Financial Risk Management of Insurance Enterprises

1. Collateralized Mortgage Obligations

2. Monte Carlo Method & Simulation

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

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

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

Cash Flows of Two-Tranche CMO

Tranche A

Time

Interest 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 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

Tranche B

Time

Interest Principal

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

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

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

Illustrative Example

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

Interest Rate EnvironmentBond Type + 300bp + 200bp + 100bp Flat - 100 bp - 200 bp - 300 bp

CMO -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 -6.21

Convexity of Bonds

Yield

Pri

cePositive Zero Negative

Numerical Illustration

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

For the option - free bond,

For the CMO, (note the relationships of prices),

V V V y b p

ConvexityV V V

y V

V V V y b p

Convexity

CMO CMO CMO

CMO

0

02

0

134 67 13184 137 59 20

2

28354

134 67 13165 137 39 20

278 46

0

. , . , . , . .

( ).

. , . , . , . .

.

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

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

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

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

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

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

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

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

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– 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

Next lectures

• Further application of binomial method and simulation techniques– Valuing interest rate options– Valuing interest rate swaps

• Introduction to Dynamic Financial Analysis (DFA)