This document discusses convertible bonds and their valuation. It begins by covering Modigliani & Miller's propositions on capital structure and the irrelevance of capital structure absent taxes. It then discusses how taxes create value from leverage through interest tax shields. Various pricing models for convertible bonds are presented, including the Black-Scholes model. The document also discusses hedging strategies using convertible bonds and their associated Greeks.

1. Financial Risk
Management
Convertible Bonds
An example of how to determine value of company by capital structure
By Philip Corsano
Gnostam Consulting
www.gnostamconsulting.com
Tel 206 384 0069

2. Firm Capital Structure
 Modigliani & Miller, [“MM”] proposition 1: Value of firm is
independent of capital structure, [i.e. you don’t create value
by capital structure alone, in absence of tax effects];
 Modigliani & Miller: It does not matter what risk preferences
are for investors.
 Assume Investors have the ability to borrow and lend for
their own account (and at the same rate as firms) so that
they can “undo” any changes in firm’s capital structure
Prepared by Philip Corsano, Gnostam Consulting
Training, Tel: 206 384 0069, pcorsano@gmail.com
1

3. M&M Proposition 2
 Bonds are almost risk-free at low debt levels
 rD is independent of leverage
 rE increases linearly with debt-equity ratios and the
increase in expected return reflects increased risk
 As firms borrow more, the risk of default rises
 rD starts to increase
 rE increases more slowly (because the holders of
risky debt bear some of the firm’s business risk)
2

4. Leverage and Returns
Expected
return on assets
rA
D
D
E
ra
rD
expected
operating
income
market val ue of all securities
E
D
E
rE
3

5. r
M&M Proposition II
rE
rA
rD
Risk free debt
Risky debt
D
E
4

6. Capital Structure
PV of Tax Shield =
D x rD x Tc
= D x Tc
rD
(assume perpetuity)
Example:
Tax benefit = 1000 x (.10) x (.40) = $40
PV of 40 perpetuity = 40 / .10 = $400
PV Tax Shield = D x Tc = 1000 x .4 = $400
5

7. Capital Structure
Firm Value =
Value of All Equity Firm + PV Tax Shield
Example
All Equity Value = 600 / .10 = 6,000
PV Tax Shield = 400
Firm Value with 1/2 Debt = $6,400
6

8. U.S. Tax Code
 Allows corporations to deduct interest payments on
debt as an expense
 Dividend payments to stockholders are not deductible
 Differential treatment results in a net benefit to financial
leverage (debt)
7

9. U.S. Tax Code
 Personal taxes bias the other way (toward equity)
 Income from bonds generally comes as interest and is
taxed at the personal income tax rate
 Income from equity comes partly from dividends and
partly from capital gains
 Capital gains are often taxed at a lower rate and the tax
is deferred until the stock is sold and the gain realized.
 If the owner of the stock dies – no capital gain tax is paid
 On balance, common stock returns are taxed at lower
rates than debt returns
8

10. U.S. Tax Rates 2013
 Top bracket (over $450,000 for a married couple)
 Personal rates: 39.6%
 Capital gains: 20% [+3.8% investment income surtax for
high income earners] (holding period of <12 mos,
otherwise taxed at marginal income tax rate)
 If stock is held for less than 1 year capital gain is taxed at the
personal rate
 If stock is held for over 1 year capital gains tax is between 20
- 23.6%, but can be less if earn less that $15,000 in taxable
income.
9

11. Financial Distress
Market Value of The Firm
Maximum value of firm
Costs of
financial distress
PV of interest
tax shields
Value of levered firm
Value of
unlevered
firm
Optimal amount
of debt
Debt/Total Assets
10

12. Financial Choices
Trade-off Theory - Theory that capital structure is
based on a trade-off between tax savings and
distress costs of debt.
Pecking Order Theory - Theory stating that firms
prefer to issue debt rather than equity if internal
finance is insufficient.
11

13. M&M with taxes and
bankruptcy
 WACC now is more hump-shaped (similar to the
traditional view – though for different reasons).
 The minimum WACC occurs where the stock price is
maximized.
 Thus, the same capital structure that maximizes stock
price also minimizes the WACC.
12

14. Introduction to Covertible
Bonds
 A convertible bond = standard corporate bond with an option
(to buy the underlying equity of the company).
 Conversion feature allows holder of the bond to convert or
exchange the bond into a predetermined number of shares of
common stock (known as the conversion ratio).
 A convertible bond [“CB”] is sensitive to the interest rate
(corporate yield curve), [duration and convexity], the credit
spread over the treasury rate [credit risk] as well as the
volatility of the underlying equity.
13

15. Capital Structure and Financial
Distress
Costs of Financial Distress - Costs arising from
bankruptcy or distorted business decisions
before bankruptcy.
Market Value =Value if all Equity Financed
+ PV Tax Shield
- PV Costs of Financial Distress
14

16. Companies: Why Issue a CB?

Advantage of convertible is that can issue shares at conversion price
which is above “current” share price. This is also referred to “premium”
above current price;

Reduces dilution, [because of premium];

Access to investor segment normally precluded from equity risk,
attractive for many bond investors;

Lower straight coupon for issuer, [because of conversion option]

Less impact on P & L statement than equity because of tax deductibility
of interest on bond.
15

17. Investors: Why they Need to Buy CB?

A CB offers lower risk, [much less volatile than equity issue];

It has a built in protection in a risky market;

A CB has a higher running yield than a share dividend;
16

18. Types of Convertible
Bonds

Callable CB: A callable CB allows an issuer to buy back the bond some time prior
to the maturity at a pre-determined price. A “soft call” means that the issuer can
only call the bond if the price of the underlying stock is above the strike price by at
least a certain percentage;

Puttable CB: A puttable CB means that the investor can sell the CB back to the
issuer within a certain timeframe before the maturity of the CB at a certain price; a
put option raises the value of the CB;

Resettable CB: If the strike price is resettable, CB investors can gain additional
exposure to the equity component; if the price of the underlying stock falls, the
parity value of the CB falls as well and therefore by resetting the strike price, or
raising the conversion ratio, the CB’s parity value increases. (Example: CBs
issued by Japanese corporations in the mid-90s; these can be analyzed by path
dependent options)
17

19. Convertible Bond Pricing Model
CB
IV
Call
n
C
IV
t 1
i
r
d2
i)
t
(1
i)
t
s
Call
d1
(1
Par
N d 1 Se
ln S K
q (T
r
t)
d1
T
2
q
T
r (T
e
t)
N d2 * K
2 T
t
t
t
18

20. Conversion Ratio & Conversion Price

Conversion ratio: # shares of common stock that the bondholder will receive from
exercising the “call” option of CB; conversion privilege may extend for all or only some
portion of the bond’s life, and the stated conversion ratio may change over time (it is
always adjusted proportionately for stock splits and stock dividends).

EXAMPLE: JBB Corp issued a convertible bond with a conversion ratio of 25.32
shares. The par value of the bonds is $1000. This means that for each $1000 of the
par value of this issue the bondholder exchanges for JBB common stock, he will
receive 25.32 shares;
Stated Conversion Price = (Par Value of the CB)/(Conv Ratio)
= $1000/25.32
= $39.49 conversion price per share, above stock price at
issue date.
19

21. Strike Price
 Further suppose that the JBB convert has a maturity of 5 years, coupon
of 6% per annum (payable annually) and that the current risk free rate
is 2.5%; the CB has no dividend yield and the credit spread is zero;
 This will give the Investment Value (IV) of the CB as $1,162.60
(discounting for 5 years at the risk free rate of interest);
 The Strike Price, K of the CB is therefore equal to $45.92 and is found
out by:
K
= (CB’s Investment Value)/(Conversion Ratio)
= $1,162.60/25.32
= $45.92
20

22. JBB Convert Pricing
(See Spreadsheet for details)
21

23. Complications: Call Provisions

Almost all CB issues are callable by the issuer;

Typically there is a “non-call” period from the time of issuance. During this time if
stock goes above the conversion price by a sufficient premium, should
convert, otherwise hold convertible as allows for accrual and collection of fixed
coupons;

Some issues have a provisional call feature that allows the issuer to call the issue
during the non-call period if the stock reaches a certain price;
22

24. Convertible Valuation as Stock-plus Method

Can value a CB as a combination of an issuer’s stock, with a relatively high yield,
plus a European put option;

Instead of viewing a CB as a fixed income instrument with an embedded call
option, because of its convertible feature one can think of it as a stock with a yield
greater than its dividend, and discount this “higher” dividend at appropriate “rate”;

The Investment Value can be looked upon as a floor, [“put], or ability to sell a put
on company assets = to credit worthiness of assets coverage, [only is net asset
value of company covers value of outstanding corporate debt]; the stock value is
simply the conversion value (stock price multiplied by the conversion ratio) and the
put value represents the fixed income value of the convertible.
23

25. Binomial Tree for Convert
Pricing
24

26. Black-Scholes Framework for
Convert Valuation
25

27. Binomial Pricing Model continued
26

28. Convertible Greeks
e
q (T
t)
N d1
N d1 e
S
v
S
q(t T )
T
T
SN d1
2 T
t
t * N d1 e
q (T t )
q (T t )
e
rKe
t
r (T t )
N d2
qSN d 1 e
q (T t )
27

29. Greeks - Continued
K (T
t )e
r (T t )
N d2
CB
o
( OAS )
CB
( FX )
upsilon
u
CB
( RR )
CB
q
28

30. Zero Coupon Convertibles

The most bond like convertible is the zero coupon CB. The zero CB doesn’t pay
any cash interest but it carries a series of (synthetic) accreting put options;

In effect the buyer has paid for a series of put options with the coupon streams
that he has forgone;

The valuation of a zero CB must include a series of puts as well as series of calls
that both the buyer and the issuer can claim as their right (the basic long stock
plus long put model helps here);

The zero retains more bond like features at issue because the put option provides
a bond floor that is close to the current value and this bond floor (put) accretes
each year , helping to reduce the downside equity risk;
29

31. Sony Zero-coupon CB Trading History
(18 June 2003 – 18 June 2004)
see spreadsheet analysis
30

32. Sony CB
(see spreadsheet analysis)
y
mx
CB
c
0 . 49 * Parity
Trading
Black
797 ,527
0 . 49
Scholes
0 . 324
0 . 001685
vega
strike
spot
2821 . 85
6 ,177 . 16
3980
31

33. CB Asset Swap
Trader / Investor
Convertible
Bond
CB Call
Option
CB Investment Value
Bond Buyer
Broker
Coupons
Coupons
LIBOR + Spread
Swap Trader
32

34. Example of a Vanilla Swap.
Break even rate = 5.5%
1
Effective Dates
3/1/Y1
9/1/Y1
3/1/Y2
9/1/Y2
3/1/Y3
9/1/Y3
3/1/Y4
* (LIBOR/2)($10,000,000)
** (.055/2)($10,000,000)
2
LIBOR
0.045
0.050
0.055
0.060
0.065
0.070
3
Floating-Rate
Payer's Payment*
4
Fixed-Rate
Payer's Payment**
5
Net Interest Received
by Fixed-Rate Payer
Column 3 - Column 4
6
Net Interest Received
by Floating-Rate Payer
Column 4 - Column 3
$225,000
$250,000
$275,000
$300,000
$325,000
$350,000
$275,000
$275,000
$275,000
$275,000
$275,000
$275,000
-$50,000
-$25,000
$0
$25,000
$50,000
$75,000
$50,000
$25,000
$0
-$25,000
-$50,000
-$75,000
33

35. Convertibles CDS
Spread Fee
Zero Payment
CDS Buyer
No credit event
CDS Seller
Credit event
Contingent Settlement
34

36. Delta and Volatility

Convertibles with very little or no call protection remaining can be subject to a
perverse effect of increased volatility;

As vol increases it has the effect of reversing the time value of an option and as
volatility decreases it has the effect of increasing the time value of the option;
2
Time
Log Trigger
Log Parity
* NORMSINV
Trigger
Call
Parity
Call
* tradingDay s
probabilit y
* 1
dt * NORMSINV
year
prob
35

37. Example

If the CB has no call protection remaining and will only be called to force conversion, then the
trader can estimate how much above the call price the parity level (trigger) should move before
it may be called with a given probability and expected volatility.

For example, if the trader has determined that the parity level must be 120% of the call price
for the company to safely call the issue, then he can estimate – using the previous formula –
the amount of time premium that should be built into the CB’s embedded option;

For example: how much time will it take with an 80% probability for the trigger level to be
reached for the CB with a parity of 102 and a trigger level of 120 and a 3-month annualized vol
of 40%;
2
Time
Log (120 )
Log (102 )
0 . 40 * 0 . 84162
Time

23 % * 255
0 . 1625
2
0 . 3366
59
Time value is equal to 23% of the number of trading days in a year or roughly 59 trading days
36

38. Example - continued

A trader can work with this formula in another way: say, a callable convertible with
a 30-day call notice period has a parity level of 102 and a 3 month vol of 60%. The
trader wants an 80% probability (of the trigger happening); then what would be the
trigger level?
Trigger
102 * (1
( 0 . 60 *
30 255 * 0 . 84162 ))
119 . 66
37

39. Delta Neutral Arbitrage using Leverage
(see spreadsheet)
38

40. Example - Amazon

Amazon CB combined with the company’s straight debt was an interesting trade in
March, 2000; Amazon 4.75% CB due 2009 was trading at 40% of par with a yield
of over 19% (but with a very little value assigned to the embedded call option);

At the same time the 10% straight bond due 2008 was trading at 58% of the par
with a YTM of 15% (the bond did not actually pay a coupon of 10%, since it was
zero coupon with a clause to start paying cash interest payment on March
1, 2003);

Traders were long 145 CB at 40.00 and short 100 straight high yield at 58 thus
creating an equal dollar offsetting investment netting to zero;

By mid-July 2000 the Amazon CB traded at 54 (gain on the long CB) and the
straight high yield traded at 66 (loss on the short position) thereby realizing a net
gain on $12,300 on an investment of zero.
39

41. Learning Outcome
 MM is not actually relevant to most corporate situations, in that
Taxes play an important role in planning;
 While the equilibrium capital structure may be defined by the point
at which increasing leverage increases risk of default such that
“firm” value decreases, for most practical purposes, a stable
capital structure usually implied a ma debt load = (1-Tax Rate);
 Convertible bonds are actually quite useful for maximizing value if
used in conjunction with an effective asset development plan.
Provide cheaper funding, though eventually will convert to
expensive equity;
 In the end it is always about identifying and properly risk managing
positive cash flow projects to be brought on stream.
40

42. Thank you
 For more information about Capital structure
consulting, please contact Philip Corsano on:
 206 384 0069, pcorsano@gmail.com
41