financial risk management of insurance enterprises 1. embedded options 2. binomial method

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Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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Page 1: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

Financial Risk Management of Insurance Enterprises

1. Embedded Options

2. Binomial Method

Page 2: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 3: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 4: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 5: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 6: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 7: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 8: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 9: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 10: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 11: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 12: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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%

Page 13: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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, ,.

..

.

Page 14: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

.

.

.

..

Page 15: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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%

Page 16: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 17: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 18: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 19: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 20: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 21: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

.

. ..

Page 22: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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. . .

Page 23: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 24: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

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

Page 25: Financial Risk Management of Insurance Enterprises 1. Embedded Options 2. Binomial Method

Next time...

• Mortgage-Backed Securities

• Embedded Option Valuation Method #2: Monte Carlo Simulation

• How to use Monte Carlo Simulation for CMOs and Callable Bonds