Commerce 4FJ3
Fixed Income Analysis
Week 9
Bonds with Options
Traditional Yield Spread
Issue
Maturity
Treasury
25
Corporate
25
Coupon
Price
YTM
8.8% $ 96.6133 9.15%
8.8% $ 87.0798 10.24%
• Traditional yield spread is 109 basis points
• Ignores term structure of interest rates
• The bond, if called in 10 years, should be
compared to a 10 year treasury
2
Static Spread
• Find the treasury spot rate term structure
using the bootstrapping method
• Find the present value of the cash flows for
the bond using the spot rate plus a spread
• Solve for the spread that gives the current
price
• Called the static or zero volatility spread
3
Static Spread Example
Period
1
2
3
4
5
6
7
8
9
10
Static Spread =
0.79%
Cash Flow
Spot Rate Present value
6
7.00%
5.7751
6
7.05%
5.5559
6
7.10%
5.3424
6
7.15%
5.1347
6
7.22%
4.9304
6
7.30%
4.7296
6
7.40%
4.5304
6
7.70%
4.3024
6
8.00%
4.0741
106
8.25%
68.1251
112.5000
4
Value of Static Spread
• Gives the return in excess of treasury over
the entire term structure
• Takes into account that there are expected
different reinvestment rates at different
points in the future
• Assumes that the required spread over
treasury is constant over time
5
Static vs. Traditional
• If yield curve is flat, no difference
• If yield curve is rising, the static spread will
be higher, with bigger differences for long
maturities and steeper term structures
• Static may be smaller if inverted yield curve
• Bigger differences in spread for amortizing
securities (MBS, asset backed securities)
6
Callable Bonds
• Two main disadvantage for buyers
– Extra reinvestment risk: if the bond is called
before the investors time horizon they will face
extra reinvestment risk, probably at a lower rate
– Price compression: if yields fall, the bond price
will not rise as much as it should because the
bond can be bought back at a fixed price
7
Traditional Valuation
• As mentioned earlier, calculate yield to each
call, yield to worst, and then decide on the
price that is reasonable
• Assumes: bond will be called at that date
• Ignores: reinvestment rates
– can be mitigated by comparing to treasuries of
the maturity of the called bond
8
Price vs. Yield for Callable
For a 10% Coupon Bond
$2,000
$1,750
Price
$1,500
$1,250
$1,000
$750
$500
0%
5%
10%
YTM
15%
20%
9
Negative Convexity
• The normal price/yield curve is convex
• With price compression the level of
convexity can become negative (technically
it is now concave)
• Price change from increasing interest rates
becomes larger than the change from falling
interest rates
10
Price vs. Call Price
• Note that the price of the bond can still be
higher than the face value plus call premium
if the bond has time before the call
• 13% bond, callable at 5% premium in one
year, market rate is 5%
6.5 100  6.5  5
Price 

 $112.56
2
1.025
1.025
11
Bonds as Bundles
• Bonds with embedded options can be seen
as a package of bonds and options
• Callable bond: package of; long an option
free bond and short a call option on bond
• Putable (retractable) bond: a package of;
long an option free bond and long a put
option on the same bond
12
Value of Options
• The option value is difficult to calculate
since most pricing models assume that the
price volatility of the underlying asset does
not change over time
• The price volatility (modified duration) of
the bond changes with time and also with
the level of interest rates
13
Another Problem
• Call options on bonds are American options
while pricing models are based on European
options
• Argument for using Euro for models is that
it is usually not worth exercizing early due
to loss of time value does not hold here
since there are intermediate cash flows
14
Interest Rate Volatility
• The major influence on the price of a bond
is interest rates
• Changes in interest rates can be measured
over time and the volatility can be estimated
• Can be used to create an interest rate model
• Textbook model is single factor, lognormal
random walk, binomial interest ladder or
lattice, estimating potential forward rates
15
Interest Rate Lattice
• A bond can be valued by taking the present
value of each cash flow, discounted by the
product of all applicable forward rates
• The model assumes that the forward rate
will take one of two equally likely values
– The higher rate = lower rate x e2s
• Rates are found for each node using trial
and error
16
Option-Free Value
• Once the interest rate lattice has been
constructed, other bonds can be analysed
• Starting with the final cash flows (since the
intermediate prices can not be determined in
advance), fill in the nodes on the lattice
• The price found should be identical to the
one found using the static spread analysis
17
Valuation with Options
• As with the option-free bond, add the value
of the bond plus coupon to each node, but if
the bond is likely to be called (greater than
call price + refunding cost), replace that
value with the call price
• As above, but replace market values below
the put price with the put price
18
Modelling Risk
• If the assumptions that the model is based
on is incorrect, the values derived from the
model will not be useful
• The volatility assumption is critical
– The higher the volatility, the higher the value of
an option, the lower the price of a callable bond
• It is important to stress test the model
19
Option Adjusted Spread
• The spread that would explain the current
price of a bond with an embedded option
• Can be constructed over the treasury term
structure or the issuer’s term structure
• Since there is disagreement between market
participants, knowing which assumption
they are using is critical
20
Option Value in Spread Terms
• If we have the OAS in terms of the treasury
forward rate structure, we can calculate the
amount of the spread that is due to the
embedded option
option value = static spread - OAS
• Main reason for spreads is because some
market participants prefer to talk about all
investments in terms of rate of return
21
Effective Duration and Convexity
• Found using the approximation formulas
P  P
approximat e duration 
2 P0 y 
P-= price if yield down
P+= price if yield up
P0= original price
P  P  2P0
approximat e convexity measure 
2
P0 y 
• Similar to modified if the option is deeply
out of the money
22
Finding P- and P+
• Five step process for binomial model
– Calculate OAS for the bond
– Shift the treasury yield curve down/up a few
basis points
– Construct the interest rate tree
– Add the OAS to each node’s interest rate
– Determine the value of the security
23
Convertible Bonds
• Another type of embedded option
• A call option on a number of the issuer’s
common share where the exercise price is
the bond, regardless of current market value
• Number of shares is conversion ratio
• Can be physical or cash settle
• Exchangeable bonds are similar options, but
on other company’s shares
24
Conversion Price
• The conversion price is simply the implied
exercise price of the option on a per share
basis
• If the bond is issued at par the conversion
price is
Par Value of Convertibl e Bond
Conversion Ratio
25
Other Features
• Conversion ratio may change over time, on
a schedule given in the issue
• Conversion ratio is adjusted for stock splits
and stock dividends
• Most convertibles are also callable, which
may trigger early conversion
• Some are putable (hard or soft put)
26
Sample Convertible
10 years  Maturity
10%  Coupon rate
50  Conversion ratio
$1,000  Par value
$950  Current market price of convertibl e bond
$17  Current market price of common stock
$1  Dividends per share
27
Minimum Price
• The bond will trade at a minimum of the
greater of the conversion value or straight
(debt) value
– conversion value: how much the stock that the
bond can be converted to is worth
– straight value: the value of the convertible if it
did not have the conversion option
28
Sample Minimum Price
• For the sample bond, conversion value
= $17 x 50 = $850
• Given a 14% yield on non-convertible
otherwise similar bonds, straight value
= PVcoupons+ PVface = $788
• This bond should trade for a minimum of
$850 since that is the higher value
29
Market Conversion Prices
• Since the exercise price is the bond, the
effective price of the common stock
changes over time
Market conversion
price
=
Market price of bond
Conversion ratio
Market conversion
premium per share
=
Market conversion price
Market conversion
premium ratio
=
Conversion premium per share
Market price of common stock
-
Current
market Price
30
Sample Conversion
• Market conversion price
= $950/50 = $19
• Market conversion premium per share
= $19 - $17 = $2
• Market conversion premium ratio
= $2/$17 = 11.8%
31
Current Income
• One reason for not converting a convertible
bond before maturity, are the coupon
payments
– FIDPS = Coupon/(conversion ratio) - dividend
– Premium payback period (break-even time) =
Market premium per share ÷
Favourable income differential per share
32
Sample Income
– Coupon interest from bond = $100
– Dividend per share = $1
– Conversion ratio = 50
• Favourable income differential per share
= $100/50 - $1 = $1
• Premium Payback Period
= $2/$1 = 2 years
33
Downside Risk
• Often measured as the premium over
straight value
= (Market value/Straight value) - 1
Sample bond = $950/$788 - 1 = 21%
• Note: the investor has more than 21%
downside risk since the YTM could
increase, decreasing the straight value
34
Jargon
• A convertible where the option is well out
of the money is called a bond equivalent or
busted convertible
• A convertible with a conversion value much
higher than its straight value is called an
equity equivalent
• Between those it is a hybrid security
35
Payoff
• Share price goes up to $34
Shareholder return = 100%
Convertible holder return = 79%
• Share price goes down to $7
Shareholder return = -59%
Convertible holder return = -17%
• The convertible is less risky
36
Call Risk
• One reason for issuing convertible bonds is
that the company would prefer to issue
equity, but considers the current price to be
too low to be worth issuing common shares
• Conversion ratio is set to reflect “reasonable
pricing”
• Call options can be used to force conversion
37
Takeover Risk
• If the issuer gets taken over before the price
of the shares make conversion reasonable,
the bond holders may be left with a bond
that pays a lower coupon than similar
corporate bonds
38
Options Approach
• Similar to callable bonds, convertibles can
be viewed as a bond and an option
• An additional problem here is that the
exercise price on the share changes over
time as the bond’s market price is affected
by changes in interest rates
• To make matters worse, most convertible
bonds are also callable
39