Financial Risk Management of Insurance EnterprisesMortgage Backed SecuritiesSecuritization of MortgagesCollateralized Mortgage Obligations (CMOs)Cash Flows of Two-Tranche CMOSlide 6Price Changes of CMOsConvexity ComparisonNegative convexity of CMOsIllustrative ExampleConvexity of BondsNumerical IllustrationMonte Carlo SimulationMonte Carlo Simulation (p.2)Option-Adjusted SpreadEffective Duration & ConvexityUsing Monte Carlo Simulation to Evaluate Mortgage-Backed SecuritiesUsing Monte Carlo Simulation to Evaluate Mortgage-Backed Securities (p.2)Applications to CMOsSimulating Callable BondsSimulating Callable Bonds (p.2)Advantages of SimulationAdvantages of Simulation (p.2)Disadvantages of SimulationTools for SimulationApplications of SimulationNext lecturesFinancial Risk Management of Insurance Enterprises1. Collateralized Mortgage Obligations2. Monte Carlo Method & SimulationMortgage 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 mortgageSecuritization 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 riskCollateralized 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 firstCash Flows of Two-Tranche CMOTranche ATimeInterest Principal•Principal is first paid to Tranche A–Amortization of principal in monthly mortgage payments–Prepayments•Once all principal is returned, the tranche no longer existsCash 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 BTranche BTimeInterest PrincipalPrice 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 occurConvexity 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 environmentNegative 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 temperedIllustrative Example•The following table illustrates the comparison of one-year returns on CMOs vs. similar TreasuriesInterest Rate EnvironmentBond Type+ 300bp + 200bp + 100bp Flat - 100 bp - 200 bp - 300 bpCMO -10.57% -3.93% 2.33% 8.21% 14.35% 19.20% 22.15%Treasury -8.83 -3.80 1.65 7.54 13.93 20.85 28.36Difference -1.74 -0.13 +0.68 +0.67 +0.42 -1.65 -6.21Convexity of BondsYieldPricePositive Zero NegativeNumerical Illustration•Let’s compare the convexity calculation of an option-free bond and a CMO92.556.. 20 ,39.137 ,65.131 ,67.134prices), of ipsrelationsh the(note CMO, For the08.167)(2.. 20 ,59.137 ,84.131 ,67.134bond, free-option For the00200CMOCMOCMOCMOConvexitypbyVVVVyVVVConvexitypbyVVVMonte 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 ratesMonte 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 pathsOption-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 dependentEffective 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 convexityUsing 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 rateUsing Monte Carlo Simulation to Evaluate Mortgage-Backed Securities (p.2)•Prepayments are also path dependent–Mortgages exposed to low refinancing rates