1. Callable Bonds Professor Anh Le

2. 0 – Plan 1.Callable bonds – what and why? 2.Yields to call, worst 3.Valuation 4.Spread due to optionality 5.Z-spread 6.Option-adjusted spread 7.Callable prices and interest rates 8.Duration and convexity

3. 1 – Callable bonds – what? Bonds that give issuer the option to call home (prepay) the bonds at some price (call price) Many bonds, call price = par value Example: Fixed rate mortgages Many, call price = par value + premium and then declines over time

4. 1 – Callable bonds – what?

5. Example: 2-yr, $100 face, 8%-coupon callable at time 1 at a call price of $100. What happens if: – The actual price of the bond at time 1 is $105? – The actual price of the bond at time 1 is $95?

6. 1 – Callable bonds – what? Callable bonds are not attractive to lenders since they can be called at very bad times. To make the bonds more attractive, some bonds have call protection period. – Example: a typical structure is “10-year noncall 5” meaning the bond has a stated maturity of 10 years and is not callable for the first 5 years

7. 1 – Callable bonds – why?

8. Hedging Callable bonds allow issuers to refinance their high-coupon-paying bonds by cheaper bonds  a means of hedging against future interest rate decreases When is hedging most needed? How can we reconcile this with the decline in callable bonds issuance after 1990?

9. 1 – Callable bonds – why? Signaling If firms issue non-callable bonds and lock in a fixed rate, they can only benefit if firms become worse in credit quality If firms issue callable bonds they may be hinting that they are confident about the prospect that their credit quality might improve in the future

10. 2 – Yields to maturity, call, worst Bond traders like yields Callable bonds don’t have fixed cash flows: – Yields-to-maturity: assuming that the bond will be held until maturity for sure – Yields-to-call: assuming that the bond will be called for sure – Yields-to-worst: the smaller of the above two

11. 2 – Yields to maturity, call, worst Example: 2-yr, $100 face, 8%-coupon callable at time 1 at a call price of $100. The bond is selling for $99.

12. 2 – Yields to maturity, call, worst Suppose:

13. 2 – Yields to maturity, call, worst The 2-yr callable (c=8%): $99 The 1-yr non-callable (c=8%): $100 The 2-yr non-callable (c=8%): $98 Do these prices look right?

14. 2 – Yields to maturity, call, worst When firm issues a 2-year bond callable at time 1: 1.The firm issues a 2-year non-callable bond but they have the option to buy the bond back at t=1 for a call price of $100; OR 2.The firm issues a 1-year non-callable bond but they have the option to extend the maturity of the bond to 2 years at time t = 1.

15. 3 - Valuation Valuation of the 2-year bond callable at time t=1 at a call price of $100 00.511.5 15.35% 11.82% 9.10%11.36% 7.00%8.74% 6.72%8.39% 6.45% 6.19%

16. 3 - Valuation Valuation of the 2-year noncallable 00.511.5 15.35% 11.82% 9.10%11.36% 7.00%8.74% 6.72%8.39% 6.45% 6.19%

17. 3 - Valuation Valuation of the 2-year non-callable 00.511.5 96.5854 95.83173 96.9007498.40869 99.2528698.79106 100.556199.8112 101.082 100.8761

18. 3 - Valuation Valuation of the 2-year callable at t=1 00.511.5 96.5854 95.83173 96.9007498.40869 99.2528698.79106 100.556199.8112 101.082 100.8761

19. 3 - Valuation Valuation of the 2-year callable at t=1.5, 1 96.5854 95.83173 96.9007498.40869 99.2528698.79106 100.556199.8112 101.082 100.8761

20. 4 – Spread due to optionality 2-yr non-callable: $99.25 2-yr callable: $99.00 00.511.5 15.35% 11.82% 9.10%11.36% 7.00%8.74% 6.72%8.39% 6.45% 6.19%

21. 4 – Spread due to optionality Without the tree, we can price the bond as follows: – 2-yr non-callable: – 2-yr callable:

22. 4 – Spread due to optionality Spread due to optionality: the extra premium added to the risk-free discount rates to account for the optionality of the bond

23. 5 – Z-spread When the investor learns: the 2-yr callable is defaultable and illiquid, he decides to pay less for it: $98. – 2-yr callable

24. 5 – Z-spread Z - spread: the extra premium added to the risk-free discount rates to account for – the optionality of the bond – the default risk of the bond – the liquidity risk of the bond Z - spread: – total spread – zero-volatility spread – static spread

25. 6 – Option-adjusted spread Question: given default and liquidity risk, if the 2-yr callable is worth $98, how much is the 2-yr non-callable worth? Z-spread = spread due to optionality + spread due to default/illiquidity How can we get rid of optionality part?

26. 6 – Option-adjusted spread To account for default and liquidity risk, we will push the risk free interest rate tree up by a constant 15.35% + s 11.82% + s 9.1% + s11.36% + s 7% + s8.74% + s 6.72% + s8.39% + s 6.45% + s 6.19% + s

27. 6 – Option-adjusted spread With s=0.006412, we have the following tree: 00.511.5 15.99% 12.46% 9.74%12.00% 7.64%9.38% 7.36%9.03% 7.09% 6.83%

28. 6 – Option-adjusted spread Use the tree to price the 2-year callable, the price is precisely $98 00.511.5 96.29866 95.26745 96.0537198.11103 98.0000098.20013 99.4383999.50501 100 100.5634

29. 6 – Option-adjusted spread 2-yr Non-callable: $99.25 + callability: $99.00 + defaultability & illiquidity: $98.00 2 price reductions: – Reduction due to callability – Reduction due to defaultability and illiquidity These two reductions occur differently in our tree!

30. 6 – Option-adjusted spread The callable feature is accounted for by physically lowering the values of the bonds when optimally called As such the spread s=0.006412 only pertains to: – Default risk – Liquidity risk

31. 6 – Option-adjusted spread Option-adjusted spread: the extra premium added to the risk-free discount rates to account for – the default risk of the bond – the liquidity risk of the bond Option-adjusted spread: the Z-spread with the optionality component taken out Z-spread = OAS + spread due to optionality

32. 6 – Option-adjusted spread So what is the price of the 2-yr noncallable? 00.511.5 96.29866 95.26745 96.0537198.11103 98.1093498.20013 99.6651399.50501 100.4702 100.5634

33. 6 – Option-adjusted spread The new tree can also price other bonds issued by the same firm 00.511.5 15.99% 12.46% 9.74%12.00% 7.64%9.38% 7.36%9.03% 7.09% 6.83%

34. 7 – Prices and interest rates 1-year bond 2-year bond callable Interest rates price Callability value negative convexity price compression

35. 7 – Prices and interest rates

38. Precisely because of the increase in duration when yields increase in the middle range, callable bonds can have negative convexity  Implications for banks who buy mortgages that have negative convexity

39. 8 – Duration and Convexity Duration for callable bonds

40. 8 – Duration and Convexity How to calculate V + 15.35% + 0.0010 11.82% + 0.0010 9.1% + 0.001011.36% + 0.0010 7% + 0.00108.74% + 0.0010 6.72% + 0.00108.39% + 0.0010 6.45% + 0.0010 6.19% + 0.0010

41. 8 – Duration and Convexity How to calculate V - 15.35% - 0.0010 11.82% - 0.0010 9.1% - 0.001011.36% - 0.0010 7% - 0.00108.74% - 0.0010 6.72% - 0.00108.39% - 0.0010 6.45% - 0.0010 6.19% - 0.0010

42. 8 – Duration and Convexity Dollar Duration = Duration = Dollar Convexity = Convexity =