callable bonds professor anh le. 0 – plan 1.callable bonds – what and why? 2.yields to call,...

Callable Bonds

Professor Anh Le

0 – Plan 1. Callable bonds – what and why?2. Yields to call, worst3. Valuation4. Spread due to optionality 5. Z-spread6. Option-adjusted spread7. Callable prices and interest rates8. 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 valueExample: Fixed rate mortgages

• Many, call price = par value + premium and then declines over time


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?


• 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?


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


• Example: 2-yr, $100 face, 8%-coupon callable at time 1 at a call price of $100. The bond is selling for $99.


• Suppose:


• 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?


• 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;

OR2. 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 $100

0 0.5 1 1.5

15.35%

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

0 0.5 1 1.5

15.35%

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

0 0.5 1 1.5

96.5854

95.83173

96.90074 98.40869

99.25286 98.79106

100.5561 99.8112

101.082

100.8761

3 - Valuation

• Valuation of the 2-year callable at t=1

0 0.5 1 1.5

96.5854

95.83173

96.90074 98.40869

99.25286 98.79106

100.5561 99.8112

101.082

100.8761

3 - Valuation

• Valuation of the 2-year callable at t=1.5, 1

96.5854

95.83173

96.90074 98.40869

99.25286 98.79106

100.5561 99.8112

101.082

100.8761

4 – Spread due to optionality

• 2-yr non-callable: $99.25• 2-yr callable: $99.00

0 0.5 1 1.5

15.35%

11.82%

9.10% 11.36%

7.00% 8.74%

6.72% 8.39%

6.45%

6.19%


• Without the tree, we can price the bond as follows:– 2-yr non-callable:

– 2-yr callable:


• 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?


• 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% + s 11.36% + s7% + s 8.74% + s

6.72% + s 8.39% + s6.45% + s

6.19% + s


• With s=0.006412, we have the following tree:

0 0.5 1 1.5

15.99%

12.46%

9.74% 12.00%

7.64% 9.38%

7.36% 9.03%

7.09%

6.83%


• Use the tree to price the 2-year callable, the price is precisely $98

0 0.5 1 1.596.29866

95.2674596.05371 98.11103

98.00000 98.2001399.43839 99.50501

100100.5634


• 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!


• 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


• 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


• So what is the price of the 2-yr noncallable?

0 0.5 1 1.5

96.29866

95.26745

96.05371 98.11103

98.10934 98.20013

99.66513 99.50501

100.4702

100.5634


• The new tree can also price other bonds issued by the same firm

0 0.5 1 1.5

15.99%

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


• 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


• How to calculate V+

15.35% + 0.0010

11.82% + 0.0010

9.1% + 0.0010 11.36% + 0.00107% + 0.0010 8.74% + 0.0010

6.72% + 0.0010 8.39% + 0.00106.45% + 0.0010

6.19% + 0.0010


• How to calculate V-

15.35% - 0.0010

11.82% - 0.0010

9.1% - 0.0010 11.36% - 0.00107% - 0.0010 8.74% - 0.0010

6.72% - 0.0010 8.39% - 0.00106.45% - 0.0010

6.19% - 0.0010


• Dollar Duration =

• Duration =

• Dollar Convexity =

• Convexity =

callable bonds professor anh le. 0 – plan 1.callable bonds – what and why? 2.yields to call,...

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