Lecture 11

Implementation Issues – Part 2

Monte Carlo Simulation

• An alternative approach to valuing embedded options is simulation

• Underlying model “simulates” future scenarios– Use stochastic interest rate model

• Generate large number of interest rate paths

• Determine cash flows along each path– Cash flows can be path dependent– Payments 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 path’s interest rates

• Repeat present value calculation over all paths– Results of calculations form a “distribution”

• Theoretical value is based on mean of distribution– Average of all paths

Option-Adjusted Spread

• Market value can be different from theoretical value determined by averaging all interest rate paths

• The Option-Adjusted Spread (OAS) is the required spread, which is added to the discount rates, to equate simulated value and market value

• “Option-adjusted” reflects the fact that cash flows can be path dependent

Using Monte Carlo Simulation to Evaluate Mortgage-Backed

Securities• Generate multiple interest rate paths

• Translate the resulting interest rate into a mortgage rate (a refinancing rate)– Include credit spreads– Add option prices if appropriate (e.g., caps)

• Project prepayments– Based on difference between original mortgage rate

and refinancing rate

Using Monte Carlo Simulation to Evaluate Mortgage-Backed

Securities (p.2)

• Prepayments are also path dependent– Mortgages exposed to low refinancing rates for the

first time experience higher prepayments

• Based on projected prepayments, determine underlying cash flow

• For each interest rate path, discount the resulting cash flows

• Theoretical value is the average for all interest rate paths

Simulating Callable Bonds

• As with mortgages, generate the interest rate paths and determine the relationship to the refunding rate

• Using simulation, the rule for when to call the bond can be very complex– Difference between current and refunding rates– Call premium (payment to bondholders if called)– Amortization of refunding costs

Simulating Callable Bonds (p.2)

• Generate cash flows incorporating call rule

• Discount resulting cash flows across all interest rate paths

• Average value of all paths is theoretical value

• If theoretical value does not equal market price, add OAS to discount rates to equate values

Effective Duration• Determine interest rate sensitivity of cash flows that

vary with interest rates by increasing and decreasing the beginning interest rate

• Generate all new interest rate paths and find cash flows along each path– Include option components

• Discount cash flows for all paths• Changes in theoretical value numerically determine

effective duration

Using Simulation to Determine the Effective Duration of Loss Reserves

Step 11. Determine a model for loss payments as a

function of interest rates2. Select an interest rate model and the

appropriate parameters3. Simulate a number of interest rate paths4. Calculate the cash flow of loss payments for

each interest rate path5. Discount each cash flow based on the

corresponding interest rate path6. The economic value of the loss reserves is

assumed to be the average discounted value

Using Simulation to Determine the Effective Duration of Loss Reserves

Step 21. Increase the starting short term interest rate by

100 basis points2. Simulate a number of interest rate paths with the

new short term interest rate3. Calculate the cash flow of loss payments for each

interest rate path4. Discount the cash flow based on the interest rate

path corresponding with each cash flow5. The 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 Reserves

Step 31. Decrease the starting short term interest rate

by 100 basis points2. Repeat points 2-5 from Step 23. Use the economic values for the interest rate

increases and decreases to determine the sensitivity of loss reserves to interest rate changes

)(2 0 rPV

PVPVED

Range of Simulated Interest Rate PathsCIR Model

Based on 1000 Simulations

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25 30

Year

Inte

res

t R

ate

5 percentile

25 percentile

50 percentile

75 percentile

95 percentile

Range of Simulated Interest Rate PathsHull-White Model

Based on 1000 Simulations

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25 30

Year

Inte

res

t R

ate

5 percentile

25 percentile

50 percentile

75 percentile

95 percentile

Advantages of Simulation

• Type of cash flow distribution may not be clear– If 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 claims

– Distribution of results provides more information than mean and variance

– Can determine tails of the distribution(e.g. 95th percentile)

Advantages of Simulation (p.2)

• Mathematical estimation may not be possible– Only numerical solutions exist for some problems

• Can be easier to explain to management

• Possible to revise values and re-run simulation to examine the effect of changes

Disadvantages of Simulation

• Computer expertise, cost, and time– Mathematical solutions may be straight forward– However, computing time is becoming cheaper

• Modeling only provides estimates of parameters and not the true values– Pinpoint accuracy may not be necessary, though

• Models are only approximately true– Simplifying assumptions are part of the model

Tools for Simulation

• Spreadsheet software (Excel, Lotus)– Include many statistical, financial functions– Macros increase programming capabilities

• Add-in packages for simulation– Crystal Ball or @RISK

• Other computing languages– FORTRAN, Pascal, C/C++, APL

• Beware of “random” number generators

Applications of Simulation

• Usefulness is unbounded

• Any stochastic variable can be modeled based on assumed process

• Interaction of variables can be captured

• Complex systems do not need to be solved analytically– Good news for insurers

Conclusion

• Simulation can be a powerful tool for interest rate modeling

• Output can be extensive and impressive• Effort involved in developing a model is

generally challenging and time consuming• Usefulness of results depends on how well the

model reflects reality• Understanding the model is essential to know

when it is reliable and when it is not