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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