Fixed Income & Debt
Valuing Bonds with Embedded Options
Embedded Bonds
Backward Induction Valuation Methods
Valuing Non-Callable Bond
• Valuing a bond using a binomial interest rate tree
• Point: start at the bond’s maturity, work backwards to time 0.
Price
$100 par
$6 coupon
Putable
Option
price
6.26%, $6
→$99.75
$100 par
$6 coupon
3.88%, $0
→102.56
Callable
4.64%, $6
→$101.30
Duration slope
$100 par
$6 coupon
Yield
y*
Today
•
•
1yr
2yr Maturity
Callable bonds:
have resistance level of call price
outperform as rates rise
underperform as rates fall due to negative convexity
Putable Bonds:
due to relative scarcity of bonds with puts, it is difficult to reach any
conclusive decision regarding their performance and valuation
additional complexity: valuation models for putable bonds fail to
incorporate credit risk: probability high yield issuer will be unable to
repurchase bonds
Relative Value Analysis
Compare “spread” on the bond to required spread
Is the bond “over” or “under” valued?
• Overvalued: bond spread < required
• Undervalued: bond spread > required
• Fairly valued: bond spread = required
REMEMBER: bigger the spread, lower the price to
compensate for risk
But where do we get the spread from?
NOTE: 50% chance of either up or down movement
also note 4.64% in 1 yr is a down move from 3.88% due to
yield curve structure
Determining Nodal Values
Value of the bond at upper node for period 1
= 50% [(100 + 6) / (1+6.26%) + (100 + 6) / (1+6.26%)] = 99.75
Value of the bond at lower node for period 1
= 50% [(100 + 6) / (1+4.64%) + (100 + 6) / (1+4.64%)] = 101.30
Value today t0:
= 50% [(99.75 + 6) / (1+3.88%) + (101.30 + 6) / (1+3.88%)] = 102.56
But Where Do the Interest Rates Come From?
• Tree is generated by computer (bootstrapped rates)
NOTE: rates in tree must be “arbitrage free” for treasury bill →
model and market price equal for on-the-run treasury securities
Valuing a Callable Bond
Same as previous example, but callable at $100 in 1 year
→ If bond price rises above $100, they will exercise Bond
Spread Measures
spread is the additional return to benchmark
3 common measures: Nominal, Z-spread, OAS
$100 Par
$6 coupon
6.26%, $6
→$99.75
Nominal spread:
3.88%, $0
bond yield less yield on benchmark (usually the comparable maturity
Treasury)
Choice of spread impacts interpretation:
• Spread on U.S. Treasury,
• Sector Spread (i.e., spread on AAA-Corp)
• Spread for Specific issuer
(-) only looks at maturity of the security ignoring liquidity, coupon,
optionality, risk, etc
→$101.92
Zero Volatility Spread (Z-Spread)
Corporate bonds have higher risk than treasuries → higher discount
rate → lower price
Take the spot rate curve on Treasury bills/bonds and shift if up until
model price = market price of Corporate bonds
The spread we have added is our Z-spread
$100 Par
$6 coupon
4.64%, $6
$101.30>100
Today
→$100 Call
$100 Par
$6 coupon
1 Year
2 Years: Maturity
Current value of the bond at node 0
= 50% [(99.755 + 6) / (1+3.88%) + (100.000 + 6) / (1+3.88%)] = 101.92
VCALL = VNON-CALLABLE BOND + VCALLABLE BOND
$102.56 – $101.92 = $0.637
Option Adjusted Spread (OAS)
For putable bonds, this value relationship is other way round
as options always positive value
part of spread between Corporate & treasury is attributed to optionality
of bond, so by removing option value, left with OAS → the spread
additional to return on Treasury Bond
VPUT = VPUTABLE BOND + VNON_PUTABLE BOND
All notes can be reproduced for educational purpose only. Copyright © 2009 – 2011 www.AnalystFormulas.com
Fixed Income & Debt
Valuing Bonds with Embedded Options
Volatility and Bond Values
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•
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•
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Volatility does not affect the value of a noncallable bond
Higher volatility decreases the value of a callable bond ...
... so higher volatility increases the value of the embedded call
option in a callable bond
Nominal and Z-spread take account of:
Nominal Spreads Represent Differences in Yields Due To:
Difference in credit risk between two bonds
Difference in the liquidity risk between two bonds
Difference in option risk between two bonds
As Option Adjusted Spread (think: option REMOVED spread) –
is adjusted and spread is only compensation for Credit risk &
Liquidity risk; a Zero volatility spread on non-callable bond
should equal OAS
Convertible Bond Basics
Convertible bonds are bonds that can be converted into
common stock. Not interest rate sensitive like callable &
putable bonds, but more sensitive to stock price.
For example, suppose you can buy a 10%, 15-year bond today
for $90 that can be converted into ten shares
The value of a comparable straight is $84
Market price of stock = $5; no dividends
•
Conversion ratio
= number or shares can convert to (10)
•
Market conversion price
= bond value / ratio (=$90 / 10 = $9)
•
Market conversion value
= market price of stock × conversion ratio (=10 × 5 = 50)
•
Straight value
= value if non-convertible ($84; given)
•
Market conversion premium per share
= conversion price – market price = 9 – 5 = $4 per share
→Per share premium if you purchased bond and immediately
converted into stock
•
Market conversion premium ratio:
Finding the Option-Adjusted Spread (OAS)
Previous example: the calculated value of the callable bond was 101.92
What if the market price is 101.21?
OAS: the spread added to each node that makes the calculated value
equal to the market value → Key point: want an arbitrage-free price
Let’s add 50 b.p. to each of the 1-year rates in the tree and re-calculate the
value of the bond
$100 Par
$6 coupon
6.26%+50bp, $6
→$99.75
3.88%+50bp, $0
$100 Par
$6 coupon
→$101.92
4.64%+50bp, $6
$101.30>100
→$100 Call
Today
=
$100 Par
$6 coupon
Think: premium stated as percentage = $4/$5 = 80%
1 Year
2 Years: Maturity
•
The OAS is the spread adjusted for option risk
Relative value analysis asks: Is the OAS great enough to compensate
for credit and liquidity risk?
→ All else equal, buy large OAS bonds!
Effective Duration and Effective Convexity
V− − V+
2V 0 ( ∆i )
Convexity =
V+ + V− − 2V0
2V0 ( ∆i ) 2
V– and V+ come from the binomial model.
We move interest rates up to get V– and down to get V+
Valuing a Putable bond
Same as before, but now putable at $100 in 1 year
If bond price falls above $100, they will exercise Bond
.
6.26%, $6
$100 Par
$6 Coupon
$99.75<100
3.88%, $0
Premium payback period
=
OAS vs. Nominal Spread
Duration =
→$100
→$102.66
$100 Par
$6 coupon
4.64%, $6
$101.30
mkt conversion premium per share
mkt price of common stock
$100 Par
$6 coupon
Today
1 Year
2 Years: Maturity
Current value of the bond at node 0
= ½ [(100.000 + 6) / (1.038796) + (101.302 + 6) / (1.038796)] = 102.668
VPUT=VPUTABLE BOND − VNON-PUTABLE BOND
= 102.668 – 102.560 = 0.108
mkt conversion premium per share
favourable income difference per share
= $4/$1 = 4 years
•
Premium over straight value
=
mkt price of convertible bond
−1
straight value
= $90/$84 – 1 = 7.14%
The greater the premium over straight value, the less attractive
the convertible bond
Option-Based Valuation Approach
Callable convertible bond value
= straight value of bond
+ value of the call option on the stock
– value of the call option on the bond
Simply a breakdown of value into straight bond plus options
Risk/Return of a Convertible Security
• If stock price falls, returns on convertible bonds exceed those
of the stock
• When stock price rises, the bond will underperform because
of the conversion premium
POINT: If the stock remains stable, return on the bond may
exceed the stock return due to the coupon payments from the
bond
All notes can be reproduced for educational purpose only. Copyright © 2009 – 2011 www.AnalystFormulas.com