Financial Risk Management of Insurance Enterprises

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

2. 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

3. 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

4. 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

5. 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

6. 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

7. A Binomial Tree Nodes F I N A L P A Y O F F S PU: Price if up scenario occurs Note: Up+Down= Down+Up Initial Price PUD=PDU PD: Price if down scenario occurs T=0 T=1 T=2 T=3 T=4

8. 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

9. 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:

10. 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

11. 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

12. Calibrating the Model (p.2) 100 Principal 4 Coupon PV1,U=97.42 4 Coupon 6.749% 100 Principal 4 Coupon PV0=97.83 4.500% PV1,D=99.05 4 Coupon 5.000% 100 Principal 4 Coupon

13. 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

14. 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

15. 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%

16. 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

17. 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

18. 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

19. 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

20. Callable Bond Example 100 Principal 4 Coupon PV1,U=99.99 4 Coupon 4.01% 100 Principal 4 Coupon PV0=99.52 4.50% PV1,D=101.00* 4 Coupon 2.97% 100 Principal 4 Coupon * Bond is called at 100

21. 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

22. 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

23. 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

24. 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

25. Next time... • Mortgage-Backed Securities • Embedded Option Valuation Method #2: Monte Carlo Simulation • How to use Monte Carlo Simulation for CMOs and Callable Bonds