HD28
.M414
no.3035-89
:^y
3
TDflO
Da5t,73TM T
BASEMB4I
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMEN
A COMMODITY LINKED BOND AS
AN OPTIMAL DEBT INSTRUMENT
IN THE PRESENCE OF MORAL HAZARD
Antonio Mello and John Parsons
WP
#3035-89-EFA-
June 1989
MASSACHUSETTS
TECHNOLOGY
50 MEMORIAL DRIVE
;AMBRIDGE, MASSACHUSETTS 02139
INSTITUTE OF
A COMMODITY LINKED BOND AS
AN OPTIMAL DEBT INSTRUMENT
IN THE PRESENCE OF MORAL HAZARD
Antonio Mello and John Parsons
WP
#3035-89-EFA-
June 1989
JUN
i
189
1989
A COMMODITY LINKED BOND AS AN OPTIMAL DEBT INSTRUMENT
IN
THE PRESENCE OF MORAL HAZARD
Antonio Mello and John Parsons*
June 1989
In this paper
strictly
The
we communicate two
prefers to issue a
bond linked
firm with a fixed rate
results.
First,
to the price of a
bond chooses
a
we
construct an example in which a firm
commodity
as
opposed
to a fixed rate bond.
suboptimal investment policy because the outstanding
nominal obligation distorts the incentives of the equity owners.
The commodity
linked
bond
resolves this moral hazard problem.
Second,
we show
that in the presence of moral hazard problems the traditional application of
contingent claims valuation techniques leads to an incorrect valuation of the the real assets and
consequently to an incorrect valuation of the financial
show how the technique can be adapted
liabilities
written against those assets.
We
to yield correct valuations.
Visiting Professor of Finance and Assistant Professor of Finance, respectively, Sloan School
of Management, M.I.T.
This research
is
supported
in part
by the Program on Investments, Finance, and Contracts
of the Center for Energy Policy Research, Energy Laboratory, M.I.T.
2
TTie use of
commodity linked debt has been advocated by Lessard (1977), primarily
instrument to improve risk sharing between
upon commodity exports and outside
would have consequences
as an
less
developed countries that are significantly dependent
investors.
Lessard also noted that the improved risk sharing
for capital budgeting.
However,
his
argument did not focus upon the
very direct nature in which a commodity linked bond could improve the equity owner's incentives
to choose the optimal investment and operating strategy.
popular, and there
prices of various
now
exist
many
commodities or
to the
improved use of the
In this paper
we
directly to export earnings.
However, almost
real assets that
There
is
all
of the literature
almost no attention given
can result from the proper design of commodity linked
Our
focus exclusively on the changes in use of the real assets.
demonstrates, therefore,
why
a
noted that the great majority of commodity linked bonds have been issued
issues
analysis
commodity linked bond may be an optimal financing instrument
a project in a developed country as well as a project in a less developed country.
and that many of the
quite
proposals for indexing of less developed country debt to the
continues to focus upon the problem of improved risk sharing.
debt.
become
Lessard's suggestion has
It
for
should be
developed countries
in
have been made by industrial corporations involved
in
mining or
processing the relevant commodities as our analysis would predict and in contradiction to the
predictions of most other explanations for the existence of these securities.
attention, moreover, to the fact that the optimal
Our
commodity linked contract must be
analysis calls
tailored to the
particular set of real assets being financed.
In
the traditional application of the contingent claim techniques
it
is
assumed
that
the
investment and op)erating decisions of the firm are given and do not vary with the parameters of
the
liabilities
sold against the firm.
describes the value of the firm
is
For example,
in
Merton (1974) the
stochastic process that
assumed given, and then the value of any debt claim written
3
against the firm
is
In recent
derived.
work on the valuation of
real options the contingent claims
techniques are used to derive the optimal investment and operating stratcgies--for example,
Brennan and Schwartz (1985).
strategy
is
in
In these models, however, the optimal investment and operating
derived under the implicit assumption that the firm
to then take the derived optimal investment
100%
is
equity owned.
It is
common
and operating strategy as the underlying determinant
of the stochastic process for the firm value and to then value the various financial claims against
this value.
However, under some circumstances the structure of
assets will impact the operating decisions of the
management so
for the firm value does not in fact obtain in equilibrium.
incorporated into the contingent claims models, and
1.
A
Consider
a firm that
owns
we reduce
mine be abandoned.
When
While the mine
is still
At any point
The
assumed stochastic process
The problem of moral hazard must be
we show how
this
can be done.
in
a
mine
that can operate for a total of three
the mine
is
future price of the commodity,
S; the price at
open
it
must produce an amount q
time the mine can be abandoned
time
t
=l
in
more
the management's discretion to a single decision:
open and operating the owner must pay
binomial method described
is
that the
Contingent-Claims Analysis of a Mine
the problem simple
A.
financial claims sold against the
s,
is
a
at
random
at
a per period
zero cost, but
variable.
We
it
years.
To make
when should
zero average
cost.'
maintenance cost of
cannot be reopened.
model the price using the
Cox, Ross and Rubinstein (1979). Accordingly, the price at time
can take on one of two values, uS and dS.
The
price at time
'
One
can alternatively consider the commodity price given
in this
t=0
t=2 can
take on three values, uuS, udS, and ddS, although conditional on the realization of the price at
of the constant per unit average cost.
the
t
=l
paper to be the price net
4
there are only two possible realizations at t=2.
Because the payoff to the firm
possible paths of prices.
when we
In Figure
we draw
the tree that defines the
path dependent
is
an important way,
in
construct a tree to represent the unfolding of uncertainty regarding price
between the node
and the node
convenience
t=2
we denote
t
s=udS and which was
time t=2 for which
at
time
at
the two nodes at
for
the four nodes at
= l; and we may
t=2 by j = 1,2,3,4;
write the price at any
The
risk-free interest rate
convenience yield which
defining
The term
r.
we assume
the
f(s) as
is
to
1
at via
s=dS
as
is
distinguish
time
at
at time
a similar notation
node
we
s=uS
arrived at via
which s=udS and which was arrived
[Insert Figure
s(t,j)c:
1,
t
t
=l
= l. For
used to denote
s(t,j).
Here]
structure of futures prices
is
determined by the
be proportional to the current price of the commodity,
one period future
price
it
follows in our
model
that f(s)= s(t,j)(l+r-
c).
A
vector
strategy,
M
where
[m(0,l);m(l,l),m(l,2);m(2,l),m(2,2),m(2,3),m(2,4)] denotes an
m(t,j)
when
in
period
=
[1;1,0;1,0,0,0];
t
=
=
(=
the price
M, =
1)
means
is s(t,j).
we have
written the
strategies.
[1;1,0;1, 1,0,0];
M, =
not the firm
is
[1;1,1;1, 1,1,0];
as the basis of
abandonment decisions
The reader should note
abandons (continues to operate) the mine
There are 5 undominated
and 4 are the interesting ones that serve
1
that the firm
that
at
abandonment
strategies:
and M^
=
M, =
[0;0,0;0,0,0,0];
Strategies 3
[1;1,1;1, 1,1,1].
comparisons made
in this
t
= l was
each node of the decision tree for both of these
under strategy
3, if
the price at
operating depends upon the past history of price:
dS, then the
operating at the
new
price
In Figure
paper.
if
t=2
udS, whether or
is
the price at
t
= l was
the mine was not abandoned and the mine will also not be abandoned at t=2; however,
at
M,
if
uS, then
the price
mine was abandoned and even though the owner might wish to be
it
is
not possible.
5
It
is
Our example
policy.
in
now
possible to derive the value of the mine and to solve for the optimal abandonment
is
essentially a
much
Brennan and Schwartz (1985)--allowing
for the discretization of the
and we solve our problem according to the
For any arbitrary strategy
M
be received by the firm
that will
it
at
These cash flows
price: x(M,t,j).
price process and the difficult
is
logic
used
how
to use the futures
in
when
that
our example--
time and contingent upon the realization of the
however, contingent upon the realization of the stochastic
problem
the proper discount to apply to these cash fiows in
is
Contingent claims analysis teaches
us,
market combined with borrowing and lending to exactly duplicate
the sequence of risky cash fiows from the mine under any given strategy.
amount of cash
in
in that paper.
recognition of the particular risk associated with each.
however,
problem
a straightforward matter to construct the set of cash flows
each point
are,
problem analyzed
simplified case of the natural resource
would be necessary
to
open
a portfolio
We
can determine the
of futures contracts and bonds which,
Under
properly rebalanced, perfectly replicates the cash flows from the mine.
assumptions about the completeness of the market
in financial
and
real assets, this
certain
must also be
the value of the mine under that strategy.
Define K(M,t,j) as the amount of futures contracts that the firm holds
price
is
at
node
j;
and define B(M,t,j)
as the
period that the firm holds in period t€{0,l}
in
period te{0,l}
amount of one period bonds paying (1+r)
if
the price
is
at
node
j.
the following set of equations which can be solved pairwise:
x(M.2,l)
=
(u+c-r) K(M,1,1)
+ (l+r)
B(M,1,1),
x(M,2,2)
=
(d+c-r) K(M,1,1)
+ (l+r)
B(M,1,1),
x(M,2,3)
=
(u+c-r) K(M,1,2)
+ (l+r)
B(M,1,2), and
x(M,2,4)
=
(d+c-r) K(M,1,2)
+ (l+r)
B(M,1,2).
is
the
in the next
For every strategy
the portfolio of futures contracts and bonds that the firm must hold at time t=l
if
M=
l,...5
the solution to
6
and bonds that the firm must hold
Similarly, the portfolio of futures contracts
at
time t=0
is
the
solution to the following pair of equations, which can be solved once the values from the previous
equations are substituted
in accordingly:
x(M,l,l)
=
(u+c-r) K(M,0,1)
+ (1+r) B(M,0,1)
-
B(M,1,1), and
x(M,l,2)
=
(d+c-r) K(M,0,1)
+ (1+r) B(M,0,1)
-
B(M,1,2).
Once
this portfolio
of futures contracts and bonds
is
opened,
as described by the solution to these equations so that
it
it
yields exactly the
the firm as would the mine operated according to the strategy
however, the firm needs to purchase the
contracts require
no cash outlays up
initial
it
is
now
front, the initial portfolio costs
a straightforward matter to
each period
free cash flow to
In order to start the strategy,
Since the futures
B(M,0,1) and
this,
then,
is
the
M.
choose the optimal abandonment strategy for the mine:
= max B(M,0,1) and M'
argmax B(M,0,1).
e
M
V
also possible to find the value of the
=
price of the commodity, T(M',t,j)
In constructing this valuation
some
realization of price
and
for
mine
x(M',t,j)
we have
some
+
at
any point
in
time as a function of the realized
B(M*,t,j).
ignored the fact that at
arbitrary planned
a negative cash account and a negative value,
This
in
simply that abandonment strategy which yields the highest value for the mine:
r
It is
M.
same
portfolio of futures and bonds.
value of the mine operated according to the strategy
It is
can be rebalanced
i.e.,
abandonment
we have
some point
in
time and for
strategy the firm might have
ignored the problem of bankruptcy.
easy to resolve."
is
To
properly incorporate the problem of bankruptcy
Note, however, that
T
is
we must
specify
what happens to the
always greater than zero, since T(1,0,1)=0
is
firm's
always feasible.
7
We
assets in this event.
cash earned
is
assume
first
retained and invested
end of period t=2
that the firm does not pay any dividends at
in
an account earning the riskless rate of interest,
a liquidating dividend
realization of the price
it
operate as planned, that
is
is
At
payed.
common knowledge whether
whether
is
it
is
The
declared bankrupt.
and the assumed sale price
is
r.
t
= l:
all
At the
the beginning of each period, given the
the firm has enough cash to continue to
can cover the maintenance cost. A, out of the revenues to
be received that period and out of any accumulated cash:
firm
t=0 and
firm, less
if it
will
any accumulated cash,
is
not have enough cash, then the
assumed to be liquidated by
sale
the value of the firm at that point in time and conditional on the
current realized commodity price, and under the assumption that the firm
is
operated from
this
point on according to the optimal strategy, T(M",t,j), derived above. Define Y(M,t,j) as the balance
that
M
would be
in the
account
and no bankruptcy;
this
is
at the
end of
a period
assuming an arbitrary abandonment strategy
the retained earnings of the firm, and at
t=2
it
is
the cash available
for distribution to equity holders:
Y(M,0,1)
=
Y(M,1,1)
= Y(M,0,1) (1+r) +
x(M,l,l),
Y(M,1,2) = Y(M,0,1) (1+r) +
x(M,l,2),
Y(M,2,1) = Y(M.1,1) (1+r) +
x(M,2,l).
Y(M,2,2) = Y(M,1,1) (1+r) +
x(M,2,2),
Y(M,23) = Y(M,1,2)
Y(M,2,4)
The
firm
is
Y(M,t,j)<0.
=
0.
(1+r)
+
x(M,2,3), and
Y(M,1,2) (1+r)
+
x(M,2,4).
bankrupt
at
the beginning of period
Define u(M,t,j)e{0,l}
gone bankrupt
at
lime
t
t
given a realization of the commodity price
as a binary operator,
where u=0
if
indicates that the firm has
or earlier given the realized path of commodity prices, and define
8
w=
w(M,t,j)e{0,l} as a binary operator, where
given the reahzation of the commodity price.
indicates that the firm
l
went bankrupt
Define Z(M,t,j) as the actual balance
account of the firm, given the intended strategy
M
at
time
in the
t
cash
and the sale of the firm when bankruptcy
occurs:
Z(M,0,1)
=
Z(M,1,1)
= Z(M,0,1) (1+r) +
x(M,l,l)u(M,l,l)
+ T(MM,l)w(M,l,l),
Z(M,1,2)
= Z(M,0,1)
x(M,l,2)u(M,l,2)
+ T(MM,2)w(M,l,2),
Z(M,2.1)
=
(1
+ r) +
Z(M,1,1) (1+r) + x(M,2.1)u(M,2,l)
+
T(M',2,l)w(M.2,l),
Z(M,2.2) = Z(M,1.1) (1+r) + x(M,2.2)u(M,2,2)
+
T(M',2,2)w(M,2,2),
Z(M,2,3)
=
Z(M,1,2) (1+r)
x(M,2,3)u(M,2,3)
+
T(M',2,3)w(M,2,3), and
Z(M,2,4)
=
Z(M,1,2) (1+r) + x(M,2,4)u(M,2,4)
+
T(M*,2,4)w(M,2,4).
The
firm
may be valued
bonds
as described
riskless
algorithm
is
the price
is
at
node
j;
+
using a portfolio hedging strategy
The only
earlier.
a recognition that
Again we define K(M,t,j)
if
+ T(M",0,l)w(M,0,l)
x(M,0,l)u(M,0,l)
as the
no cash flows
amount of
correction that must be
l,...5
futures contracts and
made
actually leave the firm until the
to the
hedging
end of period
2.
futures contracts that the firm holds in period te{0,l}
and define B(M,t,j)
as the
amount of one period bonds paying
the next period that the firm holds in period te{0,l}
M=
composed of
if
the price
is
at
node
j.
For every
(1
Z(M,2,1)
=
(u+c-r) K(M,1,1)
+ (1+r)
Z(M,2,2)
=
(d+c-r) K(M,1,1)
+
Z(M,2,3)
=
(u+c-r) K(M,1,2)
+ (1+r)
B(M,1,2),
Z(M,2,4)
=
(d+c-r) K(M,1,2)
+ (1+r)
B(M,1,2),
B(M,1,1),
(1+r) B(M,1,1),
in
strategj'
the portfolio of futures contracts and bonds that the firm must hold at time t=l
solution to the following set of equations that can be solved pairwise:
+ r)
is
the
9
Z(M,1,1)
=
(u+c-r) K(M,(),1)
+ (1+r) B(M.0,1). and
Z(M,1,2)
=
(d+c-r) K(M,0,1)
+ (1+r)
Once
B(M,0,1).
of futures contracts and bonds
this portfolio
by the solution to these equations,
it
M.
according to the strategy
In Table
can see
that
1
we
the
same
if it is
rebalanced as described
free cash flow to the firm as
would
In order to start the portfolio, however, the firm
amount B(M,0,1) and
the
in
opened, and
yields exactly the
the mine operated according to the strategy
needs to have cash
is
then,
this,
is
the value of the mine operated
M, V(M,0,1).'
display the results of this
abandonment
optimal
model
strategy
The reader
for a particular set of input values.
#3 and
is
that
the value
of the mine
is
V(M\0,1)=V(3,0,1)=$4.24.
Table
1
s(l,2)=dS, the firm
is
[Insert
At
t
= l and
j=2,
increase to udS,
it
i.e.
when
would be profitable
ddS the firm would choose
in
order to keep
this
maintenance costs A.
it
is
the firm
'
It
is
V
for the firm to operate.
alive,
the
firm
However, the cost of the option
> T when
the case that V(M,0,1)
This
is
if
because
V
>
t
the price were to
the price were to decrease to
= l,
is
The option
$0.65, the difference
costs of keeping the
is
valuable:
between the
mine open, and so
j=2.
T(M,0,1), the value of the firm calculated in the
incorporates the possibility of bankruptcy while
the event of bankruptcy takes the assets away from a
implementing a suboptimal abandonment
However,
losses.
must keep the mine open today and pay the
on the commodity and the maintenance
may be
If
This would be equivalent to choosing strategy #4.
better off abandoning the mine at
previous section.
facing an unsure future:
abandon the mine rather than incur the operating
option
worth almost $0.54.
current earnings
to
Here]
strategy.
Of
T
does not.
management team
course, V(M',0,1)=T(M',0,1).
that
is
10
2.
Valuing Corporate Liabilities
We now
suppose that the firm has outstanding a bond with a promised payment
The
with no intermediate payments due.
until the
bond
is
payed.
expenses on the mine
to pay off the
liabilities exist
The
t
firm
= l and
is
may be
restricted,
p.
and
however, against paying any dividends
allowed to use the accumulated cash to pay any operating
t=2.
The accumulated balance
bond and then the remainder
or
is
t=2,
is
in this
account
is
used
payed out as a liquidating dividend.
at
No
t=2
other
created.
definition of the cash available to the firm and the events in which bankruptcy occurs are
changed
slightly
would be
strategy
at
The
firm
at
in
M
by the existence of the outstanding debt.
Define
Y<((M,t,j) as the balance that
the firm's cash account at the end of a period assuming an arbitrary
and no bankruptcy;
this
is
the retained earnings of the firm, and at
t=2
abandonment
it
is
the cash
available for distribution to equity holders:
The
Y,(M,0,1)
=
0,
Y,(M,1,1)
=
Y,(M,0,1) (1+r)
+
x(M,l,l),
Y<,(M,1,2)
=
Y,(M,0,1) (1+r)
+
x(M,l,2),
Y,(M,2,1)
=
Y,(M,1,1) (1+r)
+
x(M,2,l)
-
p,
Y,(M,2,2)
=
Y,(M,1,1) (1+r)
+
x(M,2,2)
-
p,
Y,(M,2,3)
=
Y,(M,1,2) (1+r)
+
x(M,2,3)
-
p,
Y,(M,2,4)
=
Y,(M,1,2) (1+r)
+
x(M,2,4)
-
p.
firm
is
bankrupt
at the
as indicator functions for a
beginning of period
t
and
if
YXM,t,j)<0.
Define Ud(M,t,j) and Wj(M,t,j)
bankrupt firm and for the period of bankruptcy, just as before. Define
Zd(M,t,j) as the actual balance in the cash account of the firm, given the intended strategy
the sale of the firm
when bankruptcy
occurs:
M and
11
Z,(M,ai)= x(M,0,l)u,(M,0,l) +T(M',0,l)w,(M,0,l)
Z,(M,1,1)= Z,(M.O,l)(l+r) +x(M,l,l)u<,(M,l,l) +T(MM,l)w,(M,l,l),
Za(M,l,2)= Z,(M,0,l)(l + r) +x(M,l,2)u,(M,l,2) +T(MM,2)w<,(M,l,2),
Z,(M,2,1)= Z,(M,l,l)(l+r) +x(M,2,l)u,(M,2,l) +T(M',2,l)w,(M,2,l),
Z,(M,2,2)= Z,(M,l,l)(l+r) +x(M,2,2)u,(M,2,2) +T(M*,2,2)w,(M,2,2),
Z,(M,2,3)= Z,(M,l,2)(l+r) +x(M,23)u,(M,23) +T(M',2,3)w,(M,2,3), and
Z,(M,2,4)= Z,(M,l,2)(l+r) +x(M,2,4)u,(M,2,4) +T(M*,2,4)w,(M,2,4).
We
are
now
ready to value firm with debt outstanding and to directly value the debt and
equity claims against the firm.
The
valuation of the firm proceeds just as before, using a portfolio
hedging strategy composed of futures contracts and
riskless
bonds, Kd(M,t,j) and Bd(M,t,j):
Za(M,2,l)
=
(u+c-r) K,(M,1.1)
+
Z,(M,2,2)
=
(d+c-r) K,(M,1,1)
+ (l+r)
B,(M,1,1),
Z<,(M,2,3)
=
(u-rc-r) K,(M,1.2)
+ (l+r)
B,(M,1,2),
Z,(M,2,4)
=
(d+c-r) K,(M,1,2)
+ (l+r)
B,(M,1,2),
Z,(M,1,1)
=
(u+c-r) K,(M,0,1)
+ (l+r)
B,(M,0,1), and
Z,(M,1,2)
=
(d+c-r) K,(M,0,1)
+ (l+r)
B,(M,0,1).
Bd(M,0,l)
To
is
(1+r) B,(M,1,1),
the value of the mine operated according to the strategy
M,
Vj(M,0,l).''
value the debt and the equity claims against the firm merely requires correctly registering
the payoffs to each claim in each final realization of the commodity price path and valuing the risky
"
It may be the case that Vj(M,0,l) > V(M,0,1), the value of the firm calculated when there
no debt outstanding. This is because the possibility of bankruptcy may change the actual
abandonment strategy implemented with the real assets when these assets are taken from the
original management; and the set of events on which bankruptcy may occur will be different
depending upon the actual size of the debt obligation outstanding. This is one example of the 'free
cash flow' argument in favor of a large amount of debt that has been made by Jensen (1986). Of
is
course, V,(M',0,1)=V(M',0,1).
12
payoff using the appropriate portfolio hedging strategy composed of futures and riskless bonds.
The outstanding corporate bond promising
strategy
t=2 has the following
at
Z<,(M,2,j) w,(M,2,j),
actual payouts under
=
(Z,(M,2,j)-p) u,(M,2j)
we once
value these claims
+
j
= l,..4.
under strategy M:
firm's equity has the following payouts
E,(M,2,j)
To
pay p
M:
D,(M,2J) = p u,(M,2,j) +
The
to
max{Z,(M,2,j)-p,0} w,(M,2,j),
j
= l,..4.
again construct a hedging portfolio of futures and riskJess bonds
according to the algorithm used to value the cash flows of the firm as a whole given above.
In Table 2
in constructing
we
Table
under strategy #3
the
firm
display
never
D<,(3,0,1)=$1.076.
is
1,
some of the
The
=$4,242.
bankrupt.-
Under
strategy
total
is
value of the firm
is
less
worth $0.54 as we mentioned
is
with this size of debt outstanding
therefore
riskless;
its
value
is
Liabilities
under strategy
is
#4
than under strategy #3, the
higher, Ej(4,0,l)= $3.20
>
£^(3,0,1)= $3,166.
because the outstanding debt obligation of the firm changes the financial incentives facing
the equity owners in the event that the price
alive
the mine
Table 2 Here]
reader should note that the value of the equity
This
is
The value of
firm's equity has a value £^(3,0,1) =$3. 166.
Moral Hazard and the Revised Valuation of Corporate
Although the
for the set of input values used earlier
#3 and
The corporate debt
[Insert
3.
model
with the promised debt payment p=$1.35 at t=2.
V<,(3,0,1)
goes
results of this
earlier,
is
low.
At t=l, j=2, the abandonment option
but costs $0.65 to keep
not worthwhile and the mine should be abandoned.
And
therefore T(3,0,1)=V(3,0,1)=V,(3,0,1).
alive.
is
Keeping the option
However, the equity holder owns an
13
on
a different payoff: the returns to the
equity payoff pattern
equity
is
it
in
each event
less
not valuable to the firm as a whole.
to operate at a loss at
t
= l, j=2
in
This
the nominal debt obligation.
than the firm payoff pattern so that the option
riskier
owner even though
owner chooses
at
is
mine
is
valuable to the
Consequently, the equity
order to keep alive the hope of a large payoff
t=2, j=3.
It is
the firm
therefore incorrect to value the outstanding corporate bonds under the assumption that
is
managed according
done when various techniques
The
to strategy
#3
as
was done
previous section and as
in the
is
often
for contingent claims pricing are applied to value corporate liabilities.
actual stochastic process that the value of the firm follows
is
not independent of the nominal
obligations specified in the liabilities because these nominal obligations affect the incentives of the
equity owners.
One
cannot therefore
first
specify or derive the stochastic process for the value of
the firm and then directly calculate the value of any arbitrary financial
liability
written against that
firm.
The
usual
method
for valuing the contingent claims can
be
incorporate the consequence of moral hazard: the modification
necessary to
first
specify the nominal obligations outstanding
to value the equity
under
all
possible strategies.
The
easily
modified to correctly
as follows.
is
on the
firm,
strategy that will
and then
be chosen
Once
In general,
is
it
is
it
is
necessary
the one which
this strategy
chosen
maximizes the return to the equity given those financial
liabilities.
becomes possible
equilibrium and possible to value the firm
as a whole:
to value
all
of the financial
liabilities in
Define M' e argmax E,(M,0,1), and
V; =
V,(M',0,1),
D; =
is
D,(M',0,1), and
it
E/ =
Ea(M\0,l).
While
this
modification
is
straightforward at the level of the logic of the algorithm, in fact
significantly frustrates the use of the
it
mathematical machinery with which these problems have been
14
solved to date.
that
is
It is
no simple matter
to translate this algorithm into a robust valuation technique
easily usable for a large class of problems.
insight into certain important questions as
4.
A Commodity
is
evidenced
in
possible to use
it is
in
which the firm promises to pay
only debt instrument outstanding on the firm, and assuming
we can
previous example, then
D,',
=
calculate
the value of the
and the value of the equity,
E,',
all
at
#3 and #4
strategies
0.1231
s(2,j).
The important
thing to note
liability
abandonment
is
strategy.
this
V;=
$4.24
is
commodity linked bond
linked debt
greater than the firm value
bond, D/, and therefore
is
is
is,
firm,
V', the value of this debt
higher
when
when
abandonment
for
when
the
strategy #3, the first-best
therefore, an optimal debt contract which
a fixed rate debt contract
say that the firm value
same
$3.20.
is
used,
The
V/=
firm
$4.18.
as the value of the fixed rate
is
the
is
increased by changing the type of debt
the firm has the commodity linked bond outstanding than
bond outstanding: E;= $3.26 > £„'=
results
3.
contract while holding constant the value of the debt outstanding at t=0.
is
The
fixed rate debt contracts for this firm.
worth D,'= $0.98 which
we can
the
Table 3 Here]
problem associated with
This commodity linked bond
is
other values as given for the
that the optimal strategy for the equity holders
The commodity
resolves the moral hazard
value
is
If this
using the hedging portfolio constructions used
are displayed in Table
[Insert
outstanding
some
time t=2 an amount
above and acknowledging the moral hazard problem as discussed above.
abandonment
to gain
the next section.
contingent upon the realized price of the commodity: p(2,j)
instrument,
it
Linked Bond as an Optimal Debt Instrument
Consider a corporate debt obligation
that
is
Nevertheless,
The value of
when
it
the equity
has the fixed rate
15
The commodity
linked
financial incentives for
linked
bond increases the value of the
abandonment when the
bond outstanding
it
is
no longer
price of the
commodity
is
low.
beneficial for the equity holder to
option open at t=l, j=2 and therefore the equity holder
The
firm by changing the equity owner's
will
With the commodity
keep the abandonment
implement the
first
best strategy #3.
increase in the firm value that follows from this accrues to the equity.*
worth emphasizing that
It is
this result
could not be obtained using the conventional techniques
of contingent claims analysis, precisely because these techniques do not recognize the moral hazard
problems associated with various financial
variety of contingent claims that
now
liabilities.
exist--for
The techniques
that
dominate
in
valuing the
example the models of Schwartz (1982) and Rajan
(1988) developed for valuing commodity linked debt-ignore the impact that the issuance of a given
type of security has on the stochastic process which describes the firm's value.
before, while the logic of the algorithms
is
easily reconstructed to incorporate this
mathematical machinery with which the original algorithms have beeii so
so easily reconstructed.
Even once
a
method
is
in future
problem, the
fruitfully applied
devised to value a given set of financial
acknowledging the problem of moral hazard, a more significant problem
by which to find the optimal set of financial
As mentioned
liabilities.
It is
lies in
to these tasks that
not
liabilities,
devising a
we
is
means
intend to return
research on this problem.
Although the particular commodity linked bond that we have analyzed here achieves the
There is a large set of
it is not the uniquely optimal commodity linked instrument.
commodity price contingent debt securities that would also achieve the first-best. In the case at
hand, another debt instrument with an option feature would have also achieved the first best. In
fact, one would not generally want to make the payment due contingent only upon the final
realized price, as we have done in the case of the commodity linked bond used in this example.
One would rather have the total value of the promised payment contingent upon the full path of
The key
prices in such a fashion that it hedged the total value of production from the mine.
characteristic of any optimal debt contract for our problem would be that it would lower the
required payment in the worst outcome, s(2,4), so that the equity holder would not be tempted to
*
first-best,
keep the mine open through time
f)eriod
t
= l.
16
References
Brennan, Michael and Eduardo Schwartz, 1985, "Evaluating Natural Resource Investmens," Journal
of Business 58:135-157.
.
Cox,
J.,
S.
Ross and M. Rubinstein, 1979, "Option
Economics 7:229-63.
Financial
Pricing:
A
Simplified Approach," Journal of
,
Jensen, Michael, 1986, "Agency Costs of Free Cash Flow, Corporate Finance and Takeovers,
American Economic Review 76:323-29.
,
An
Lessard, Donald, 1977, "Commodity-Linked
Bonds from Less-Developed Countries:
Opportunity?" Proceedings, Seminar on the Analysis of Security Prices
Chicago.
,
Investment
University of
Merton, Robert, 1974, "On the Pricing of Corporate Debt: the Risk Structure of Interest Rates,"
Journal of Finance 29:449-70.
.
Rajan, Raghuram, 1988, "Pricing
Commodity Bonds Using Binomial Option
and Research Working Paper, The World Bank.
Pricing," Policy Planning
Schwartz, Eduardo, 1982, "The Pricing of Commodity-Linked Bonds," Journal of Finance 37:525,
39.
Figure
1:
the Decision Tree and Associated Parameter Values
t^
t=2
Table
1:
Contingent Claim Valuations of the Firm
Model Input Parameters
s(0,l)= 10.00
u1.25
d0.80
rc=
A-
0.12
0.12
8.65
Model Results
t.j
0,1
s(t,j)
f(t,j)
V(l,t,j)
V(2,t,j)
V(3,t.j)
V(4,t,j)
V(5,t,j)
Table
2:
Fixed Rate Debt- -Firm. Equity and Debt Values
Model Input Parameters
Table
3:
Commodity Linked Debt- -Firm. Equity and Debt Values
Model Input Parameters
s(0,l)= 10.00
u=
1.25
d0.80
r0.12
c=
0.12
A8.65
p=
0.1231 s(2,j)
Model Results
t.j