Financial Risk Management of Insurance Enterprises

1. Financial Risk Management of Insurance Enterprises 1. Collateralized Mortgage Obligations 2. Monte Carlo Method & Simulation

2. Mortgage Backed Securities • Mortgage-backed securities (MBS) are good examples of instruments with embedded options • Individual mortgages are risky to banks or other lenders • Options that are given to borrower are forms of prepayment risk • If interest rates decrease, borrower can refinance • If borrower dies, divorces, or moves, she pays off mortgage

3. Securitization of Mortgages • Lenders pool similar loans in a package and sell to the financial markets creating a mortgage-backed security • Investors become the “owners” of the underlying mortgages by receiving the monthly interest and principal payments made by the borrowers • All prepayment risks are transferred to investors • Yields on MBS are higher to compensate for risk

4. Collateralized Mortgage Obligations (CMOs) • Investors liked the MBS but had different maturity preferences • CMOs create different maturities from the same package of mortgages • Maturities of investors are grouped in “tranches” • Typically, a CMO issue will have 4-5 tranches • The first tranch receives all underlying mortgage principal repayments until it is paid off • Longer tranches receive only interest

5. Cash Flows of Two-Tranche CMO • Principal is first paid to Tranche A • Amortization of principal in monthly mortgage payments • Prepayments • Once all principal is returned, the tranche no longer exists

6. Cash Flows of Two-Tranche CMO • Only interest is paid until first tranche is paid off • There is a lower limit for the time until principal repayments • Then, principal is paid to tranche B

7. Price Changes of CMOs • Prepayments are based on level of interest rates • Prepayments affect short term tranches less • Principal is paid on all mortgages even if rates increase through amortization payments • Interest rates over short term are “less volatile” • The average life of a tranch is correlated with interest rate movements • As interest rates increase, prepayments decrease and average life increases • Average life decreases when prepayments do occur

8. Convexity Comparison • Option-free bonds exhibit positive convexity • For a fixed change in interest rates, the price increase due to an interest rate decline exceeds the loss when interest rates increase • Callable bonds exhibit negative convexity when interest rates are “low” • Positive convexity when interest rates are “high” • CMOs are negatively convex in any interest rate environment

9. Negative convexity of CMOs • Increasing interest rates • Prepayments decrease and average life increases • Relative to option-free bond, duration is therefore higher • Price decline is magnified • Decreasing interest rate environments • Prepayments increase and average life decreases • Relative to option-free bond, duration is therefore higher • Price increase is tempered

10. Illustrative Example • The following table illustrates the comparison of one-year returns on CMOs vs. similar Treasuries

11. Convexity of Bonds

12. Numerical Illustration • Let’s compare the convexity calculation of an option-free bond and a CMO

13. Monte Carlo Simulation • The second numerical 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

14. 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

15. 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

16. Effective Duration & Convexity • Determine interest rate sensitivity of option-embedded cash flows 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 duration and convexity • Also called option-adjusted duration and convexity

17. 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

18. 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

19. Applications to CMOs • When applying the simulation method to CMOs, the distribution of results is useful • Short-term tranches have smaller standard deviations • Short-term tranches are less sensitive to prepayments • Longer term tranches are more sensitive to prepayments • Distribution will be less compact

20. 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

21. 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

22. 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 90th percentile of distribution

23. 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

24. 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

25. Tools for Simulation • Spreadsheet software • 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

26. 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

27. Next lectures • Further application of binomial method and simulation techniques • Valuing interest rate options • Valuing interest rate swaps • Introduction to Dynamic Financial Analysis (DFA)