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1. Bonds with embedded options

2. Callable Bonds • A problem with traditional pricing is that it ignores options imbedded in the bond such as a call option. • The call feature increases reinvestment rate risk since the bond will be called if rates are low, especially if lower than the coupon rate. • As the yield decreases the price increase lessens because there is a higher probability of the bond being called (price compression)

3. Valuing a Callable Bond • You can think of a callable bond as having two components: A noncallable bond and a call option. The price of the callable bond would then be equal to the noncallable bond price minus the call option price.

4. Valuation Model Review • Remember from before that the appropriate rate to use is not a single rate, but the zero spot rate or the forward rates (example on next slide) • The value of the callable bond will be tied directly to the volatility of interest rates. To price the bond we will use a binomial tree model.

5. 5.25% coupon bond, 3 years to maturity, yearly payments • Assume you have the following observed yield curve, spot rates, and forward rates. Maturity YTM Market Value Spot Rate Forward Rate 1 yr 3.5% 100 3.5% 3.5% 2 yr 4.0% 100 4.01% 4.5225% 3 yr 4.5% 100 4.531% 5.5792%

6. Binomial Tree Model • We are going to represent the two possible paths of interest rates in a tree structure. • Let each time be denoted as a decision Node N with a subscript denoting whether it represent the higher or lower interest rate environment. • The current level of interest rates is r*, it may increase to state H or decrease to state L If the level of interest rates increases to H the bond will have a value of VH in the next period. Likewise if rates decrease to L the value will be VL

7. Binomial Tree Model l VH NH V r* l N l VL NL Time 0 Time 1

8. Value at Time 0 • The value of the bond at time 0 can be calculated assuming that there is an equal chance of obtaining either state. • The expected value at time 0 is then equal to the PV of total amount you would receive in each state multiplied by the probability of the state (in this case 1/2) • The total amount you would receive is the value or the bond plus any other cash flows received (For example the coupon payment)

9. Binomial Tree Model VH CH l NH V0 r* l N VL CL l NL Time 0 Time 1

10. Simple Example • Assume that the current level of interest rates is 3.5% as in our previous example. • If the higher interest rate price is \$98 with a coupon of \$5 and the lower interest rate price is \$102 with the same \$5 of coupon the value at time 0 would be

11. Binomial Tree Model continued • Starting today we want to think about the future path of interest rates. • Start with the time 0 (today) and think about the future path of interest rates. We will assume that over the next year there are two possible outcomes for the one year rate at time 1. • Let the lower rate be r1L and the higher rate be r1H

12. Binomial Tree Method continued • Assuming the lower rate in the next period the higher rate can be found from the equation. r1H=e2sr1L where: e = natural logarithm 2.71828 (x = lny y=ex) For example let r1L=4.5% and s=10% then r1H = .045e2(.10)=.054963

13. Binomial Tree Model VH CH r1H=.05496 l NH V0 r* l N VL CL r1L=.045 l NL Time 0 Time 1

14. Extending the model • It is easy to extend the model to add a second year. • From each of the decision nodes NH and NL you can just repeat the same tree.

15. Binomial Tree Model VHH CHH r2HH l VH CH r1H=.05496 NHH l VHL CHL r2HL NH V0 r=.035 l l N NHL VL CL r1L=.045 l NL VLL CLL r2LL l NLL Time 0 Time 1 Time 2

16. The Two Year Bond • Using the observed yield curve from before, the two year bond would have a 4% coupon rate implying \$4 coupon payments each year. • At maturity the bond will have a value of \$100 • Substitute the value in for VHH, VHL,and VLL. Let the coupon be \$4 in each period.

17. Binomial Tree Model VHH = 100 CHH = 4 r2HH l VH CH = 4 r1H=.05496 NHH l VHL = 100 CHL = 4 r2HL NH V0 r=.035 l l N NHL VL CL = 4 r1L=.045 l NL VLL= 100 CLL= 4 r2LL l NLL Time 0 Time 1 Time 2

18. Finding VH and VL • The values of the bond can be found at time 1 by applying the earlier formula

19. Binomial Tree Model VHH = 100 CHH = 4 r2HH l VH=98.852 CH = 4 r1H=.05496 NHH l VHL = 100 CHL = 4 r2HL NH V0 r=.035 l l N NHL VL=99.522 CL = 4 r1L=.045 l NL VLL= 100 CLL= 4 r2LL l NLL Time 0 Time 1 Time 2

20. Iterative Procedure • We assumed an interest rate of 4.5% for r1L this is the correct rate IF V0 can be found given the current values it the tree and V0 is equal to the market price of 100 • Since the price from the tree is too low, the rate r1L must be lower to increase the price. Try a new price and repeat until the correct price 4.074% is found

21. Iterative procedure • Given the rate of 4.074 the expected value of the two possible changes in interest rates is equal to the current value, in other words it is “fairly priced.” • The change in rates requires finding new values for r1H and for VH and VL

22. Binomial Tree Model VHH = 100 CHH = 4 r2HH l VH=99.071 CH = 4 r1H=.04976 NHH l VHL = 100 CHL = 4 r2HL NH V0 r=.035 l l N NHL VL=99.929 CL = 4 r1L=.04074 l NL VLL= 100 CLL= 4 r2LL l NLL Time 0 Time 1 Time 2

23. Interpretations • r1H and r1Lare a set of forward rates from time 1 to time 2 or 1f1 as it was previously called. • Notice that since the change in rates makes a difference in the value of the bond, for each forward rate we will have a different value of the bond.

24. The next step, time 3 • The model could be extended again to include the next year. The YTM for the three year treasury was 4.5% so the coupons at every time period become 4.50. • The goal is to find a value for r2LL that will allow us to move from right to left through the tree to produce a value of 100 again at V0 • r2HL=r2LLe2s as before and r2HH=r2HLe2s=r2LLe4s • the correct rate is then 4.53%

25. Binomial Tree Model VHH = 97.886 CHH = 4.50 r2HH=.06757 l VH =98.074 CH= 4.50 r1H=.04976 NHH l VHL = 99.022 CHL= 4.50 r2HL=.05532 NH V0 r=.035 l l N NHL VL=99.926 CL= 4.50 r1L=.04074 l NL VLL= 100 CLL= 4.50 r2LL=.0453 l NLL Time 0 Time 1 Time 2

26. Valuing an Option Free Bond • The rates in the binomial tree now represent the correct rates for the on the run treasury yield curve that we started with. It is now possible to use it to value the three year 5.25% coupon bond. • Starting with year three the values VHH, VHL, and VLL can be found then we can work right to left through the tree

27. Binomial Tree Model VHH = 98.588 CHH = 5.25 r2HH=.06757 l VH =99.461 CH = 5.25 r1H=.04976 NHH l VHL= 99.732 CHL= 5.25 r2HL =.05532 NH V0 r=.035 l l N NHL VL=101.333 CL= 5.25 r1L=.04074 l NL VLL= 100.689 CLL= 5.25 r2LL=.0453 l NLL Time 0 Time 1 Time 2

28. Value of the bond The same value as we calculated before!!!

29. Valuing a Call Option • Assume that the the bond can be called at the end of the first year or later for a call price of \$100. • If the value at a node is greater than \$100 then the bond will be called (the yield is less than the coupon) and the firm can refinance at a lower rate. Starting on the right, if the value exceeds 100 it needs to be replaced, then the tree is worked right to left again.

30. Binomial Tree Model VHH = 98.588 CHH = 5.25 r2HH=.06757 l VH =99.461 CH = 5.25 r1H=.04976 NHH l VHL = 99.732 CHL = 5.25 r2HL =.05532 NH V0 r=.035 l l N NHL VL=101.001 VL=100 CL = 5.25 r1L=.04074 l NL VLL= 100.689 VLL =100 CLL= 5.25 r2LL=.0453 l NLL Time 0 Time 1 Time 2

31. Value of callable bond • The value has decreased because of the call option • The value of the call option is then 102.075-101.4302=0.6448

32. Put option • The same model could be used to value a put option. • Now, you look at increases in rates that lower the price. • If the value of the bond at the node is less than the puttable value then the option would be exercised and the value of the bond becomes the put value. • Assume that our bond has a put option after year one with the puttable value being \$100

33. Binomial Tree Model (Put) VHH = 98.588 VHH=100 CHH = 5.25 r2HH=.06757 l VH =100.261 CH = 5.25 r1H=.04976 NHH l VHL = 99.732 VHL=100 CHL = 5.25 r2HL =.05532 NH V0 r=.035 l l N NHL VL=101.461 CL = 5.25 r1L=.04074 l NL VLL= 100.689 CLL= 5.25 r2LL=.0453 l NLL Time 0 Time 1 Time 2

34. Value of puttable bond • The value has increased because of the put option. • The value of the put option is 102.075 – 102.523= -0.448

35. Modeling Risk • Modeling risk is the risk that the valuation model has produced an incorrect result due to assumptions used in the model. • Higher volatility lowers the value of the call option (raises value of put) • Lower volatility raises the value of the a call option (lower the value of a put)

36. Option Adjusted Spread • The constant spread that when added to all the forward rates in the binomial tree will make the theoretical value equal to the market price.

37. OAS Intuition • Converts the difference between the valuation and the market price into a spread measure. • The key is the inputs in the model • If the tree uses the treasury spot curve, the OAS represents the richness or cheapness of the security plus a credit spread • If the tree uses issuer’s spot rate curve, then the credit risk is already incorporated.

38. OAS and total yield spreads • The OAS is attempting to separate the amount of the nominal spread that is the result of option risk. Therefore it reports a spread that is adjusted for the option. • Example: Assume you have calculated the OAS of a BBB callable corporate bond compared to non callable treasuries to be 120 Bp. would imply that the BBB pays 120 Bp more because of the liquidity and credit risk etc. the spread has removed the portion of the spread attributable to the option.

39. OAS and benchmarks • In the previous example the OAS was a representation of credit and liquidity risks. If instead of using the on the run treasury as a benchmark we used the on the run issues for the same issuer (the issuer of the BBB). Then credit risk is also not part of the spread, only liquidity and other factors. • OAS is the spread after adjusting for options, what it actually represents however depends upon the benchmark being used…

40. OAS in our example • Assume that the 5.25% callable three year coupon bond is currently selling for \$101.17 • Previously we found the price to be \$101.4302 • The OAS would be the additional yield added to the binomial interest rate tree at every yield that produced a value for the bond of \$101.17, in this case the OAS is 45 Bp

41. Binomial Tree Model VHH = 102.6309 VHH = 100 CHH = 5.25 r2HH=.07208 l VH =99.833 CH = 5.25 r1H=.005426 NHH l VHL = 103.817 VHL = 100 CHL = 5.25 r2HL =.0598 NH V0 r=.035 l l N NHL VL=100.6946 VL=100 CL = 5.25 r1L=.04524 l NL VLL= 104.809 VLL =100 CLL= 5.25 r2LL=.0498 l NLL Time 0 Time 1 Time 2

42. Value of callable bond (with OAS)

43. Funding Cost as a Benchmark • Often the on the run issues of the LIBOR is used as the benchmark. • LIBOR is used as a benchmark borrowing rate that the institution pays to obtain funds. It can then compare its cost of funding to LIBOR by looking at the spread above LIBOR it pays to obtain funds. • As long as the assets spread relative to LIBOR is greater than the spread the institution must pay to obtain funding, it is covering the funding cost.

44. Effective Duration and Convexity • Effective Duration (option adjusted duration) allows a yield change to change the expected future cash flows.

45. Quick approximation of duration and convexity P- = the price if yield is decreased by x Bp P+ = the price if yield is increased by x Bp P0 = the initial price Dy=change in rate (x Bp in dec form)

46. Calculating P+ and P- • Calculate the OAS • Shift the on the run yield curve by a small basis points • Construct a binomial interest rate tree based on the new yield curve • To each of the short rates add the OAS to adjust the tree • Use the adjusted tree to determine the value of P.

47. Valuing a Step Up Callable Note • The Binomial model can be expanded to cover other types of options. • One possibility is a note whose coupon rte changes over the life of the note. In this case the coupon rate may increase in the future. • Initially, the procedure is the same as before. • After developing the interest rate tree, the bond is valued using the coupon rates that correspond to what the bond will pay.

48. Valuing a Floating Rate Note • On a floating rate bond the payment at the end of the year is determined by the rate at the beginning of the year. • Therefore the coupon payment for each node will be based off of the interest rate for that node (the rate at the beginning of the period determines the coupon at the end of the period). • The valuation therefore uses the coupon for that node as the payment at the next point in time.

49. A Capped Floater • If the floating rate bond has a cap, then whenever the coupon is above the cap the value of the coupon will be based off of the cap.

50. Analysis of Convertible Bonds • A convertible bond can allows the holder to convert the bond into a predetermined number of shares of common stock of the issuer. • It may also be callable and puttable. • Exchangeable securities allow conversion to the stock of another firm. • The conversion privilege may extend over the entire life of the of the issue or a portion of the issue