This package uses Monte Carlo simulation to estimate the fair market value of a large portfolio of synthetic variable annuities. The portfolio of variable annuities under consideration is generated based on realistic features of common types of guarantee riders in North America. The Monte Carlo simulation engine generates sample paths of asset prices based on Black-Scholes model. In this vignette, we will demonstrate the functionalities provided in this package.
For illustrative purposes, we will use few scenarios to valuate a pool of two variable annuities. Users may obtain a more robust valuation result by increasing the amount of risk-neutral scenarios.
In this step, we exploit Newton’s method to calculate discount factors and forward rates at different tenor based on given swap rates using buildCurve().
# Initialize required inputs to boostrap a curve
swap <- c(0.69, 0.77, 0.88, 1.01, 1.14, 1.38, 1.66, 2.15)*0.01
tenor <- c(1, 2, 3, 4, 5, 7, 10, 30)
fixFreq <- 6
fixDCC <- "Thirty360"
fltFreq <- 6
fltDCC <- "ACT360"
calendar <- "NY"
bdc <- "Modified_Foll"
curveDate <- "2016-02-08"
numSetDay <- 2
yieldCurveDCC <- "Thirty360"
# Bootstrap a forward curve
buildCurve(swap, tenor, fixFreq, fixDCC, fltFreq, fltDCC, calendar, bdc,
curveDate, numSetDay, yieldCurveDCC)
#> obsDate discountFac zeroRate forwardCurve dayCount
#> 1 2016-02-08 1.0000000 0.000000000 0.006912035 0.000000
#> 2 2017-02-10 0.9930975 0.006888125 0.008520036 1.005556
#> 3 2018-02-10 0.9847078 0.007683825 0.011060713 2.005556
#> 4 2019-02-10 0.9739354 0.008787170 0.014146403 3.005556
#> 5 2020-02-10 0.9603499 0.010100373 0.016846310 4.005556
#> 6 2021-02-10 0.9444396 0.011420029 0.020451281 5.005556
#> 7 2023-02-10 0.9073275 0.013882093 0.024514485 7.005556
#> 8 2026-02-10 0.8451708 0.016812319 0.032098279 10.005556
#> 9 2046-02-10 0.5147311 0.022132923 0.032098279 30.005556In the following example, we first simulate the index movements using genIndexScen(). Three of the inputs to genIndexScen() are stored as default data under variable names “mCov”, “indexNames”, and “cForwardCurve” respectively. For illustration purposes, we will simulate 100 scenarios for 360 steps with a step length dT = 1/12 and seed = 1.
The underlying model utilizes the multivariate Black-Scholes model. All the simulated index movements are stored in a 3D-array with dimensions [number of Scenarios, number of Steps, number of Indices]
| US | SMALL | INT | FIXED | MONEY |
|---|---|---|---|---|
| 0.15 | 0.0 | 0.00 | 0.000 | 0.0000 |
| 0.00 | 0.2 | 0.00 | 0.000 | 0.0000 |
| 0.00 | 0.0 | 0.17 | 0.000 | 0.0000 |
| 0.00 | 0.0 | 0.00 | 0.029 | 0.0000 |
| 0.00 | 0.0 | 0.00 | 0.000 | 0.0061 |
| Index Names |
|---|
| US |
| SMALL |
| INT |
| FIXED |
| MONEY |
# We will show the index simulated path for five months of the first scenario
indexScen <- genIndexScen(mCov, 100, 360, indexNames, 1 / 12, cForwardCurve, 1)
indexScen[1, 1:5, ]
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.9280933 0.7108209 1.0542468 1.0391611 1.0050411
#> [2,] 1.0160765 1.1770017 0.9065351 0.9935556 0.9305910
#> [3,] 0.9066403 0.9151064 1.2239807 1.0199796 0.9939919
#> [4,] 1.1897872 0.9397168 0.9853616 1.0115470 1.0117871
#> [5,] 1.0327826 0.9718774 0.8855638 1.0016170 0.9723979Then we use genFundScen() to map the index movements to funds according to different allocations of capital using a fund map (stored as default data under variable “fundMap”). The fund movements are also stored in a 3D-array with dimension [number of Scenarios, number of Steps, number of Funds]
# Again, we show the fund simulated path for five months of the first scenario
fundScen <- genFundScen(fundMap, indexScen)
fundScen[1, 1:5, ]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.9280933 0.7108209 1.0542468 1.0391611 1.0050411 0.8411844 0.9911700
#> [2,] 1.0160765 1.1770017 0.9065351 0.9935556 0.9305910 1.0804466 0.9613058
#> [3,] 0.9066403 0.9151064 1.2239807 1.0199796 0.9939919 0.9100268 1.0653105
#> [4,] 1.1897872 0.9397168 0.9853616 1.0115470 1.0117871 1.0897591 1.0875744
#> [5,] 1.0327826 0.9718774 0.8855638 1.0016170 0.9723979 1.0084206 0.9591732
#> [,8] [,9] [,10]
#> [1,] 0.9836272 0.9512190 0.9474726
#> [2,] 1.0048160 0.9876751 1.0047520
#> [3,] 0.9633100 1.1313184 1.0119398
#> [4,] 1.1006671 0.9716682 1.0276399
#> [5,] 1.0171998 0.9114579 0.9728478Perhaps the most value-added step in this package is the generation of synthetic portfolio of variable annuities that has realistic charateristic features. Using the fuction genPortInception(), users can generate a synthetic variable annuity portfolio of desirable size. The function genPortInception() has certain predetermined default values based on the research in the package reference. We recommend users to change these default values, such as maturity and issue range, to match their portfolio characteristics. In the current version, there are a few constraints for the portfolio being generated: The issue range must be later than the first date of historical scenario; The maturity range should also be set after the valuation date to be meaningful.
# For illustration purposes, we will only simulate one guarantee contract for each of the 19 guarantee types. Please note that due to randomness the generated portfolio under this code block may not align with the default VAPort under lazy data.
if(capabilities("long.double")) {
genPortInception(issueRng = c("2001-08-01", "2014-01-01"), numPolicy = 1)
}
#> recordID survivorShip gender productType issueDate matDate birthDate
#> DBRP 1 1 F DBRP 2013-11-01 2031-11-01 1954-02-01
#> DBRU 2 1 F DBRU 2009-09-01 2029-09-01 1957-06-01
#> DBSU 3 1 F DBSU 2010-08-01 2030-08-01 1978-08-01
#> ABRP 4 1 M ABRP 2007-07-01 2022-07-01 1969-05-01
#> ABRU 5 1 M ABRU 2008-07-01 2025-07-01 1960-07-01
#> ABSU 6 1 M ABSU 2011-05-01 2030-05-01 1966-05-01
#> IBRP 7 1 F IBRP 2001-11-01 2024-11-01 1978-03-01
#> IBRU 8 1 F IBRU 2002-07-01 2026-07-01 1968-11-01
#> IBSU 9 1 M IBSU 2013-07-01 2039-07-01 1964-07-01
#> MBRP 10 1 M MBRP 2009-08-01 2026-08-01 1964-11-01
#> MBRU 11 1 M MBRU 2009-03-01 2033-03-01 1975-04-01
#> MBSU 12 1 M MBSU 2013-12-01 2041-12-01 1953-05-01
#> WBRP 13 1 F WBRP 2008-10-01 2027-10-01 1972-09-01
#> WBRU 14 1 F WBRU 2008-12-01 2026-12-01 1968-06-01
#> WBSU 15 1 M WBSU 2004-10-01 2031-10-01 1950-02-01
#> DBAB 16 1 F DBAB 2007-12-01 2034-12-01 1971-05-01
#> DBIB 17 1 F DBIB 2012-05-01 2028-05-01 1951-10-01
#> DBMB 18 1 F DBMB 2007-09-01 2033-09-01 1956-09-01
#> DBWB 19 1 F DBWB 2004-04-01 2033-04-01 1963-07-01
#> currentDate baseFee riderFee rollUpRate gbAmt gmwbBalance wbWithdrawalRate
#> DBRP 2013-11-01 0.02 0.0025 5e-04 0 0 5e-04
#> DBRU 2009-09-01 0.02 0.0035 5e-04 0 0 5e-04
#> DBSU 2010-08-01 0.02 0.0035 5e-04 0 0 5e-04
#> ABRP 2007-07-01 0.02 0.0050 5e-04 0 0 5e-04
#> ABRU 2008-07-01 0.02 0.0060 5e-04 0 0 5e-04
#> ABSU 2011-05-01 0.02 0.0060 5e-04 0 0 5e-04
#> IBRP 2001-11-01 0.02 0.0060 5e-04 0 0 5e-04
#> IBRU 2002-07-01 0.02 0.0070 5e-04 0 0 5e-04
#> IBSU 2013-07-01 0.02 0.0070 5e-04 0 0 5e-04
#> MBRP 2009-08-01 0.02 0.0050 5e-04 0 0 5e-04
#> MBRU 2009-03-01 0.02 0.0060 5e-04 0 0 5e-04
#> MBSU 2013-12-01 0.02 0.0060 5e-04 0 0 5e-04
#> WBRP 2008-10-01 0.02 0.0065 5e-04 0 0 5e-04
#> WBRU 2008-12-01 0.02 0.0075 5e-04 0 0 5e-04
#> WBSU 2004-10-01 0.02 0.0075 5e-04 0 0 5e-04
#> DBAB 2007-12-01 0.02 0.0075 5e-04 0 0 5e-04
#> DBIB 2012-05-01 0.02 0.0085 5e-04 0 0 5e-04
#> DBMB 2007-09-01 0.02 0.0075 5e-04 0 0 5e-04
#> DBWB 2004-04-01 0.02 0.0090 5e-04 0 0 5e-04
#> withdrawal fundNum1 fundNum2 fundNum3 fundNum4 fundNum5 fundNum6 fundNum7
#> DBRP 0 1 2 3 4 5 6 7
#> DBRU 0 1 2 3 4 5 6 7
#> DBSU 0 1 2 3 4 5 6 7
#> ABRP 0 1 2 3 4 5 6 7
#> ABRU 0 1 2 3 4 5 6 7
#> ABSU 0 1 2 3 4 5 6 7
#> IBRP 0 1 2 3 4 5 6 7
#> IBRU 0 1 2 3 4 5 6 7
#> IBSU 0 1 2 3 4 5 6 7
#> MBRP 0 1 2 3 4 5 6 7
#> MBRU 0 1 2 3 4 5 6 7
#> MBSU 0 1 2 3 4 5 6 7
#> WBRP 0 1 2 3 4 5 6 7
#> WBRU 0 1 2 3 4 5 6 7
#> WBSU 0 1 2 3 4 5 6 7
#> DBAB 0 1 2 3 4 5 6 7
#> DBIB 0 1 2 3 4 5 6 7
#> DBMB 0 1 2 3 4 5 6 7
#> DBWB 0 1 2 3 4 5 6 7
#> fundNum8 fundNum9 fundNum10 fundValue1 fundValue2 fundValue3 fundValue4
#> DBRP 8 9 10 0.000 0.00 0.000 105809.638
#> DBRU 8 9 10 0.000 31553.25 31553.248 31553.248
#> DBSU 8 9 10 27721.639 27721.64 27721.639 27721.639
#> ABRP 8 9 10 0.000 0.00 71421.693 71421.693
#> ABRU 8 9 10 22052.253 22052.25 22052.253 22052.253
#> ABSU 8 9 10 0.000 21364.51 21364.506 21364.506
#> IBRP 8 9 10 0.000 0.00 205836.574 0.000
#> IBRU 8 9 10 0.000 0.00 0.000 0.000
#> IBSU 8 9 10 48171.917 48171.92 48171.917 48171.917
#> MBRP 8 9 10 0.000 0.00 52395.296 52395.296
#> MBRU 8 9 10 38515.487 38515.49 38515.487 38515.487
#> MBSU 8 9 10 9882.453 0.00 9882.453 9882.453
#> WBRP 8 9 10 0.000 55944.97 55944.968 55944.968
#> WBRU 8 9 10 6677.476 0.00 6677.476 0.000
#> WBSU 8 9 10 50324.322 0.00 50324.322 0.000
#> DBAB 8 9 10 57902.770 57902.77 0.000 0.000
#> DBIB 8 9 10 66714.406 66714.41 66714.406 66714.406
#> DBMB 8 9 10 0.000 0.00 0.000 0.000
#> DBWB 8 9 10 70521.621 0.00 0.000 70521.621
#> fundValue5 fundValue6 fundValue7 fundValue8 fundValue9 fundValue10
#> DBRP 0.000 0.000 0.000 0.000 105809.638 0.000
#> DBRU 31553.248 31553.248 31553.248 31553.248 31553.248 31553.248
#> DBSU 27721.639 27721.639 27721.639 27721.639 0.000 27721.639
#> ABRP 71421.693 71421.693 71421.693 71421.693 0.000 0.000
#> ABRU 22052.253 22052.253 22052.253 22052.253 22052.253 0.000
#> ABSU 21364.506 21364.506 0.000 0.000 0.000 21364.506
#> IBRP 0.000 0.000 0.000 0.000 205836.574 0.000
#> IBRU 0.000 131955.783 0.000 0.000 131955.783 0.000
#> IBSU 48171.917 48171.917 48171.917 48171.917 48171.917 48171.917
#> MBRP 0.000 0.000 52395.296 52395.296 0.000 52395.296
#> MBRU 38515.487 38515.487 38515.487 38515.487 38515.487 38515.487
#> MBSU 9882.453 9882.453 9882.453 9882.453 9882.453 9882.453
#> WBRP 55944.968 55944.968 55944.968 0.000 0.000 55944.968
#> WBRU 6677.476 6677.476 6677.476 6677.476 6677.476 6677.476
#> WBSU 50324.322 50324.322 0.000 50324.322 0.000 50324.322
#> DBAB 0.000 0.000 0.000 0.000 0.000 0.000
#> DBIB 0.000 0.000 0.000 66714.406 66714.406 0.000
#> DBMB 0.000 91520.653 91520.653 91520.653 0.000 91520.653
#> DBWB 70521.621 0.000 70521.621 70521.621 70521.621 0.000
#> fundFee1 fundFee2 fundFee3 fundFee4 fundFee5 fundFee6 fundFee7 fundFee8
#> DBRP 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> DBRU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> DBSU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> ABRP 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> ABRU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> ABSU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> IBRP 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> IBRU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> IBSU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> MBRP 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> MBRU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> MBSU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> WBRP 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> WBRU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> WBSU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> DBAB 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> DBIB 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> DBMB 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> DBWB 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> fundFee9 fundFee10
#> DBRP 0.0047 0.0046
#> DBRU 0.0047 0.0046
#> DBSU 0.0047 0.0046
#> ABRP 0.0047 0.0046
#> ABRU 0.0047 0.0046
#> ABSU 0.0047 0.0046
#> IBRP 0.0047 0.0046
#> IBRU 0.0047 0.0046
#> IBSU 0.0047 0.0046
#> MBRP 0.0047 0.0046
#> MBRU 0.0047 0.0046
#> MBSU 0.0047 0.0046
#> WBRP 0.0047 0.0046
#> WBRU 0.0047 0.0046
#> WBSU 0.0047 0.0046
#> DBAB 0.0047 0.0046
#> DBIB 0.0047 0.0046
#> DBMB 0.0047 0.0046
#> DBWB 0.0047 0.0046After generating the above required elements for Monte Carlo valuation, we can now proceed to calculate the fair market price of the portfolio by calling the function valuatePortfolio(). Under the current version of the package, all the annuity contracts in the portfolio are assumed to be valuated on the same date, i.e. the first date of our simulated fund scenario. Users can either use the default mortality table by calling “mortTable”, or input a mortality table to project liability cash flows.
# In this vignette, we will arbitrarily use the first two scenarios from fundSen to valuate a portfolio of two guarantees to speed up the execution of the example.
# The input cForwardCurve is a vector of 0.02 with dimension 360. It could also be a forward curve calculated using the buildCurve() function.
valuatePortfolio(VAPort[1:2, ], mortTable, fundScen[1, , ], 1 / 12, cForwardCurve)
#> $portVal
#> [1] 0
#>
#> $portRC
#> [1] 597.3302
#>
#> $vecVal
#> [1] 0 0
#>
#> $vecRC
#> [1] 347.5886 249.7416Note that users can also “age” the portfolio, calling the function agePortfolio(), to a particular valuatioon date by incorporating the historical fund movements prior to that date.
# Again, we will arbitrarily age a portfolio of two guarantees to speed up the execution.
targetDate <- "2016-01-01"
# Here we generate historical fund scenarios using default index data stored under "histIdxScen"
histFundScen <- genFundScen(fundMap, histIdxScen)
# Perform aging
agePortfolio(VAPort[1:2, ], mortTable, histFundScen, histDates, dT = 1 / 12, targetDate, cForwardCurve)
#> recordID survivorShip gender productType issueDate matDate birthDate
#> DBRP 1 1 M DBRP 2010-11-01 2036-11-01 1965-07-01
#> DBRU 2 1 M DBRU 2013-02-01 2034-02-01 1969-11-01
#> currentDate baseFee riderFee rollUpRate gbAmt gmwbBalance wbWithdrawalRate
#> DBRP 2016-01-01 0.02 0.0025 5e-04 0 0 5e-04
#> DBRU 2016-01-01 0.02 0.0035 5e-04 0 0 5e-04
#> withdrawal fundNum1 fundNum2 fundNum3 fundNum4 fundNum5 fundNum6 fundNum7
#> DBRP 0 1 2 3 4 5 6 7
#> DBRU 0 1 2 3 4 5 6 7
#> fundNum8 fundNum9 fundNum10 fundValue1 fundValue2 fundValue3 fundValue4
#> DBRP 8 9 10 120788.3 135115.39 90082.35 56505.24
#> DBRU 8 9 10 0.0 54975.59 0.00 44099.64
#> fundValue5 fundValue6 fundValue7 fundValue8 fundValue9 fundValue10
#> DBRP 47943.95 126814.8 0.00 0.00 103083.3 85152.89
#> DBRU 42005.60 0.0 51996.72 51055.69 0.0 0.00
#> fundFee1 fundFee2 fundFee3 fundFee4 fundFee5 fundFee6 fundFee7 fundFee8
#> DBRP 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> DBRU 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> fundFee9 fundFee10
#> DBRP 0.0047 0.0046
#> DBRU 0.0047 0.0046Though the primary purpose of this package is to valuate a portfolio of variable annuities, users can also use the valuateOnePolicy() and the ageOnePolicy() functions to perform fair market valuation on a single variable annuity as demonstrated below.
# Similarly, users can age this single policy before pricing it. We use the same target date and historical fund scenario as generated before
exPolicy <- VAPort[1, ]
ageOnePolicy(exPolicy, mortTable, histFundScen, histDates, dT = 1 / 12, targetDate, cForwardCurve)
#> recordID survivorShip gender productType issueDate matDate birthDate
#> DBRP 1 1 M DBRP 2010-11-01 2036-11-01 1965-07-01
#> currentDate baseFee riderFee rollUpRate gbAmt gmwbBalance wbWithdrawalRate
#> DBRP 2016-01-01 0.02 0.0025 5e-04 0 0 5e-04
#> withdrawal fundNum1 fundNum2 fundNum3 fundNum4 fundNum5 fundNum6 fundNum7
#> DBRP 0 1 2 3 4 5 6 7
#> fundNum8 fundNum9 fundNum10 fundValue1 fundValue2 fundValue3 fundValue4
#> DBRP 8 9 10 120788.3 135115.4 90082.35 56505.24
#> fundValue5 fundValue6 fundValue7 fundValue8 fundValue9 fundValue10
#> DBRP 47943.95 126814.8 0 0 103083.3 85152.89
#> fundFee1 fundFee2 fundFee3 fundFee4 fundFee5 fundFee6 fundFee7 fundFee8
#> DBRP 0.003 0.005 0.006 0.008 0.001 0.0038 0.0045 0.0055
#> fundFee9 fundFee10
#> DBRP 0.0047 0.0046