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©2009, The McGraw-Hill Companies, All Rights Reserved 3-1 McGraw-Hill/Irwin Chapter Three Interest Rates and Security Valuation

Financial Markets & Institutions Ch03

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Page 1: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-1McGraw-Hill/Irwin

Chapter ThreeInterest Rates

and Security

Valuation

Page 2: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-2McGraw-Hill/Irwin

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

Page 3: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-3McGraw-Hill/Irwin

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

Page 4: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-4McGraw-Hill/Irwin

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

Page 5: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-5McGraw-Hill/Irwin

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

Page 6: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-6McGraw-Hill/Irwin

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

Page 7: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-7McGraw-Hill/Irwin

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

Page 8: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-8McGraw-Hill/Irwin

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 /

Page 9: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-9McGraw-Hill/Irwin

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(

Page 10: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-10McGraw-Hill/Irwin

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

Page 11: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-11McGraw-Hill/Irwin

Relation between InterestRates and Bond Values

Interest Rate

Bond Value

Interest Rate

Bond Value

12%

10%

8%

874.50 1,000 1,152.47

Page 12: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-12McGraw-Hill/Irwin

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

Page 13: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-13McGraw-Hill/Irwin

Impact of Coupon Rates onInterest Rate Sensitivity

Bond Value

Bond Value

Interest Rate

Low-Coupon Bond

High-Coupon Bond

Page 14: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-14McGraw-Hill/Irwin

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

Page 15: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-15McGraw-Hill/Irwin

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(

Page 16: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-16McGraw-Hill/Irwin

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(

Page 17: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-17McGraw-Hill/Irwin

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

Page 18: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-18McGraw-Hill/Irwin

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

Page 19: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-19McGraw-Hill/Irwin

Figure 3-7

Page 20: Financial Markets & Institutions Ch03

©2009, The McGraw-Hill Companies, All Rights Reserved

3-20McGraw-Hill/Irwin

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