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 Initial Price P U : Price if up scenario occurs P D : Price if down scenario occurs Note: Up+Down= Down+Up FINALPAYOFFSFINALPAYOFFS Nodes T=0T=1T=4T=3T=2 P UD =P DU

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, P UD =P DU 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 100 Principal 4 Coupon 100 Principal 4 Coupon PV 1,U =97.42 4 Coupon 6.749% PV 1,D =99.05 4 Coupon 5.000% PV 0 =97.83 4.500%

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 100 Principal 4 Coupon 100 Principal 4 Coupon PV 1,U =99.99 4 Coupon 4.01% PV 1,D =101.00* 4 Coupon 2.97% PV 0 =99.52 4.50% * 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