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

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 41

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 42

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 tranche receives all underlying mortgage principal repayments until it is paid off
Longer tranches receive only interest at first

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 tranche is correlated with interest rate movements
As interest rates increase, prepayments decrease and average life increases
Average life decreases when prepayments do occur 2

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 1

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
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 25

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 26

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 27

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 28

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 29

18. Using Monte Carlo Simulation to Evaluate Mortgage-Backed Securities (p
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 30

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 31

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 32

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 33

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 34

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 35

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
Banks are now finding out just how poor their CMO models were
Models are only approximately true
Simplifying assumptions are part of the model 36

25. Tools for Simulation Spreadsheet software
Include many statistical, financial functions
Macros increase programming capabilities
Add-in packages for simulation
Crystal Ball
Other computing languages
FORTRAN, Pascal, C/C++, APL
Beware of “random” number generators 37

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 38

27. Next lectures Collateralized Debt Obligations (CDO)
Further application of binomial method and simulation techniques
Valuing interest rate options
Valuing interest rate swaps
Interest rate sensitivity of insurance liabilities
Introduction to Dynamic Financial Analysis (DFA)