financial risk management of insurance enterprises 1. embedded options 2. binomial method
TRANSCRIPT
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Financial Risk Management of Insurance Enterprises
1. Embedded Options
2. Binomial Method
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Embedded Options• Up to this point, we have considered cash flows which are
fixed• Insurers’ liabilities are not fixed due to options given to the
policyholder• Frequently, asset cash flows are not fixed either
– Callable bonds or defaults on bonds can cause payments to differ
• Embedded options are features which can alter the payments of an otherwise fixed cash flow– Embedded options may be part of assets or liabilities
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Evaluating Option-Embedded Cash Flows
• Cash flows with embedded options can be simplified by separating into two components– Fixed cash flow– Option cash flow
• Evaluating the fixed cash flow and its sensitivity to interest is easy
• To estimate the option’s cash flows, we need to consider a variety of possible future scenarios
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Valuation Methods
• Today and next lecture, we will discuss two popular approaches in developing future scenarios to predict option cash flows– Binomial method– Monte Carlo method or simulation
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Binomial Method• As its name suggests, the binomial method
models future periods with two distinct scenarios– Usually described by an “up” scenario and a
“down” scenario
• The tree “grows” by repeating this assumption at every point in time
• This binomial process continues until maturity
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Binomial Method (p.2)
• Typically, the binomial method is used for stock prices or interest rates– Stock prices go up or down– Interest rates go up or down
• The volatility of the stock price or interest rate is based on the difference between an up movement and a down movement– Higher volatility requires a bigger difference
between up and down movements
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A Binomial Tree
InitialPrice
PU: Price if up scenario occurs
PD: Price if down scenario occurs
Note: Up+Down= Down+Up
FINAL
PAYOFFS
Nodes
T=0 T=1 T=4T=3T=2
PUD=PDU
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Notes to Binomial Trees• A simple tree or lattice is recombining
– An up-down movement has the same ending value as a down-up movement
– In the example, PUD=PDU
• For a t-period tree, there are t+1 final payoffs
• By decreasing the time interval between nodes, the binomial method increases the number of possible future states of the world that occur in any finite period
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Up and Down Movements
• We will consider interest rate binomial models
• At each node, the up and down movements of the interest rate are related by the following:
r r eu d
2
Where Assumed volatility
of one - period rate
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Building the Tree - An Overview
• Our objective is to value non-fixed cash flows
• We must first “calibrate” our model– Valuing a non-callable bond with the binomial tree
must replicate its market value
• Similar to bootstrap method, we must build the tree one period at a time
• At any node, the value of the bond depends on future cash flows and the one-period interest rate
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Calibrating the Model
• Assume the following information is given:– The one year spot rate is 4.5%– Two-year, annual coupon bonds are selling at
par and yield 4%– One-year interest rate volatility is 15%
• To determine the one-year forward rates, one year from now, consider the cash flows on the two year bond
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Calibrating the Model (p.2)100 Principal
4 Coupon
100 Principal
4 Coupon
100 Principal
4 Coupon
PV1,U=97.42
4 Coupon
6.749%
PV1,D=99.05
4 Coupon
5.000%
PV0=97.83
4.500%
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The Calculations
• The coupons use the two-year bond
• Guess an interest rate for the “down” scenario– In the example this guess is 5%
• The “up” interest rate is .05e(.15)(2)=.06749
• Begin at the bond’s maturity and work backward– Discount by the assumed one-year interest rate
PV PVU D1 1
104
10674997 42
104
10599 05, ,.
..
.
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The Calculations (p.2)
• After calculating the values at time 1, include the coupon payment and discount to time 0
• The value of the bond is the average present value
PV0
97 42 4
1045
99 05 4
10452 97 83
.
.
.
..
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Adjusting the Initial Guess
• Our interest rate process does not reproduce the two-year bond market value
• Since the PV is too low, the guess of 5% is too high
• Use trial-and-error (or a Solver) to find the correct rate
• In the example, the correct rate is 2.97%
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Binomial “Bootstrap”
• Once the model is calibrated through two years, we can continue the process for three years– Keep the “calibrated” two year rates for the three-
period tree
• For each period, the unknown interest rate that we must determine is the one-year interest rate corresponding to all down movements
• All other rates are related to this guess
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The Completed Tree• Assume that we have found all of the nodes
needed for valuing a cash flow– Interest rate binomial tree is completed through the
last payment date
• Bonds can be valued using the completed tree– For option-free bonds, results should be identical to
valuation using spot rates or implied forward rates– Bonds with options may also be valued using the tree
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The Option in Callable Bonds
• Many bonds are callable
• Option is owned by the issuer and gives the right to buy the bond at a fixed price at any time– However, there may be some period of call
protection
• Issuer will call an issue if the market yield is below the coupon– At this point, the bond will sell at a premium
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Valuing Callable Bonds• Using the interest rate tree, the value of a callable
bond can be determined• At nodes where the present value exceeds par, the
issuer will call at par– Coupon will exceed interest rate. too– This may occur in the part of the tree where interest
rates decline– Holder of bond only gets the call value at that node
and the present value of future cash flows is irrelevant
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Callable Bond Example100 Principal
4 Coupon
100 Principal
4 Coupon
100 Principal
4 Coupon
PV1,U=99.99
4 Coupon
4.01%
PV1,D=101.00*
4 Coupon
2.97%
PV0=99.52
4.50%
* Bond is called at 100
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Callable Bond Calculations
• At (1,D), note that the present value exceeds 100 and the issuer calls the bond– In effect, issuer buys bond at less than market value– Holder still receives the coupon payment
• To get the value at time zero, we use the value assuming the bond is called
PV0
99 99 4
1045
100 4
10452 99 52
.
. ..
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Call Option Value
• From the examples above, the value of the call option is the difference between the non-callable bond and the callable bond
• The two-year non-callable sells at par
• The callable bond sells at 99.52
Call Option Value 100 00 99 52 0 48. . .
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Note about Callable Bonds
• Using the binomial model, it can be seen that a callable bond has a “ceiling” value– Issuer calls bond in
good scenarios
• Bonds of this type exhibit negative convexity– Not good for assets
Callables vs. Non-Callables
Yield
Pri
ce
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Extensions
• Embedded options come in all shapes and sizes
• For nodes where the option is exercised, incorporate the effects on cash flows
• Potential uses:– Putable bonds– Options on bonds
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Next time...
• Mortgage-Backed Securities
• Embedded Option Valuation Method #2: Monte Carlo Simulation
• How to use Monte Carlo Simulation for CMOs and Callable Bonds