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Bond pricing theoremsBond pricing theorems
Bond convexityBond convexity
The mathematical relationship between bond yields and prices
DurationDuration
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 bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Exemplification: A 6% coupon bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Observation:
Bond prices and yields move inversely.
Exemplification: A 6% coupon bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Observation:
Dollar changes in bond prices are not symmetrical for a given basis point increase/decrease in YTM, other things constant
2.6% decrease in price2.6% decrease in price
2.5% increase in price2.5% increase in price
Exemplification: A 6% coupon bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Observation
The longer the maturity, the longer the duration, other things held constant.
Exemplification: A 6% coupon bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Exemplification: A 6% coupon bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Observation
Longer maturity bonds are more sensitive to yield changes than shorter maturity bonds, other things held constant
3.4% decrease in price3.4% decrease in price
3.2% increase in price3.2% increase in price
0.94% decrease 0.94% decrease in pricein price
0.95%increase 0.95%increase in pricein price
Exemplification: A 6% coupon bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Exemplification: A 6% coupon bondExemplification: 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
ModifiedDuration
3.46 2.67 1.83 0.94 0
Observation
As maturity approaches, bond prices converge towards their face value at an
increasing rate, other things held constant.
0.7% 0.7% decrease decrease in pricein price
0.77% 0.77% decrease decrease in pricein price
0.79% 0.79% decrease decrease in pricein price
0.94% 0.94% decrease decrease in pricein 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
ModifiedDuration
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
ModifiedDuration
3.5 2.69 1.84 0.94 0
A 6% coupon bond
A 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
ModifiedDuration
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
ModifiedDuration
3.5 2.69 1.84 0.94 0
A 6% coupon bond
A 5% coupon bond
Observation
The 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
ModifiedDuration
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
ModifiedDuration
3.5 2.69 1.84 0.94 0
A 6% coupon bond
A 5% coupon bond
Observation
Lower coupon bonds are more sensitive to yield changes than higher coupon bonds
3.4% decrease in price3.4% decrease in price
3.2% increase in price3.2% increase in price
3.43% decrease in price3.43% decrease in price
3.6% increase in price3.6% increase in price
Bond Pricing Theorems: A SummaryBond 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 SummaryDuration Theorems: A Summary
I. 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 changesUsing 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).