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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
1 – Callable bonds – what?
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?
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;
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%
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% + s 11.36% + s7% + s 8.74% + s
6.72% + s 8.39% + s6.45% + s
6.19% + s
6 – Option-adjusted spread
• 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%
6 – Option-adjusted spread
• 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
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!
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
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?
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
6 – Option-adjusted spread
• 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
7 – Prices and interest rates
7 – Prices and interest rates
7 – Prices and interest rates
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+
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
8 – Duration and Convexity
• 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
8 – Duration and Convexity
• Dollar Duration =
• Duration =
• Dollar Convexity =
• Convexity =