copyright 2014 by diane s. docking1 duration & convexity

Copyright 2014 by Diane S. Docking 1

Duration & Convexity

Learning Objectives

Know how to the calculate duration of a security.

Know how to calculate the convexity of a security.

Understand the economic meaning of duration.


Duration

Duration allows for the comparison of securities of different coupons, maturities, etc.

Duration measures the weighted average life of an instrument. It equals the average time necessary to recover the initial cost. E.g..: A bond with 4 years until final maturity with a duration of 3.5 years indicates that

an investor would recover the initial cost of the bond in 3.5 years, on average, regardless of intervening interest rate changes.

Duration measures of the price sensitivity of a financial asset with fixed cash flows to interest rate changes.

3Copyright 2014 by Diane S.

Docking

Macaulay DurationUsing annual compounding, Macaulay duration (D) is:

where:CF = the interest and/or principal payment that occurs in period t,t = the time period in which the coupon and/or principal payment

occurs,i = the current market rate or current market yield on the security

N

tt

t

N

ttt

i

CFi

CFt

D

1

1

1

1

Numerator is:PV of Future CFs weighted by period of

receipt

Denominator is:PV of Future

CFs = Current Price

Duration is in

years or fraction of years

4Copyright 2014 by Diane S.

Docking


Duration For interest

rate increases, Duration overestimates the price decrease.

For interest rate decreases, Duration underestimates the price increase

Change in price predicted by durationyield

Price

Actual change in price

i0 i1


Example 1:Calculating Duration: 10-yr, 10% Coupon Bond; rm =10%

(1) (2) (3) (4) (5)

Year CF ($) PVCF @ rm% t x PVCF Duration

1 $100 $90.91 $90.91

2 $100 82.64 165.293 $100 75.13 225.394 $100 68.30 273.215 $100 62.09 310.466 $100 56.45 338.687 $100 51.32 359.218 $100 46.65 373.219 $100 42.41 381.6910 $100 38.55 385.5410 $1,000 385.54 3,855.43

Total 1,000.00$ 6,759.02$ 6.7590 yrs

< 10 years

This is current Price = P0

6,759.021,000.00



(1) (2) (3) (4) (5)Year CF ($) PVCF @ rm% t x PVCF Duration

1 $100 $83.33 $83.332 $100 $69.44 138.893 $100 $57.87 173.614 $100 $48.23 192.905 $100 $40.19 200.946 $100 $33.49 200.947 $100 $27.91 195.368 $100 $23.26 186.059 $100 $19.38 174.4310 $100 $16.15 161.5110 $1,000 $161.51 1,615.06

Total 580.75$ 3,323.01$ 5.7219 yrs

< 10 years


Verify Price:FV = 1,000n = 10 yrs.Pmt = $100i = 20% PV = 580.75

3,323.01580.75




1 $200 $181.82 $181.822 $200 $165.29 330.583 $200 $150.26 450.794 $200 $136.60 546.415 $200 $124.18 620.926 $200 $112.89 677.377 $200 $102.63 718.428 $200 $93.30 746.419 $200 $84.82 763.3810 $200 $77.11 771.0910 $1,000 $385.54 3,855.43

Total 1,614.46$ 9,662.61$ 5.9851 yrs

< 10 years


Verify Price:FV = 1,000n = 10 yrs.Pmt = $200i = 10% PV = 1,614.46

9,662.611,614.46




1 $100 $90.91 $90.912 $100 $82.64 165.293 $100 $75.13 225.394 $100 $68.30 273.215 $100 $62.09 310.465 $1,000 $620.92 3,104.61

Total 1,000.00$ 4,169.87$ 4.1699 yrs

< 5 years


4,169.871,000.00


Example 5:Calculating Duration: 5-yr, Zero-Coupon Bond; rm =10%

(1) (2) (3) (4) (5)

Year CF ($) PVCF @ rm% t x PVCF Duration

1 $0 $0.00 $0.002 $0 $0.00 0.003 $0 $0.00 0.004 $0 $0.00 0.005 $0 $0.00 0.005 $1,000 $620.92 3,104.61

Total 620.92$ 3,104.61$ 5.0000 yrs

This is current Price = P0 D=maturity

Verify Price:FV = 1,000n = 5yrs.Pmt = 0i = 10% PV = 620.92

3,104.61620.92


Example 6:Calculating Duration: 10-yr, Zero-Coupon Bond; rm =20%



1 $0 $0.00 $0.002 $0 $0.00 0.003 $0 $0.00 0.004 $0 $0.00 0.005 $0 $0.00 0.006 $0 $0.00 0.007 $0 $0.00 0.008 $0 $0.00 0.009 $0 $0.00 0.0010 $0 $0.00 0.0010 $1,000 $161.51 1,615.06

Total $161.51 1,615.06$ 10.0000 yrs

1615.06161.506

Verify Price:FV = 1,000n = 10 yrs.Pmt = 0i = 20% PV = 161.51

D=maturity


Asset Properties and Duration

Maturity Coupon Rm Duration

Ex. 1 10 yrs. 10% 10% 6.759 yrs.

Ex. 2 10 yrs. 10% 20% 5.722 yrs.

Ex. 3 10 yrs. 20% 10% 5.985 yrs.

Ex. 4 5 yrs. 10% 10% 4.170 yrs.

Ex. 5 5 yrs. 0% 10% 5.000 yrs.

Ex. 6 10 yrs. 0% 20% 10.000 yrs.


Asset Properties and Duration

1. For bonds with the same coupon rate and the same yield, the bond with the longer maturity will have the________ duration. (Ex.1 vs. Ex.4)

2. For bonds with the same maturity and the same yield, the bond with the lower coupon rate will have the ________ duration. (Ex. 1 vs. Ex. 3; Ex. 2 & 6; Ex. 4 vs. Ex. 5)

3. When interest rates rise, the duration of a coupon bond ________. (Ex. 1 vs. Ex. 2)

4. The lower the initial yield, the _______ the duration for a given bond. (Ex. 1 vs. Ex. 2)


Key facts about duration

1. The longer a bond’s duration, the its sensitivity to interest rate changes

2. The duration of a______________ bond = bond’s term to maturity

3. The Macaulay duration of any coupon bond is always ______ than the bond’s term to maturity

4. Duration is the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each.

5. The more frequently a security pays interest or principal, the its duration.

Duration


Macaulay Duration Duration – annual interest pmts:

Duration – semi-annual interest pmts:

2

;

21

21

1

1yrsdouble

yrsN

tt

t

N

ttt

yrsdouble

DD

i

CF

i

CFt

D

N

tt

t

N

ttt

yrs

i

CFi

CFt

D

1

1

1

1


Modified Duration

Modified duration – annual interest pmts:

Modified duration – semi-annual interest pmts:

212

1

21yrs.)(in mod i

D

i

DD yrsyrsdouble

iD

D yrs

1yrs.)(in mod


Duration and Price Sensitivity So, an estimate of the percentage change in the

price of a financial asset is:

So, an estimate of $ price change in the price of a financial asset is:

or

00 1 i

iD

P

P

001P

i

iDP


Duration estimates of price change

If annual interest payments:

If semi-annual interest payments:

001P

i

iDP yrs

00

00

21or

212P

ii

DPiiD

P yrsyrsdouble


Duration For interest rate

increases, Duration overestimates the price decrease.



Price



Problem 1: Duration – Annual Payments

Assume today is September 1, 2XX1. Midwest Bank owns the following security:G.E. Corporate bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid annually.

1. What is the security’s YTM?2. If market interest rates increase 1%, what will be the security’s

price?3. What is the security’s Macaulay Duration? Modified Duration?4. What is the dollar change in price for a 1% increase in market rates

as estimated by Duration?5. Compare the actual price change to the estimated price change.

Problem 1 Solution: Duration - Annual Payments

1. YTM: PV = 124 3/32 = 124.09375 = $1,240.94

FV = 1,000

Pmt = 107.50

n = 5 yrs.

therefore Rm = YTM =___________

2. Rm increases 1% to 6.1603%:

FV = 1,000

Rm = 6.1603%

Pmt = 107.50

n = 5 yrs.

therefore PV = ___________


Makes sense: rates increase, price decreases


Problem 1 Solution: Duration - Annual Payments

Calculating Duration on a Bond - Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.1603%

(1) (2) (3) (4)Year CF ($) PVCF @ rm% t x PVCF Duration

1 $107.50 $102.22 $102.222 $107.50 $97.21 $194.423 $107.50 $92.44 $277.324 $107.50 $87.90 $351.615 $107.50 $83.59 $417.955 $1,000 $777.57 $3,887.86

Total $1,240.94 $5,231.38

Macaulay 4.2157 years

Modified 4.0088 years

5,231.381,240.94 4.2157

1.051603

Problem 1 Solution: Duration - Annual Payments (cont.)

4. From Duration:

Macaulay

or

Modified


$49.751,240.941.051603

.012157.4ΔP duration todue

$49.751,240.9401.0088.4ΔP duration todue

Problem 1 Solution: Duration - Annual Payments (cont.)

5. Original Price when (Rm = 5.1603%) $1,240.94

Duration est. of price change < 49.75 >

Duration est. of new Price $1,191.19

vs.

$1,192.49


Diff = <$1.30>

Duration overestimated

price decrease.


Problem 2: Duration – Semi-annual payments

Assume today is September 1, 2XX1. Midwest Bank owns the following security:U.S. Treasury bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid semi-annually.


price?3. What is the security’s Macaulay Duration? Modified Duration?4. What is the dollar change in price for a 1% increase in market rates

as estimated by Duration?5. Compare the actual price change to the estimated price change.

Problem 2 Solution: Duration – Semi-annual Payments

1. YTM: PV = 124 3/32 = 124.09375 = $1,240.94

FV = 1,000

Pmt = 107.50/2 = 53.75

n = 5 yrs. x 2 = 10

therefore Rm = YTM = 2.6069% semi-annual; 5.2137% annual

2. Rm increases 1% to 6.2137% annual/ 2 = 3.1069%

FV = 1,000

Rm = 3.1069%

Pmt = 53.75

n = 10

therefore PV = $1,192.42



Problem 2 Solution: Duration – Semi-annual Payments (cont.)

Calculating Duration on a Bond - Semi-Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.2137% 0.026069

(1) (2) (3) (4)

Year CF ($) PVCF @ rm%/2 t x PVCF Duration

1 $53.75 $52.38 $52.382 $53.75 $51.05 $102.113 $53.75 $49.76 $149.274 $53.75 $48.49 $193.975 $53.75 $47.26 $236.306 $53.75 $46.06 $276.367 $53.75 $44.89 $314.238 $53.75 $43.75 $349.999 $53.75 $42.64 $383.7410 $53.75 $41.55 $415.5410 $1,000 $773.10 $7,731.01

Total $1,240.94 $10,204.90

8.2235 dbl-yrs

Macaulay 4.1118 yrs.

Modified 4.0073 yrs.

10,204.901240.94

8.22352

4.1118(1.026069)


4. From Duration (in years):

Macaulay

or

Modified


$49.731,240.941.026069


$49.731,240.9401.0073.4ΔP duration todue




Duration est. of new Price $1,191.21

vs.

$1,192.42


Diff = <$1.21>


price decrease.


Duration

For interest rate increases, Duration overestimates the price decrease.



Price


ConvexityTaking convexity

into account: For interest rate

increases, the actual reduction in price will be less than that predicted by duration

For interest rate decreases, the actual increase in price will be more than that predicted by duration


Price


Convexity adds amount back

31Copyright 2014 by Diane S. Docking


Convexity Convexity – annual interest pmts:

Convexity – semi-annual interest pmts:

2

22

2 2)1(

'

2

iyrsdouble

yrs

DCX

CX

2

'

1 i

DCX yrs


Example 1A:Calculating Macaulay Convexity: 10-yr 10% Coupon Bond; rm =10%Calculating Macaulay Convexity on a Bond - Annual Payments

Face = $1,000Maturity = 10 yearsCoupon Rate = 10.00%Market Rate = 10.00%

(1) (2) (3) (5) (6) (7)

Year CF ($) PVCF @ rm% (t2+t) (t2+t)PVCF D' Convexity1 $100 $90.91 2 $181.822 $100 $82.64 6 $495.873 $100 $75.13 12 $901.584 $100 $68.30 20 $1,366.035 $100 $62.09 30 $1,862.766 $100 $56.45 42 $2,370.797 $100 $51.32 56 $2,873.698 $100 $46.65 72 $3,358.859 $100 $42.41 90 $3,816.8810 $100 $38.55 110 $4,240.9810 $1,000 $385.54 110 $42,409.76

Total 1,000.00$ $63,879.00

63.8790 52.7926 "yrs."

63.8790(1.10)2


Example 2A:Calculating Macaulay Convexity: 10-yr 10% Coupon Bond; rm =20%

(1) (2) (3) (5) (6) (7)

Year CF ($) PVCF @ rm% (t2+t) (t2+t)PVCF D' Convexity

1 $100 $83.33 2 $166.672 $100 $69.44 6 $416.673 $100 $57.87 12 $694.444 $100 $48.23 20 $964.515 $100 $40.19 30 $1,205.636 $100 $33.49 42 $1,406.577 $100 $27.91 56 $1,562.868 $100 $23.26 72 $1,674.499 $100 $19.38 90 $1,744.2610 $100 $16.15 110 $1,776.5610 $1,000 $161.51 110 $17,765.61

Total 580.75$ $29,378.27

50.5865 35.1295 "yrs."

50.5865(1.20)2



(1) (2) (3) (5) (6) (7)


1 $200 $181.82 2 $363.642 $200 $165.29 6 $991.743 $200 $150.26 12 $1,803.164 $200 $136.60 20 $2,732.055 $200 $124.18 30 $3,725.536 $200 $112.89 42 $4,741.587 $200 $102.63 56 $5,747.378 $200 $93.30 72 $6,717.719 $200 $84.82 90 $7,633.7610 $200 $77.11 110 $8,481.9510 $1,000 $385.54 110 $42,409.76

Total 1,614.46$ $85,348.24

52.8650 43.6901 "yrs."

52.8650(1.10)2



(1) (2) (3) (5) (6) (7)


1 $100 $90.91 2 $181.822 $100 $82.64 6 $495.873 $100 $75.13 12 $901.584 $100 $68.30 20 $1,366.035 $100 $62.09 30 $1,862.765 $1,000 $620.92 30 $18,627.64

Total 1,000.00$ $23,435.69

23.4357 19.3683 "yrs."

23.4357(1.10)2


Example 5A:Calculating Macaulay Convexity: 5-yr Zero-Coupon Bond; rm =10%(assume annual payments)

(1) (2) (3) (5) (6) (7)


1 $0 $0.00 2 $0.002 $0 $0.00 6 $0.003 $0 $0.00 12 $0.004 $0 $0.00 20 $0.005 $0 $0.00 30 $0.005 $1,000 $620.92 30 $18,627.64

Total 620.92$ $18,627.64

30.0000 24.7934 "yrs."

18,627.64620.92

30.0000(1.10)2


Asset Properties and Convexity

1. For bonds with the same coupon rate and the same yield,

the the bond with the longer maturity will have the greater convexity. (Ex.1A vs. Ex.4A)

2. For bonds with the same maturity and the same yield,the the bond with the lower coupon rate will have the greater convexity. (Ex. 1A vs. Ex. 3A)

3. When interest rates rise,the convexity of a coupon bond falls. (Ex. 1A vs. Ex. 2A)

4. The greater the initial yield,the less the convexity for a given bond. (Ex. 1A vs. Ex. 2A)


Key facts about convexity

1. Convexity increases with bond maturity

2. Given the same maturity, coupon bonds are _____convex than zero-coupon bonds.

Convexity


Price change explained by Convexity: If annual interest payments:

If semi-annual interest payments:

02

21 PiCXP yrsconvexitytodue

0

2

02

221or2

1 PiCXPiCXP yrsdoubleyrsCXtodue


Problem 1A: Duration and Convexity – Annual Payments

Assume today is September 1, 2XX1. Midwest Bank owns the following security:G.E. Corporate bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid annually.


price?3. What is the security’s Macaulay Duration? Modified Duration?4. What is the security’s Convexity?5. What is the dollar change in price for a 1% increase in

market rates:a) From Duration?b) From Convexity?

6. Compare the actual price change to the estimated price change.


Problem 1A Solution: Duration & Convexity Annual Payments

3. 4.

Calculating Macaulay Duration & Convexity on a Bond - Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.1603%

(1) (2) (3) (4) (5) (6) (7)

Year CF ($) PVCF @ rm% t x PVCF (t2+t) (t2+t)PVCF D' Duration Convexity

1 $107.50 $102.22 $102.22 2 $204.452 $107.50 $97.21 $194.42 6 $583.253 $107.50 $92.44 $277.32 12 $1,109.264 $107.50 $87.90 $351.61 20 $1,758.055 $107.50 $83.59 $417.95 30 $2,507.675 $1,000 $777.57 $3,887.86 30 $23,327.18

Total $1,240.94 $5,231.38 $29,489.86

23.7642 4.2157 years 21.4892 "years"

5,231.381,240.94

29,489.861,240.94

23.7642(1.051603)2

Problem 1A Solution: Duration & Convexity - Annual Payments (cont.)

5. a) From Macaulay Duration:

6. b) From Convexity:


33.1$1,240.9401.4892.2121ΔP 2

convexity todue

$49.751,240.941.051603


Problem 1A Solution: Duration & Convexity - Annual Payments (cont.)



Duration est. of new Price $1,191.19 vs. $1,192.49

Convexity est. of price change

not explained by duration + 1.33

Duration + Convexity est. of new Price $1,192.52

vs.

$1,192.49


Diff = <$1.30>


price decrease.

Diff = <$.03>


Problem 2A: Duration and Convexity – Semi-annual payments

Assume today is September 1, 2XX1. Midwest Bank owns the following security:U.S. Treasury bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid semi-annually.

1. What is the security’s YTM?2. If market interest rates increase 1%, what will be the security’s price?3. What is the security’s Duration?4. What is the security’s Convexity?5. What is the dollar change in price for a 1% increase in market rates:

a) From Duration?b) From Convexity?

6. Compare the actual price change to the estimated price change.


Problem 2A Solution: Duration & Convexity Semi-annual Payments

3. 4.

Calculating Macaulay Duration & Convexity on a Bond - Semi-Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.2137%

(1) (2) (3) (4) (5) (6) (7)

Year CF ($) PVCF @ rm%/2 t x PVCF (t2+t) (t2+t)PVCF D' Duration Convexity1 $53.75 $52.38 $52.38 2 $104.772 $53.75 $51.05 $102.11 6 $306.323 $53.75 $49.76 $149.27 12 $597.084 $53.75 $48.49 $193.97 20 $969.855 $53.75 $47.26 $236.30 30 $1,417.816 $53.75 $46.06 $276.36 42 $1,934.507 $53.75 $44.89 $314.23 56 $2,513.818 $53.75 $43.75 $349.99 72 $3,149.929 $53.75 $42.64 $383.74 90 $3,837.3710 $53.75 $41.55 $415.54 110 $4,570.9610 $1,000 $773.10 $7,731.01 110 $85,041.15

Total $1,240.94 $10,204.90 $104,443.54

84.1650 8.2235 dbl-yrs 79.9427 "dbl-yrs"

4.1118 yrs. 19.9857 "yrs"

10,204.901240.94

104,443.541,240.94

84.1650

(1.026069)2

8.22352

79.9427

22

Problem 2A Solution: Duration & Convexity – Semi-annual Payments (cont.)

5. a) From Macaulay Duration:

6. b) From Convexity (in years):


24.1$1,240.9401.9857.1921ΔP 2

convexity todue

$49.731,240.941.026069


Problem 2A Solution: Duration & Convexity – Semi-annual Payments (cont.)



Duration est. of new Price $1,191.21 vs. $1,192.42

Convexity est. of price change

not explained by duration + 1.24

Duration + Convexity est. of new Price $1,192.45

vs.

$1,192.42


Diff = <$1.21>


price decrease.

Diff = <$.03>

copyright 2014 by diane s. docking1 duration & convexity

Documents

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current price duration

duration measures

docking slide

price sensitivity

price decrease

macaulay duration d

duration underestimate