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Chapter ThreeInterest Rates
and Security
Valuation
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Various Interest Rate Measures
• Coupon rate– periodic cash flow a bond issuer contractually
promises to pay a bond holder• Required rate of return (rrr)
– rates used by individual market participants to calculate fair present values (PV)
• Expected rate of return (Err)– rates participants would earn by buying securities at
current market prices (P)• Realized rate of return (rr)
– rates actually earned on investments
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Required Rate of Return
• The fair present value (PV) of a security is determined using the required rate of return (rrr) as the discount rate
CF1 = cash flow in period t (t = 1, …, n)~ = indicates the projected cash flow is uncertainn = number of periods in the investment horizon
nn
rrrFC
rrrFC
rrrFC
rrrFCPV
)1(
~...
)1(
~
)1(
~
)1(
~3
32
21
1
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Expected Rate of Return
• The current market price (P) of a security is determined using the expected rate of return (Err) as the discount rate
CF1 = cash flow in period t (t = 1, …, n)~ = indicates the projected cash flow is uncertainn = number of periods in the investment horizon
nn
ErrFC
ErrFC
ErrFC
ErrFCP
)1(
~...
)1(
~
)1(
~
)1(
~3
32
21
1
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Realized Rate of Return
• The realized rate of return (rr) is the discount rate that just equates the actual purchase price ( ) to the present value of the realized cash flows (RCFt) t (t = 1, …, n)P
nn
rrRCF
rrRCF
rrRCF
rrRCFP
)1(...
)1()1()1( 33
22
11
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Bond Valuation
• The present value of a bond (Vb) can be written as:
M = the par value of the bondINT = the annual interest (or coupon) paymentT = the number of years until the bond maturesi = the annual interest rate (often called yield to maturity (ytm))
)()(2
)2/1()2/1(1
2
2,2/2,2/
2
12
TiTi
T
tT
d
t
db
ddPFIVMPVIFAINT
iM
iINTV
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Bond Valuation
• A premium bond has a coupon rate (INT)greater then the required rate of return (rrr) and the fair present value of the bond (Vb) is greater than the face value (M)
• Discount bond: if INT < rrr, then Vb < M• Par bond: if INT = rrr, then Vb = M
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Equity Valuation
• The present value of a stock (Pt) assuming zero growth in dividends can be written as:
D = dividend paid at end of every yearPt = the stock’s price at the end of year tis = the interest rate used to discount future cash flows
st iDP /
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Equity Valuation
• The present value of a stock (Pt) assuming constant growth in dividends can be written as:
D0 = current value of dividendsDt = value of dividends at time t = 1, 2, …, ∞
g = the constant dividend growth rate
giD
gigDP
s
t
s
t
t
10 )1(
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Equity Valuation
• The return on a stock with zero dividend growth, if purchased at price P0, can be written as:
• The return on a stock with constant dividend growth, if purchased at price P0, can be written as:
gPDg
PgDis
0
1
0
0 )1(
0/ PDis
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Relation between InterestRates and Bond Values
Interest Rate
Bond Value
12%
10%
8%
874.50 1,000 1,152.47
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Impact of Maturity onInterest Rate Sensitivity
Absolute Value of Percent Change in aBond’s Price for a
Given Change inInterest Rates
Time to Maturity
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Impact of Coupon Rates onInterest Rate Sensitivity
Bond Value
Interest Rate
Low-Coupon Bond
High-Coupon Bond
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Duration
• Duration is the weighted-average time to maturity (measured in years) on a financial security
• Duration measures the sensitivity (or elasticity) of a fixed-income security’s price
to small interest rate changes
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Duration
• Duration (D) for a fixed-income security that pays interest annually can be written as:
t = 1 to T, the period in which a cash flow is receivedT = the number of years to maturityCFt = cash flow received at end of period tR = yield to maturity or required rate of returnPVt = present value of cash flow received at end of period t
T
tt
T
tt
T
tt
t
T
tt
t
PV
tPV
RCF
RtCF
D
1
1
1
1
)1(
)1(
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Duration
• Duration (D) (measured in years) for a fixed-income security in general can be written as:
m = the number of times per year interest is paid
T
mtmt
t
T
mtmt
t
mRCF
mRtCF
D
/1
/1
)/1(
)/1(
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Duration
• Duration and coupon interest– the higher the coupon payment, the lower the bond’s
duration• Duration and yield to maturity
– the higher the yield to maturity, the lower the bond’s
duration• Duration and maturity
– duration increases with maturity at a decreasing rate
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Duration and Modified Duration
• Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price is found by rearranging the durationformula:
MD = modified duration = D/(1 + R)
RMDR
RDPP
1
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Convexity
• Convexity (CX) measures the change in slope of the price-yield curve around interest rate level R
• Convexity incorporates the curvature of the price-yield curve into the estimated percentage price change of a bond given an interest rate change:
22 )(21)(
21
1RCXRMDRCX
RRD
PP