McGraw-Hill/Irwin

Chapter Three

Interest Rates

and

Security Valuation

Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Various Interest Rate Measures

   

Coupon rate

 periodic cash flow a bond issuer contractually promises to pay a bond holder

Required rate of return (r)

 rates used by individual market participants to calculate

fair present values (PV) Expected rate of return or E(r)

 rates participants would earn by buying securities at

current market prices (P) Realized rate of return ( r )

 rate actually earned on investments 3-2

Required Rate of Return

 The

fair present value (PV)

of a security is determined using the

required rate of return (r)

as the discount rate PV  ( 1 C F 1  r ) 1  ( 1  F 2 r ) 2  ( 1  F 3 r ) 3  ...

 ( 1  F n r ) n

CF 1

= cash flow in period

t

~

(

t

= 1, …,

n

) = indicates the projected cash flow is uncertain

n

= number of periods in the investment horizon 3-3

Expected Rate of Return

 The

current market price (P)

of a security is determined using the

expected rate of return

or

E(r)

as the discount rate P  ( 1  F 1 E ( r )) 1  ( 1  F 2 E ( r )) 2  ( 1  F 3 E ( r )) 3  ...

 ( 1  F n E ( r )) n

CF 1

~

= cash flow in period

t

(

t

= 1, …,

n

) = indicates the projected cash flow is uncertain

n

= number of periods in the investment horizon 3-4

Realized Rate of Return

 The

realized rate of return ( r )

is the discount rate that just equates the

actual purchase price

P

the present value of the realized cash flows (

RCF t

)

t

(

t

= 1, …,

n

) P  ( RCF 1 1  r ) 1  ( RCF 2 1  r ) 2  ( RCF 3 1  r ) 3  ...

 ( RCF n 1  r ) n 3-5

Bond Valuation

 The

present value of a bond

(

V b

) can be written as:

V b



INT

2  INT 2

t

2

T

   1  ( 1  ( 1

r

/ 2 ))  

t

   1  1 

Par

( 1  (

r

/ 2 )) 2

T

(1  (r (r 2 ) 2 )) 2T     Par (1  (r/2)) 2T

Par

= the par or face value of the bond, usually $1,000

INT

= the annual interest (or coupon) payment

T

= the number of years until the bond matures

r

= the annual interest rate (often called

yield to maturity (ytm)

) 3-6

Bond Valuation

 A

premium bond

has a coupon rate (

INT)

greater than the required rate of return (

r

) and the fair present value of the bond (

V b

) is greater than the face or par value (Par)   

Premium bond:

If

INT

>

r

; then

V b

>

Par

Discount bond:

if

INT

<

r

, then

V b

<

Par

Par bond:

if

INT

=

r,

then

V b

=

Par

3-7

Equity Valuation

 The

present value of a stock

(

P t

) assuming zero growth in dividends can be written as: P t  D / r s

D

= dividend paid at end of every year

P t

= the stock’s price at the end of year

t r s =

the interest rate used to discount future cash flows 3-8

Equity Valuation

 The

present value of a stock

(

P t

) assuming constant growth in dividends can be written as: P t  D 0 ( 1  r s  g g ) t  r D t s   1 g

D

0

D t

= current value of dividends = value of dividends at time t = 1, 2, …, ∞

g

= the constant dividend growth rate 3-9

Equity Valuation

 The return on a stock with zero dividend growth, if purchased at current price

P

0

, can be written as: r s  D / P 0  The return on a stock with constant dividend growth, if purchased at price

P

0

, can be written as: r s  D 0 ( 1  g )  g P 0  D 1  g P 0 3-10

Relation between Interest Rates and Bond Values

Interest Rate 12% 10% 8% Bond Value 874.50

1,000 1,152.47

3-11

Impact of Maturity on Price Volatility (a)

Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates Time to Maturity

3-12

Impact of Maturity on Price Volatility (b)

Coupon Par yield rate old yield rate change yield rate new 6.00% 7.00% 0.50% 7.50% Maturity 1 5 10 15 20 25 30 35 40 45 50 55 60 Price old Price new $ 986.05

$ 939.31

$ 897.04

$ 867.59

$ 847.08

$ 832.80

$ 822.84

$ 815.91

$ 811.08

$ 807.72

$ 805.38

$ 803.75

$ 802.61

65 70 75 80 85 90 $ $ $ $ $ $ 801.82

801.27

800.88

800.61

800.43

800.30

95 100 105 $ $ $ 800.21

800.14

800.10

Predicted limit price change = 1 - (r old / r new) 6.67% IM Figure 3.1 Absolute Rate of change 0.47% 2.05% 3.52% 4.55% 5.25% 5.74% 6.06% 6.27% 6.42% 6.51% 6.57% 6.61% 6.63% 6.65% 6.66% 6.66% 6.66% 6.67% 6.67% 6.67% 6.67% 6.67% 3-13

Impact of Coupon Rates on Price Volatility

Bond Value High-Coupon Bond Low-Coupon Bond Interest Rate

3-14

Impact of Coupon on Price Volatility (b)

Coupon Par rate old rate change rate new Coupon rate 6.00% 5.50% 5.00% 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% Varies $1,000 7.00% -0.50% 6.50% Price old $ 929.76 $ 894.65 $ 859.53 $ 824.41 $ 789.29 $ 754.17 $ 719.06 $ 683.94 $ 648.82 $ 613.70 $ 578.59 $ 543.47 Price new $ 964.06 $ 928.11 $ 892.17 $ 856.22 $ 820.28 $ 784.34 $ 748.39 $ 712.45 $ 676.50 $ 640.56 $ 604.61 $ 568.67 Maturity 10 years Absolute Rate of change 3.69% 3.74% 3.80% 3.86% 3.93% 4.00% 4.08% 4.17% 4.27% 4.38% 4.50% 4.64% 0.00% $ 508.35 $ 532.73 IM Figure 2 Coupons and Price Volatility 4.80% 3-15

Impact of r on Price Volatility

Bond Price How does volatility change with interest rates?

Price volatility is inversely related to the level of the initial interest rate

Interest Rate r 3-16

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

captures the coupon and maturity effects on volatility.

3-17

Duration



Duration (Dur)

for a fixed-income security that pays interest annually can be written as:

T CF Dur



t

  1 ( 1 

P

0

t



t r

)

t



t T

  1

PV t P

0 

t

t

P 0

= Current price of the security = 1 to

T

, the period in which a cash flow is received

T

= the number of years to maturity

CF t r

= cash flow received at end of period

t

= yield to maturity or required rate of return

PV t

= present value of cash flow received at end of period

t

3-18

Duration and Volatility

9% Coupon, 4 year maturity annual payment bond with a 8% ytm Dur  t T   1 CF t (1   r) t t P 0

Year (T)

1 2 3 4 Totals

Cash Flow

$ 90 90 90 $1090

PV @8% CF T /(1+r) T

$ 83.33 77.16 71.45 $ 801.18 $1033.12

% of Value PV/Price

8.06% 7.47% 6.92% 77.55% 100.00%

Weighted % of Value (PV/Price)*T

0.0806 0.1494 0.2076 3.1020 3.5396 Duration = 3.5396 years 3-19

Duration



Duration (Dur)

(measured in years) for a fixed income security, in general, can be written as:

Dur



t T

  1 /

m

( 1 

CF t



t r

/

m

)

mt P

0

m

= the number of times per year interest is paid, the sum term is incremented in m units 3-20

Closed form duration equation:

PVIFA r, N  1  (1  r)  N r

Dur

 N     INT (P o  r)   N  ((1  r)  PVIFA r, N )     • P 0 = Price • INT= Periodic cash flow in dollars, normally the semiannual coupon on a bond

or

the periodic monthly payment on a loan. • r = periodic interest rate = APR / m, where m = # compounding periods per year • N = Number of compounding or payment periods (or the number of years * m) • Dur = Duration = # Compounding or payment periods; Duration period what you actually get from the formula is 3-21

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 but at a decreasing rate 3-22

Duration and Modified Duration

 Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price given by  P P   Dur   1   r r   3-23

Duration and Modified Duration

  Modified duration (Dur Mod ) can be used to predict price changes for non-annual payment loans or securities: It is found as: where r period = APR/m Dur Mod  Dur Annual (1  r period )  Using modified duration to predict price changes: ΔP   Dur Mod  Δr annual P 3-24

Duration Based Prediction Errors

3-25

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:  P P   Dur   1  r  r    1 CX (  r ) 2 2 3-26

Practice Problem

P 0 $30     1.035

0.035

 10    $1000 1.035

10  $ 958 .

42 Dur  N     $C (P o  r) Dur  10  $30 ($958.42

 0.035)   10  (1.035

 8.316605)  ΔP P P  Dur semi 1 Predicted    (1  Δr semi r old semi ) $958.42

 (1   8 .

7548   0.021147)  0 .

0025 1 .

035  $ 938   2 .

1147 %   N  ((1  r)  PVIFA r, N )      month periods Using Modified Duration Dur Mod  Dur Annual (1  r period )  ( 8 .

7548 / 2 ) 1 .

035  4 1 .

.

3774 035  4 .

Predicted Price Change Using Modified Duration ΔP P   Dur Mod  Δr annual   4 .

2294  0 .

0050   2 .

1147 3-27