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Bond pricing theorems Bond pricing theorems
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Bond pricing theorems
Bond convexity
The mathematical relationship between bond yields and prices
Duration
A measure of the average maturity of the stream of payments generated by a financial asset
D = [ (1)CF1/(1+ ytm) + (2)CF2/(1+ ytm)2 + ....... + (t)CFt/(1+ ytm)t ] /(Price)
Very often used:Modified duration: D* = D/(1+ytm)
Exemplification: A 6% coupon bond
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
Exemplification: A 6% coupon bondObservation:Bond prices and yields move inversely.
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
Exemplification: A 6% coupon bondObservation: Dollar changes in bond prices are not symmetrical for a given basis point increase/decrease in YTM, other things constant2.6% decrease in price2.5% increase in price
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
Exemplification: A 6% coupon bondObservationThe longer the maturity, the longer the duration, other things held constant.
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
Exemplification: A 6% coupon bond
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
Exemplification: A 6% coupon bondObservationLonger maturity bonds are more sensitive to yield changes than shorter maturity bonds, other things held constant3.4% decrease in price3.2% increase in price0.94% decrease in price0.95%increase in price
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
Exemplification: A 6% coupon bond
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
Exemplification: A 6% coupon bondObservationAs maturity approaches, bond prices converge towards their face value at an increasing rate, other things held constant.
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
A 6% coupon bondA 5% coupon bond
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
ytm
t = - 4
t = - 3
t = - 2
t = - 1
t = 0
7%
$ 932.25
$ 947.51
$ 963.84
$ 981.31
$ 1,000
6%
$ 965.34
$ 973.27
$ 981.67
$ 990.57
$ 1,000
5%
$1,000
$1,000
$1,000
$1,000
$ 1,000
Duration ( at 6%)
3.71
2.85
1.952
1
0
Modified Duration
3.5
2.69
1.84
0.94
0
A 6% coupon bondA 5% coupon bondObservationThe lower the coupon rate the longer the duration
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
ytm
t = - 4
t = - 3
t = - 2
t = - 1
t = 0
7%
$ 932.25
$ 947.51
$ 963.84
$ 981.31
$ 1,000
6%
$ 965.34
$ 973.27
$ 981.67
$ 990.57
$ 1,000
5%
$1,000
$1,000
$1,000
$1,000
$ 1,000
Duration ( at 6%)
3.71
2.85
1.952
1
0
Modified Duration
3.5
2.69
1.84
0.94
0
A 6% coupon bondA 5% coupon bondObservationLower coupon bonds are more sensitive to yield changes than higher coupon bonds 3.4% decrease in price3.2% increase in price3.43% decrease in price3.6% increase in price
ytm
t = - 4
t = - 3
t = - 2
t = -1
t = 0
7%
$ 966.13
$ 973.76
$ 981.92
$ 990.65
$ 1,000
6%
$ 1,000
$ 1,000
$ 1,000
$1,000
$ 1,000
5%
$ 1,032.54
$ 1,025.24
$ 1,017.57
$ 1,009.52
$ 1,000
Duration ( at 6%)
3.67
2.83
1.94
1
0
Modified Duration
3.46
2.67
1.83
0.94
0
ytm
t = - 4
t = - 3
t = - 2
t = - 1
t = 0
7%
$ 932.25
$ 947.51
$ 963.84
$ 981.31
$ 1,000
6%
$ 965.34
$ 973.27
$ 981.67
$ 990.57
$ 1,000
5%
$1,000
$1,000
$1,000
$1,000
$ 1,000
Duration ( at 6%)
3.71
2.85
1.952
1
0
Modified Duration
3.5
2.69
1.84
0.94
0
Bond Pricing Theorems: A Summary I. Bond prices and yields move inversely.
II. As maturity approaches, bond prices converge towards their face value at an increasing rate, other things held constant.
III. Dollar changes in bond prices are not symmetrical for a given basis point increase/decrease in YTM, other things constant.
IV. Lower coupon bonds are more sensitive to yield changes than higher coupon bonds, other things held constant.
V. Longer maturity bonds are more sensitive to yield changes than shorter maturity bonds, other things held constant.
Duration Theorems: A SummaryI. The duration of a zero coupon bond always equals its time to maturity.
II. The lower the coupon rate the longer the duration, other things held constant.
III. The longer the maturity, the longer the duration, other things held constant.
IV. The lower the yield to maturity, the longer the duration, other things held constant
Using duration to approximate bond price changes The following formula approximates the change in bond prices for small changes in yields: (P1 - P0)/P0 = - D* (ytm1- ytm0) A better approximation is given by the following formula: (P1 - P0)/P0 = - D*(ytm1- ytm0) + (0.5)(Convexity)(ytm1- ytm0)2
Convexity The rate of change of the rate of change of the bond price (the curvature of the relationship between yields and prices).