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

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

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

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

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

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

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

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

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

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

Convexity of Bonds

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

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

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

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

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

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

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

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

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

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

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

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

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

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 37

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

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)