```Lecture 11
Implementation Issues – Part 2
Monte Carlo Simulation
• An alternative 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
• Market value can be different from theoretical
value determined by averaging all interest rate
paths
rates, to equate simulated value and market
value
• “Option-adjusted” reflects the fact that cash
flows can be path dependent
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)
– 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
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
Effective Duration
• Determine interest rate sensitivity of cash flows
that vary with interest rates 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 effective duration
Using Simulation to Determine the
Effective Duration of Loss Reserves
Step 1
1. Determine a model for loss payments as a
function of interest rates
2. Select an interest rate model and the
appropriate parameters
3. Simulate a number of interest rate paths
4. Calculate the cash flow of loss payments for
each interest rate path
5. Discount each cash flow based on the
corresponding interest rate path
6. The economic value of the loss reserves is
assumed to be the average discounted value
Using Simulation to Determine the
Effective Duration of Loss Reserves
Step 2
1. Increase the starting short term interest rate
by 100 basis points
2. Simulate a number of interest rate paths with
the new short term interest rate
3. Calculate the cash flow of loss payments for
each interest rate path
4. Discount the cash flow based on the interest
rate path corresponding with each cash flow
5. The economic value of the loss reserves if
interest rates were to change in this direction
is assumed to be the average discounted value
Using Simulation to Determine the
Effective Duration of Loss Reserves
Step 3
1. Decrease the starting short term interest rate
by 100 basis points
2. Repeat points 2-5 from Step 2
3. Use the economic values for the interest rate
increases and decreases to determine the
sensitivity of loss reserves to interest rate
changes
ED 
PV   PV 
2 PV 0 (  r )
Range of Simulated Interest Rate Paths
CIR Model
Based on 1000 Simulations
0.14
0.12
Interest Rate
0.1
0.08
5 percentile
25 percentile
0.06
50 percentile
75 percentile
0.04
95 percentile
0.02
0
0
5
10
15
Year
20
25
30
Range of Simulated Interest Rate Paths
Hull-White Model
Based on 1000 Simulations
0.14
0.12
Interest Rate
0.1
0.08
5 percentile
0.06
25 percentile
50 percentile
0.04
75 percentile
0.02
95 percentile
0
0
5
10
15
-0.02
Year
20
25
30
• 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
than mean and variance
– Can determine tails of the distribution
(e.g. 95th percentile)
• Mathematical estimation may not be possible
– Only numerical solutions exist for some problems
• Can be easier to explain to management
• Possible to revise values and re-run simulation
to examine the effect of changes
• 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
– Include many statistical, financial functions
– Macros increase programming capabilities
– 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
Conclusion
• Simulation can be a powerful tool for interest
rate modeling
• Output can be extensive and impressive
• Effort involved in developing a model is
generally challenging and time consuming
• Usefulness of results depends on how well the
model reflects reality
• Understanding the model is essential to know
when it is reliable and when it is not
```