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CHAPTER10-fins5536代写
时间：2023-07-25
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CHAPTER 10: AGENCY AND CORPORATE DEBT
SECURITIES
CORPORATE DEBT MARKET
Credit Risk of Issuers (Fabozzi, F.J., Bond Markets, Analysis
and Strategies, 3rd Ed. Prentice-Hall, p. 143).
Summary of Corporate Bond Ratings Systems and Symbols
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Moody’s S&P Fitch D&P* Definition
Investment Grade: High Creditworthiness
Aaa AAA AAA AAA Best quality
Aa AA AA AA High quality
A A A A Upper medium grade
Baa BBB BBB BBB Lower medium grade
Distinctly Speculative: Low Creditworthiness (Junk Bonds)
Ba BB BB BB Low grade, speculative
B B B B Highly speculative
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Predominantly Speculative: Substantial Risk or in Default
Caa CCC CCC CCC Substantial risk, in poor
standing
Ca CC CC May be in default,
extremely speculative
D DD or D DD Default
*Note: D&P stands for Duff & Phelps.
Commercial Paper
These are short-term (usually less than 270 days) zero-
coupon corporate bonds. (See text.)
10 - 4
Contractual Provisions of Corporate Debt Contracts
Call features: Enable issuers to buy back the debt, after a
call protection period, for the call price. Call (exercise)
price is typically = par + coupon, but declines to par over
time. We can find the value of the call option using, e.g.,
Black’s model, etc.
Rationale:
Effect on bond price:
NOTE: junk bonds are more likely to have call features.
10 - 5
Sinking fund provisions: Require the issuer to retire an
outstanding debt issue periodically. Some bonds may be
either retired at par or through open market purchases.
Rationale:
Effect on bond price:
Putable bonds: Provide bondholders with the right to sell the
debt back to the issuer.
Effect on bond price:
10 - 6
Convertible bonds: Provide bondholders with the right to
convert bonds into a specified number of shares of
common stock.
Effect on bond price:
Bonds with warrants: [Note: this is the U.S. definition of a
warrant. In Australia this would be called a “company
issued call”.] The buyer of debt is also provided with
warrants that give the right to buy the issuer’s shares of
common stock. (Similar to a call option on the firm’s
stock, sold to the bondholder.)
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Effect on bond price:
Agency Problems and callable debt
Asymmetric Information: If a firm has favourable info, they
can issue callable debt; later, when the info is released,
the price of the firm’s debt rises, and they can call it back;
and assuming their credit rating increases, they can issue
new debt at a lower interest rate. Note also: when a firm
issues more equity (i.e., a seasoned equity issue) it’s
taken as a signal that its stock is overpriced, and so the
price drops. Another way of thinking about this is that
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issuing equity is “too easy”. If you issue debt, you’re
saying, We will pay this back or go bankrupt. When you
issue equity, you can only say that you intend to give
them dividends at times.
Convertible debt and warrants are ways of getting around this
effect.
Risk-shifting: Shareholders have an incentive to take risks at
the expense of bondholders. (See the option
interpretation below.) If debt is convertible, there’s more
incentive to keep the price of bonds high.
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Protective Covenants:
- Aggregate debt limitations: Firm can’t let its total debt rise
above a certain level.
- Restrictions on dividend payments.
- Restrictions on mergers, consolidations, and the sale of
assets.
- Credible third party guarantees and credit enhancements.
- Minimum standards on working capital levels.
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ASIDE: In the US, some people are pushing for fewer bond
features. This would mean more standardization, which
could lead to more liquidity.
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Leveraged Buy-Outs and Management Buy-Outs
Here, the buyer issues debt (junk bonds) in order to buy
controlling interest in the firm.
Both LBOs and MBOs lead to high debt levels.
But, MBOs also give managers large numbers of shares.
Pro:
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Con:
THEORIES OF CORPORATE-DEBT PRICING
The Merton Model (used by KMV)
Suppose a firm has total equity worth $S, and zero coupon
bonds with a face value of $F, and a market price of $B,
and maturity date T. Then, the value of the firm is
V = S + B
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At maturity, if the value of the firm is VT > F, then the
bondholders receive F, while the shareholders receive the
“surplus”, VT F. If, at maturity, VT < F, then the firm is
in default. That means that the bondholders’ payoff is VT,
and the shareholders receive nothing. We can summarize
this as follows:
Bondholders’ payoff = min{F, VT},
Shareholders’ payoff = max{0, VT F},
where the shareholders’ payoff is consistent with their
limited liability.
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The second equation is the payoff for a (European) call
option on the value of the firm with strike price F, i.e.,
buying the firm’s equity is equivalent to buying a call
option on the value of the firm. So, the price of equity is
given by the Black-Scholes equation:
S = VN(d1) Fer(T t)N(d2),where
d1 =
21
2ln( ) ( )( )V
V
V r T t
F
T t
,
d2 = d1 V T t ,
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T t is the time to maturity of the bond, and V is the
volatility of V.
ASIDE for those who are interested.
More on The KMV approach (ref: Caouette, Altman, and
Narayanan, Managing Credit Risk: The Next Great
Financial Challenge, Wiley, 1998.
1. Solving for the unknowns, V and V.
Note that our equation for S as a call option on V has two
unknowns: V and = V, the volatility of V. So, at the
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moment we have one equation and two unknowns. For
the second equation, we consider the volatility of equity,
which I’ll denote by S. Now, S is a function of V, so
let’s write S = c(V, t), where c is the usual call function,
given by Black & Scholes. Note that c
V
= N(d1). Note
also that we’re assuming that V is a lognormal process;
that is, (in a risk-neutral world) it satisfies the stochastic
differential equation
dV = rVdt + VVdZ,
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where Z is a standard Wiener process (also known as
Brownian motion). From a mathematical result known as
Itô’s lemma (which is similar to Taylor’s theorem) we
have
dS =
2
2
2
1 ( )
2
c c cdV dV dt
V tV
= N(d1)VVdZ + (…)dt.
On the other hand, when we think of the volatility of equity,
we write the stock price as
dS = SSdZ + SSdt.
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Equating the coefficients of the dZ term, we find
1( )S VS N d V , or
1( )
S
V
S
N d V
.
Because we can easily measure the volatility of equity, S,
this gives us a second equation, so now we have two
equations and two unknowns.
2. The distance from default.
It is common to define the distance from default to be “the
number of standard deviations that the asset value, V,
must drop in order to reach the default point”.
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In Caouette, et al., they define this distance from default as
(expected mkt value of assets default point)
(expected mkt value of assets)( )V
.
Alternatively, some authors write the distance from
default as
ln[expected mkt value of assets] ln[default point]
V
.
(See Schönbucher, p. 277.)
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However, because we assume that V is lognormal, we can
show by analogy to Black & Scholes’ result, that the
(actual) probability of default is N( 2d ), where
2d
=
21
2ln( / ) ( )( )V V
V
V F T t
T t
,
(while the risk-neutral probability of default would be
N(d2), where we replace V with the risk-free rate). So,
some people define 2d
as the distance to default.
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Merton’s Model and the value of Debt
We’ve seen how Merton’s model values equity. There are
two ways of looking at the value of the firm’s debt, B.
1. First, note that
B = V S
= V[1 N(d1)] + Fer(T t)N(d2)
= Fer(T t)[ V
Fe r T t ( )
N(d1) + N(d2)]
since 1 N(d1) = N(d1). Note that Fer(T t) represents
the price of an otherwise identical risk-free bond.
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Next, we let y(t, T) = the yield to maturity of the corporate
discount bond at date t, and call (t) the default (or
credit) spread at time t: (t) = y(t, T) r.
Then B = Fey(t, T)(T t) = Fe[r + (t)](T t) = Fer(T t)e(t)(T t)
= Fer(T t)[ V
Fe r T t ( )
N(d1) + N(d2)].
Solving for (t) gives
(t) = 1
T t
ln[ V
Fe r T t ( )
N(d1) + N(d2)].
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NOTE: the credit spread depends on the ratio d = Fe
V
r T t ( )
known as the leverage factor. As d increases, the credit
spread increases.
Also, as Vincreases, the credit spread increases. Recall that
we argued earlier that shareholders will take risks at the
expense of bondholders. Because the shareholders have
effectively bought a call option, it is in their interests to
increase V. (Why?) This leads to increased credit
spreads, i.e., lower bond prices for the bondholders.
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FACT: In general, if an asset’s volatility increases, the value
of a call option on that asset also increases. In addition,
if an asset’s volatility increases, the value of a put option
on that asset increases.
2. The second way to interpret the value of debt is to write
the bondholders’ payoff as
min{F, VT} = F + min{0, VT F} = F max{0, F VT}.
The last term above is the payoff of a put option on the value
of the firm. So, buying a risky bond is equivalent to
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buying a risk-free bond and selling a put on the value of
the firm. The put is sold to the equity-holders.
How does an increase in V affect the debt-holders?
Finally, note how put-call parity relates to this model:
p + V = c + FerT,
or solving for V:
V = c + FerT – p = S + B.
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Problems with this model:
- The sizes of the credit spreads implied by this model are too
small.
- Coupon-bonds are hard to value using this model.
- Bankruptcy may occur before the bond’s maturity (e.g., if a
coupon payment can’t be met.)
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Subordinated Corporate Debt (Black and Cox model)
Suppose a firm has two classes of zero-coupon bonds. The
senior debt has face value F1, and current market value
B1, while the subordinated (junior) debt has face value F2,
and market value B2. (What’s the difference between the
two types?)
Now, V = S + B1 + B2.
Assume all debt matures at date T. Here are the possible
outcomes:
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Outcome/Payoffs: Senior Debt Junior Debt Equity
VT > (F1 + F2) F1 F2 VT (F1 + F2)
F1 < VT < (F1 + F2) F1 VT F1 0
VT < F1 VT 0 0
(These are called absolute priority rules.)
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Financial Distress
In the event of financial distress, i.e., a firm’s inability to pay
principal or interest, there are three ways it can be
handled:
1. Assets can be liquidated, i.e., sold in order to pay
debtholders.
2. The firm can renegotiate loan payments, and either
- reduce coupons and/or principal obligations
- increase the maturity of the debt, or
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- ask debtholders to accept equity in lieu of debt
payments.
3. The firm can issue new debt in order to pay the old debt.
Factors Affecting the Probability of Default
(Reference: Zmijewski, J. Accounting Research, 1984, pp
59-82.)
Profitability = ratio of net income to total assets.
Liquidity = the ratio of current assets to current liabilities.
Leverage = the ratio of total debt to total assets.
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The author uses a probit model to estimate the probability of
default.
Other Factors Affecting Bond Ratings
Interest coverage = EBIT/(interest payments),
Total Debt/Total Capital,
Cash Flow/Total Debt,
Return on Assets,
Current Ratio = current assets (i.e.,maturing within one year)
current liabilities (i.e.,due within one year)
.
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Recovery Rates
In most defaults bondholders recover some portion of their
investment. This recovery may take the form of cash,
securities (debt or equity) or occasionally assets of the
business.
The recovery rate = the percentage of the par value of the
security that is recovered by the bondholder.
Moody’s recovery rates 1989-1996 (Reference: Das, Credit
Derivatives, Wiley, 1998, p. 204.)
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Classes of debt Recovery rate (%) St. dev. (%)
senior secured bank debt 71 21
senior secured public debt 63 26
senior unsecured public debt 48 26
senior subordinated public debt 38 25
subordinated public debt 28 20
junior subordinated public debt 15 9
(secured debt refers to debt with collateral.)
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A Common Pricing Model (Not in the text)
Consider a defaultable zero coupon bond with maturity, T,
and face value $1. The risk-neutral probability of default
is Q(T), and the recovery rate in the event of default is
0 < RD < 1. We also denote the writedown (also known
as the loss given default) by w = 1 – RD. In other words,
the defaultable bond has payoff at time T given by
Payoff =
1 if nodefault
if defaultDR
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=
1 with prob. 1 ( )
with prob. ( )D
Q T
R Q T
.
Suppose the price of an otherwise identical risk-free bond is
denoted b(0, T). Then, the price of a defaultable zero
coupon bond is given by (assuming the probability of
default and the write-down are independent of the risk-
free rate)
B*(0,T) = b(0,T)[1 – Q(T) + RDQ(T)]
= b(0,T)[1 – (1 – RD)Q(T)]
= b(0,T)[1 – wQ(T)].
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Example: A 5-year zero-coupon bond has a (risk-neutral)
probability of default equal to 5%. Its recovery rate is
likely to be 60%. The 5-year risk-free rate is 7% with
annual compounding. Find the yield on the risky bond;
then find the credit spread.
b(0, 5) = (1.07)5 = 0.71299;
B*(0, T) = 0.71299(1 0.400.05) = 0.712990.98
= 0.69873;
But, B*(0, T) = (1 + y)5; y = B*(0, T)1/5 1 = 7.43%.
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The credit spread is then y – r = 7.43% 7.00% = 43 basis
points.
Note that we sometimes write 0 (0, )B T for the price of an
otherwise equivalent risky bond having 0DR , or w = 1.
It is not hard to see that
* 0(0, ) (0, ) (0, )DB T R b t wB T ,
where (0, )b t is a risk-free bond, as usual; if there is a
default, the payoff is DR , if there is no default, the payoff
is 1DR w .
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The price of a coupon bond with coupon payment C and N
payments remaining is just
P = CB*(0, t1) + CB*(0, t2) + … + (100 + C)B*(0, tN).
Note that to calculate each zero coupon bond price we
need to find the risk-neutral probability of default, Q(ti),
corresponding to that cash-flow.
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HYBRIDS
Convertible Bonds
Convertible debt can be exchanged for equity. Convertible
debt usually includes a call provision. Issuers tend to
have moderate to low credit ratings.
Rationale: Issuing equity is costly (equity prices drop.)
Issuing debt may lead to bankruptcy. Convertible debt is
a sort of middle ground. The call provision allows the
firm to force conversion when necessary.
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To see this, suppose that, if a bond is converted to equity, you
receive n shares. Now suppose B > X, and nS > X, (here,
X is the call exercise price) and the firm exercises their
call option on the debt. What would a bondholder do?
10 - 41
Convertible Debt and Dilution
Some definitions:
Conversion ratio: the number of shares into which a bond
may be converted.
Dilution: when convertible debt is converted into equity, the
company actually issues more equity. If the total value of
the firm’s equity doesn’t change, but there are more
shares outstanding, the price per share must drop.
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Notation
N = # of shares outstanding (before conversion)
M = # of convertible bonds outstanding
n = # of shares each bond can be converted into
(conversion ratio),
TE = value of the firm’s equity at time T.
If all bonds are converted, then there are N + nM shares
outstanding. So, for the original shareholders, their price
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per share before conversion = TE
N
= beforeTS . The price per
share after conversion becomes (since beforeT TE NS )
afterTS = (1 )
before beforeT
T T
E N S S
N nM N nM
< beforeTS .
The total value of the original equity is afterTN S =
(1 )T T
N E E
N nM
(after conversion) and the value of
the newly converted equity is afterTnM S = T
nM E
N nM
=
TE .
10 - 44
(Note: value of original equity + value of new equity = TE ).
The ratio nM
N nM
is called the dilution factor associated
with the conversion. (Here, is the dilution factor from
the bondholders’ point of view; that is, if all the bonds
are converted to equity, the total value of that new equity
is TE .) So, the original shareholders now own (1 ) TE ,
and the new shareholders (i.e., the bondholders) own TE .
10 - 45
The Valuation of Convertible Debt
We’ll discuss two approaches to valuing convertible debt.
The first approach is called a structural approach. This
approach is fairly intuitive, but harder to implement. The
second approach is called the reduced form approach.
We’ll have a spreadsheet that uses the reduced form
approach.
The Structural Model
The total value of all the convertible bonds at time T is the
maximum of their value as straight bonds or their
10 - 46
conversion value. The value of the straight bond at date
T is min[F + c, VT] (we’re including the possibility of
default, and TV is the value of the firm at date T).
Here F = face value of all bonds, and c = the coupon
payment. Recall that nM
N nM
is the dilution factor.
Then, the value of the convertible bond at date T is
max{min[ , ], }T TF c V E .
value if don’t convert value if convert
QUESTION: would you convert if there were a default?
10 - 47
ASIDE: If we assume that the convertible debt is the firm’s
only debt, then if every bondholder converts to equity, the
total value of equity becomes the total value of firm; i.e.,
E = V. So, we can construct the binomial tree by
assuming that V is lognormal. In this case, the payoff at
maturity can be written
max{min[ , ], }T TF c V V .
10 - 48
ASIDE: If this bond is also callable with strike price X, then
the payoff is given by,
max{min[ , , ], }T TF c V X V .
So, for example, if the value of the firm is high, so that
min[ , ]TX F c V , and TX V , then the firm can
exercise their call option and force conversion.
We now work backwards through a binomial tree to find the
current value of the convertible debt. In earlier periods,
say at time s < T, the value is given by
max{ , }s sV P c ,
10 - 49
where Ps + c is the value of not converting at date s,
(given by risk-neutral pricing) and TV is the value of
converting at date s. Note that c may be equal to zero at
date s. This is an American style convertible.
For example, at time T 1, if VT = uV in the “up” state and
VT = dV in the “down” state, and similarly for VT, then the
value of convertible debt is given by
1TP =
1
1 R
[pmax{min[F + c, uV], uV}
+ (1 p)max{min[F + c, dV], dV}],
10 - 50
where again, p = the risk-neutral probability of an “up”
state. This would be the value of the convertible bond if
the decision to convert was of the European type. For an
American-style convertible, the price of the convertible
bond would be
1max{ , }TV P c ,
where 1TV V , and again, c may be equal to zero at date
1T .
Much like American options, we can now work backwards.
QUESTION: what else should be random in this model?
10 - 51
The Reduced Form Approach
In this section we discuss a trinomial tree for convertible
debt which is based on Hull, Section 27.4. Hull points
out that this model is widely used in industry. (While
technically a trinomial tree it basically has the form of a
binomial tree.)
As mentioned above, convertible debt is usually associated
with firms with relatively low credit ratings, so it is
important to model default. The approach we will
10 - 52
consider here is sometimes known as the reduced form
approach, makes it easier to model default.
For the reduced form approach, we assume that for each
time interval, t , there is a small probability,
1 te t , of the firm defaulting. The parameter
is known as the risk-neutral hazard rate, or default
intensity. We can then construct a trinomial tree for the
stock price and the calculated bond price. If a default
occurs, the stock price drops to zero, and the bond price
becomes equal to some constant percentage of the bond’s
10 - 53
face value, which we denote by DR , with 0 1DR ,
often called the recovery rate.
More notation: (1) Write r for the risk-free rate, (2) is the
stock’s volatility, and (3) we let q equal the stock’s
dividend yield; that is, we assume that the stock pays a
small dividend yield, tS q t , every t years. This
approach is used for options/futures on indices. (It is
easier than determining the actual date of a dividend
payment. Technically, q is a continuously compounded
yield as a percentage of the current stock price.)
10 - 54
The risk-neutral probabilities for the trinomial tree are
( )r q t t
u
e dep
u d
,
( )t r q t
d
ue ep
u d
, 1 tfp e
,
where
2( ) tu e and 1d
u
. Here, up represents the
risk-neutral probability of the stock price going up, dp is
the risk-neutral probability of the stock price going down,
and fp represents the risk-neutral probability that a
default occurs at that step of the tree.
ASIDE: First, note that
10 - 55
u d fp p p
( ) ( )
1
r q t t t r q t
te de ue e e
u d u d
( )(1 )
t t tue de u d e
u d
( ) ( ) ( ) 1
t tu d e u d u d e u d
u d u d
.
Next, note that, using these probabilities, we have,
t tE S 0u t d t fp uS p dS p
10 - 56
( ) ( )r q t t t r q t
t t
e de ue euS dS
u d u d
( ) ( )r q t t t r q t
t t
ue ude due deS S
u d u d
( ) ( )
( )
r q t r q t
r q t
t t
ue de S S e
u d
,
which would be the expected value in a risk-neutral world
on a stock paying dividend yield, q.
One final calculation: suppose a short-term (i.e., maturity
date is t ) default-risky zero coupon bond pays $1 in
10 - 57
states u and d, but $0 in the default state; that is, assume
the recovery rate is equal to zero. We denote this bond
price by 0 (0, )B t . Then,
0 (0, )B t 0r t u d fe p p p
( ) ( )r q t t t r q t
r t e de ue ee
u d u d
t t
r t ue dee
u d
( )r te .
10 - 58
So, ends up being the credit spread when the recovery
rate is equal to zero.
Conversely, if we want to derive up , dp , and fp , we start
with the fact that 1 tfp e
. Next, the fact that
1u d fp p p implies that
t
u dp p e
. We also
require that ( )r q tu dp u p d e
. These two equations
now allow us to solve for up and dp :
( )r q te u dp u p d
( )tu up u e p d
10 - 59
( ) tup u d e d
,
which implies that
( )r q t t
u
e dep
u d
, as desired.
Let’s assume that the bond is callable as well as convertible.
As Hull points out, the bond price at each node is of the
form,
ˆmax min , ,B B K nS ,
where
Bˆ = value of bond if it’s neither converted nor called,
10 - 60
= ( ) r tu u d d f Dp B p B p R FV e
,
(FV = the bond’s Face Value); at maturity, Bˆ FV c ,
where c is the final coupon payment.
K = the call price,
nS = value if conversion takes place; here n is the
conversion ratio, and S is the stock price per share at that
node.
Because DR FV is a constant, we really only need to
construct a binomial tree.
10 - 61
Note that we have ignored coupon payments, for these
calculations, and we will ignore the dilution effect. The
spreadsheet includes coupon payments.
Example: A 9-month zero coupon bond has a face value of
$100. It can be exchanged for 2 shares of the company’s
stock at any time during the 9 months, the initial stock
price is $50, its volatility is 30% per annum, and the stock
pays no dividends. Assume it is also callable for $113 at
any time. The hazard rate is 1% , and the risk-free
rate for all maturities is 5%, with continuous
10 - 62
compounding. In the event of a default, the bond is worth
$40 (i.e., the recovery rate is 40%). Find the price of the
convertible bond.
Here, we have, 9 /12 3 / 4t T , S = 50, n = 2, K = 113,
0.01 , 0.40DR . So,
2( ) (0.09 0.01)3/4 1.2776tu e e , d = 0.7827,
( ) 0.05 0.75 1.0382r q te e , 0.01 0.75 0.9925te e
( ) 1.0382 (0.7827)(0.9925) 0.5281
1.2776 0.7827
r q t t
u
e dep
u d
10 - 63
( ) (1.2776)(0.9925) 1.0382 0.4644
1.2776 0.7827
t r q t
d
ue ep
u d
1 0.0075tfp e
.
Next, 50(1.2776) 63.88Su , 50(0.7827) 39.135Sd ,
and so, since ˆ ˆ 100u dB B at the bond’s maturity, T,
uB ˆmax min , ,uB K nSu
max min 100,113 ,2 63.88
max min 100,113 ,127.76 127.76 ,
10 - 64
dB ˆmax min , ,dB K nSd
max min 100,113 ,2 39.135
max min 100,113 ,78.27 100 .
Thus, at date t = 0,
Bˆ ( ) r tu u d d f Dp B p B p R FV e
(0.5281)127.76 (0.4644)100 (0.0075)40 (0.9632)
= 110.01.
So, the price of the bond is
10 - 65
B ˆmax min , ,B K nS
max min 110.01,113 ,100 110.01 .
More Terminology
One more term that is often used with convertible debt is the
conversion price which is equal to
ratioconversion
bondpervalueface .
Note that the face value per bond is
M
F = 1000 (usually) and
the conversion ratio is n, so we can write
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conversion price =
ratioconversion
bondpervalueface =
nM
F .
Example: Suppose the face value per bond =
M
F = 1000,
and the conversion ratio is n = 40 shares. Then the
“conversion price” =
40
1000 = $25.
Intuitively, suppose we assume that the total of all the bond
prices remains constant total face value = F. In this
case the option to convert debt to equity becomes
equivalent to the option to (recall, M = number of bonds)
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buy n shares for each bond, where the price per bond =
M
F .
For each bond, then, it is as if you own n call options on
equity with exercise price X =
nM
F . This “exercise price”
is the conversion price.
10 - 68
Debt with Warrants
Warrants (this is U.S. terminology) are like call options on
the firm’s equity, except that they are sold by the firm,
and when exercised, the firm issues additional shares of
the common stock.
Let N = # shares of common stock before warrants’ exercise
M = # of warrants that can be exercised at date T,
S = market price of a share of common stock,
W = market price of each warrant,
X = exercise price
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E = total value of firm’s equity.
QUESTION: when would issuing a warrant be
a better signal?
a worse signal?
Before the warrants are issued, the total value of equity is NS.
After the warrants are sold, the total value of the firm’s
equity becomes (still at t = 0)
E = NS + MW.
If at time T, all warrants are exercised, the total value of
equity becomes ET + MX. This value is distributed
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among the new number of shares outstanding, N + M.
So, the share price immediately after exercise becomes
afterTS = T
E MX
N M
.
If exercised, the payoff to the warrant holder is therefore,
afterTS X = T
E MX X
N M
= TE MX
N M
MN
MNX
)(
= TE NX
N M
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= ( )TEN X
N M N
= N
N M
( beforeTS X),
so the payoff to the warrant holder can be written
max{ ,0}TEN X
N M N
.
NOTE: TE
N
= beforeTS is the price per share just before the
warrant is exercised, so that the exercise decision is the
same as for a call option. So, at time of exercise, the
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payoff is equal to N
N M
call options (i.e., there’s a sort
of “dilution” to the warrant holder).
One other difference is that ET is the future (uncertain) value
of equity assuming that the warrants were sold (at date
t = 0). In terms of today’s share price (at the time of the
sale of the warrants) we have
E = NS + MW
where S is the stock price (before the sale of the warrants)
and W is the warrant price, so that (at t = 0)
10 - 73
E
N
= S + M
N
W.
So, we can use the Black and Scholes formula to calculate
the warrant price if
1. The stock price S is replaced by S + M
N
W.
2. The volatility, , is the volatility of the value of the
shares plus the warrants, not just the shares.
3. The formula is multiplied by N
N M
.
10 - 74
When these adjustments are made, we end up with a formula
for W as a function of W. To see this, let c(S) be the usual
Black-Scholes formula for a European call option. For
the warrant, we have W = N
N M
c(S + M
N
W). Here, W
changes the price per share, and we need the price per
share to calculate W. This can be solved numerically.
10 - 75
ASIDE: we’ve seen a sort of “dilution” to the call holder, but
is there dilution in the stock price when a warrant is
exercised? We exercise the warrant if TE X
N
. Now,
afterTS = T
E MX
N M
<
( )TT
EE M
N
N M
= ( )TE N M
N N M
= TE
N
= beforeTS .
So, yes, the stock price drops, and the size of the drop
depends on X:
10 - 76
ST = T
EM X
N M N
;
i.e., the lower the exercise price, the greater the drop in
the stock price. This is drop understandable, though,
because at the time the warrant is sold (t = 0), there is an
increase in the value of equity, and the increase at that
date is higher for lower exercise prices. (On the other
hand, as discussed in class, a low exercise price may send
the market a bad signal; but that signal is unrelated to the
simple option pricing/dilution effect.)
10 - 77
Here are a few definitions that were in the news during the
global financial crisis.
Credit Default Swaps (CDSs)
A Credit Default Swap basically provides insurance against
defaults on credit-risky coupon bonds. If a particular
firm defaults, then the CDS pays L(1 – RD) = Lw, where L
is the notional principal on the firm’s bond, RD is the
recovery rate, and w is the write-down, also known as the
loss given default. (If there is no default then the CDS
10 - 78
makes no payment.) Note that the timing of the default is
random.
One unusual feature of this “default insurance” is that you do
not pay for it in a lump sum; rather, you pay for it with
periodic premium payments. If the bond defaults, then
the premium payments stop—at the random default time.
Note that CDSs are zero sum games, much like standard
options and futures.
CDSs are often used in empirical corporate finance. As
explained in Zhang, et al. (2009),
10 - 79
Compared with corporate bond spreads, CDS spreads
have two important advantages. First, CDS spreads
provide relatively pure pricing of the default risk of the
underlying entity and are typically traded on
standardized terms. However, bond spreads are more
likely to be affected by differences in contractual
arrangements, such as differences related to seniority,
coupon rates, embedded options, and guarantees.
Second, as shown in Blanco, et al. (2005), and Zhu
(2006), CDS spreads tend to respond more quickly to
changes in credit conditions in the short run, which
may be partly due to the absence of funding and short-
sale restrictions in the derivatives market.
10 - 80
ASIDE: Pricing a CDS is a bit more complicated than other
assets that we’ve seen. Suppose we buy the CDS. First,
we need to price the cashflow we receive at the random
default time. In the simplest model, the price of this
cashflow is
0
(1 ) (0, ) ( )
T
DL R b t dQ t ,
where Q(t) is the risk-neutral probability that firm has
defaulted by date t. This assumes that the risk-free rate
and the default probability are non-random functions of t.
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(If the short rate, r, is constant, and if we also assume that
the probability of default is given by Q(t) = 1 et,
where is known as the default intensity, then we can
calculate these integrals explicitly.)
Now, let us discuss our premium payments. We pay for this
insurance (i.e., the CDS) with premium payments, $Ls
per year, for as long as the risky bond survives. Note that
s, known as the CDS rate or CDS spread, is an annual
percentage of the face value, and so looks much like an
interest rate. (More on that below.) Let’s assume that
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there are m premium payments per year, so that the
periodic payment is L
m
s . Suppose the bond has N
payments, made at dates T1, T2,…, TN. Again, in the
simplest model, the price of the premium payments is
1
(0, )[1 ( )]
N
i i
i
sL b T Q T
m
,
where 1 – Q(Ti) is the probability that the bond survives
until time Ti; i.e., if the bond defaults, we stop paying the
premium.
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In addition, it is usually assumed that, when the default
occurs, we make one more premium payment. If the final
cashflow (i.e., the default) occurs at date t, with Tk < t <
Tk+1, then it is accrued as
L
1
k
k k
t Ts
m T T
,
where L
m
s is the periodic premium payment and
1
k
k k
t T
T T
is the fraction of the period that has elapsed since the last
payment. This simplifies further if we assume that
10 - 84
Tk+1 Tk =
1
m
. This final cashflow then becomes
Ls(t – Tk).
Let us ignore the accrual term for the moment. Its price is
slightly more complicated than those given above. Now,
the question is, “What should s be?” The CDS rate, s, is
chosen so that the value of the premium payments is
equal to the value of the cashflow received in the event of
default; that is,
10 - 85
1
(0, )[1 ( )]
N
i i
i
sL b T Q T
m
=
0
(1 ) (0, ) ( )
T
DL R b t dQ t ,
or,
s = 0
1
(1 ) (0, ) ( )
1 (0, )[1 ( )]
T
D
N
i i
i
R b t dQ t
b T Q T
m
,
or more generally,
10 - 86
s = 0
1
(1 ) (0, ) ( )
1 (0, )[1 ( )]
T
D
N
i i
i
R b t dQ t
b T Q T accrual
m
.
It can be shown that s behaves much like a credit spread; that
is, the spread between the yield on the firm’s risky debt
minus the risk-free rate.
10 - 87
Collateralized Debt Obligations (CDOs)
A CDO begins with a portfolio of assets, such as corporate
bonds or mortgage-backed securities (or one could even
use the premium payments from CDSs), and then this
portfolio is divided into tranches. The difference is that,
with a CDO, the tranches are based on defaults.
With a CDO, the first tranche incurs all the defaults until that
tranche becomes worthless. This is known as the equity
tranche, and is very risky. Often it is kept by the firm
that organized the CDO. As an example, suppose the
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portfolio consists of 100 corporate bonds, and suppose
the first tranche becomes worthless after seven defaults.
The second tranche incurs no losses until the first tranche
is worthless. Again, as an example, the second tranche
may become worthless after four more defaults occur,
and so on. The last tranche is usually rated AAA.
Typically, the last tranche is unaffected until, say, 20
firms default, and only then does it begin to lose value. It
never becomes worthless—all 100 firms would have to
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default in order for it to become worthless—but it can
lose some value if 20 or more firms default.
Even the AAA tranches performed poorly during the GFC.
“A chart … prepared by the Senate Permanent
Subcommittee on Investigations showed that 91 percent
of the triple-A-rated subprime residential mortgage-
backed securities issued in 2007, and 93 percent of those
issued in 2006, were subsequently downgraded to junk
status.” (Source: “All the Devils are Here”, Bethany
McLean and Joe Nocera, footnote on p. 305.)
10 - 90
The ratings agencies played an important role here, as “[t]he
triple-As were easy to sell because investors around the
globe that were legally confined to conservative
investments, or didn’t want to hold the capital against a
higher-risk investment, embraced their higher yield
relative to their super-safe rating. The triple-B and B-
minus tranches were harder to sell….” (M&N p. 121.)
In many cases these riskier tranches, known as mezzanine
tranches, got recycled into CDO squareds, which were
CDOs composed of tranches of other CDOs—
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particularly, tranches of other CDOs that nobody would
buy in the first place.
There is no closed form solution for pricing CDOs. One
must use Monte Carlo simulations.
Note that CDOs are not zero sum games. Like a portfolio of
stocks, if their prices go down, most people lose.
One final definition: a synthetic CDO is a CDO composed of
a portfolio of CDSs. Synthetic CDOs were also quite
popular during the subprime crisis. A synthetic CDO is a
zero sum game.
10 - 92
ASIDE: Comments on the Global Financial Crisis
1. CDSs and synthetic CDOs are zero sum games. There are
pros and cons to zero sum games.
Con: a naked position in a zero sum game tends to be quite
risky. They are often highly levered; for a small initial
investment, you can get huge gains or losses. This is why
there have been so many fiascos in the derivatives market
over the last 20 years or so.
10 - 93
Pro: on the other hand, with a zero sum game, if one party
loses a billion dollars, somebody else is making a billion
dollars, so it shouldn’t affect the economy as a whole.
There is, however, “counterparty risk”, which means that,
if the losing party can’t pay their losses, then money does
disappear from the economy.
2. In the book “The Big Short” by Michael Lewis, Lewis
argues that many of the subprime loans had conditions
that made default very likely. The loans had low rates at
10 - 94
the beginning, but the interest rates jumped later in the
loan. Some of the individuals who read these contracts
could predict the month in which the loans would start to
default.
3. Bank regulation is important. There was a serious bubble
and crash in the dot.com industry in the early 2000s.
Trillions of dollars were wiped out. However, we hardly
talk about that crash now. What was the difference
between the dot.com crash and the subprime crash? The
subprime crisis involved banks. Banks play an important
10 - 95
role in the economy, and so they have to be carefully
regulated. Canada and Australia seem to have had fairly
effective bank regulation.
4. One contributor to the crisis was the fact that the U.S.
Federal Reserve kept interest rates low. This had three
effects on the crisis. (a) Low interest rates helped keep
house prices up. If rates are low, people don’t mind
paying a little more for houses. (There did seem to be a
bubble in house prices: apparently house prices were high
compared to rental income.) (b) Low rates meant that
10 - 96
AAA bonds were not as attractive, which meant that
people went looking for more exotic investments. (c)
Lower rates meant that it was cheaper to borrow money
and invest in risky assets such as CDOs, increasing firms’
leverage.
5. Some have argued that U.S. government policy lead to an
increase in subprime loans. I would argue, however, that
the market became a fad. The interest in synthetic CDOs
shows that there was so much demand for this product,
10 - 97
that essentially, people ran out of mortgages. They had to
create synthetic CDOs out of nothing.
References:
All the Devils are Here, Bethany McLean and Joe Nocera,
Portfolio/Penguin Press, 2010.
The Big Short: Inside the Doomsday Machine, Michael
Lewis, W.W. Norton & Company, 2010.
For a more optimistic book about financial innovation, see
the following:
10 - 98
Smart Money: How High Stakes Financial Innovation is
Reshaping Our World for the Better, Andrew Palmer,
Basic Books, 2015.
CHAPTER 10: AGENCY AND CORPORATE DEBT
SECURITIES
CORPORATE DEBT MARKET
Credit Risk of Issuers (Fabozzi, F.J., Bond Markets, Analysis
and Strategies, 3rd Ed. Prentice-Hall, p. 143).
Summary of Corporate Bond Ratings Systems and Symbols
10 - 2
Moody’s S&P Fitch D&P* Definition
Investment Grade: High Creditworthiness
Aaa AAA AAA AAA Best quality
Aa AA AA AA High quality
A A A A Upper medium grade
Baa BBB BBB BBB Lower medium grade
Distinctly Speculative: Low Creditworthiness (Junk Bonds)
Ba BB BB BB Low grade, speculative
B B B B Highly speculative
10 - 3
Predominantly Speculative: Substantial Risk or in Default
Caa CCC CCC CCC Substantial risk, in poor
standing
Ca CC CC May be in default,
extremely speculative
D DD or D DD Default
*Note: D&P stands for Duff & Phelps.
Commercial Paper
These are short-term (usually less than 270 days) zero-
coupon corporate bonds. (See text.)
10 - 4
Contractual Provisions of Corporate Debt Contracts
Call features: Enable issuers to buy back the debt, after a
call protection period, for the call price. Call (exercise)
price is typically = par + coupon, but declines to par over
time. We can find the value of the call option using, e.g.,
Black’s model, etc.
Rationale:
Effect on bond price:
NOTE: junk bonds are more likely to have call features.
10 - 5
Sinking fund provisions: Require the issuer to retire an
outstanding debt issue periodically. Some bonds may be
either retired at par or through open market purchases.
Rationale:
Effect on bond price:
Putable bonds: Provide bondholders with the right to sell the
debt back to the issuer.
Effect on bond price:
10 - 6
Convertible bonds: Provide bondholders with the right to
convert bonds into a specified number of shares of
common stock.
Effect on bond price:
Bonds with warrants: [Note: this is the U.S. definition of a
warrant. In Australia this would be called a “company
issued call”.] The buyer of debt is also provided with
warrants that give the right to buy the issuer’s shares of
common stock. (Similar to a call option on the firm’s
stock, sold to the bondholder.)
10 - 7
Effect on bond price:
Agency Problems and callable debt
Asymmetric Information: If a firm has favourable info, they
can issue callable debt; later, when the info is released,
the price of the firm’s debt rises, and they can call it back;
and assuming their credit rating increases, they can issue
new debt at a lower interest rate. Note also: when a firm
issues more equity (i.e., a seasoned equity issue) it’s
taken as a signal that its stock is overpriced, and so the
price drops. Another way of thinking about this is that
10 - 8
issuing equity is “too easy”. If you issue debt, you’re
saying, We will pay this back or go bankrupt. When you
issue equity, you can only say that you intend to give
them dividends at times.
Convertible debt and warrants are ways of getting around this
effect.
Risk-shifting: Shareholders have an incentive to take risks at
the expense of bondholders. (See the option
interpretation below.) If debt is convertible, there’s more
incentive to keep the price of bonds high.
10 - 9
Protective Covenants:
- Aggregate debt limitations: Firm can’t let its total debt rise
above a certain level.
- Restrictions on dividend payments.
- Restrictions on mergers, consolidations, and the sale of
assets.
- Credible third party guarantees and credit enhancements.
- Minimum standards on working capital levels.
10 - 10
ASIDE: In the US, some people are pushing for fewer bond
features. This would mean more standardization, which
could lead to more liquidity.
10 - 11
Leveraged Buy-Outs and Management Buy-Outs
Here, the buyer issues debt (junk bonds) in order to buy
controlling interest in the firm.
Both LBOs and MBOs lead to high debt levels.
But, MBOs also give managers large numbers of shares.
Pro:
10 - 12
Con:
THEORIES OF CORPORATE-DEBT PRICING
The Merton Model (used by KMV)
Suppose a firm has total equity worth $S, and zero coupon
bonds with a face value of $F, and a market price of $B,
and maturity date T. Then, the value of the firm is
V = S + B
10 - 13
At maturity, if the value of the firm is VT > F, then the
bondholders receive F, while the shareholders receive the
“surplus”, VT F. If, at maturity, VT < F, then the firm is
in default. That means that the bondholders’ payoff is VT,
and the shareholders receive nothing. We can summarize
this as follows:
Bondholders’ payoff = min{F, VT},
Shareholders’ payoff = max{0, VT F},
where the shareholders’ payoff is consistent with their
limited liability.
10 - 14
The second equation is the payoff for a (European) call
option on the value of the firm with strike price F, i.e.,
buying the firm’s equity is equivalent to buying a call
option on the value of the firm. So, the price of equity is
given by the Black-Scholes equation:
S = VN(d1) Fer(T t)N(d2),where
d1 =
21
2ln( ) ( )( )V
V
V r T t
F
T t
,
d2 = d1 V T t ,
10 - 15
T t is the time to maturity of the bond, and V is the
volatility of V.
ASIDE for those who are interested.
More on The KMV approach (ref: Caouette, Altman, and
Narayanan, Managing Credit Risk: The Next Great
Financial Challenge, Wiley, 1998.
1. Solving for the unknowns, V and V.
Note that our equation for S as a call option on V has two
unknowns: V and = V, the volatility of V. So, at the
10 - 16
moment we have one equation and two unknowns. For
the second equation, we consider the volatility of equity,
which I’ll denote by S. Now, S is a function of V, so
let’s write S = c(V, t), where c is the usual call function,
given by Black & Scholes. Note that c
V
= N(d1). Note
also that we’re assuming that V is a lognormal process;
that is, (in a risk-neutral world) it satisfies the stochastic
differential equation
dV = rVdt + VVdZ,
10 - 17
where Z is a standard Wiener process (also known as
Brownian motion). From a mathematical result known as
Itô’s lemma (which is similar to Taylor’s theorem) we
have
dS =
2
2
2
1 ( )
2
c c cdV dV dt
V tV
= N(d1)VVdZ + (…)dt.
On the other hand, when we think of the volatility of equity,
we write the stock price as
dS = SSdZ + SSdt.
10 - 18
Equating the coefficients of the dZ term, we find
1( )S VS N d V , or
1( )
S
V
S
N d V
.
Because we can easily measure the volatility of equity, S,
this gives us a second equation, so now we have two
equations and two unknowns.
2. The distance from default.
It is common to define the distance from default to be “the
number of standard deviations that the asset value, V,
must drop in order to reach the default point”.
10 - 19
In Caouette, et al., they define this distance from default as
(expected mkt value of assets default point)
(expected mkt value of assets)( )V
.
Alternatively, some authors write the distance from
default as
ln[expected mkt value of assets] ln[default point]
V
.
(See Schönbucher, p. 277.)
10 - 20
However, because we assume that V is lognormal, we can
show by analogy to Black & Scholes’ result, that the
(actual) probability of default is N( 2d ), where
2d
=
21
2ln( / ) ( )( )V V
V
V F T t
T t
,
(while the risk-neutral probability of default would be
N(d2), where we replace V with the risk-free rate). So,
some people define 2d
as the distance to default.
10 - 21
Merton’s Model and the value of Debt
We’ve seen how Merton’s model values equity. There are
two ways of looking at the value of the firm’s debt, B.
1. First, note that
B = V S
= V[1 N(d1)] + Fer(T t)N(d2)
= Fer(T t)[ V
Fe r T t ( )
N(d1) + N(d2)]
since 1 N(d1) = N(d1). Note that Fer(T t) represents
the price of an otherwise identical risk-free bond.
10 - 22
Next, we let y(t, T) = the yield to maturity of the corporate
discount bond at date t, and call (t) the default (or
credit) spread at time t: (t) = y(t, T) r.
Then B = Fey(t, T)(T t) = Fe[r + (t)](T t) = Fer(T t)e(t)(T t)
= Fer(T t)[ V
Fe r T t ( )
N(d1) + N(d2)].
Solving for (t) gives
(t) = 1
T t
ln[ V
Fe r T t ( )
N(d1) + N(d2)].
10 - 23
NOTE: the credit spread depends on the ratio d = Fe
V
r T t ( )
known as the leverage factor. As d increases, the credit
spread increases.
Also, as Vincreases, the credit spread increases. Recall that
we argued earlier that shareholders will take risks at the
expense of bondholders. Because the shareholders have
effectively bought a call option, it is in their interests to
increase V. (Why?) This leads to increased credit
spreads, i.e., lower bond prices for the bondholders.
10 - 24
FACT: In general, if an asset’s volatility increases, the value
of a call option on that asset also increases. In addition,
if an asset’s volatility increases, the value of a put option
on that asset increases.
2. The second way to interpret the value of debt is to write
the bondholders’ payoff as
min{F, VT} = F + min{0, VT F} = F max{0, F VT}.
The last term above is the payoff of a put option on the value
of the firm. So, buying a risky bond is equivalent to
10 - 25
buying a risk-free bond and selling a put on the value of
the firm. The put is sold to the equity-holders.
How does an increase in V affect the debt-holders?
Finally, note how put-call parity relates to this model:
p + V = c + FerT,
or solving for V:
V = c + FerT – p = S + B.
10 - 26
Problems with this model:
- The sizes of the credit spreads implied by this model are too
small.
- Coupon-bonds are hard to value using this model.
- Bankruptcy may occur before the bond’s maturity (e.g., if a
coupon payment can’t be met.)
10 - 27
Subordinated Corporate Debt (Black and Cox model)
Suppose a firm has two classes of zero-coupon bonds. The
senior debt has face value F1, and current market value
B1, while the subordinated (junior) debt has face value F2,
and market value B2. (What’s the difference between the
two types?)
Now, V = S + B1 + B2.
Assume all debt matures at date T. Here are the possible
outcomes:
10 - 28
Outcome/Payoffs: Senior Debt Junior Debt Equity
VT > (F1 + F2) F1 F2 VT (F1 + F2)
F1 < VT < (F1 + F2) F1 VT F1 0
VT < F1 VT 0 0
(These are called absolute priority rules.)
10 - 29
Financial Distress
In the event of financial distress, i.e., a firm’s inability to pay
principal or interest, there are three ways it can be
handled:
1. Assets can be liquidated, i.e., sold in order to pay
debtholders.
2. The firm can renegotiate loan payments, and either
- reduce coupons and/or principal obligations
- increase the maturity of the debt, or
10 - 30
- ask debtholders to accept equity in lieu of debt
payments.
3. The firm can issue new debt in order to pay the old debt.
Factors Affecting the Probability of Default
(Reference: Zmijewski, J. Accounting Research, 1984, pp
59-82.)
Profitability = ratio of net income to total assets.
Liquidity = the ratio of current assets to current liabilities.
Leverage = the ratio of total debt to total assets.
10 - 31
The author uses a probit model to estimate the probability of
default.
Other Factors Affecting Bond Ratings
Interest coverage = EBIT/(interest payments),
Total Debt/Total Capital,
Cash Flow/Total Debt,
Return on Assets,
Current Ratio = current assets (i.e.,maturing within one year)
current liabilities (i.e.,due within one year)
.
10 - 32
Recovery Rates
In most defaults bondholders recover some portion of their
investment. This recovery may take the form of cash,
securities (debt or equity) or occasionally assets of the
business.
The recovery rate = the percentage of the par value of the
security that is recovered by the bondholder.
Moody’s recovery rates 1989-1996 (Reference: Das, Credit
Derivatives, Wiley, 1998, p. 204.)
10 - 33
Classes of debt Recovery rate (%) St. dev. (%)
senior secured bank debt 71 21
senior secured public debt 63 26
senior unsecured public debt 48 26
senior subordinated public debt 38 25
subordinated public debt 28 20
junior subordinated public debt 15 9
(secured debt refers to debt with collateral.)
10 - 34
A Common Pricing Model (Not in the text)
Consider a defaultable zero coupon bond with maturity, T,
and face value $1. The risk-neutral probability of default
is Q(T), and the recovery rate in the event of default is
0 < RD < 1. We also denote the writedown (also known
as the loss given default) by w = 1 – RD. In other words,
the defaultable bond has payoff at time T given by
Payoff =
1 if nodefault
if defaultDR
10 - 35
=
1 with prob. 1 ( )
with prob. ( )D
Q T
R Q T
.
Suppose the price of an otherwise identical risk-free bond is
denoted b(0, T). Then, the price of a defaultable zero
coupon bond is given by (assuming the probability of
default and the write-down are independent of the risk-
free rate)
B*(0,T) = b(0,T)[1 – Q(T) + RDQ(T)]
= b(0,T)[1 – (1 – RD)Q(T)]
= b(0,T)[1 – wQ(T)].
10 - 36
Example: A 5-year zero-coupon bond has a (risk-neutral)
probability of default equal to 5%. Its recovery rate is
likely to be 60%. The 5-year risk-free rate is 7% with
annual compounding. Find the yield on the risky bond;
then find the credit spread.
b(0, 5) = (1.07)5 = 0.71299;
B*(0, T) = 0.71299(1 0.400.05) = 0.712990.98
= 0.69873;
But, B*(0, T) = (1 + y)5; y = B*(0, T)1/5 1 = 7.43%.
10 - 37
The credit spread is then y – r = 7.43% 7.00% = 43 basis
points.
Note that we sometimes write 0 (0, )B T for the price of an
otherwise equivalent risky bond having 0DR , or w = 1.
It is not hard to see that
* 0(0, ) (0, ) (0, )DB T R b t wB T ,
where (0, )b t is a risk-free bond, as usual; if there is a
default, the payoff is DR , if there is no default, the payoff
is 1DR w .
10 - 38
The price of a coupon bond with coupon payment C and N
payments remaining is just
P = CB*(0, t1) + CB*(0, t2) + … + (100 + C)B*(0, tN).
Note that to calculate each zero coupon bond price we
need to find the risk-neutral probability of default, Q(ti),
corresponding to that cash-flow.
10 - 39
HYBRIDS
Convertible Bonds
Convertible debt can be exchanged for equity. Convertible
debt usually includes a call provision. Issuers tend to
have moderate to low credit ratings.
Rationale: Issuing equity is costly (equity prices drop.)
Issuing debt may lead to bankruptcy. Convertible debt is
a sort of middle ground. The call provision allows the
firm to force conversion when necessary.
10 - 40
To see this, suppose that, if a bond is converted to equity, you
receive n shares. Now suppose B > X, and nS > X, (here,
X is the call exercise price) and the firm exercises their
call option on the debt. What would a bondholder do?
10 - 41
Convertible Debt and Dilution
Some definitions:
Conversion ratio: the number of shares into which a bond
may be converted.
Dilution: when convertible debt is converted into equity, the
company actually issues more equity. If the total value of
the firm’s equity doesn’t change, but there are more
shares outstanding, the price per share must drop.
10 - 42
Notation
N = # of shares outstanding (before conversion)
M = # of convertible bonds outstanding
n = # of shares each bond can be converted into
(conversion ratio),
TE = value of the firm’s equity at time T.
If all bonds are converted, then there are N + nM shares
outstanding. So, for the original shareholders, their price
10 - 43
per share before conversion = TE
N
= beforeTS . The price per
share after conversion becomes (since beforeT TE NS )
afterTS = (1 )
before beforeT
T T
E N S S
N nM N nM
< beforeTS .
The total value of the original equity is afterTN S =
(1 )T T
N E E
N nM
(after conversion) and the value of
the newly converted equity is afterTnM S = T
nM E
N nM
=
TE .
10 - 44
(Note: value of original equity + value of new equity = TE ).
The ratio nM
N nM
is called the dilution factor associated
with the conversion. (Here, is the dilution factor from
the bondholders’ point of view; that is, if all the bonds
are converted to equity, the total value of that new equity
is TE .) So, the original shareholders now own (1 ) TE ,
and the new shareholders (i.e., the bondholders) own TE .
10 - 45
The Valuation of Convertible Debt
We’ll discuss two approaches to valuing convertible debt.
The first approach is called a structural approach. This
approach is fairly intuitive, but harder to implement. The
second approach is called the reduced form approach.
We’ll have a spreadsheet that uses the reduced form
approach.
The Structural Model
The total value of all the convertible bonds at time T is the
maximum of their value as straight bonds or their
10 - 46
conversion value. The value of the straight bond at date
T is min[F + c, VT] (we’re including the possibility of
default, and TV is the value of the firm at date T).
Here F = face value of all bonds, and c = the coupon
payment. Recall that nM
N nM
is the dilution factor.
Then, the value of the convertible bond at date T is
max{min[ , ], }T TF c V E .
value if don’t convert value if convert
QUESTION: would you convert if there were a default?
10 - 47
ASIDE: If we assume that the convertible debt is the firm’s
only debt, then if every bondholder converts to equity, the
total value of equity becomes the total value of firm; i.e.,
E = V. So, we can construct the binomial tree by
assuming that V is lognormal. In this case, the payoff at
maturity can be written
max{min[ , ], }T TF c V V .
10 - 48
ASIDE: If this bond is also callable with strike price X, then
the payoff is given by,
max{min[ , , ], }T TF c V X V .
So, for example, if the value of the firm is high, so that
min[ , ]TX F c V , and TX V , then the firm can
exercise their call option and force conversion.
We now work backwards through a binomial tree to find the
current value of the convertible debt. In earlier periods,
say at time s < T, the value is given by
max{ , }s sV P c ,
10 - 49
where Ps + c is the value of not converting at date s,
(given by risk-neutral pricing) and TV is the value of
converting at date s. Note that c may be equal to zero at
date s. This is an American style convertible.
For example, at time T 1, if VT = uV in the “up” state and
VT = dV in the “down” state, and similarly for VT, then the
value of convertible debt is given by
1TP =
1
1 R
[pmax{min[F + c, uV], uV}
+ (1 p)max{min[F + c, dV], dV}],
10 - 50
where again, p = the risk-neutral probability of an “up”
state. This would be the value of the convertible bond if
the decision to convert was of the European type. For an
American-style convertible, the price of the convertible
bond would be
1max{ , }TV P c ,
where 1TV V , and again, c may be equal to zero at date
1T .
Much like American options, we can now work backwards.
QUESTION: what else should be random in this model?
10 - 51
The Reduced Form Approach
In this section we discuss a trinomial tree for convertible
debt which is based on Hull, Section 27.4. Hull points
out that this model is widely used in industry. (While
technically a trinomial tree it basically has the form of a
binomial tree.)
As mentioned above, convertible debt is usually associated
with firms with relatively low credit ratings, so it is
important to model default. The approach we will
10 - 52
consider here is sometimes known as the reduced form
approach, makes it easier to model default.
For the reduced form approach, we assume that for each
time interval, t , there is a small probability,
1 te t , of the firm defaulting. The parameter
is known as the risk-neutral hazard rate, or default
intensity. We can then construct a trinomial tree for the
stock price and the calculated bond price. If a default
occurs, the stock price drops to zero, and the bond price
becomes equal to some constant percentage of the bond’s
10 - 53
face value, which we denote by DR , with 0 1DR ,
often called the recovery rate.
More notation: (1) Write r for the risk-free rate, (2) is the
stock’s volatility, and (3) we let q equal the stock’s
dividend yield; that is, we assume that the stock pays a
small dividend yield, tS q t , every t years. This
approach is used for options/futures on indices. (It is
easier than determining the actual date of a dividend
payment. Technically, q is a continuously compounded
yield as a percentage of the current stock price.)
10 - 54
The risk-neutral probabilities for the trinomial tree are
( )r q t t
u
e dep
u d
,
( )t r q t
d
ue ep
u d
, 1 tfp e
,
where
2( ) tu e and 1d
u
. Here, up represents the
risk-neutral probability of the stock price going up, dp is
the risk-neutral probability of the stock price going down,
and fp represents the risk-neutral probability that a
default occurs at that step of the tree.
ASIDE: First, note that
10 - 55
u d fp p p
( ) ( )
1
r q t t t r q t
te de ue e e
u d u d
( )(1 )
t t tue de u d e
u d
( ) ( ) ( ) 1
t tu d e u d u d e u d
u d u d
.
Next, note that, using these probabilities, we have,
t tE S 0u t d t fp uS p dS p
10 - 56
( ) ( )r q t t t r q t
t t
e de ue euS dS
u d u d
( ) ( )r q t t t r q t
t t
ue ude due deS S
u d u d
( ) ( )
( )
r q t r q t
r q t
t t
ue de S S e
u d
,
which would be the expected value in a risk-neutral world
on a stock paying dividend yield, q.
One final calculation: suppose a short-term (i.e., maturity
date is t ) default-risky zero coupon bond pays $1 in
10 - 57
states u and d, but $0 in the default state; that is, assume
the recovery rate is equal to zero. We denote this bond
price by 0 (0, )B t . Then,
0 (0, )B t 0r t u d fe p p p
( ) ( )r q t t t r q t
r t e de ue ee
u d u d
t t
r t ue dee
u d
( )r te .
10 - 58
So, ends up being the credit spread when the recovery
rate is equal to zero.
Conversely, if we want to derive up , dp , and fp , we start
with the fact that 1 tfp e
. Next, the fact that
1u d fp p p implies that
t
u dp p e
. We also
require that ( )r q tu dp u p d e
. These two equations
now allow us to solve for up and dp :
( )r q te u dp u p d
( )tu up u e p d
10 - 59
( ) tup u d e d
,
which implies that
( )r q t t
u
e dep
u d
, as desired.
Let’s assume that the bond is callable as well as convertible.
As Hull points out, the bond price at each node is of the
form,
ˆmax min , ,B B K nS ,
where
Bˆ = value of bond if it’s neither converted nor called,
10 - 60
= ( ) r tu u d d f Dp B p B p R FV e
,
(FV = the bond’s Face Value); at maturity, Bˆ FV c ,
where c is the final coupon payment.
K = the call price,
nS = value if conversion takes place; here n is the
conversion ratio, and S is the stock price per share at that
node.
Because DR FV is a constant, we really only need to
construct a binomial tree.
10 - 61
Note that we have ignored coupon payments, for these
calculations, and we will ignore the dilution effect. The
spreadsheet includes coupon payments.
Example: A 9-month zero coupon bond has a face value of
$100. It can be exchanged for 2 shares of the company’s
stock at any time during the 9 months, the initial stock
price is $50, its volatility is 30% per annum, and the stock
pays no dividends. Assume it is also callable for $113 at
any time. The hazard rate is 1% , and the risk-free
rate for all maturities is 5%, with continuous
10 - 62
compounding. In the event of a default, the bond is worth
$40 (i.e., the recovery rate is 40%). Find the price of the
convertible bond.
Here, we have, 9 /12 3 / 4t T , S = 50, n = 2, K = 113,
0.01 , 0.40DR . So,
2( ) (0.09 0.01)3/4 1.2776tu e e , d = 0.7827,
( ) 0.05 0.75 1.0382r q te e , 0.01 0.75 0.9925te e
( ) 1.0382 (0.7827)(0.9925) 0.5281
1.2776 0.7827
r q t t
u
e dep
u d
10 - 63
( ) (1.2776)(0.9925) 1.0382 0.4644
1.2776 0.7827
t r q t
d
ue ep
u d
1 0.0075tfp e
.
Next, 50(1.2776) 63.88Su , 50(0.7827) 39.135Sd ,
and so, since ˆ ˆ 100u dB B at the bond’s maturity, T,
uB ˆmax min , ,uB K nSu
max min 100,113 ,2 63.88
max min 100,113 ,127.76 127.76 ,
10 - 64
dB ˆmax min , ,dB K nSd
max min 100,113 ,2 39.135
max min 100,113 ,78.27 100 .
Thus, at date t = 0,
Bˆ ( ) r tu u d d f Dp B p B p R FV e
(0.5281)127.76 (0.4644)100 (0.0075)40 (0.9632)
= 110.01.
So, the price of the bond is
10 - 65
B ˆmax min , ,B K nS
max min 110.01,113 ,100 110.01 .
More Terminology
One more term that is often used with convertible debt is the
conversion price which is equal to
ratioconversion
bondpervalueface .
Note that the face value per bond is
M
F = 1000 (usually) and
the conversion ratio is n, so we can write
10 - 66
conversion price =
ratioconversion
bondpervalueface =
nM
F .
Example: Suppose the face value per bond =
M
F = 1000,
and the conversion ratio is n = 40 shares. Then the
“conversion price” =
40
1000 = $25.
Intuitively, suppose we assume that the total of all the bond
prices remains constant total face value = F. In this
case the option to convert debt to equity becomes
equivalent to the option to (recall, M = number of bonds)
10 - 67
buy n shares for each bond, where the price per bond =
M
F .
For each bond, then, it is as if you own n call options on
equity with exercise price X =
nM
F . This “exercise price”
is the conversion price.
10 - 68
Debt with Warrants
Warrants (this is U.S. terminology) are like call options on
the firm’s equity, except that they are sold by the firm,
and when exercised, the firm issues additional shares of
the common stock.
Let N = # shares of common stock before warrants’ exercise
M = # of warrants that can be exercised at date T,
S = market price of a share of common stock,
W = market price of each warrant,
X = exercise price
10 - 69
E = total value of firm’s equity.
QUESTION: when would issuing a warrant be
a better signal?
a worse signal?
Before the warrants are issued, the total value of equity is NS.
After the warrants are sold, the total value of the firm’s
equity becomes (still at t = 0)
E = NS + MW.
If at time T, all warrants are exercised, the total value of
equity becomes ET + MX. This value is distributed
10 - 70
among the new number of shares outstanding, N + M.
So, the share price immediately after exercise becomes
afterTS = T
E MX
N M
.
If exercised, the payoff to the warrant holder is therefore,
afterTS X = T
E MX X
N M
= TE MX
N M
MN
MNX
)(
= TE NX
N M
10 - 71
= ( )TEN X
N M N
= N
N M
( beforeTS X),
so the payoff to the warrant holder can be written
max{ ,0}TEN X
N M N
.
NOTE: TE
N
= beforeTS is the price per share just before the
warrant is exercised, so that the exercise decision is the
same as for a call option. So, at time of exercise, the
10 - 72
payoff is equal to N
N M
call options (i.e., there’s a sort
of “dilution” to the warrant holder).
One other difference is that ET is the future (uncertain) value
of equity assuming that the warrants were sold (at date
t = 0). In terms of today’s share price (at the time of the
sale of the warrants) we have
E = NS + MW
where S is the stock price (before the sale of the warrants)
and W is the warrant price, so that (at t = 0)
10 - 73
E
N
= S + M
N
W.
So, we can use the Black and Scholes formula to calculate
the warrant price if
1. The stock price S is replaced by S + M
N
W.
2. The volatility, , is the volatility of the value of the
shares plus the warrants, not just the shares.
3. The formula is multiplied by N
N M
.
10 - 74
When these adjustments are made, we end up with a formula
for W as a function of W. To see this, let c(S) be the usual
Black-Scholes formula for a European call option. For
the warrant, we have W = N
N M
c(S + M
N
W). Here, W
changes the price per share, and we need the price per
share to calculate W. This can be solved numerically.
10 - 75
ASIDE: we’ve seen a sort of “dilution” to the call holder, but
is there dilution in the stock price when a warrant is
exercised? We exercise the warrant if TE X
N
. Now,
afterTS = T
E MX
N M
<
( )TT
EE M
N
N M
= ( )TE N M
N N M
= TE
N
= beforeTS .
So, yes, the stock price drops, and the size of the drop
depends on X:
10 - 76
ST = T
EM X
N M N
;
i.e., the lower the exercise price, the greater the drop in
the stock price. This is drop understandable, though,
because at the time the warrant is sold (t = 0), there is an
increase in the value of equity, and the increase at that
date is higher for lower exercise prices. (On the other
hand, as discussed in class, a low exercise price may send
the market a bad signal; but that signal is unrelated to the
simple option pricing/dilution effect.)
10 - 77
Here are a few definitions that were in the news during the
global financial crisis.
Credit Default Swaps (CDSs)
A Credit Default Swap basically provides insurance against
defaults on credit-risky coupon bonds. If a particular
firm defaults, then the CDS pays L(1 – RD) = Lw, where L
is the notional principal on the firm’s bond, RD is the
recovery rate, and w is the write-down, also known as the
loss given default. (If there is no default then the CDS
10 - 78
makes no payment.) Note that the timing of the default is
random.
One unusual feature of this “default insurance” is that you do
not pay for it in a lump sum; rather, you pay for it with
periodic premium payments. If the bond defaults, then
the premium payments stop—at the random default time.
Note that CDSs are zero sum games, much like standard
options and futures.
CDSs are often used in empirical corporate finance. As
explained in Zhang, et al. (2009),
10 - 79
Compared with corporate bond spreads, CDS spreads
have two important advantages. First, CDS spreads
provide relatively pure pricing of the default risk of the
underlying entity and are typically traded on
standardized terms. However, bond spreads are more
likely to be affected by differences in contractual
arrangements, such as differences related to seniority,
coupon rates, embedded options, and guarantees.
Second, as shown in Blanco, et al. (2005), and Zhu
(2006), CDS spreads tend to respond more quickly to
changes in credit conditions in the short run, which
may be partly due to the absence of funding and short-
sale restrictions in the derivatives market.
10 - 80
ASIDE: Pricing a CDS is a bit more complicated than other
assets that we’ve seen. Suppose we buy the CDS. First,
we need to price the cashflow we receive at the random
default time. In the simplest model, the price of this
cashflow is
0
(1 ) (0, ) ( )
T
DL R b t dQ t ,
where Q(t) is the risk-neutral probability that firm has
defaulted by date t. This assumes that the risk-free rate
and the default probability are non-random functions of t.
10 - 81
(If the short rate, r, is constant, and if we also assume that
the probability of default is given by Q(t) = 1 et,
where is known as the default intensity, then we can
calculate these integrals explicitly.)
Now, let us discuss our premium payments. We pay for this
insurance (i.e., the CDS) with premium payments, $Ls
per year, for as long as the risky bond survives. Note that
s, known as the CDS rate or CDS spread, is an annual
percentage of the face value, and so looks much like an
interest rate. (More on that below.) Let’s assume that
10 - 82
there are m premium payments per year, so that the
periodic payment is L
m
s . Suppose the bond has N
payments, made at dates T1, T2,…, TN. Again, in the
simplest model, the price of the premium payments is
1
(0, )[1 ( )]
N
i i
i
sL b T Q T
m
,
where 1 – Q(Ti) is the probability that the bond survives
until time Ti; i.e., if the bond defaults, we stop paying the
premium.
10 - 83
In addition, it is usually assumed that, when the default
occurs, we make one more premium payment. If the final
cashflow (i.e., the default) occurs at date t, with Tk < t <
Tk+1, then it is accrued as
L
1
k
k k
t Ts
m T T
,
where L
m
s is the periodic premium payment and
1
k
k k
t T
T T
is the fraction of the period that has elapsed since the last
payment. This simplifies further if we assume that
10 - 84
Tk+1 Tk =
1
m
. This final cashflow then becomes
Ls(t – Tk).
Let us ignore the accrual term for the moment. Its price is
slightly more complicated than those given above. Now,
the question is, “What should s be?” The CDS rate, s, is
chosen so that the value of the premium payments is
equal to the value of the cashflow received in the event of
default; that is,
10 - 85
1
(0, )[1 ( )]
N
i i
i
sL b T Q T
m
=
0
(1 ) (0, ) ( )
T
DL R b t dQ t ,
or,
s = 0
1
(1 ) (0, ) ( )
1 (0, )[1 ( )]
T
D
N
i i
i
R b t dQ t
b T Q T
m
,
or more generally,
10 - 86
s = 0
1
(1 ) (0, ) ( )
1 (0, )[1 ( )]
T
D
N
i i
i
R b t dQ t
b T Q T accrual
m
.
It can be shown that s behaves much like a credit spread; that
is, the spread between the yield on the firm’s risky debt
minus the risk-free rate.
10 - 87
Collateralized Debt Obligations (CDOs)
A CDO begins with a portfolio of assets, such as corporate
bonds or mortgage-backed securities (or one could even
use the premium payments from CDSs), and then this
portfolio is divided into tranches. The difference is that,
with a CDO, the tranches are based on defaults.
With a CDO, the first tranche incurs all the defaults until that
tranche becomes worthless. This is known as the equity
tranche, and is very risky. Often it is kept by the firm
that organized the CDO. As an example, suppose the
10 - 88
portfolio consists of 100 corporate bonds, and suppose
the first tranche becomes worthless after seven defaults.
The second tranche incurs no losses until the first tranche
is worthless. Again, as an example, the second tranche
may become worthless after four more defaults occur,
and so on. The last tranche is usually rated AAA.
Typically, the last tranche is unaffected until, say, 20
firms default, and only then does it begin to lose value. It
never becomes worthless—all 100 firms would have to
10 - 89
default in order for it to become worthless—but it can
lose some value if 20 or more firms default.
Even the AAA tranches performed poorly during the GFC.
“A chart … prepared by the Senate Permanent
Subcommittee on Investigations showed that 91 percent
of the triple-A-rated subprime residential mortgage-
backed securities issued in 2007, and 93 percent of those
issued in 2006, were subsequently downgraded to junk
status.” (Source: “All the Devils are Here”, Bethany
McLean and Joe Nocera, footnote on p. 305.)
10 - 90
The ratings agencies played an important role here, as “[t]he
triple-As were easy to sell because investors around the
globe that were legally confined to conservative
investments, or didn’t want to hold the capital against a
higher-risk investment, embraced their higher yield
relative to their super-safe rating. The triple-B and B-
minus tranches were harder to sell….” (M&N p. 121.)
In many cases these riskier tranches, known as mezzanine
tranches, got recycled into CDO squareds, which were
CDOs composed of tranches of other CDOs—
10 - 91
particularly, tranches of other CDOs that nobody would
buy in the first place.
There is no closed form solution for pricing CDOs. One
must use Monte Carlo simulations.
Note that CDOs are not zero sum games. Like a portfolio of
stocks, if their prices go down, most people lose.
One final definition: a synthetic CDO is a CDO composed of
a portfolio of CDSs. Synthetic CDOs were also quite
popular during the subprime crisis. A synthetic CDO is a
zero sum game.
10 - 92
ASIDE: Comments on the Global Financial Crisis
1. CDSs and synthetic CDOs are zero sum games. There are
pros and cons to zero sum games.
Con: a naked position in a zero sum game tends to be quite
risky. They are often highly levered; for a small initial
investment, you can get huge gains or losses. This is why
there have been so many fiascos in the derivatives market
over the last 20 years or so.
10 - 93
Pro: on the other hand, with a zero sum game, if one party
loses a billion dollars, somebody else is making a billion
dollars, so it shouldn’t affect the economy as a whole.
There is, however, “counterparty risk”, which means that,
if the losing party can’t pay their losses, then money does
disappear from the economy.
2. In the book “The Big Short” by Michael Lewis, Lewis
argues that many of the subprime loans had conditions
that made default very likely. The loans had low rates at
10 - 94
the beginning, but the interest rates jumped later in the
loan. Some of the individuals who read these contracts
could predict the month in which the loans would start to
default.
3. Bank regulation is important. There was a serious bubble
and crash in the dot.com industry in the early 2000s.
Trillions of dollars were wiped out. However, we hardly
talk about that crash now. What was the difference
between the dot.com crash and the subprime crash? The
subprime crisis involved banks. Banks play an important
10 - 95
role in the economy, and so they have to be carefully
regulated. Canada and Australia seem to have had fairly
effective bank regulation.
4. One contributor to the crisis was the fact that the U.S.
Federal Reserve kept interest rates low. This had three
effects on the crisis. (a) Low interest rates helped keep
house prices up. If rates are low, people don’t mind
paying a little more for houses. (There did seem to be a
bubble in house prices: apparently house prices were high
compared to rental income.) (b) Low rates meant that
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AAA bonds were not as attractive, which meant that
people went looking for more exotic investments. (c)
Lower rates meant that it was cheaper to borrow money
and invest in risky assets such as CDOs, increasing firms’
leverage.
5. Some have argued that U.S. government policy lead to an
increase in subprime loans. I would argue, however, that
the market became a fad. The interest in synthetic CDOs
shows that there was so much demand for this product,
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that essentially, people ran out of mortgages. They had to
create synthetic CDOs out of nothing.
References:
All the Devils are Here, Bethany McLean and Joe Nocera,
Portfolio/Penguin Press, 2010.
The Big Short: Inside the Doomsday Machine, Michael
Lewis, W.W. Norton & Company, 2010.
For a more optimistic book about financial innovation, see
the following:
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Smart Money: How High Stakes Financial Innovation is
Reshaping Our World for the Better, Andrew Palmer,
Basic Books, 2015.
