monte carlo - implementation issues

Lecture 11

Implementation Issues Part 2

Monte Carlo SimulationAn alternative approach to valuing embedded options is simulationUnderlying model simulates future scenariosUse stochastic interest rate modelGenerate large number of interest rate pathsDetermine cash flows along each pathCash flows can be path dependentPayments may depend not only on current level of interest but also the history of interest rates

Monte Carlo Simulation (p.2)Discount the path dependent cash flows by the paths interest rates Repeat present value calculation over all pathsResults of calculations form a distributionTheoretical value is based on mean of distributionAverage of all paths

Option-Adjusted SpreadMarket value can be different from theoretical value determined by averaging all interest rate pathsThe Option-Adjusted Spread (OAS) is the required spread, which is added to the discount rates, to equate simulated value and market valueOption-adjusted reflects the fact that cash flows can be path dependent

Using Monte Carlo Simulation to Evaluate Mortgage-Backed SecuritiesGenerate multiple interest rate pathsTranslate the resulting interest rate into a mortgage rate (a refinancing rate)Include credit spreadsAdd option prices if appropriate (e.g., caps)Project prepaymentsBased on difference between original mortgage rate and refinancing rate

Using Monte Carlo Simulation to Evaluate Mortgage-Backed Securities (p.2)Prepayments are also path dependentMortgages exposed to low refinancing rates for the first time experience higher prepaymentsBased on projected prepayments, determine underlying cash flowFor each interest rate path, discount the resulting cash flowsTheoretical value is the average for all interest rate paths

Simulating Callable BondsAs with mortgages, generate the interest rate paths and determine the relationship to the refunding rateUsing simulation, the rule for when to call the bond can be very complexDifference between current and refunding ratesCall premium (payment to bondholders if called)Amortization of refunding costs

Simulating Callable Bonds (p.2)Generate cash flows incorporating call ruleDiscount resulting cash flows across all interest rate pathsAverage value of all paths is theoretical valueIf theoretical value does not equal market price, add OAS to discount rates to equate values

Effective DurationDetermine interest rate sensitivity of cash flows that vary with interest rates by increasing and decreasing the beginning interest rateGenerate all new interest rate paths and find cash flows along each pathInclude option componentsDiscount cash flows for all pathsChanges in theoretical value numerically determine effective duration

Using Simulation to Determine the Effective Duration of Loss ReservesStep 1Determine a model for loss payments as a function of interest ratesSelect an interest rate model and the appropriate parametersSimulate a number of interest rate pathsCalculate the cash flow of loss payments for each interest rate pathDiscount each cash flow based on the corresponding interest rate pathThe economic value of the loss reserves is assumed to be the average discounted value

Using Simulation to Determine the Effective Duration of Loss ReservesStep 2Increase the starting short term interest rate by 100 basis pointsSimulate a number of interest rate paths with the new short term interest rateCalculate the cash flow of loss payments for each interest rate pathDiscount the cash flow based on the interest rate path corresponding with each cash flowThe economic value of the loss reserves if interest rates were to change in this direction is assumed to be the average discounted value

Using Simulation to Determine the Effective Duration of Loss ReservesStep 3Decrease the starting short term interest rate by 100 basis pointsRepeat points 2-5 from Step 2Use the economic values for the interest rate increases and decreases to determine the sensitivity of loss reserves to interest rate changes

Sheet1

Ten Year Zero Coupon Bond ($1 Million Face Value)Ten Year 10% Annual Coupon Bond ($1 Million Face Value)Zero Coupon Bond: Maturity A Function of Interest RatesCash Flow is a Function of the Interest Rate

Interest RatePresent ValuePresent ValuePresent ValuePresent Value

0$1,000,000.00$2,000,000.00$1,000,000.00$1,000,000.00

0.01$905,286.95$1,852,417.41$990,099.01$1,158,843.84

0.02$820,348.30$1,718,606.80$961,168.78$1,336,260.94

0.03$744,093.91$1,597,114.20$915,141.66$1,533,601.04

0.04$675,564.17$1,486,653.75$854,804.19$1,752,239.47

0.05$613,913.25$1,386,086.75$783,526.17$1,993,573.42

0.06$558,394.78$1,294,403.48$704,960.54$2,259,018.31

0.07$508,349.29$1,210,707.45$622,749.74$2,550,004.16

0.08$463,193.49$1,134,201.63$540,268.88$2,867,971.99

0.09$422,410.81$1,064,176.58$460,427.78$3,214,370.31

0.1$385,543.29$1,000,000.00$385,543.29$3,590,651.69

0.11$352,184.48$941,107.68$317,283.31$3,998,269.46

0.12$321,973.24$886,995.54$256,675.09$4,438,674.48

0.13$294,588.35$837,212.70$204,164.50$4,913,312.05

0.14$269,743.81$791,355.37$159,709.99$5,423,618.98

0.15$247,184.71$749,061.57$122,894.49$5,971,020.72

0.16$226,683.60$710,006.35$93,040.53$6,556,928.75

0.17$208,037.38$673,897.75$69,317.09$7,182,738.01

0.18$191,064.47$640,473.10$50,830.44$7,849,824.54

0.19$175,602.38$609,495.86$36,695.07$8,559,543.22

0.2$161,505.58$580,752.79$26,084.05$9,313,225.75

0.21$148,643.63$554,051.42$18,260.27$10,112,178.65

0.22$136,899.45$529,217.88$12,591.68$10,957,681.55

0.23$126,167.90$506,094.90$8,554.26$11,850,985.50

0.24$116,354.49$484,540.12$5,726.37$12,793,311.51

0.25$107,374.18$464,424.51$3,777.89$13,785,849.18

0.26$99,150.42$445,631.03$2,456.78$14,829,755.50

0.27$91,614.20$428,053.38$1,575.07$15,926,153.75

0.28$84,703.29$411,594.98$995.68$17,076,132.55

0.29$78,361.53$396,167.90$620.72$18,280,745.02

0.3$72,538.15$381,692.10$381.68$19,541,008.06

Figure 1

2000000

1852417.40776315

1718606.80049938

1597114.19857431

1486653.7467613

1386086.74645924

1294403.48205659

1210707.44622798

1134201.62797883

1064176.57701159

1000000

941107.679888589

886995.539431783

837212.695721414

791355.374148257

749061.568707289

710006.351292552

673897.746058655

640473.096406047

609495.861973634

580752.791444923

554051.424203123

529217.88008393

506094.901821882

484540.119393985

464424.50944

445631.026373478

428053.384768577

411594.975180634

396167.897789445

381692.100190937

Interest Rate

Bond Value

Figure 1Present Value of a 10% Annual Coupon BondFace Value of $1 Million

Sheet2

k =0.25

m =0.25

T20

n =410.5

00.250.250.25

10.2500031250.2750.3618033989

20.250050.30.408113883

30.2502531250.3250.4436491673

40.25080.350.4736067977

50.2519531250.3750.5

60.254050.40.5238612788

70.2575031250.4250.5458039892

80.26280.450.566227766

90.2705031250.4750.5854101966

100.281250.50.6035533906

110.2957531250.5250.6208099244

120.31480.550.6372983346

130.3392531250.5750.6531128874

140.370050.60.6683300133

150.4082031250.6250.6830127019

160.45480.650.6972135955

170.5110031250.6750.7109772229

180.578050.70.724341649

190.6572531250.7250.7373397172

200.750.750.75

Figure 2

0.250.250.25

0.2500031250.2750.3618033989

0.250050.30.408113883

0.2502531250.3250.4436491673

0.25080.350.4736067977

0.2519531250.3750.5

0.254050.40.5238612788

0.2575031250.4250.5458039892

0.26280.450.566227766

0.2705031250.4750.5854101966

0.281250.50.6035533906

0.2957531250.5250.6208099244

0.31480.550.6372983346

0.3392531250.5750.6531128874

0.370050.60.6683300133

0.4082031250.6250.6830127019

0.45480.650.6972135955

0.5110031250.6750.7109772229

0.578050.70.724341649

0.6572531250.7250.7373397172

0.750.750.75

n>1

n=1

n

Advantages of SimulationType of cash flow distribution may not be clearIf one statistical distribution is used for the number of claims and another distribution determines the size of claims, statistical theory may not be helpful to determine distribution of total claimsDistribution of results provides more information than mean and varianceCan determine tails of the distribution(e.g. 95th percentile)

Advantages of Simulation (p.2)Mathematical estimation may not be possibleOnly numerical solutions exist for some problemsCan be easier to explain to managementPossible to revise values and re-run simulation to examine the effect of changes

Disadvantages of SimulationComputer expertise, cost, and timeMathematical solutions may be straight forwardHowever, computing time is becoming cheaperModeling only provides estimates of parameters and not the true valuesPinpoint accuracy may not be necessary, thoughModels are only approximately trueSimplifying assumptions are part of the model

Tools for SimulationSpreadsheet software (Excel, Lotus)Include many statistical, financial functionsMacros increase programming capabilitiesAdd-in packages for simulationCrystal Ball or @RISKOther computing languagesFORTRAN, Pascal, C/C++, APLBeware of random number generators

Applications of SimulationUsefulness is unboundedAny stochastic variable can be modeled based on assumed processInteraction of variables can be capturedComplex systems do not need to be solved analyticallyGood news for insurers

ConclusionSimulation can be a powerful tool for interest rate modelingOutput can be extensive and impressiveEffort involved in developing a model is generally challenging and time consumingUsefulness of results depends on how well the model reflects realityUnderstanding the model is essential to know when it is reliable and when it is not

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