Author: Hengxin Li, Mingbin Feng, Mingyi Jiang
Date: 2020-02-26

# A Comprehensive Monte Carlo Valuation of Variable Annuities

#### Hengxin Li, Mingbin Feng, Mingyi Jiang

#### 2020-02-26

# Package Info

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.

## Yield Curve Generation

In this step, we exploit Secant 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 <- "General"
    bdc <- "Modified_Foll"
    curveDate <- "2016-02-08"
    numSetDay <- 2
    yieldCurveDCC <- "Thirty360"
    holidays <- NULL

    # Bootstrap a forward curve
    buildCurve(swap, tenor, fixFreq, fixDCC, fltFreq, fltDCC, calendar, bdc,
                curveDate, numSetDay, yieldCurveDCC, holidays)
    #>      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 30.005556

## Generate index scenarios and fund scenarios

In 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]

### Default covariance matrix

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

### Default index names

Index Names
US
SMALL
INT
FIXED
MONEY

### Risk-neutral path simulation for 5 indices

    # 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 0.9723979

Then 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]

### Risk-neutral path simulation for 10 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 0.9728478

## Generate a synthetic portfolio of variable annuities

Perhaps 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")) {
    VAport <- genPortInception(issueRng = c("2001-08-01", "2014-01-01"), numPolicy =  1)
    }

## Monte Carlo Valuation

After 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.
    valuatePortfolio(VAPort[1:5, ], mortTable, fundScen[1, , ], 1 / 12, cForwardCurve)
    #> $portVal
    #> [1] 605.5763
    #>
    #> $portRC
    #> [1] 1246.656
    #>
    #> $vecVal
    #> [1]   0.0000   0.0000 605.5763   0.0000   0.0000
    #>
    #> $vecRC
    #> [1] 347.58464 249.73990 575.59559  45.55381  28.18163

Note that users can also “age” the portfolio, calling the function `agePortfolio()`, to a particular valuation 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    0.0046

## Closing Remarks

Though 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.

### Valuation of one variable annuity

    exPolicy <- VAPort[1, ]
    valuateOnePolicy(exPolicy, mortTable, fundScen[1:2, , ], 1 / 12, cForwardCurve)
    #> $policyValue
    #> [1] 0
    #>
    #> $riskCharge
    #> [1] 454.1367

### Aging of one variable annuity

    # 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