BOND VALUATION

All bonds have the following characteristics:

1. A maturity date- typically 20-25 years.

2. A coupon rate- the rate of interest that the issuing company pays to the holder.

3. A face value- usually $1000 or $5000.

%818

BOND VALUATION

The value of a bond is the sum of the present value of the annual interest payments plus the present value of the face value;

nn r

Face

r

Interestpv

)1()1(

Where; interest = coupon rate x face value r = discount rate n = years to maturity

BOND VALUATION

nn rFace

rCouponPV

)1(

1

)1(

1

Where 1/(1+r) = discount rate

BOND VALUATION

EXAMPLE

Find the value of a 20 year, 10%, $1000 face value bond.The interest payment is given by: .10 x $1000 = $100/year

THE FORMULA IS:

PV =

20

120)1(

1000$

)1(

100$

NN rr

PV = $100(6.145) + $1000(.386) = $614.50 + $386 = $1000

BOND VALUATION

if the coupon rate is 8%, then the formula for the value of the bond is;

20

20

1 )1(

1000$

)1(

80$

rrPV

nn

PV = $80(6.145) + $1000(.386) = $877.60

THE BOND SELLS AT A DISCOUNT

BOND VALUATION

if the coupon rate is 12%, then the formula for the value of the bond is;

20

20

1 )1(

1000$

)1(

120$

rrPV

nn

PV = $120(6.145) + $1000(.386) = $1123.40

THE BOND SELLS AT A PREMIUM

BOND THEOREMS

In this section we will look at the relationship between changes in bond prices and changes in term to maturity, coupon rate, and discount rates (market yields).

$886.

.

.

2572 50

1

1000

1

9 55%

1

7

7

y y

y

TT

7 1/4 %, due 1995, $1000 Face 8/8

$1032.

.

.

50103 75

1

1000

1

9 71%

1

7

7

y y

y

TT

10 3/8 %, due 1995, $1000 Face 8/8

$882.

.

.

5072 50

1

1000

1

9 63%

1

7

7

y y

y

TT

7 1/4 %, due 1995, $1000 Face 8/8

$1027.

.

.

50103 75

1

1000

1

9 81%

1

7

7

y y

y

TT

10 3/8 %, due 1995, $1000 Face 8/8

Change in Bond Prices

• Price of 7 1/4 bond fell by $3.75 or .42%

• Price of 10 3/8 bond fell by $5.00 or .48%

• When market yields fall unexpectedly, the prices of financial assets rise and vice-versa

Theorem I

Consider two Bonds with 12% coupon of equal risk, one 5 year term, the other 15

year term

$931

. .

120

114

1000

1141

5

5TT

$877

. .

120

114

1000

1141

15

15TT

% in 5 year bond is :1000 931

1000069

.

% in 15 year bond is :1072 1000

1000072

.

% in 5 year bond is :1037 1000

1000037

.

% in 15 year bond is :1000 877

1000123

.

If yields fall to 11%:

Theorem II

Holding coupon rate constant, for a given change in market yields, percentage changes in bond prices are greater the longer the term to maturity.

$1151.. .

72120

110

1000

1101

15

15 TT

$1123.. .

40120

110

1000

1101

10

10 TT

$1242.. .

32120

109

1000

1091

15

15 TT

$1192.. .

16120

109

1000

1091

10

10 TT

% in 15 year bond is :1242 32 115172

1151720787

. .

..

% in 10 year bond is :1192 16 1123 4

1123 400612

. .

..

. . .0787 0612 0175(% change in 15 - % change in 10)

$1075.. .

92120

110

1000

1101

5

5 TT

$1116.. .

80120

109

1000

1091

5

5 TT

% in 5 year bond is :1116 80 1075 92

1075 920380

. .

..

. . .0612 0380 0232(% change in 10 - % change in 5)

Theorem III

The percentage price changes described in Theorem II increase at a decreasing rate as N increases.

- Slopes are percentage changes.

Consider: 12%, 8 year, $1000 coupon bond

If yields move from 12% to 14%, price falls to 907.

%1000 907

1000093

.=

If yields fall to 10%, price is 1107

%1107 1000

1000107

.=

Theorem IVHolding N constant and starting from same market yield, equal yield changes up or down do not result in equal percentage price changes. A decrease in yield increases prices more than an equal increase in yield decreases prices. Price changes are asymmetric with respect to changes in yield.

$1123.. .

40120

110

1000

1101

10

10 TT

$1000.. .

00100

110

1000

1101

10

10 TT

$1192.. .

16120

109

1000

1091

10

10 TT

$1063.. .

80100

109

1000

1091

10

10 TT

% in 12% coupon =1192 16 1123 40

1123 40061

. .

..

% in 10% coupon = 106380 1000

10000638

..

Theorem V

Holding N constant and starting from the same yield,the greater the coupon rate, the smaller the

percentage change in price for a given change in yield.

DURATION AND BOND PRICES

The relationship between duration and the expected percentage price change expected from a change in market yield is closely approximated by:

% P0 = -DUR0

0

P

P

)1( Y

Y

Percentage price changes accompanying the change in market yields between August 8th and August 10th can be estimated:

% 4/71P = -5.6409 X 0955.1

08.0= -.41%

8/103P = -5.3366 X 0971.1

010% = -.49%

ESTIMATING INTEREST RATE ELASTICITY

E = -DUR

)1( Y

Y

E =

=

YY

PP

0

0

Y

YY

YDur

1 =

Y

YDUR

Y

Y

Y

YDUR

11

Y

P

%

% 0=

YY

PP

0

0