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Given Data Number of years
n 20 The number of semi-annual periods 10C 0 The semi-annual coupon rateF 100 The par value of the bondy 0.09 The annualized bond yield ratesdelta-y -0.001 Change in the value of yield
P $41.46429 DMOD 9.5693780conv-1 $0.02 conv-2 0conv-3 0conv-4 15947.4371986
Convexity 96.1516448799
R 0.005024
The value of R describes the relative importance of convexity and duration in
explaining a bond's percentage price change
Duration
20
Given Data Number of years
n 20 The number of semi-annual periods 10
C 0 The semi-annual coupon rateF 100 The par value of the bond 0.005024 4
y 0.09 The annualized bond yield rates 0.50
delta-y -0.001 Change in the value of yield 1.001.50
P 41.46428597 2.00
DMOD 9.56937799 2.50
conv-1 0.02411714 3.00
conv-2 0 3.50
conv-3 0 4.00
conv-4 15947.4372 4.50
Convexity 96.15164488 5.00
R 0.005023923 5.50
6.006.507.007.508.008.509.009.50
10.0010.5
Semi-annual Periods4
Semi-annual coupon rates
0.50 0.001191.00 0.001191.50 0.001192.00 0.001182.50 0.001183.00 0.001183.50 0.001184.00 0.001174.50 0.001175.00 0.001175.50 0.00116
The value of R describes the relative importance of convexity and duration in
explaining a bond's percentage price change
Semi-annual coupon rates6.00 0.001166.50 0.001167.00 0.001167.50 0.001168.00 0.001158.50 0.001159.00 0.001159.50 0.00115
10.00 0.0011410.50 0.00114
Duration
20
8 12 16 20 24 28 32
Semi-annual Periods8 12 16 20 24 28 32
0.00214 0.00307 0.00400 0.00491 0.00580 0.00667 0.00751
0.00212 0.00304 0.00393 0.00480 0.00564 0.00645 0.007220.00211 0.00301 0.00388 0.00471 0.00551 0.00626 0.006980.00210 0.00298 0.00383 0.00463 0.00539 0.00611 0.006780.00209 0.00295 0.00378 0.00456 0.00529 0.00598 0.006610.00207 0.00293 0.00373 0.00449 0.00520 0.00586 0.006470.00206 0.00290 0.00369 0.00444 0.00512 0.00576 0.006350.00205 0.00288 0.00366 0.00438 0.00505 0.00568 0.006250.00204 0.00286 0.00362 0.00433 0.00499 0.00560 0.006160.00203 0.00284 0.00359 0.00429 0.00493 0.00553 0.006070.00202 0.00282 0.00356 0.00425 0.00488 0.00546 0.00600
0.00201 0.00280 0.00353 0.00421 0.00483 0.00541 0.005940.00201 0.00279 0.00351 0.00417 0.00479 0.00536 0.005880.00200 0.00277 0.00348 0.00414 0.00475 0.00531 0.005830.00199 0.00275 0.00346 0.00411 0.00471 0.00527 0.005780.00198 0.00274 0.00344 0.00408 0.00468 0.00523 0.005730.00197 0.00273 0.00342 0.00405 0.00464 0.00519 0.005690.00197 0.00271 0.00340 0.00403 0.00461 0.00515 0.005660.00196 0.00270 0.00338 0.00401 0.00459 0.00512 0.005620.00195 0.00269 0.00336 0.00398 0.00456 0.00509 0.005590.00195 0.00268 0.00335 0.00396 0.00453 0.00507 0.00556
36 40
Semi-annual Periods36 40
0.00833 0.00912
0.00794 0.008620.00764 0.008250.00740 0.007960.00720 0.007730.00703 0.007540.00689 0.007390.00677 0.007250.00667 0.007140.00658 0.007040.00650 0.00695
0.50
1.50
2.50
3.50
4.50
5.50
6.50
7.50
8.50
9.50
10.50
0.00000
0.00100
0.00200
0.00300
0.00400
0.00500
0.00600
0.00700
0.00800
0.00900
0.01000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in Coupon rates and change in bond maturity
Semi-annual Coupon Rates (%)
Valu
e of
R
Semi-annual periods(n)
0.00643 0.006880.00636 0.006810.00631 0.006750.00625 0.006690.00621 0.006640.00616 0.006600.00612 0.006560.00609 0.006520.00605 0.006480.00602 0.00645
0.50
1.50
2.50
3.50
4.50
5.50
6.50
7.50
8.50
9.50
10.50
0.00000
0.00100
0.00200
0.00300
0.00400
0.00500
0.00600
0.00700
0.00800
0.00900
0.01000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in Coupon rates and change in bond maturity
Semi-annual Coupon Rates (%)
Valu
e of
R
Semi-annual periods(n)
0.50
1.50
2.50
3.50
4.50
5.50
6.50
7.50
8.50
9.50
10.50
0.00000
0.00100
0.00200
0.00300
0.00400
0.00500
0.00600
0.00700
0.00800
0.00900
0.01000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in Coupon rates and change in bond maturity
Semi-annual Coupon Rates (%)
Valu
e of
R
Semi-annual periods(n)
0.50
1.50
2.50
3.50
4.50
5.50
6.50
7.50
8.50
9.50
10.50
0.00000
0.00100
0.00200
0.00300
0.00400
0.00500
0.00600
0.00700
0.00800
0.00900
0.01000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in Coupon rates and change in bond maturity
Semi-annual Coupon Rates (%)
Valu
e of
R
Semi-annual periods(n)
Given Data Number of years
n 20 The number of semi-annual periods 10C 0 The semi-annual coupon rateF 100 The par value of the bond 0.005024 4
y 0.09 The annualized bond yield rates -0.001
delta-y -0.001 Change in the value of yield -0.002-0.003
P 41.464285968 -0.004
DMOD 9.5693779904 -0.005
conv-1 0.0241171402 -0.006
conv-2 0 -0.007
conv-3 0 -0.008
conv-4 15947.437199 -0.009
Convexity 96.15164488 -0.010
R 0.0050239234 -0.011
-0.012-0.013-0.014-0.015-0.016-0.017-0.018-0.019-0.020-0.021-0.022-0.023-0.024-0.025
Semi-annual Periods4
-0.001 0.001196-0.002 0.002392-0.003 0.003589-0.004 0.004785-0.005 0.005981-0.006 0.007177-0.007 0.008373-0.008 0.009569-0.009 0.010766
The value of R describes the relative importance of convexity and duration
in explaining a bond's percentage price change
Change in annualized yield to maturity rates (%)
-0.010 0.011962-0.011 0.013158-0.012 0.014354-0.013 0.015550-0.014 0.016746-0.015 0.017943-0.016 0.019139-0.017 0.020335-0.018 0.021531-0.019 0.022727-0.020 0.023923-0.021 0.025120-0.022 0.026316-0.023 0.027512-0.024 0.028708-0.025 0.029904
Change in annualized yield to maturity rates (%)
Duration
20
8 12 16 20 24 28 32
Semi-annual Periods8 12 16 20 24 28 32
0.002153 0.003110 0.004067 0.005024 0.005981 0.006938 0.0078950.004306 0.006220 0.008134 0.010048 0.011962 0.013876 0.0157890.006459 0.009330 0.012201 0.015072 0.017943 0.020813 0.0236840.008612 0.012440 0.016268 0.020096 0.023923 0.027751 0.0315790.010766 0.015550 0.020335 0.025120 0.029904 0.034689 0.0394740.012919 0.018660 0.024402 0.030144 0.035885 0.041627 0.0473680.015072 0.021770 0.028469 0.035167 0.041866 0.048565 0.0552630.017225 0.024880 0.032536 0.040191 0.047847 0.055502 0.0631580.019378 0.027990 0.036603 0.045215 0.053828 0.062440 0.071053
0.021531 0.031100 0.040670 0.050239 0.059809 0.069378 0.0789470.023684 0.034211 0.044737 0.055263 0.065789 0.076316 0.0868420.025837 0.037321 0.048804 0.060287 0.071770 0.083254 0.0947370.027990 0.040431 0.052871 0.065311 0.077751 0.090191 0.1026320.030144 0.043541 0.056938 0.070335 0.083732 0.097129 0.1105260.032297 0.046651 0.061005 0.075359 0.089713 0.104067 0.1184210.034450 0.049761 0.065072 0.080383 0.095694 0.111005 0.1263160.036603 0.052871 0.069139 0.085407 0.101675 0.117943 0.1342110.038756 0.055981 0.073206 0.090431 0.107656 0.124880 0.1421050.040909 0.059091 0.077273 0.095455 0.113636 0.131818 0.1500000.043062 0.062201 0.081340 0.100478 0.119617 0.138756 0.1578950.045215 0.065311 0.085407 0.105502 0.125598 0.145694 0.1657890.047368 0.068421 0.089474 0.110526 0.131579 0.152632 0.1736840.049522 0.071531 0.093541 0.115550 0.137560 0.159569 0.1815790.051675 0.074641 0.097608 0.120574 0.143541 0.166507 0.1894740.053828 0.077751 0.101675 0.125598 0.149522 0.173445 0.197368
36 40
Semi-annual Periods36 40
0.008852 0.0098090.017703 0.0196170.026555 0.0294260.035407 0.0392340.044258 0.0490430.053110 0.0588520.061962 0.0686600.070813 0.0784690.079665 0.088278
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-0.007
-0.008
-0.009
-0.010
-0.011
-0.012
-0.013
-0.014
-0.015
-0.016
-0.017
-0.018
-0.019
-0.020
-0.021
-0.022
-0.023
-0.024
-0.025
0.000000
0.050000
0.100000
0.150000
0.200000
0.250000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in annualized yield to maturity rates and change in bond maturity
Change in annual-ized yield to ma-turity (delta y)
Valu
e of
R
Semi annual periods to maturity
0.088517 0.0980860.097368 0.1078950.106220 0.1177030.115072 0.1275120.123923 0.1373210.132775 0.1471290.141627 0.1569380.150478 0.1667460.159330 0.1765550.168182 0.1863640.177033 0.1961720.185885 0.2059810.194737 0.2157890.203589 0.2255980.212440 0.2354070.221292 0.245215
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-0.007
-0.008
-0.009
-0.010
-0.011
-0.012
-0.013
-0.014
-0.015
-0.016
-0.017
-0.018
-0.019
-0.020
-0.021
-0.022
-0.023
-0.024
-0.025
0.000000
0.050000
0.100000
0.150000
0.200000
0.250000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in annualized yield to maturity rates and change in bond maturity
Change in annual-ized yield to ma-turity (delta y)
Valu
e of
R
Semi annual periods to maturity
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-0.007
-0.008
-0.009
-0.010
-0.011
-0.012
-0.013
-0.014
-0.015
-0.016
-0.017
-0.018
-0.019
-0.020
-0.021
-0.022
-0.023
-0.024
-0.025
0.000000
0.050000
0.100000
0.150000
0.200000
0.250000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in annualized yield to maturity rates and change in bond maturity
Change in annual-ized yield to ma-turity (delta y)
Valu
e of
R
Semi annual periods to maturity
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-0.007
-0.008
-0.009
-0.010
-0.011
-0.012
-0.013
-0.014
-0.015
-0.016
-0.017
-0.018
-0.019
-0.020
-0.021
-0.022
-0.023
-0.024
-0.025
0.000000
0.050000
0.100000
0.150000
0.200000
0.250000
4 8 12 16 20 24 28 32 36 40
Change in the value of R with respect to change in annualized yield to maturity rates and change in bond maturity
Change in annual-ized yield to ma-turity (delta y)
Valu
e of
R
Semi annual periods to maturity
Inferences - Hands On Project 1
Overview
General Inferences
Takeaways for the investor
General Inferences
Takeaways for the investor
Inferences - Hands On Project 1
Que (b)
Que (c)
SUMMARY
1. The value of R gives the relative importance of convexity and duration in explaining a bond's percentage change in price from a given change in yield.2. Duration and convexity are two metrics used to help investors understand how the price of a bond will be affected by changes in interest rates.3. Duration gives the first order approximation of the sensitivity of the bond prices to changes in yield while convexity gives the second order approximation of the same.
1. For a given semi-annual coupon rate, the value of R increases with the increase in maturity. Hence, the impact of convexity is more pronounced in bonds with higher maturity periods.
2. For a given maturity period, the value of R decreases with an increase in the coupon rate. The effect of convexity on the sensitivity of the bond prices hence decreases with an incerase in coupon rate.
1. If the investor has invested in a fixed-rate high maturity bond, then he must consider the effect of the second-order convexity term while estimating the impact of change in yields on the prices of his bond. Estimating the changes in bond prices purely on the basis of the first-order duration term would lead to a gross error in estimation.
2. If the investor has invested in a floating-rate bond and expects the yields to decrease (and hence the coupon rates to decrease), the effect of convexity would be high in determining the changes in bond prices,especially for floating-rate bonds with a high maturity period. The investor must hence consider the impact of convexity in addition to the bond duration for a more accurate estimation.
1. For a zero-coupon bond, the value of R increases as the decrease in the annualized yield to maturity becomes more steep. This increase in R is more pronounced for bonds with higher maturity.
1. If the investor has invested in a zero-coupon bond and faces a situation wherein he expects the market yields to decrease sharply, then he must consider the impact of convexity to estimate the sensitivity of bond prices, especially when the maturity period of the bond is high.
The bond investor must not resort to a linear hedging strategy (hedging strategy based on the first order, duration based estimation) in the following scenarios:1. Low coupon high maturity bonds2. In case the forecasted yields are bound to decrease steeply
