Download Convertible bonds

Part 5 - Convertible bonds as an asset class
• Various features in convertible bonds
• Issuance of convertibles - perspectives of corporate treasurers
- conversion into shares
- call (hard and soft provisions)
- put
- reset on conversion number
- dilution protection
• Delayed call phenomena
• Decomposition of convertibles into different components
• Valuation of convertibles
- interest rate sensitivities (duration analysis)
- binomial tree calculations
1
Combination of bonds and equities - bond plus a conversion option
* Bondholder has the right to convert the bond into common shares at
some contractual price (conversion number may change over time).
Conversion value: stock price x conversion number
Conversion premium: (bond price – conversion value) / conversion value
Bond floor value: sum of present value of coupon and par
convertible
bond price
conversion
premium
conversion value
straight bond value
stock price
2
Perspectives on convertibles
• To take advantage of the upside potential growth of the
underlying stock (participation into equity).
Conversion option that allows the investor to exchange the
straight bond for fixed number of shares.
• Swapping the variable stock dividends in return for fixed
coupon payments until the earlier of the maturity date and the
conversion date.
• Provides the “bond floor” value as downside protection.
3
Analytics of convertible bonds
stock price
stock dividend
convertible market price
coupon rate
maturity
conversion price
$30.00 per share
$0.50 per share
$1,000
7.00%
20 years
$36.37
Stock dividend yield = annual dividend rate
/ current stock price
= $0.50 / $30.00 = 1.67%
4
Conversion ratio
= number of shares for which one bond may be
exchanged
= par / conversion price
= $1,000 / $36.37 = 27.50 shares
Conversion value
= equity value or stock value of the convertible
= stock price x conversion ratio
= $30.00 x 27.50 = $825.00
5
Premium for conversion right
• An investor who purchases a convertible bond
rather than the underlying stock typically pays a
premium over the current market price of the stock.
• Why would someone be willing to pay a premium
to buy this stock? The market conversion premium
per share is related to the price of a call option –
limit the downside risk of the convertible bond.
6
Conversion premium
= (convertible price – conversion value)
/ conversion value
= ($1,000 – $825) / $825.00 = 21.21%
Dollar premium
= (convertible price – conversion value)
/convertible price
= ($1,000 – $825) / $1,000 x 100%
= 17.50 points
7
Capture of accrued interest upon
conversion
• Upon conversion to stock, any accrued interest
will be lost. This is the infamous screw clause.
• Trick - sell an equivalent amount of stock short
against the convertible bond, hold the
short sale until the interest payment date;
then convert the bond and deliver the stock
against the short sale.
8
Break even calculations
Break even (years)
= conversion premium / (convertible yield – stock yield)
= 21.21 / (7.00 – 1.67) = 3.98 (years)
Number of years necessary for the stock investor to recover
the conversion premium from the convertible’s higher
income relative to an instrument of an equivalent amount in
the stock.
 After 3.98 years, the convertible has made up, in income
alone, the amount of the conversion premium.
9
Break-even calculations (cont’d)
Dollar maintenance
market price – conversion value
=
coupon - market price stock dividend
stock price
The time it takes for the convertible yield advantage to
pay for its premium compared to an equivalent dollar
amount purchased of the underlying stock.
• May use conversion ratio instead of market
price/ stock price.
10
Weaknesses of break-even analysis
• It ignores the main advantage of convertible:
protection on downside risk on the underlying
equity.
• It ignores the margin of safety offered by the
convertible with the payment of principal at
maturity.
11
Call terms
Issuer has the right to call back the bond at a pre-specified call price prior
to final maturity, usually with a notice period requirement. Upon call, the
holder can either convert the bond or redeem at the call price.
Issuer’s perspective on the call right
• Refinancing
To have the flexibility to call if they think they can refinance the debt
more cheaply.
• Managing the debt / equity balance
To force bondholders to convert debt into equity, which can reduce debt
levels. The issuer has the flexibility to shift debt into equity to reduce
the leverage of the firm.
12
Call protection
Hard (or absolute):
To protect the bond from being called for a certain period of time.
Soft (or provisional):
The issuer is allowed to call only when certain conditions are satisfied.
For example, the closing price of stock has been in excess of 150% of
the conversion price on any 20 trading days within 30 consecutive days.
Investor’s protection
To preserve the value of the equity option for the bondholders. While
waiting for the stock price to increase, convertibles typically provide
more income than the stock. Without the call protection, this income
stream could be called away at any time. Hard call protection with the
longest possible duration is the most desirable for the investors.
13
Background for make-whole provision
1. Provisional call protection give issuers the option to retire the debt
early, while the make-whole payment just what would have had to
make anyway.
2. They are designed to protect investors from sharp premium
contraction in a rapidly rising equity market.
Make-whole payments should become a price variable at the point
where a convertible without a make-whole provision, given a rising
common stock, would begin to lose premium.
14
Make-whole provision
Premium make-whole provision
• First issuance of Amazon.com 4.75% convertible note in January 1999
• Assures that holders will recover the amount of original premium paid
at issuance, less any interest already paid.
Soft (or provisional):
The issuer is allowed to call only when certain conditions are satisfied.
For example, the closing price of stock has been in excess of 150% of
the conversion price on any 20 trading days within 30 consecutive days.
15
Put feature
Allows the holder to sell back the bond to the issuer in return for a fixed
sum. Usually, the put right lasts for a much shorter time period than the
maturity date of the bond.
•
The holder is compensated for the lesser amount of coupons
received in case the equity portion of the convertible has low
value.
•
It protects the holder against rising interest rates by effectively
reducing the year to maturity. The convertible’s price then
becomes less sensitive to interest rate.
The put feature may shorten the maturity of the bond (par paid earlier)
and thus effectively raises its investment value and lower the sensitivity
to interest rate fluctuation.
16
Reset feature in convertible bonds
In most cases, the reset on conversion price is downward and this makes
the bond more valuable. For example, the conversion number is reset
by dividing the par by the prevailing stock price.
Floor limit
The extent of downward reset cannot be below a certain multiplier of
the first conversion price.
17
Why reset convertible bonds are popular in Japan in mid1990’s?
• Japanese banks were considered quite risky as they had large
real estates exposures.
• To raise capital
a. equity issuance was out of the question since the stock
markets were depressed;
b. straight bond issues would have required a high coupon yield.
Reset feature was included in convertible bonds to give investors some
sort of insurance against bank’s stock decline.
18
Pricing difficulties
There are many possible conversion prices since they depend on the past
history of the stock price.
Impact on bond price
At high stock price (not likely to reset) or low stock price (low equity
value) regions, the reset premium is low. The reset premium is
significant only at intermediate stock price level.
Nightmare for the issuers
The feature is too sweet for the investors and harmful to the issuer.
• When the stock price drops, the investors are compensated.
• When the stock price rises, the conversion premium becomes more
expensive.
These structures have fallen from popularity in recent years.
19
Takeover clauses
An unexpected takeover bid can have the effect of
eliminating any conversion premium if the takeover
price is below the conversion price.
Poison put is added as protection in the change-ofcontrol clause:
triggered as a result of a hostile takeover and
would allow the bondholder to put the bonds back
to the company at par.
Alternatively, the issuer can prevent the conversion
premium from evaporating by increasing the conversion
ratio to restore the premium to the average level seen
before the takeover announcement.
20
Dilution protection
(extraordinary dividends)
Convertible investors are not compensated for normal
dividend payment. However, if a company makes an
extraordinary payment, resembling a return of capital,
convertible holders want to be compensated.
Dilution protection clause may be added:
If the dividend payment yields more than a specified
amount, any amount over the threshold will be
compensated.
21
Convertible bond issued by the Bank of East Asia
US$250,000,000
2.00 percent Convertible Bonds due 2003
Issue date
July 19, 1996
Issue price
100 percent of the principal amount of the Bonds,
plus accrued interest, if any, from July 19, 1996
(in denominations of US$1,000 each)
Conversion period From and including September 19, 1996 up to and
including July 7, 2003
22
Conversion feature
Conversion price HK$31.40 per Share and with a fixed rate of
exchange on conversion of HK$7.7405 = US$1.00.
Dilution protection The Conversion Price will be subject to adjustment
clause
for, among other things, subdivision or consolidation
of the Shares, bonus issues, right issues and other
dilutive events.
23
Call feature
Redemption at the
option of the
bondholders
On or after July 19, 1998, the Issuer may redeem
the Bonds at any time in whole or in part at the
principal amount of each Bond, together with
accrued interest, if for each of 30 consecutive
Trading Days, the last of which Trading Days is
not less than five nor more than 30 days prior to
the day upon which the notice of redemption is
first published, the closing price of the Shares as
quoted on the Hong Kong Stock Exchange shall
have at least 130 percent of the Conversion Price
in effect on such Trading Day.
24
Soft call protection
Parisian feature
The closing price has to be above 130 percent of the conversion price on
consecutive 30 trading days.
• On the date of issuance of the notice of redemption (treated as day 0),
the Issuer looks back 5 to 30 days (corresponds to [-30,-5] time
interval) to check whether the history of the stock price path
satisfies the Parisian constraint. That is, the last of the 30 trading days
falls in [-30,-5] time interval.
• From Issuer’s perspective, when the Parisian constraint has been
satisfied, the Issuer has 5 to 30 days to make the decision on
redemption or not.
25
Put feature
Redemption at the
option of the
bondholders
On July 19, 2001, the Bonds may be redeemed at
the option of the Bondholders in US dollars at the
redemption price equal to 127.25 percent of the
principal amount of the Bonds, together with
accrued interest.
The investors are protected to have 27.25% returns on the bond
investment upon early redemption by the issuer.
26
Put above par value or premium redemption at maturity
Renong Berhad (a Malaysian company) issued a 5-year bond with
a 2.5 percent coupon with yield-to-put at 7.5 percent and a put
price of 129.7. This is above the par of 100 used in the calculation
of conversion into stock. Also, this results in increased downside
protection in case the equity portion has low value.
Investors’ perspective
Even if the conversion turns out to be unprofitable, they are
guaranteed a 7.5 percent return to the time of the put.
27
Examples of reset convertibles
• United Artists Communications (1987) issued convertibles that after
a fixed period of time, the bonds were evaluated by an independent
investment banker. This is to determine the coupon rate that would
allow the bonds to trade at 101 plus accrued interest.
• Mitsuibishi Bank issued (Oct., 1995) $2 billion of 7-year bond with
annual reset of the conversion ratio. It offers investors more shares if
the stock price declines, with the goal of keeping the bond’s equity
value at par.
28
Casino operator brings ringgit convertible
• Malaysia's only casino operator, Resorts World, has
raised M$1.1 billion ($300 million) from a convertible
bond that was well received despite offering a negative
yield.
• Desire to see bonds convert prompts Resorts World to
use rare negative yield structure – less bond
characteristics.
• Reset feature added to prompt higher propensity of
conversion.
29
Bond indenture
• Negative yield
Issuing the zero-coupon bonds at par and setting the
redemption price at 99%, which results in a yield to
maturity of -0.5%.
• Lower conversion price
The conversion price was fixed at launch at 10%
over yesterday's (September 7, 2006) volume weighted
average price of M$11.593, giving an initial
conversion price of M$12.75.
30
• Forced conversion
There is an issuer call after one year, subject to a 120%
hurdle, to force conversion in case investors drag their
feet.
• The reset mechanism has a floor at 90.9% of the original
conversion price, which is high compared with the
typical reset floor at 80-85%.
• The bonds were priced assuming a credit spread of 40
basis points over the Malaysian interest rate curve, a
dividend yield of 2.2% or 120% of the previous year's,
and a stock borrow cost of 5%.
31
Issuer’s perspectives
• While common a few years back when interest rates were
much lower, negative yields are rarely seen on CBs
nowadays but highlights the issuer's desire to have the
bonds convert in order to get equity on its balance sheet.
• The bonds have a short maturity of only two years, a
conversion premium of only 10% and two conversion
price resets - after the first year and 60 days before
maturity - making it all but inevitable that the bonds will
convert.
32
Investor’s perspectives
The bond floor was set at 90.7%, which one observer
says is "reasonably attractive" given the strong focus on
conversion and the implied volatility is 24%.
Market background
• The share price is up a modest 4.5% this year to Thursday's
closing price of M$11.70, which compares with a 6.2% gain
in the Kuala Lumpur Composite Index.
•
The expected appreciation of the ringgit makes the bonds a
reasonable proposition.
•
Of the 19 analysts that cover the company, according to
Bloomberg data, 16 have a "buy" or "overweight“
recommendation.
33
Types of companies as convertible issuers
Companies that are characterized by strong performing, highvisibility, sub-investment grade, high-growth potential have
comparative advantage in the convertible market versus the
fixed income market.
 They lack a long-term track record and have volatile capital
structures – high coupon must be offered.
 They can transform the high volatility into a benefit since the
warrant is more expensive.
 When the company grows, they may call the bonds. This in
turn will strengthen the company’s equity base at the moment
34
when it is most needed.
Convertibles as backdoor
equity financing
Delayed equity
 Convertibles provide a way of selling common stock
at a price above the existing market.
 They are employed as deferred common stock
financing.
The call feature is important since it gives the
company the means to shift debt to equity.
Convertibles offer a means to control the
debt/equity ratio.
35
Reason(s) for offering
My firm chose convertible as its financing source …
a.
b.
c.
d.
e.
f.
g.
because of the lower coupon rate versus straight debt.
because management felt that the stock was undervalued
at the time.
because management felt that the stock was overvalued
at the time.
as "delayed equity" financing, expecting that the debt
would be converted.
because the conversion feature provides bondholders with
protection against unfavorable actions by stockholders or
management.
because our investment banker recommended it over
other forms of financing.
because other firms had recently made successful
convertible offerings.
Strongly
Disagree
1
4.7%
2
4.7%
Neither
Agree Nor
Disagree
3
4.7%
16.9%
14.5%
24.1%
25.3%
19.3%
42.7%
19.5%
31.7%
3.7%
2.4%
7.0%
3.5%
5.8%
46.5%
37.2%
45.1%
20.7%
28.0%
6.1%
0.0%
10.7%
8.3%
45.2%
29.8%
6.0%
10.8%
10.8%
33.7%
42.2%
2.4%
4
50.6%
Strongly
Agree
5
35.5%
36
Reasons for offering (cont’d)
By order of importance, please rank the following factors on a scale of
1 to 6 on the extent to which it influenced your firm’s decision to issue
convertible; 1 = most influential, 2 = next most influential, etc.)
Lower coupon versus straight debt
stock was undervalued, so we couldn't issue equity
Stock was overvalued, so we took the opportunity
to lock in a favorable conversion premium
Firm wished to issue "delayed equity"
Investment banker recommended it
Other firms had recently issued convertible debt successfully
1
48.3%
15.7%
2
25.0%
22.6%
3
13.3%
7.8%
4
5.6%
11.3%
5
5.6%
7.0%
6
1.2%
28.0%
3.4%
4.8%
2.2%
7.0% 26.8%
22.5% 31.0% 20.0% 12.7% 14.1%
7.9%
4.8% 27.8% 38.0% 18.3%
2.20% 11.90% 28.90% 25.40% 28.20%
52.4%
2.4%
7.3%
8.50%
37
Environment in which issuance occurred
1. In retrospect, my firm’s stock was _____ valued around the time
of the convertible debt offering.
under
1
46.4%
correctly
2
41.6%
over
3
12.0%
Mean
1.66
Median
2.00
2. Around the time of the convertible debt offering, my firm’s
management expected future earning to be _____ the market was
expecting.
2
3.6%
about the same as
3
49.4%
4
35.1%
much higher than
5
11.9%
Mean
3.55
Median
3.00
3. At the time of the convertible debt offering, prospects for my
firm’s short-term (1-2 years) performance relative to its industry
were:
poor
1
2.4%
fair
2
5.9%
good
3
34.1%
very good
4
47.1%
excellent
5
10.6%
Mean
3.58
Median
4.00
38
Delayed call phenomena
• Firms wait until the value of the bond value is much
higher that the call price times the conversion number
before issuing the call. In Ingersoll’s study of
179 convertibles, the percentage amount exceeded had
a median of 43.9%.
• How to explain such delayed call phenomena?
• Several possible theories are proposed for the rationales
of delayed call.
39
Safety premium
• Since convertible bond covenants typically require 30-day
notice before the bonds are redeemed, and during that
period, the stock price may fall (likely to fall given the
normal negative reaction to the call announcement),
forcing the firm to redeem for cash.
• There would be a higher cost to raise funds shortly so that
the calling firms require a safety premium on the risk that
the stock price may drop significantly over the notice
period causing the bond to become out-of-the-money.
40
Tax advantage
• Some firms enjoy an advantage of paying less in
after-tax interest than they would pay in dividends
were the bond converted.
• Therefore, firms have the incentive to keep the
bond alive even if the net present value of coupons
to be paid somewhat exceeds the net present value
of the dividends that would be paid to the former
bondholder following conversion.
41
Stripping different components of
convertible bonds
A convertible bond consists of 3 components
• bond component – interest rate risk
• equity component – equity risk (long volatility)
• credit quality component - credit risk
Fixed income
investor
coupon payment
principal
Financial
institution
holding a
convertible bond
periodic
payment
payment
contingent
upon default
Default risk
protection
provider
42
Contingent claims approach
Wide spread use of the option pricing theory for pricing
convertibles.
 The underlying state variable is the issuer’s firm value process.
The firm value is not the total value of assets owned by the
firm. It is the takeover value when the firm is sold.
 Overall assessment of the price impact of different features in a
convertible.
43
Dominant factors in structural models
for risky convertibles
1.
2.
3.
4.
5.
6.
Issuer’s firm value process - volatility.
Issuer’s capital structure – debt/equity ratio.
Loss given default.
Terms and conditions of the debt issue.
Interest rate process.
Correlation between the interest rate and asset value.
•
Difficult to estimate the parameter values when implementing
the models.
44
Structural models for pricing
convertible bonds
Assumptions
• Non-callable and can be converted only at maturity.
• No transaction costs and no bankruptcy costs.
• Capital structure consists of common shares and non-
callable convertibles.
45
N = number of common shares
M = number of convertibles
f = face value per convertible
F = face value of the convertible issue = Mf
r = conversion ratio
Upon conversion, the convertible holders will possess
l fraction of equity, where
Mr
l
.
N  Mr
The parameter l is called the dilution factor of the
convertible issue.
46
Let VT denote the value of the firm at maturity after
the last coupon has been paid.
The holder will convert if and only if
l VT > F
Value of the convertible bond at maturity
 VT

 F
lV
 T
if
VT  F
if
F  VT  F / l
if
VT  F / l
 F - max( F - VT ,0)  l max( VT - F / l ,0).
47
payoff to
convertible
straight bond
F
F/l
Decomposition into a straight bond
plus l units of call with strike F/l
and short a put to issuer
VT
48
Intrinsic value of convertibles
The intrinsic value of a convertible bond is the greater of
1. Conversion value
2. Bond investment value – value as a corporate bond
without the conversion option (based on the convertible
bond’s cash flow if not converted).
• To estimate the bond investment value, one has to
determine the required yield on a non-convertible bond
with the same quality rating and similar investment
characteristics.
• If the convertible bond does not sell for the greater of
these two values, arbitrage profits could be realized.
49
Convertible = bond + warrant
Factors that affect the bond component
•
•
•
•
Interest rates
Credit rating/spreads
Coupon
Duration
Factors that affect the warrant component

•
•
•
•
Stock performance
Embedded strike price
Common dividend yield and dividend growth rate
Stock volatility
Life of warrant / call protection
50
Put plus stock plus yield advantage
Applying the put-call parity:
call + bond = put + stock.
Here, put is the right to sell the stock for bond
One may treat a convertible bond as yield-enhanced stock
plus a put option.
• The put option represents the bond floor protection.
The strike price is the bond investment value.
51
Bond investment value
• Present value of the interest and principal payments discounted at the
straight (non-convertible) bond interest rate
n
C
P

bond interest value = 
t
n
(
1

r
)
(
1

r
)
t 1
where P = par value, r = discount rate, C = coupon rate,
n = number of periods to maturity.
Years
1 - 20
20
present
value
payment
$80
$1,000
present
value
factor
8.514
0.149
$681.12
$149.00
$830.12
take r = 10%
52
Estimation of the discount rate
Use the yield-to-maturity of a similar nonconvertible bond as a proxy.
• The apparent deterioration of the creditworthiness
of an issue will not be reflected in the convertible
price because the value of the conversion option
may be rising due to higher share price volatility.
53
Valuation of convertible debts
List of parameters
 Coupon rate
 Creditworthiness of the issuer
 Maturity date
 Conversion premium
 Ratio of conversion price to current stock price
 Volatility of the stock price
 Dividend yield of the stock price
 Presence of other embedded option features, like callability
and puttability
 Prevailing risk free interest rate and volatility of interest rate
 Correlation of the stock price with the interest rate
54
Duration
Duration is the weighted average of the times that the
principal and interest payments are made.
n
t
tC
/(
1

i
)
 t
duration =
t 1
n
t
C
/(
1

i
)
 t
t 1
where t is the time of payment
Ct is the coupon and/or principal payment
i is the market yield.
Duration analysis provides a measure how bond
values change with changing interest rates.
55
Duration analysis applied to convertibles
The approximation for the convertible bond’s interest
rate sensitivity
 C/I 
D  Dadj 1 
2 

cv
where C = conversion value and I = investment value.
• The equity component of the convertible bond may
dampen the convertible’s interest rate sensitivity,
depending on the bond’s equity participation.
Hence, convertibles trading high above their
investment value will be less sensitive to interest
56
rates.
Duration and coupon
 For non-convertible bonds, the duration decreases
as their coupon increases. This is because higher
coupon bonds deliver more cash flows near the
start of bond’s life.
 With convertible feature, the higher coupon rate
may lead to lower propensity to convert. The CB
then has a longer life, so this leads to higher
duration.
These two effects are counteracting.
57
Interest rate sensitivity
1. The exercise price is a function of the investment
value. An increase in interest rates will lower the
investment value.
2. However, the exercise price of the embedded call
is reduced. A lower exercise price will increase the
value of the warrant.
58
Interest rate sensitivity (cont’d)
Basic
Int rate
Change Int rate
Change
price
+ 1%
-1%
_______________________________________________________
Investment value $847.84 $812.75 -$35.09 884.74 $36.90
Warrant value
$337.66 $362.58 +24.92
Total
$1185.50 $1175.33 -$10.17 $1197.47 +$11.92
Percent change
-1.02%
$312.72 -$24.94
1.19%
59
Correlation with interest rates
Consider the impact of an increase on interest rate
 The future share price is expected to be higher because of
higher drift rate.
 Due to negative correlation between interest rate and share
price (say, the S&P 500-stock index has a correlation of about
minus 0.5), the share price drops first.
Negative correlations normally lower CB value;
positive correlations make the CB worth more.
In some situation, CBs may have price differences in the range
of 10-15% when correlation moves from 1.0 to –1.0.
60
Pricing of risky convertible bonds
One-factor binomial model
* stock price process follows binomial random walk
* interest rates to be deterministic
Two discount rates
1. If the convertible is certain to remain a bond, it is appropriate
to use a discount rate corresponding to the creditworthiness
of the issuer - risky rate.
2. Suppose the bond is certain to be converted, it is then
appropriate to use the riskfree rate.
At maturity, the holder will choose the maximum between
the par value and the value of stocks received upon
conversion.
61
How to account for the creditworthiness of the issuer?
The discount rate to be used when we roll back is given by
pwu + (1 - p)wd.
Here, p is the probability to a node where the discount rate is
wu and (1 - p) is probability to a node with wd. The appropriate
discount rate is the weighted average of the discounted rates
at the nodes in the next time step.
62
conv = value of stocks received if conversion takes place
call = call price
roll = value given by the rollback
(neither converted nor recalled)
At each node, the optimal strategy of the holder is exemplified
by taking the maximum of min(roll, call) and conv.
• The maximum reflects the conversion right, which persists
with or without recall by the issuer.
• min(roll, call) means the bond value can never shoot beyond
the call price.
Dynamic programming procedure:
max(min(roll, call), conv)
63
Alternative dynamic programming procedure:
min(max(roll, conv), max(call, conv))
• The term max(roll, conv) represents the optimal strategy of
the holder.
• Upon recall, the holder chooses to accept the call price or
convert into shares. This can be represented by
max(call, conv).
The issuer chooses to recall or to abstain from recalling in
order to minimize the option value.
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Example
A 9-month discount bond issued XYZ company with a face
value of $100. Assume that it can be exchanged for 2 shares
of company’s stock at any time during the 9 months.
* It is callable for $115 at any time.
* Initial stock price = $50, s = 30% per annum and no
dividend; risk-free yield curve to be flat at 10% per annum.
* Yield curve corresponding to bonds issued by the company
to be flat at 15%.
* Tree parameters are: u = 1.1618, d = 0.8607, p = 0.5467,
R = e0.1Dt = 1.0253.
* At maturity, the convertible is worth max (100, 2ST).
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Binomial tree for pricing a risky convertible bond
67.49
10%
58.09
11.03%
D
134.98
B
50.00
12.27%
116.18
50.00
11.59%
104.85
equity 78.42
10%
156.84
A
43.04
13.51%
E
105.56
bond
C
98.00
upper figure: stock price
middle figure: discount rate
lower figure: value of convertible
equity 58.09
10%
116.18
37.04
15%
F
96.32
43.04
15%
100.00
bond 31.88
15%
66
100.00
At node D
Roll back gives the bond value
(0.5467  156.84 + 0.4533  116.18)e-0.1  0.25 = 134.98.
The bondholder is indifferent to conversion or hold, also the
issuer is also indifferent as to whether the bond is called;
the correct discount rate at node D is 10%.
At node F
The correct discount rate is 15% since the convertible is
contain not to be converted if node E is reached.
At node E
The correct discount rate is
0.5467  10% + 0.4533  15% = 12.27%.
The value of convertible at E
(0.5467  116.18 + 0.4533  100)e-0.1227  0.25 = 105.56.
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The bond should be neither converted nor called.
At node B
The discount rate is
0.5467  10% + 0.4533  12.27% = 11.03%
and value of convertible is
(0.5467  134.99 + 0.4533  105.56)e-0.1103  0.25 = 118.34.
It is optimal to call the bond at node B so that it causes
immediate conversion and leads to $116.18. The discount
rate at node B should be taken to be 10%, since conversion
takes place at this node.
At node A
The discount rate is
0.5467  10% + 0.4533  13.51% = 11.59%.
The convertible value at node A is
(0.5467  116.18 + 0.4533  98.00)e-0.1159  0.25 = 104.85.
If the bond has no conversion option, its value is
e-0.75  0.15 = 89.36.
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The value of conversion option = 104.85 - 89.36 = 15.49.