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 mortgagebacked 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 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
Tranche A
Interest
Principal
Time
• 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
Tranche B
Interest
Principal
• 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
Time
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
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
Bond Type
CMO
Treasury
Difference
+ 300bp + 200bp
-10.57% -3.93%
-8.83
-3.80
-1.74
-0.13
Interest Rate Environment
+ 100bp
Flat - 100 bp
2.33% 8.21% 14.35%
1.65
7.54
13.93
+0.68 +0.67
+0.42
- 200 bp
19.20%
20.85
-1.65
- 300 bp
22.15%
28.36
-6.21
Convexity of Bonds
Zero
Price
Positive
Yield
Negative
Numerical Illustration
• Let’s compare the convexity calculation of
an option-free bond and a CMO
For the option - free bond,
V0  134.67, V  131.84, V  137.59, y  20 b. p.
V  V  2V0
Convexity 
 167.08
2
(y ) V0
For the CMO, (note the relationsh ips of prices),
V 0CMO  134.67, V CMO  131.65, V CMO  137.39, y  20 b. p.
ConvexityCMO  556.92
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 optionembedded 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
– Banks are now finding out just how poor their
CMO models were
• 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
• 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)