©2009, The McGraw-Hill Companies, All Rights Reserved 3-1 McGraw-Hill/Irwin Chapter Three Interest Rates and Security Valuation
Jan 28, 2015
©2009, The McGraw-Hill Companies, All Rights Reserved
3-1McGraw-Hill/Irwin
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
• 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
• 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
n
n
rrr
FC
rrr
FC
rrr
FC
rrr
FCPV
)1(
~...
)1(
~
)1(
~
)1(
~
3
3
2
2
1
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
• 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
n
n
Err
FC
Err
FC
Err
FC
Err
FCP
)1(
~...
)1(
~
)1(
~
)1(
~
3
3
2
2
1
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)
• 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
n
n
rr
RCF
rr
RCF
rr
RCF
rr
RCFP
)1(...
)1()1()1( 3
3
2
2
1
1
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Bond Valuation
• The present value of a bond (Vb) can be written as:
M = the par value of the bond
INT = the annual interest (or coupon) payment
T = the number of years until the bond matures
i = the annual interest rate (often called yield to maturity (ytm))
• The present value of a bond (Vb) can be written as:
M = the par value of the bond
INT = the annual interest (or coupon) payment
T = the number of years until the bond matures
i = 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
d
b
ddPFIVMPVIFA
INT
i
M
i
INTV
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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
• 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 year
Pt = the stock’s price at the end of year t
is = the interest rate used to discount future cash flows
• The present value of a stock (Pt) assuming zero growth in dividends can be written as:
D = dividend paid at end of every year
Pt = the stock’s price at the end of year t
is = 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
• 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
gi
D
gi
gDP
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:
• 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:
g
P
Dg
P
gDis
0
1
0
0 )1(
0/ PDis
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Relation between InterestRates and Bond Values
Interest Rate
Bond Value
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 aGiven Change inInterest Rates
Absolute Value of Percent Change in aBond’s Price for aGiven Change inInterest Rates
Time to Maturity
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Impact of Coupon Rates onInterest Rate Sensitivity
Bond Value
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
• 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 maturity
CFt = cash flow received at end of period tR = yield to maturity or required rate of return
PVt = present value of cash flow received at end of period t
• 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 maturity
CFt = cash flow received at end of period tR = yield to maturity or required rate of return
PVt = 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
• 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
mR
CFmR
tCF
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
• 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 duration formula:
MD = modified duration = D/(1 + R)
• Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price is found by rearranging the duration formula:
MD = modified duration = D/(1 + R)
RMDR
RD
P
P
1
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Figure 3-7
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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:
• 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 )(2
1)(
2
1
1RCXRMDRCX
R
RD
P
P