Bond pricing theorems. Bond convexity The mathematical relationship between bond yields and prices

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).