Course page: https://www.coursera.org/learn/financial-engineering-1
- Lesson 3.1: Mechanics of the Futures Margin Account
- Lesson 3.2: Stock Price Dynamics in the Binomial Model
- Lesson 3.4: Option pricing in the 1-period binomial model
- Lesson 4.1: Option pricing in the multi-period binomial model
- Lesson 4.2: Pricing American Options
- Lesson 4.4: Pricing a European Put on a Futures Contract
- Lesson 5.1: Pricing a ZCB
- Lesson 5.2: Fixed-Income Derivatives
- Lesson 5.3: Caplets, Floorlets, Swaps, Swaptions
- Lesson 5.4: The Forward Equations (Elementary Securities)
- Lesson 6.1: Black-Derman-Toy (BDT) Model Calibration
- Lesson 6.2: Pricing using Calibrated BDT Model
- Lesson 6.3: Pricing Defaultable Bonds, Calibrating Hazard Rates
- Lesson 6.5: Pricing Credit Default Swaps
- Lesson 7.1: Mortgage mathematics
- Lesson 7.2: MBS Pass Thru Mathematics
- Lesson 7.4: MBS Pass-Thru Sequential CMO with Tranches
source("fin-eng-1.R") spot.prices <- c(690.25,
693.50,
705.00,
709.50,
719.50,
724.25,
723.00,
732.00,
708.00,
711.50,
725.25,
734.50,
741.00,
741.25,
732.75,
730.50) / 100
margin.schedule <- buildMarginAccountSchedule( spot.price.vec=spot.prices,
num.contracts=5000,
initial.margin=1688,
maintenance.margin=1250 )
data.frame(margin.schedule)## spot.price profit margin.account margin.call
## 1 6.9025 0.0 1688.0 0
## 2 6.9350 162.5 1850.5 0
## 3 7.0500 575.0 2425.5 0
## 4 7.0950 225.0 2650.5 0
## 5 7.1950 500.0 3150.5 0
## 6 7.2425 237.5 3388.0 0
## 7 7.2300 -62.5 3325.5 0
## 8 7.3200 450.0 3775.5 0
## 9 7.0800 -1200.0 2575.5 0
## 10 7.1150 175.0 2750.5 0
## 11 7.2525 687.5 3438.0 0
## 12 7.3450 462.5 3900.5 0
## 13 7.4100 325.0 4225.5 0
## 14 7.4125 12.5 4238.0 0
## 15 7.3275 -425.0 3813.0 0
## 16 7.3050 -112.5 3700.5 0
sum(margin.schedule$profit)## [1] 2012.5
# (random) spot prices, quoted in dollars per bushel
spot.prices <- c(690.25,
686.81,
686.59,
668.51,
662.08,
674.89,
661.97,
709.88,
719.20,
709.36,
725.84,
743.18,
733.77,
743.12,
776.84,
796.17) / 100
margin.schedule <- buildMarginAccountSchedule( spot.price.vec=spot.prices,
num.contracts=5000,
initial.margin=1688,
maintenance.margin=1250 )
data.frame(margin.schedule)## spot.price profit margin.account margin.call
## 1 6.9025 0.0 1688.0 0
## 2 6.8681 -172.0 1516.0 0
## 3 6.8659 -11.0 1505.0 0
## 4 6.6851 -904.0 1688.0 1087
## 5 6.6208 -321.5 1366.5 0
## 6 6.7489 640.5 2007.0 0
## 7 6.6197 -646.0 1361.0 0
## 8 7.0988 2395.5 3756.5 0
## 9 7.1920 466.0 4222.5 0
## 10 7.0936 -492.0 3730.5 0
## 11 7.2584 824.0 4554.5 0
## 12 7.4318 867.0 5421.5 0
## 13 7.3377 -470.5 4951.0 0
## 14 7.4312 467.5 5418.5 0
## 15 7.7684 1686.0 7104.5 0
## 16 7.9617 966.5 8071.0 0
sum(margin.schedule$profit)## [1] 5296
stock.price.lattice <- buildAssetPriceLattice(S0=100,
n=3,
u=1.07,
d=1/1.07)
stock.price.lattice## [,1] [,2] [,3] [,4]
## [1,] 100 107.00000 114.49000 122.50430
## [2,] NA 93.45794 100.00000 107.00000
## [3,] NA NA 87.34387 93.45794
## [4,] NA NA NA 81.62979
u = 1.07
d = 1/u
R = 1.01
q = (R - d) / (u - d)
option.price.lattice <- buildOptionPriceLattice(asset.price.lattice=stock.price.lattice,
n=1,
K=102,
R=1.01,
q=q,
is.call.option=T,
is.amer.option=F)
option.price.lattice ## [,1] [,2]
## [1,] 2.757108 5
## [2,] NA 0
option.price.lattice <- buildOptionPriceLattice(asset.price.lattice=stock.price.lattice,
n=3,
K=100,
R=1.01,
q=q,
is.call.option=T,
is.amer.option=F)
option.price.lattice ## [,1] [,2] [,3] [,4]
## [1,] 6.574383 10.22933 15.480099 22.5043
## [2,] NA 2.12846 3.859951 7.0000
## [3,] NA NA 0.000000 0.0000
## [4,] NA NA NA 0.0000
option.price.lattice <- buildOptionPriceLattice(asset.price.lattice=stock.price.lattice,
n=3,
K=100,
R=1.01,
q=q,
is.call.option=F,
is.amer.option=T)
option.price.lattice ## [,1] [,2] [,3] [,4]
## [1,] 3.823931 1.258939 0.000000 0.000000
## [2,] NA 7.134456 2.869852 0.000000
## [3,] NA NA 12.656127 6.542056
## [4,] NA NA NA 18.370212
shouldExerciseOption( asset.price.lattice = stock.price.lattice,
option.price.lattice = option.price.lattice,
K=100,
is.call.option = F )## [,1] [,2] [,3] [,4]
## [1,] FALSE FALSE FALSE FALSE
## [2,] NA FALSE FALSE FALSE
## [3,] NA NA TRUE TRUE
## [4,] NA NA NA TRUE
T = 0.50
sigma = 20/100
n = 10
r = 2/100
c = 1/100
bin.model <- calibrateBinomialModel(T=T,
n=n,
r=r,
c=c,
sigma=sigma)
rbind(bin.model)## q u d R n r sigma T c
## bin.model 0.4944112 1.045736 0.9562639 1.001001 10 0.02 0.2 0.5 0.01
stock.price.lattice <- buildAssetPriceLattice(S0=100,
n=n,
u=bin.model$u,
d=bin.model$d)
stock.price.lattice## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 100 104.57364 109.35647 114.35804 119.58837 125.05792 130.77762
## [2,] NA 95.62639 100.00000 104.57364 109.35647 114.35804 119.58837
## [3,] NA NA 91.44406 95.62639 100.00000 104.57364 109.35647
## [4,] NA NA NA 87.44466 91.44406 95.62639 100.00000
## [5,] NA NA NA NA 83.62017 87.44466 91.44406
## [6,] NA NA NA NA NA 79.96295 83.62017
## [7,] NA NA NA NA NA NA 76.46568
## [8,] NA NA NA NA NA NA NA
## [9,] NA NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA NA
## [11,] NA NA NA NA NA NA NA
## [,8] [,9] [,10] [,11]
## [1,] 136.75892 143.01379 149.55473 156.39483
## [2,] 125.05792 130.77762 136.75892 143.01379
## [3,] 114.35804 119.58837 125.05792 130.77762
## [4,] 104.57364 109.35647 114.35804 119.58837
## [5,] 95.62639 100.00000 104.57364 109.35647
## [6,] 87.44466 91.44406 95.62639 100.00000
## [7,] 79.96295 83.62017 87.44466 91.44406
## [8,] 73.12137 76.46568 79.96295 83.62017
## [9,] NA 69.92333 73.12137 76.46568
## [10,] NA NA 66.86515 69.92333
## [11,] NA NA NA 63.94073
futures.price.lattice <- buildFuturesPriceLattice(asset.price.lattice=stock.price.lattice,
n=n,
q=bin.model$q)
futures.price.lattice## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 100.5013 105.04529 109.79477 114.75900 119.94768 125.37096 131.03944
## [2,] NA 96.05768 100.40080 104.94029 109.68503 114.64430 119.82779
## [3,] NA NA 91.81057 95.96167 100.30045 104.83540 109.57540
## [4,] NA NA NA 87.75125 91.71881 95.86575 100.20020
## [5,] NA NA NA NA 83.87141 87.66354 91.62714
## [6,] NA NA NA NA NA 80.16311 83.78758
## [7,] NA NA NA NA NA NA 76.61877
## [8,] NA NA NA NA NA NA NA
## [9,] NA NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA NA
## [11,] NA NA NA NA NA NA NA
## [,8] [,9] [,10] [,11]
## [1,] 136.96422 143.15688 149.62953 156.39483
## [2,] 125.24565 130.90847 136.82732 143.01379
## [3,] 114.52971 119.70802 125.12046 130.77762
## [4,] 104.73062 109.46588 114.41524 119.58837
## [5,] 95.76994 100.10005 104.62594 109.35647
## [6,] 87.57592 91.53555 95.67422 100.00000
## [7,] 80.08298 83.70383 87.48839 91.44406
## [8,] 73.23113 76.54219 80.00294 83.62017
## [9,] NA 69.99329 73.15794 76.46568
## [10,] NA NA 66.89859 69.92333
## [11,] NA NA NA 63.94073
option.price.lattice <- buildOptionPriceLattice(asset.price.lattice=futures.price.lattice,
n=n,
K=100,
R=bin.model$R,
q=bin.model$q,
is.call.option=F,
is.amer.option=F)
option.price.lattice## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 5.212449 3.151526 1.612738 0.6284893 0.1420516 0.0000000
## [2,] NA 7.238124 4.662531 2.5784187 1.1054165 0.2812439
## [3,] NA NA 9.771099 6.7097950 4.0239583 1.9135557
## [4,] NA NA NA 12.7840590 9.3495311 6.0956671
## [5,] NA NA NA NA 16.1679546 12.5499604
## [6,] NA NA NA NA NA 19.7379567
## [7,] NA NA NA NA NA NA
## [8,] NA NA NA NA NA NA
## [9,] NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA
## [11,] NA NA NA NA NA NA
## [,7] [,8] [,9] [,10] [,11]
## [1,] 0.0000000 0.000000 0.000000 0.000000 0.000000
## [2,] 0.0000000 0.000000 0.000000 0.000000 0.000000
## [3,] 0.5568267 0.000000 0.000000 0.000000 0.000000
## [4,] 3.2440769 1.102445 0.000000 0.000000 0.000000
## [5,] 8.8962770 5.344781 2.182699 0.000000 0.000000
## [6,] 16.1477031 12.386861 8.447534 4.321461 0.000000
## [7,] 23.2878962 19.857355 16.263609 12.499104 8.555936
## [8,] NA 26.688679 23.410946 19.977073 16.379831
## [9,] NA NA 29.946761 26.815231 23.534319
## [10,] NA NA NA 33.068321 30.076673
## [11,] NA NA NA NA 36.059268
short.rate.lattice <- buildShortRateLattice(r0=0.06,
n=5,
u=1.25,
d=0.9)
short.rate.lattice## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.06 0.075 0.09375 0.1171875 0.1464844 0.18310547
## [2,] NA 0.054 0.06750 0.0843750 0.1054688 0.13183594
## [3,] NA NA 0.04860 0.0607500 0.0759375 0.09492188
## [4,] NA NA NA 0.0437400 0.0546750 0.06834375
## [5,] NA NA NA NA 0.0393660 0.04920750
## [6,] NA NA NA NA NA 0.03542940
zcb.price.lattice <- buildZCBPriceLattice(short.rate.lattice = short.rate.lattice,
face.value = 100,
n=4,
q=0.5)
zcb.price.lattice## [,1] [,2] [,3] [,4] [,5]
## [1,] 77.21774 79.26800 83.07635 89.51049 100
## [2,] NA 84.43361 87.34985 92.21902 100
## [3,] NA NA 90.63619 94.27292 100
## [4,] NA NA NA 95.80930 100
## [5,] NA NA NA NA 100
option.price.lattice <- buildOptionPriceLattice(asset.price.lattice=zcb.price.lattice,
n=2,
K=84,
short.rate.lattice = short.rate.lattice,
q=0.5,
is.call.option=T,
is.amer.option=F)
option.price.lattice ## [,1] [,2] [,3]
## [1,] 2.969474 1.558072 0.000000
## [2,] NA 4.737214 3.349855
## [3,] NA NA 6.636192
option.price.lattice <- buildOptionPriceLattice(asset.price.lattice=zcb.price.lattice,
n=3,
K=88,
short.rate.lattice = short.rate.lattice,
q=0.5,
is.call.option=F,
is.amer.option=T)
option.price.lattice ## [,1] [,2] [,3] [,4]
## [1,] 10.78226 8.731999 4.9236527 0
## [2,] NA 3.566392 0.6501451 0
## [3,] NA NA 0.0000000 0
## [4,] NA NA NA 0
shouldExerciseOption( asset.price.lattice = zcb.price.lattice,
option.price.lattice = option.price.lattice,
K=88,
is.call.option = F )## [,1] [,2] [,3] [,4]
## [1,] TRUE TRUE TRUE FALSE
## [2,] NA TRUE TRUE FALSE
## [3,] NA NA FALSE FALSE
## [4,] NA NA NA FALSE
bond.price.lattice <- buildBondPriceLattice( short.rate.lattice = short.rate.lattice,
face.value = 100,
coupon.rate = 0.10,
n = 6,
q = 0.5)
bond.price.lattice## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 124.1371 115.8298 108.9798 104.0300 101.6554 102.9757 110
## [2,] NA 126.1409 118.5542 112.4884 108.4429 107.1872 110
## [3,] NA NA 126.2709 119.2747 113.8289 110.4638 110
## [4,] NA NA NA 124.5687 117.9973 112.9631 110
## [5,] NA NA NA NA 121.1626 114.8410 110
## [6,] NA NA NA NA NA 116.2361 110
## [7,] NA NA NA NA NA NA 110
# ex-coupon at t=4:
ex.coupon.price <- bond.price.lattice[,4+1] - 100 * 0.10
t(t(ex.coupon.price))## [,1]
## [1,] 91.65536
## [2,] 98.44286
## [3,] 103.82894
## [4,] 107.99732
## [5,] 111.16255
## [6,] NA
## [7,] NA
forward.price.lattice <- buildBinomialPriceLattice( final.price.vec = ex.coupon.price,
short.rate.lattice = short.rate.lattice,
n = 4,
q = 0.5)
forward.price.lattice## [,1] [,2] [,3] [,4] [,5]
## [1,] 79.82696 79.99109 81.52936 85.07893 91.65536
## [2,] NA 89.24207 90.45149 93.26654 98.44286
## [3,] NA NA 97.67079 99.84740 103.82894
## [4,] NA NA NA 104.98777 107.99732
## [5,] NA NA NA NA 111.16255
zcb.price.lattice <- buildZCBPriceLattice(short.rate.lattice = short.rate.lattice,
face.value = 1,
n=4,
q=0.5)
zcb.price.lattice## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.7721774 0.7926800 0.8307635 0.8951049 1
## [2,] NA 0.8443361 0.8734985 0.9221902 1
## [3,] NA NA 0.9063619 0.9427292 1
## [4,] NA NA NA 0.9580930 1
## [5,] NA NA NA NA 1
forward.price.lattice[1,1] / zcb.price.lattice[1,1]## [1] 103.379
futures.price.lattice <- buildFuturesPriceLattice( final.price.vec = ex.coupon.price,
n = 4,
q = 0.5)
futures.price.lattice## [,1] [,2] [,3] [,4] [,5]
## [1,] 103.222 100.8085 98.09251 95.04911 91.65536
## [2,] NA 105.6355 103.52452 101.13590 98.44286
## [3,] NA NA 107.74653 105.91313 103.82894
## [4,] NA NA NA 109.57994 107.99732
## [5,] NA NA NA NA 111.16255
K <- 0.02
# final cash flow is discounted (paid in arrears)
caplet.final.price <- pmax( (short.rate.lattice[,6] - K) / (1 + short.rate.lattice[,6]), 0 )
caplet.price.lattice <- buildBinomialPriceLattice(final.price.vec = caplet.final.price,
short.rate.lattice = short.rate.lattice,
n = 5, # technically expires at t=6
q=0.5)
caplet.price.lattice ## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.04204522 0.05150267 0.06367244 0.08004766 0.10321618 0.13786215
## [2,] NA 0.03763321 0.04705830 0.05923580 0.07564031 0.09880932
## [3,] NA NA 0.03227250 0.04123367 0.05282733 0.06842669
## [4,] NA NA NA 0.02644821 0.03464991 0.04525112
## [5,] NA NA NA NA 0.02056019 0.02783768
## [6,] NA NA NA NA NA 0.01490145
swap.price.lattice <- buildInterestRateSwapPriceLattice( short.rate.lattice = short.rate.lattice,
n = 5, # technically expires at t=6
q = 0.5,
K = 0.05)
swap.price.lattice## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.09900443 0.14025187 0.16860896 0.179274211 0.16475480
## [2,] NA 0.04963752 0.08293256 0.102057879 0.10143601
## [3,] NA NA 0.01370333 0.040003133 0.05115201
## [4,] NA NA NA -0.008464509 0.01221463
## [5,] NA NA NA NA -0.01736413
## [6,] NA NA NA NA NA
## [,6]
## [1,] 0.112505159
## [2,] 0.072303710
## [3,] 0.041027471
## [4,] 0.017170269
## [5,] -0.000755332
## [6,] -0.014072036
swaption.price.lattice <- buildOptionPriceLattice(asset.price.lattice = swap.price.lattice,
n=3,
K=0,
short.rate.lattice = short.rate.lattice,
q=0.5,
is.call.option=T,
is.amer.option=F)
swaption.price.lattice## [,1] [,2] [,3] [,4]
## [1,] 0.06197181 0.09076654 0.12860896 0.17927421
## [2,] NA 0.04061369 0.06653912 0.10205788
## [3,] NA NA 0.01907454 0.04000313
## [4,] NA NA NA 0.00000000
elem.price.lattice <- buildElementarySecurityPriceLattice(short.rate.lattice = short.rate.lattice,
n = 6,
q=0.5)
elem.price.lattice## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 1 0.4716981 0.2193945 0.1002946 0.0448871 0.01957598 0.008273132
## [2,] NA 0.4716981 0.4431602 0.3078638 0.1868416 0.10408383 0.054253219
## [3,] NA NA 0.2237657 0.3142665 0.2900886 0.21931522 0.146131169
## [4,] NA NA NA 0.1066974 0.1992471 0.22926638 0.207450993
## [5,] NA NA NA NA 0.0511130 0.11904756 0.164032044
## [6,] NA NA NA NA NA 0.02458855 0.068605732
## [7,] NA NA NA NA NA NA 0.011873599
# all ZCB bond prices, w/ expirations t=0 thru t=6
zcb.prices <- colSums( elem.price.lattice, na.rm=T) * 100
zcb.prices## [1] 100.00000 94.33962 88.63204 82.91223 77.21774 71.58775 66.06199
# spot rates
(100/zcb.prices)^(1/0:6) - 1## [1] 0.00000000 0.06000000 0.06219594 0.06445458 0.06676984 0.06913428
## [7] 0.07153919
swap.coupon.lattice <- buildInterestRateSwapCouponPaymentLattice( short.rate.lattice = short.rate.lattice,
n = 2, # technically expires at t=3
forward.start = 1,
K = 0.07)
swap.coupon.lattice## [,1] [,2] [,3]
## [1,] 0 0.005 0.02375
## [2,] NA -0.016 -0.00250
## [3,] NA NA -0.02140
discounted.swap.coupon.lattice <- swap.coupon.lattice / (1 + short.rate.lattice[1:3,1:3])
discounted.swap.coupon.lattice ## [,1] [,2] [,3]
## [1,] 0 0.004651163 0.02171429
## [2,] NA -0.015180266 -0.00234192
## [3,] NA NA -0.02040816
sum( discounted.swap.coupon.lattice * elem.price.lattice[1:3,1:3], na.rm=T ) * 1e6## [1] -5807.057
q <- 0.5
n <- 13
s <- c(7.3, 7.62, 8.1, 8.45, 9.2, 9.64, 10.12, 10.45, 10.75, 11.22, 11.55, 11.92, 12.2, 12.32) / 100
a <- rep(5.0/100,n+1) # starting values for a (to be calibrated)
b <- rep(0.5/100,n+1) # fixed b for all time periods
o <- calibrateBDTShortRateLattice(a=a,
b=b,
s=s,
q=q,
n=n, # num periods in calibrated BDT short-rate lattice
optim.rounds=1)
o## $par
## [1] 0.07299988 0.07921192 0.09020980 0.09435880 0.12130371 0.11719397
## [7] 0.12849958 0.12566351 0.12918755 0.15192559 0.14538660 0.15636858
## [13] 0.15151140 0.13449896
##
## $value
## [1] 7.680024e-08
##
## $counts
## function gradient
## 116 116
##
## $convergence
## [1] 1
##
## $message
## [1] "NEW_X"
##
## $bdt
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.07299988 0.07960897 0.09111642 0.09578485 0.1237542 0.1201608
## [2,] NA 0.07921192 0.09066198 0.09530712 0.1231370 0.1195614
## [3,] NA NA 0.09020980 0.09483177 0.1225228 0.1189651
## [4,] NA NA NA 0.09435880 0.1219117 0.1183718
## [5,] NA NA NA NA 0.1213037 0.1177814
## [6,] NA NA NA NA NA 0.1171940
## [7,] NA NA NA NA NA NA
## [8,] NA NA NA NA NA NA
## [9,] NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA
## [11,] NA NA NA NA NA NA
## [12,] NA NA NA NA NA NA
## [13,] NA NA NA NA NA NA
## [14,] NA NA NA NA NA NA
## [,7] [,8] [,9] [,10] [,11] [,12]
## [1,] 0.1324130 0.1301396 0.1344598 0.1589184 0.1528407 0.1652098
## [2,] 0.1317526 0.1294905 0.1337892 0.1581258 0.1520784 0.1643858
## [3,] 0.1310954 0.1288447 0.1331219 0.1573371 0.1513199 0.1635659
## [4,] 0.1304416 0.1282021 0.1324579 0.1565524 0.1505652 0.1627501
## [5,] 0.1297910 0.1275627 0.1317973 0.1557716 0.1498143 0.1619384
## [6,] 0.1291437 0.1269265 0.1311400 0.1549947 0.1490671 0.1611307
## [7,] 0.1284996 0.1262934 0.1304859 0.1542217 0.1483236 0.1603271
## [8,] NA 0.1256635 0.1298351 0.1534525 0.1475838 0.1595274
## [9,] NA NA 0.1291875 0.1526871 0.1468478 0.1587318
## [10,] NA NA NA 0.1519256 0.1461154 0.1579401
## [11,] NA NA NA NA 0.1453866 0.1571524
## [12,] NA NA NA NA NA 0.1563686
## [13,] NA NA NA NA NA NA
## [14,] NA NA NA NA NA NA
## [,13] [,14]
## [1,] 0.1608803 0.1435318
## [2,] 0.1600779 0.1428159
## [3,] 0.1592796 0.1421036
## [4,] 0.1584851 0.1413949
## [5,] 0.1576947 0.1406897
## [6,] 0.1569082 0.1399880
## [7,] 0.1561256 0.1392898
## [8,] 0.1553469 0.1385951
## [9,] 0.1545721 0.1379038
## [10,] 0.1538012 0.1372160
## [11,] 0.1530341 0.1365317
## [12,] 0.1522709 0.1358507
## [13,] 0.1515114 0.1351731
## [14,] NA 0.1344990
forward.start.swap.lattice <- buildInterestRateSwapPriceLattice( short.rate.lattice=o$bdt,
n=9, # technically expires at t=10
forward.start=2,
K=11.65/100 )
forward.start.swap.lattice ## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0009495206 0.0023548143 0.003985117 0.03107983 0.05601786
## [2,] NA -0.0003171433 0.001099440 0.02838377 0.05352607
## [3,] NA NA -0.001783970 0.02569051 0.05103758
## [4,] NA NA NA 0.02300009 0.04855244
## [5,] NA NA NA NA 0.04607066
## [6,] NA NA NA NA NA
## [7,] NA NA NA NA NA
## [8,] NA NA NA NA NA
## [9,] NA NA NA NA NA
## [10,] NA NA NA NA NA
## [,6] [,7] [,8] [,9] [,10]
## [1,] 0.05680504 0.06092997 0.05386226 0.04780411 0.03660172
## [2,] 0.05458715 0.05901009 0.05230756 0.04666041 0.03594238
## [3,] 0.05237310 0.05709434 0.05075695 0.04552029 0.03528543
## [4,] 0.05016293 0.05518274 0.04921046 0.04438375 0.03463087
## [5,] 0.04795667 0.05327530 0.04766807 0.04325077 0.03397869
## [6,] 0.04575432 0.05137202 0.04612979 0.04212135 0.03332889
## [7,] NA 0.04947294 0.04459563 0.04099550 0.03268146
## [8,] NA NA 0.04306559 0.03987322 0.03203640
## [9,] NA NA NA 0.03875449 0.03139371
## [10,] NA NA NA NA 0.03075337
swaption.price.lattice <- buildOptionPriceLattice( forward.start.swap.lattice,
K=0,
n=2,
short.rate.lattice=o$bdt )
swaption.price.lattice## [,1] [,2] [,3]
## [1,] 0.001334663 0.0023548143 0.003985117
## [2,] NA 0.0005093718 0.001099440
## [3,] NA NA 0.000000000
swaption.price.lattice[1,1] * 1e6## [1] 1334.663
Note: answer is slightly off from lecture notes, due to diffs in calibration
b <- rep(1.0/100,n+1)
o <- calibrateBDTShortRateLattice(a=a,
b=b,
s=s,
q=q,
n=n,
optim.rounds=1)
o## $par
## [1] 0.07299996 0.07901313 0.08976069 0.09364855 0.12009703 0.11573303
## [7] 0.12658456 0.12347876 0.12663376 0.14855930 0.14179059 0.15211842
## [13] 0.14706940 0.13019601
##
## $value
## [1] 1.937905e-08
##
## $counts
## function gradient
## 115 115
##
## $convergence
## [1] 0
##
## $message
## [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
##
## $bdt
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.07299996 0.07980722 0.09157398 0.09650057 0.1249983 0.1216668
## [2,] NA 0.07901313 0.09066280 0.09554038 0.1237545 0.1204562
## [3,] NA NA 0.08976069 0.09458973 0.1225232 0.1192576
## [4,] NA NA NA 0.09364855 0.1213040 0.1180710
## [5,] NA NA NA NA 0.1200970 0.1168962
## [6,] NA NA NA NA NA 0.1157330
## [7,] NA NA NA NA NA NA
## [8,] NA NA NA NA NA NA
## [9,] NA NA NA NA NA NA
## [10,] NA NA NA NA NA NA
## [11,] NA NA NA NA NA NA
## [12,] NA NA NA NA NA NA
## [13,] NA NA NA NA NA NA
## [14,] NA NA NA NA NA NA
## [,7] [,8] [,9] [,10] [,11] [,12]
## [1,] 0.1344121 0.1324320 0.1371807 0.1625498 0.1567028 0.1698065
## [2,] 0.1330747 0.1311143 0.1358157 0.1609324 0.1551436 0.1681169
## [3,] 0.1317506 0.1298097 0.1344644 0.1593311 0.1535999 0.1664441
## [4,] 0.1304396 0.1285180 0.1331264 0.1577457 0.1520716 0.1647879
## [5,] 0.1291417 0.1272393 0.1318018 0.1561761 0.1505584 0.1631483
## [6,] 0.1278568 0.1259732 0.1304903 0.1546221 0.1490603 0.1615249
## [7,] 0.1265846 0.1247197 0.1291919 0.1530836 0.1475772 0.1599177
## [8,] NA 0.1234788 0.1279065 0.1515604 0.1461088 0.1583265
## [9,] NA NA 0.1266338 0.1500523 0.1446549 0.1567511
## [10,] NA NA NA 0.1485593 0.1432156 0.1551914
## [11,] NA NA NA NA 0.1417906 0.1536472
## [12,] NA NA NA NA NA 0.1521184
## [13,] NA NA NA NA NA NA
## [14,] NA NA NA NA NA NA
## [,13] [,14]
## [1,] 0.1658203 0.1482709
## [2,] 0.1641703 0.1467956
## [3,] 0.1625368 0.1453350
## [4,] 0.1609196 0.1438888
## [5,] 0.1593184 0.1424571
## [6,] 0.1577331 0.1410397
## [7,] 0.1561637 0.1396363
## [8,] 0.1546098 0.1382469
## [9,] 0.1530714 0.1368713
## [10,] 0.1515483 0.1355094
## [11,] 0.1500404 0.1341611
## [12,] 0.1485475 0.1328261
## [13,] 0.1470694 0.1315045
## [14,] NA 0.1301960
forward.start.swap.lattice <- buildInterestRateSwapPriceLattice( short.rate.lattice=o$bdt,
n=9, # technically expires at t=10
forward.start=2,
K=11.65/100 )
forward.start.swap.lattice ## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0009533597 0.003694839 0.006876445 0.03513094 0.06101718
## [2,] NA -0.001648929 0.001102983 0.02973339 0.05602387
## [3,] NA NA -0.004661416 0.02434696 0.05104364
## [4,] NA NA NA 0.01897199 0.04607677
## [5,] NA NA NA NA 0.04112352
## [6,] NA NA NA NA NA
## [7,] NA NA NA NA NA
## [8,] NA NA NA NA NA
## [9,] NA NA NA NA NA
## [10,] NA NA NA NA NA
## [,6] [,7] [,8] [,9] [,10]
## [1,] 0.06237145 0.06672334 0.05934663 0.05243030 0.03961101
## [2,] 0.05792043 0.06286306 0.05621267 0.05011779 0.03827301
## [3,] 0.05348467 0.05901918 0.05309504 0.04781958 0.03694464
## [4,] 0.04906436 0.05519182 0.04999379 0.04553568 0.03562587
## [5,] 0.04465971 0.05138108 0.04690897 0.04326606 0.03431666
## [6,] 0.04027089 0.04758710 0.04384061 0.04101070 0.03301697
## [7,] NA 0.04380997 0.04078874 0.03876959 0.03172676
## [8,] NA NA 0.03775340 0.03654271 0.03044599
## [9,] NA NA NA 0.03433005 0.02917463
## [10,] NA NA NA NA 0.02791262
swaption.price.lattice <- buildOptionPriceLattice( forward.start.swap.lattice,
K=0,
n=2,
short.rate.lattice=o$bdt )
swaption.price.lattice## [,1] [,2] [,3]
## [1,] 0.001959901 0.0036948393 0.006876445
## [2,] NA 0.0005111073 0.001102983
## [3,] NA NA 0.000000000
swaption.price.lattice[1,1] * 1e6## [1] 1959.901
Note: answer is slightly off from lecture notes, due to diffs in calibration
delta = 1/2 # coupon payments every 6 mos.
h <- rep(0.02,10+1) # fixed hazard rate
bond.1yr <- buildDefaultableBondSchedule( face.value = 100,
hazard.rate.vec = h,
r = 0.05,
recovery.rate = 0.10,
coupon.rate = 0.05,
delta = delta,
n = 1/delta )
data.frame(bond.1yr)## hazard.rate survival.probability default.probability discount.rate
## 1 0.02 1.0000 0.0000 1.0000000
## 2 0.02 0.9800 0.0200 0.9756098
## 3 0.02 0.9604 0.0196 0.9518144
## coupon.payment principal.payment recovery.payment all.payment
## 1 0.000000 0.00000 0.0000000 0.00000
## 2 4.780488 0.00000 0.1951220 4.97561
## 3 4.570613 91.41225 0.1865556 96.16942
bond.1yr.price <- sum(bond.1yr$all.payment)
bond.1yr.price## [1] 101.145
bond.5yr <- buildDefaultableBondSchedule( face.value = 100,
hazard.rate.vec = h,
r = 0.05,
recovery.rate = 0.20,
coupon.rate = 0.10,
delta = delta,
n = 5 / delta )
data.frame(bond.5yr)## hazard.rate survival.probability default.probability discount.rate
## 1 0.02 1.0000000 0.00000000 1.0000000
## 2 0.02 0.9800000 0.02000000 0.9756098
## 3 0.02 0.9604000 0.01960000 0.9518144
## 4 0.02 0.9411920 0.01920800 0.9285994
## 5 0.02 0.9223682 0.01882384 0.9059506
## 6 0.02 0.9039208 0.01844736 0.8838543
## 7 0.02 0.8858424 0.01807842 0.8622969
## 8 0.02 0.8681255 0.01771685 0.8412652
## 9 0.02 0.8507630 0.01736251 0.8207466
## 10 0.02 0.8337478 0.01701526 0.8007284
## 11 0.02 0.8170728 0.01667496 0.7811984
## coupon.payment principal.payment recovery.payment all.payment
## 1 0.000000 0.0000 0.0000000 0.000000
## 2 9.560976 0.0000 0.3902439 9.951220
## 3 9.141225 0.0000 0.3731112 9.514337
## 4 8.739903 0.0000 0.3567307 9.096634
## 5 8.356200 0.0000 0.3410694 8.697270
## 6 7.989343 0.0000 0.3260956 8.315438
## 7 7.638591 0.0000 0.3117792 7.950370
## 8 7.303238 0.0000 0.2980914 7.601330
## 9 6.982608 0.0000 0.2850044 7.267613
## 10 6.676055 0.0000 0.2724920 6.948547
## 11 6.382960 63.8296 0.2605290 70.473086
bond.5yr.price <- sum(bond.5yr$all.payment)
bond.5yr.price## [1] 145.8158
# these are just random values
market.prices <- c(101.2179,
92.5828,
107.3675,
104.0419,
145.9155)
#
# compute 1-5 yr bond prices using the given hazard rates
#
computeBondPrices <- function(h) {
delta = 1/2 # coupon payments every 6 mos.
bond.1yr <- computeDefaultableBondPricing( face.value = 100,
hazard.rate.vec = h,
r = 0.05,
recovery.rate = 0.10,
coupon.rate = 0.05,
delta = delta,
n = 1/delta )
bond.2yr <- computeDefaultableBondPricing( face.value = 100,
hazard.rate.vec = h,
r = 0.05,
recovery.rate = 0.25,
coupon.rate = 0.02,
delta = delta,
n = 2 / delta )
bond.3yr <- computeDefaultableBondPricing( face.value = 100,
hazard.rate.vec = h,
r = 0.05,
recovery.rate = 0.50,
coupon.rate = 0.05,
delta = delta,
n = 3 / delta )
bond.4yr <- computeDefaultableBondPricing( face.value = 100,
hazard.rate.vec = h,
r = 0.05,
recovery.rate = 0.10,
coupon.rate = 0.05,
delta = delta,
n = 4 / delta )
bond.5yr <- computeDefaultableBondPricing( face.value = 100,
hazard.rate.vec = h,
r = 0.05,
recovery.rate = 0.20,
coupon.rate = 0.10,
delta = delta,
n = 5 / delta )
return( list(bond.1yr=bond.1yr,
bond.2yr=bond.2yr,
bond.3yr=bond.3yr,
bond.4yr=bond.4yr,
bond.5yr=bond.5yr) )
}
#
# define objective function (to be minimized)
#
objectiveFn <- function(h) {
bonds <- computeBondPrices(h)
model.prices <- c(bonds$bond.1yr$price,
bonds$bond.2yr$price,
bonds$bond.3yr$price,
bonds$bond.4yr$price,
bonds$bond.5yr$price )
sq.diffs <- (market.prices - model.prices)^2
sum(sq.diffs) # to minimize
}
o <- optim(par=h,
fn=objectiveFn,
method = "L-BFGS-B")
o## $par
## [1] 0.01957532 0.01962483 0.02014341 0.02016492 0.02010088 0.02012203
## [7] 0.02008368 0.02010715 0.01943048 0.01948716 0.02000000
##
## $value
## [1] 2.994097e-11
##
## $counts
## function gradient
## 23 23
##
## $convergence
## [1] 0
##
## $message
## [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
bonds <- computeBondPrices(o$par)
rbind(model.prices=c(bonds$bond.1yr$price, bonds$bond.2yr$price, bonds$bond.3yr$price, bonds$bond.4yr$price, bonds$bond.5yr$price),
market.prices)## [,1] [,2] [,3] [,4] [,5]
## model.prices 101.2179 92.5828 107.3675 104.0419 145.9155
## market.prices 101.2179 92.5828 107.3675 104.0419 145.9155
#
# using a constraint: allow positive values only.
#
# ui %*% theta - ci >= 0
# kxp px1 kx1
#
ui <- matrix(1,nrow=1,ncol=length(h))
ci <- c(0)
ui %*% h - ci ## [,1]
## [1,] 0.22
o1 <- constrOptim( theta=h,
f=objectiveFn,
grad=NULL,
ui=ui,
ci=ci)
o1## $par
## [1] 0.01952914 0.01967337 0.02010897 0.02019854 0.01995501 0.02028295
## [7] 0.02007349 0.02011269 0.01898095 0.01999905 0.02161238
##
## $value
## [1] 2.202676e-13
##
## $counts
## function gradient
## 864 NA
##
## $convergence
## [1] 0
##
## $message
## NULL
##
## $outer.iterations
## [1] 3
##
## $barrier.value
## [1] 5.538853e-05
bonds <- computeBondPrices(o1$par)
rbind(model.prices=c(bonds$bond.1yr$price, bonds$bond.2yr$price, bonds$bond.3yr$price, bonds$bond.4yr$price, bonds$bond.5yr$price),
market.prices)## [,1] [,2] [,3] [,4] [,5]
## model.prices 101.2179 92.5828 107.3675 104.0419 145.9155
## market.prices 101.2179 92.5828 107.3675 104.0419 145.9155
#
# using a constraint: pair-wise compare hazard rates
# to ensure non-decreasing
#
# ui %*% theta - ci >= 0
# kxp px1 kx1
#
ui <- matrix(c(
-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1 ),
nrow = 10,
ncol = 11,
byrow=T)
ci <- rep(0,10)
h1 <- h + cumsum(rep(0.0001,length(h))) - 0.0005
ui %*% h1 - ci ## [,1]
## [1,] 1e-04
## [2,] 1e-04
## [3,] 1e-04
## [4,] 1e-04
## [5,] 1e-04
## [6,] 1e-04
## [7,] 1e-04
## [8,] 1e-04
## [9,] 1e-04
## [10,] 1e-04
o2 <- constrOptim( theta=h1,
f=objectiveFn,
grad=NULL,
ui=ui,
ci=ci)
o2## $par
## [1] 0.01953676 0.01986096 0.01990686 0.01997758 0.02003733 0.02007377
## [7] 0.02007377 0.02007377 0.02007377 0.02007379 0.02107375
##
## $value
## [1] 0.003323694
##
## $counts
## function gradient
## 5518 NA
##
## $convergence
## [1] 1
##
## $message
## NULL
##
## $outer.iterations
## [1] 12
##
## $barrier.value
## [1] 1.313746e-06
bonds <- computeBondPrices(o2$par)
rbind(model.prices=c(bonds$bond.1yr$price, bonds$bond.2yr$price, bonds$bond.3yr$price, bonds$bond.4yr$price, bonds$bond.5yr$price),
market.prices)## [,1] [,2] [,3] [,4] [,5]
## model.prices 101.2006 92.59752 107.3832 104.0717 145.8746
## market.prices 101.2179 92.58280 107.3675 104.0419 145.9155
# copied from spreadsheet
h <- c(0.010070286520099,
0.010070286520099,
0.010070286520099,
0.010070286520099,
0.009721551669331,
0.009721551669331,
0.009721551669331,
0.009721551669331,
0.009721551669331) # for time periods t=0 thru n
cds <- buildCDSSchedule( N = 1e6, # principal
hazard.rate.vec = h,
n = 8,
delta = 1/4,
r = 0.01,
recovery.rate = 0.45 )
data.frame(cds)## hazard.rate.vec survival.probability default.probability discount.rate
## 1 0.010070287 1.0000000 0.000000000 1.0000000
## 2 0.010070287 0.9899297 0.010070287 0.9975062
## 3 0.010070287 0.9799608 0.009968876 0.9950187
## 4 0.010070287 0.9700924 0.009868486 0.9925373
## 5 0.009721552 0.9603232 0.009769108 0.9900622
## 6 0.009721552 0.9509874 0.009335832 0.9875932
## 7 0.009721552 0.9417423 0.009245073 0.9851304
## 8 0.009721552 0.9325871 0.009155197 0.9826737
## 9 0.009721552 0.9235209 0.009066194 0.9802231
## S.par premium.payment accrued.interest protection.payment
## 1 0.02188895 0.000 0.00000 0.000
## 2 0.02188895 5403.622 27.48479 5524.845
## 3 0.02188895 5335.867 27.14016 5455.570
## 4 0.02188895 5268.961 26.79985 5387.163
## 5 0.02188895 5202.893 26.46381 5319.613
## 6 0.02188895 5139.465 25.22703 5071.002
## 7 0.02188895 5076.809 24.91949 5009.181
## 8 0.02188895 5014.917 24.61569 4948.114
## 9 0.02188895 4953.780 24.31560 4887.791
## premium.leg protection.leg
## 1 0.000 0.000
## 2 5431.107 5524.845
## 3 10794.114 10980.415
## 4 16089.875 16367.578
## 5 21319.232 21687.191
## 6 26483.923 26758.194
## 7 31585.652 31767.375
## 8 36625.185 36715.489
## 9 41603.280 41603.280
mort <- buildMortgageSchedule(mortgage.balance = 20000,
monthly.mortgage.rate = 0.05/12,
n = 18)
data.frame(mort)## begin.balance monthly.payment mortgage.interest principal.payment
## 1 20000.000 1155.611 83.333333 1072.277
## 2 18927.723 1155.611 78.865511 1076.745
## 3 17850.977 1155.611 74.379073 1081.232
## 4 16769.746 1155.611 69.873941 1085.737
## 5 15684.009 1155.611 65.350038 1090.261
## 6 14593.748 1155.611 60.807285 1094.803
## 7 13498.945 1155.611 56.245604 1099.365
## 8 12399.580 1155.611 51.664916 1103.946
## 9 11295.634 1155.611 47.065142 1108.546
## 10 10187.089 1155.611 42.446203 1113.164
## 11 9073.924 1155.611 37.808017 1117.803
## 12 7956.121 1155.611 33.150506 1122.460
## 13 6833.661 1155.611 28.473589 1127.137
## 14 5706.524 1155.611 23.777184 1131.834
## 15 4574.691 1155.611 19.061211 1136.549
## 16 3438.141 1155.611 14.325588 1141.285
## 17 2296.856 1155.611 9.570234 1146.040
## 18 1150.816 1155.611 4.795065 1150.816
## end.balance
## 1 18927.723
## 2 17850.977
## 3 16769.746
## 4 15684.009
## 5 14593.748
## 6 13498.945
## 7 12399.580
## 8 11295.634
## 9 10187.089
## 10 9073.924
## 11 7956.121
## 12 6833.661
## 13 5706.524
## 14 4574.691
## 15 3438.141
## 16 2296.856
## 17 1150.816
## 18 0.000
computeMortgagePresentValue(monthly.payment = mort$monthly.payment[1],
monthly.market.rate = 0.05/12,
n = 18)## [1] 20000
cpr.rate <- buildCPRSchedule( psa.multiplier = 100,
seasoning = 0,
n = 30 )
mbs <- buildMBSPassThruSchedule( mortgage.balance = 400,
monthly.mortgage.rate = 8.125/100/12,
monthly.passthru.rate = 7.500/100/12,
cpr.rate = cpr.rate[4:20],
n = 17)
data.frame(mbs)## cpr.rate smm.rate begin.balance monthly.payment mortgage.interest
## 1 0.008 0.0006691237 400.00000 24.98903 2.7083333
## 2 0.010 0.0008371774 377.46656 24.97231 2.5557632
## 3 0.012 0.0010055425 354.75277 24.95140 2.4019719
## 4 0.014 0.0011742204 331.86930 24.92631 2.2470317
## 5 0.016 0.0013432122 308.82696 24.89705 2.0910159
## 6 0.018 0.0015125192 285.63674 24.86360 1.9339988
## 7 0.020 0.0016821426 262.30979 24.82600 1.7760559
## 8 0.022 0.0018520835 238.85738 24.78424 1.6172635
## 9 0.024 0.0020223433 215.29093 24.73833 1.4576990
## 10 0.026 0.0021929233 191.62199 24.68830 1.2974405
## 11 0.028 0.0023638246 167.86220 24.63416 1.1365670
## 12 0.030 0.0025350486 144.02335 24.57593 0.9751581
## 13 0.032 0.0027065965 120.11730 24.51363 0.8132942
## 14 0.034 0.0028784696 96.15600 24.44728 0.6510563
## 15 0.036 0.0030506693 72.15149 24.37691 0.4885257
## 16 0.038 0.0032231967 48.11587 24.30255 0.3257845
## 17 0.040 0.0033960532 24.06130 24.22422 0.1629151
## passthru.interest principal.payment principal.prepayment
## 1 2.5000000 22.28070 2.527409e-01
## 2 2.3591660 22.41655 2.972398e-01
## 3 2.2172048 22.54943 3.340446e-01
## 4 2.0741831 22.67928 3.630572e-01
## 5 1.9301685 22.80603 3.841868e-01
## 6 1.7852296 22.92960 3.973496e-01
## 7 1.6394362 23.04994 4.024692e-01
## 8 1.4928586 23.16697 3.994766e-01
## 9 1.3455683 23.28063 3.883107e-01
## 10 1.1976374 23.39086 3.689179e-01
## 11 1.0491388 23.49760 3.412526e-01
## 12 0.9001460 23.60078 3.052771e-01
## 13 0.7507331 23.70034 2.609618e-01
## 14 0.6009750 23.79623 2.082854e-01
## 15 0.4509468 23.88839 1.472348e-01
## 16 0.3007242 23.97676 7.780508e-02
## 17 0.1503831 24.06130 7.118471e-16
## total.principal end.balance
## 1 22.53344 3.774666e+02
## 2 22.71379 3.547528e+02
## 3 22.88348 3.318693e+02
## 4 23.04234 3.088270e+02
## 5 23.19022 2.856367e+02
## 6 23.32695 2.623098e+02
## 7 23.45241 2.388574e+02
## 8 23.56645 2.152909e+02
## 9 23.66894 1.916220e+02
## 10 23.75978 1.678622e+02
## 11 23.83885 1.440234e+02
## 12 23.90605 1.201173e+02
## 13 23.96130 9.615600e+01
## 14 24.00451 7.215149e+01
## 15 24.03562 4.811587e+01
## 16 24.05457 2.406130e+01
## 17 24.06130 2.096101e-13
avg.life <- 1 / 12 / 400 * sum( 1:17 * mbs$total.principal )
avg.life## [1] 0.75815
tranches <- buildMBSTrancheSchedule( mbs=mbs,
tranche.begin.balances = c(194.5, 36, 96.5, 73),
n=17 )
data.frame(tranches[1])## begin.balance total.principal passthru.interest end.balance
## 1 194.50000 22.53344 1.21562500 171.96656
## 2 171.96656 22.71379 1.07479101 149.25277
## 3 149.25277 22.88348 0.93282984 126.36930
## 4 126.36930 23.04234 0.78980811 103.32696
## 5 103.32696 23.19022 0.64579349 80.13674
## 6 80.13674 23.32695 0.50085464 56.80979
## 7 56.80979 23.45241 0.35506118 33.35738
## 8 33.35738 23.56645 0.20848362 9.79093
## 9 9.79093 9.79093 0.06119331 0.00000
## 10 0.00000 0.00000 0.00000000 0.00000
## 11 0.00000 0.00000 0.00000000 0.00000
## 12 0.00000 0.00000 0.00000000 0.00000
## 13 0.00000 0.00000 0.00000000 0.00000
## 14 0.00000 0.00000 0.00000000 0.00000
## 15 0.00000 0.00000 0.00000000 0.00000
## 16 0.00000 0.00000 0.00000000 0.00000
## 17 0.00000 0.00000 0.00000000 0.00000
data.frame(tranches[2])## begin.balance total.principal passthru.interest end.balance
## 1 36.00000 0.00000 0.2250000 36.00000
## 2 36.00000 0.00000 0.2250000 36.00000
## 3 36.00000 0.00000 0.2250000 36.00000
## 4 36.00000 0.00000 0.2250000 36.00000
## 5 36.00000 0.00000 0.2250000 36.00000
## 6 36.00000 0.00000 0.2250000 36.00000
## 7 36.00000 0.00000 0.2250000 36.00000
## 8 36.00000 0.00000 0.2250000 36.00000
## 9 36.00000 13.87801 0.2250000 22.12199
## 10 22.12199 22.12199 0.1382624 0.00000
## 11 0.00000 0.00000 0.0000000 0.00000
## 12 0.00000 0.00000 0.0000000 0.00000
## 13 0.00000 0.00000 0.0000000 0.00000
## 14 0.00000 0.00000 0.0000000 0.00000
## 15 0.00000 0.00000 0.0000000 0.00000
## 16 0.00000 0.00000 0.0000000 0.00000
## 17 0.00000 0.00000 0.0000000 0.00000
data.frame(tranches[3])## begin.balance total.principal passthru.interest end.balance
## 1 96.50000 0.000000 0.6031250 96.50000
## 2 96.50000 0.000000 0.6031250 96.50000
## 3 96.50000 0.000000 0.6031250 96.50000
## 4 96.50000 0.000000 0.6031250 96.50000
## 5 96.50000 0.000000 0.6031250 96.50000
## 6 96.50000 0.000000 0.6031250 96.50000
## 7 96.50000 0.000000 0.6031250 96.50000
## 8 96.50000 0.000000 0.6031250 96.50000
## 9 96.50000 0.000000 0.6031250 96.50000
## 10 96.50000 1.637796 0.6031250 94.86220
## 11 94.86220 23.838850 0.5928888 71.02335
## 12 71.02335 23.906052 0.4438960 47.11730
## 13 47.11730 23.961300 0.2944831 23.15600
## 14 23.15600 23.156002 0.1447250 0.00000
## 15 0.00000 0.000000 0.0000000 0.00000
## 16 0.00000 0.000000 0.0000000 0.00000
## 17 0.00000 0.000000 0.0000000 0.00000
data.frame(tranches[4])## begin.balance total.principal passthru.interest end.balance
## 1 73.00000 0.0000000 0.4562500 7.300000e+01
## 2 73.00000 0.0000000 0.4562500 7.300000e+01
## 3 73.00000 0.0000000 0.4562500 7.300000e+01
## 4 73.00000 0.0000000 0.4562500 7.300000e+01
## 5 73.00000 0.0000000 0.4562500 7.300000e+01
## 6 73.00000 0.0000000 0.4562500 7.300000e+01
## 7 73.00000 0.0000000 0.4562500 7.300000e+01
## 8 73.00000 0.0000000 0.4562500 7.300000e+01
## 9 73.00000 0.0000000 0.4562500 7.300000e+01
## 10 73.00000 0.0000000 0.4562500 7.300000e+01
## 11 73.00000 0.0000000 0.4562500 7.300000e+01
## 12 73.00000 0.0000000 0.4562500 7.300000e+01
## 13 73.00000 0.0000000 0.4562500 7.300000e+01
## 14 73.00000 0.8485105 0.4562500 7.215149e+01
## 15 72.15149 24.0356219 0.4509468 4.811587e+01
## 16 48.11587 24.0545675 0.3007242 2.406130e+01
## 17 24.06130 24.0613000 0.1503831 3.232969e-13