Callable Bonds Professor Anh Le

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

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

1 – Callable bonds – what?

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?

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

1 – Callable bonds – why?

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?

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

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

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.

2 – Yields to maturity, call, worst Suppose:

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?

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.

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

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

3 - Valuation Valuation of the 2-year non-callable

3 - Valuation Valuation of the 2-year callable at t=

3 - Valuation Valuation of the 2-year callable at t=1.5,

4 – Spread due to optionality 2-yr non-callable: $ yr callable: $ % 11.82% 9.10%11.36% 7.00%8.74% 6.72%8.39% 6.45% 6.19%

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

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

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

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

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?

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

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

6 – Option-adjusted spread Use the tree to price the 2-year callable, the price is precisely $

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

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= only pertains to: – Default risk – Liquidity risk

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

6 – Option-adjusted spread So what is the price of the 2-yr noncallable?

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

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

7 – Prices and interest rates

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

8 – Duration and Convexity Duration for callable bonds

8 – Duration and Convexity How to calculate V % % % % % % % % % %

8 – Duration and Convexity How to calculate V % % % % % % % % % %

8 – Duration and Convexity Dollar Duration = Duration = Dollar Convexity = Convexity =