected
Expected rate rate of
of return
return (Err)
(Err) isis the
the interest
interest rate
rate that
that
equates
equates thethe current
current market
market price
priceto to the
thepresent
present
value
value ofof all
all promised
promised cashcash flows
flows received
received over
over the
the
life
life of
of the
the bond
bond
–– Realized
Realized raterateofof return
return (rr)
(rr) on
on aa bond
bond isis the
the actual
actual
return
return earned
earned on on aa bond
bond investment
investment that that has
has already
already
taken
taken place
place
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 COUPON BOND

• Bonds that pay interest based on a stated

coupon rate. The interest, or coupon,
payments per year are generally constant
over the life of the bond.

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 zero-coupon bonds

• Bonds that do not pay interest.

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 Bond Valuation Formula
VVbb == INT/2
INT/2 ++ INT/2 INT/2 ++ ......++ INT/2
INT/2__ __
(1
(1++iidd/2)
/2)11 (1
(1++iidd/2)
/2)22 (1
(1++iidd/2)
/2)2N2N
++ M_M_ ____
(1
(1++iidd/2)
/2)2N2N
Where:
Where: VVbb == Present
Presentvalue
valueof
ofthe
thebond
bond
MM == Par Paror
orface
facevalue
valueofofthe
thebond
bond
INT
INT== Annual
Annualinterest
interest(or
(orcoupon)
coupon)payment
paymentper peryear
year
on
onthe
thebond;
bond;equals
equalsthe
thepar
parvalue
valueofofthe
thebond
bond
times
timesthe
the(percentage)
(percentage)coupon
couponrate
rate
NN == Number
Numberyearsyearsuntil
untilthe
thebond
bondmatures
matures
idid == Interest
Interestrate
rateused
usedtotodiscount
discountcash
cashflows
flowsononthe
thebond
bond

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 Bond Valuation Example

V
Vbb == 1,000(.1)
1,000(.1) (PVIFA
(PVIFA8%/2, ) + 1,000(PVIF 8%/2,12(2)
12(2)) + 1,000(PVIF8%/2,
8%/2,12(2)
)
12(2))
22

Where:
Where: VVbb ==$1,152.47
$1,152.47(solution)
(solution)
MM == $1,000
$1,000
INT
INT ==$100
$100per peryear
year(10%
(10%ofof$1,000)
$1,000)
NN == 1212years
years
iid ==8%
8%(rrr)
(rrr)
d
PVIF
PVIF==Present
Presentvalue
valueinterest
interestfactor
factorof
ofaalump
lumpsum
sumpayment
payment
PVIFA
PVIFA==present
presentvalue
valueinterest
interestfactor
factorof
ofan
anannuity
annuitystream
stream

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 Description of a Premium, Discount,
and Par Bond

•• Premium
Premium bond—when
bond—when the
the coupon
coupon rate,
rate, INT,
INT, isis
greater
greater then
then the
the required
required rate
rate of
of return,
return, rrr,rrr, the
the
fair
fair present
present value
value of of the
the bond
bond (V(Vbb)) isis greater
greater than
than
its
its face
face value
value (M)
(M)
•• Discount
Discount bond—
bond—whenwhen INT<rrr,
INT<rrr, then
then V
Vbb<M
<M
•• Par
Par bond—
bond—when
when INT=rrr,
INT=rrr, then
then V
Vbb=M
=M

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 How do you decide?

• investors make the decision to buy or sell

by comparing the bond’s present value to its
current market price.

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 Yield to Maturity

The
The return
return oror yield
yield the
the bond
bond holder
holder will
will earn
earn on
on
the
the bond
bond ifif he
he or
or she
she buys
buys itit at
at its
its current
current market
market
price,
price, receives
receives all
all coupon
coupon and
and principal
principal payments
payments
as
as promised,
promised, andand holds
holds the
the bond
bond until
until maturity
maturity

V
Vbb == INT
INT (PVIFA
(PVIFAytm/m, ) + M(PVIF ytm/m,Nm))
Nm) + M(PVIFytm/m,Nm
ytm/m,Nm
m
m

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 • The yield to maturity calculation implicitly
assumes that all coupon payments
periodically received by the bond holder
can be reinvested at the same rate

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 • YTM= 12%

• Current bond selling at rrr of 11%?

• Current bond selling at rrr of 13%?

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 Summary of Factors that Affect Security
Prices and Price Volatility when Interest
Rates Change

•• Interest
Interest Rate
Rate
•• Time
Time Remaining
Remaining to
to Maturity
Maturity
•• Coupon
Coupon Rate
Rate

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 Impact of Interest Rate Changes on
Security Values
Interest
Interest
Rate
Rate
12%

10%

8%

Bond
BondValue
Value
874.50 1,000 1,152.47

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 Balance sheet of an FI before and
after an Interest Rate Increase
(a)
(a)Balance
BalanceSheet
Sheetbefore
beforethe
theInterest
InterestRate
RateIncrease
Increase
Assets Liabilities and Equity
Bond $1,152.47 Bond $1,000
(8% required (10% required
rate of return) rate of return)
Equity $152.47

(b) Balance Sheet after 2% increase in the Interest Rate Increase

Assets Liabilities and Equity
Bond $1,000 Bond $874.50
(10% required (12% required
rate of return) rate of return)
Equity $125.50

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 Impact of Maturity on Security Values

12
12Years
YearstotoMaturity
Maturity 16 Years to Maturity
Required Percentage Percentage
Rate of Fair Price Price Fair Price Price
Return Price* Change Change Price* Change Change

8% $1,152.47 $1,178.74
-$152.47 -13.23% -$178.74 -15.16%
10% $1,000.00 $1,000.00
-$125.50 -12.55% -$140.84 -14.08%

12% $874.50 $859.16

*The bond pays 10% coupon interest compounded semiannually and has a face value
of $1,000

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 Impact of a Bond’s Maturity
on its Interest Rate Sensitivity
Absolute
AbsoluteValue
Valueofof
Percent
PercentChange
Changein inaa
Bond’s
Bond’sPrice
Pricefor
foraa
Given
GivenChange
Changein in
Interest
InterestRates
Rates

Time to Maturity

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McGraw-Hill/Irwin 3-35 ©2007, The McGraw-Hill Companies, All Rights Reserved
 Impact of a Bond’s Coupon Rate
on Its Interest Rate Sensitivity
Interest
Interest
Rate
Rate
High-Coupon Bond

Low-Coupon Bond

Bond Value

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 Duration: A Measure of
Interest Rate Sensitivity
The
The weighted-average
weighted-average time
time to
to maturity
maturity on
on an
an
investment
investment
NN NN


 CFt t  tt
CF 
 PVt t  tt
PV
t t==11
(1 + R)
(1 + R)
tt t t==11

D
D == NN == NN


 CF
CFt t 
 PV
PVt t
t t==11
(1 + R)
(1 + R)
tt t t==11

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 Features of the Duration Measure

•• Duration
Duration and
and Coupon
Coupon Interest
Interest
–– the
the higher
higher the
the coupon
coupon payment,
payment, the
thelower
lower isis aa
bond’s
bond’s duration
duration
•• Duration
Duration and
and Yield
Yield to
to Maturity
Maturity
–– duration
duration increases
increases as
as yield
yield to
to maturity
maturity increases
increases
•• Duration
Duration and
and Maturity
Maturity
–– Duration
Duration increases
increases with
with the
the maturity
maturity of
of aa bond
bond but
but
at
at aa decreasing
decreasing rate
rate
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 Discrepancy Between Maturity and Duration on a
Coupon Bond

7
6
5
4
3
2
1
0
1 2 3 4 5 6
Maturity Duration

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 Economic Meaning of Duration

•• Measure
Measure of of the
the average
average life
life of
of aa bond
bond
•• Measure
Measure of of aa bond’s
bond’s interest
interest rate
rate
sensitivity
sensitivity (elasticity)
(elasticity)

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