9 A case study: The Argentine Par bond
Tải bản đầy đủ - 0trang
credit risk Chapter 40
Figure 40.5 Speciﬁcations of the Argentinian Par bond.
and to be uncorrelated with the spot interest rate. In this model the risk of default is mean
reverting. I have not included any interest rate dependence in this because, provided f > j 2 /2,
this precludes the possibility of negative risk of default. It also makes the yield-curve ﬁtting
very simple because of the decomposition of the present value of each cashﬂow into the product
of risk-free bond value and a function of just p and t. In other words, an interest rate model is
not needed if a risk-free yield curve is given. The speed of reversion is determined by h. The
parameters were chosen to be h = 0.5, f = 0.045 and j = 0.03. This choice was made partly
so that the resulting time series for the implied p had the right theoretical properties (deduced
from (40.6)) and partly using common sense.
Finally, the value for p was chosen daily so that the market price of the bonds and their
theoretical price coincided.
The period chosen (end December 1993 to end September 1996) is a particularly exciting
one because of the ‘Tequila Effect’ and it could easily be argued that there was a dramatic
change of market conditions (and hence model parameters) at that time. However, I have kept
the same parameters for the whole of this period since it was risk of default causing the Tequila
effect and this should therefore be accounted for in these parameters.
659
Part Four credit risk
The Tequila effect took place in December 1994 but its consequences lasted much longer,
in some countries up to three and four months. In the case of Argentina, we can observe in
Figure 40.6 the minimum price of the Par bond at the end of March 1995, dipping below $35.
Since then a steady recovery can be seen.
In Figure 40.7 we can observe that the Tequila effect was accompanied by a sharp increase
in long rates in the US, which knocked Brady bond prices even further. The highest long rate
over this period was 8% and it occurred in March 1995.
80
70
60
Market price
50
40
30
20
10
0
19/08/1993
07/03/1994
23/09/1994
11/04/1995
28/10/1995
15/05/1996
01/12/1996
Time
Figure 40.6 Market price of the Argentinian Par bond from end December 1993 to end September
1996.
1
5
Maturity
7/10/96
4/17/96
1/24/96
8/9/95
5/17/95
2/22/95
11/1/95
Date
11/30/94
9/9/94
6/17/94
30
3/25/94
Yield
9
8
7
6
5
4
3
2
1
0
12/31/93
660
Figure 40.7 US yield curve from end December 1993 to end September 1996.
credit risk Chapter 40
0.5
0.45
Implied risk of default
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
19/08/1993
07/03/1994
23/09/1994
11/04/1995
28/10/1995
15/05/1996
01/12/1996
Figure 40.8 The implied risk of default for the Argentinian Par, see text for description of the
stochastic model.
In Figure 40.8 is shown the implied instantaneous risk of default for Argentinian Par bonds
over the period end December 1993 to end September 1996. As expected, the highest probability
of default took place at the end of March 1995 when the Tequila Effect was at its worst. Since
then there has been a steady, but obviously not monotonic, decrease in the risk of default
implied by this model.
40.10 HEDGING THE DEFAULT
In the above we used riskless bonds to hedge the random movements in the spot interest rate.
Can we introduce another risky bond or bonds into the portfolio to help with hedging the default
risk? To do this we must assume that default in one bond automatically means default in the
other.
Assuming that the risk of default p is constant for simplicity, consider the portfolio
=V −
Z−
1 V1 ,
where both V and V1 are risky.
The choices
1
V
=
V1
and
=
V1
∂V
∂V1
−V
∂r
∂r
∂Z
V1
∂r
661
662
Part Four credit risk
eliminate both default risk and spot rate risk. The analysis results in
∂V
∂ 2V
∂V
+ 12 w2 2 + (u − λw)
− (r + λ1 (r, t)p)V = 0.
∂t
∂r
∂r
Observe that the ‘market price of default risk’ λ1 is now where the probability of default
appeared before, thus we have a risk-neutral probability of default. This equation is the riskneutral version of Equation (40.4).
Can you imagine what happens if the risk of default is stochastic? There are actually three
sources of randomness:
• the spot rate (the random movement in r);
• the probability of default (the random movement in p);
• the event of default (the Poisson process kicking in).
This means that we must hedge with three bonds, two other risky bonds and a risk-free bond,
say. Where will you ﬁnd market prices of risk? Can you derive a risk-neutral version of Equation
(40.5)?
40.11 IS THERE ANY INFORMATION CONTENT
IN THE MARKET PRICE?
A bond has a nominal value of 100, an amount to be received in one day’s time. Yet the value
in the market is 96. What does this mean? The implied p from this is an astronomical 1000%.
Conclusion: Some people in the market know something. Default is certain.
Assume that a certain fraction of the market ‘knows’ what a mess the company is in. Call
that small fraction . Assume that this fraction of the market assigns a value of zero to the
bond, while the remaining 1 − give it the default-free value. Equate the average value with
the value in the market to get
× 0 + (1 − ) × e−r(T −t) = e−(r+p)(T −t) .
From this we get
p=−
log(1 − )
≈
T −t
T −t
if
is small.
Don’t worry about the details of this concept, just observe how p grows inversely with time to
maturity.
40.11.1
Implied Hazard Rate and Duration
There is an interesting relationship between the implied hazard rate and the duration of a
coupon-bearing bond.
Again, suppose that a fraction of the market are ‘in the know.’ Equate the average value
with the market value assuming a constant hazard rate model
N
× 0 + (1 − ) ×
i=1
ci e−r(ti −t) =
N
ci e−(r+p)(ti −t) .
i=1
(The notation is obvious, and I’ve lumped the principal in as just another coupon.)
credit risk Chapter 40
If p is small we can write the right-hand side as
N
ci e
−r(ti −t)
i=1
N
−p
ci (ti − t)e−r(ti −t) + · · · .
i=1
Equating the two sides we ﬁnd that p is just
assumed risk free.
divided by the duration of the bond when
40.12 CREDIT RATING
There are many Credit Rating Agencies who compile data on individual companies or countries
and estimate the likelihood of default. The most famous of these are Standard & Poor’s and
Moody’s. These agencies assign a credit rating or grade to ﬁrms as an estimate of their
credit-worthiness. Standard & Poor’s rate businesses as one of AAA, AA, A, BBB, BB, B,
CCC or Default. Moody’s use Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C. Both of these companies
also have ﬁner grades within each of these main categories. The Moody grades are described
in Table 40.2.
In Figure 40.9 is shown the percentage of defaults over the past eighty years, sorted according
to their Moody’s credit rating.
The credit rating agencies continually gather data on individual ﬁrms and will, depending
on the information, grade/regrade a company according to well-speciﬁed criteria. A change of
rating is called a migration and has an important effect on the price of bonds issued by the
company. Migration to a higher rating will increase the value of a bond and decrease its yield,
since it is seen as being less likely to default.
Clearly there are two stages to modeling risky bonds under the credit-rating scenario. First
we must model the migration of the company from one grade to another and second we must
price bonds taking this migration into account.
Figure 40.10 shows the credit rating for Eastern European countries by rating agency.
Table 40.2 The meaning of Moody’s ratings.
Aaa
Aa
A
Baa
Ba
B
Caa
Ca
C
Bonds of best quality. Smallest degree of risk.
Interest payments protected by a large or stable margin.
High quality. Margin of protection lower than Aaa.
Many favorable investment attributes. Adequate security of principal and
interest.
May be susceptible to impairment in the future.
Neither highly protected nor poorly secured. Adequate security for the
present.
Lacking outstanding investment characteristics. Speculative features.
Speculative elements. Future not well assured.
Lack characteristics of a desirable investment.
Poor standing. May be in default or danger with respect to principal or
interest.
High degree of speculation. Often in default.
Lowest-rated class. Extremely poor chance of ever attaining any real
investment standing.
663
664
Part Four credit risk
Figure 40.9 Percentage of defaults according to rating. Source: Moody’s. Reproduced by permission of Moody’s Investors Services.
Figure 40.10
Ratings for Eastern European countries by rating agency. Source: Bloomberg L.P.
credit risk Chapter 40
40.13 A MODEL FOR CHANGE
OF CREDIT RATING
Company XYZ is currently rated A by Standard & Poor’s. What
is the probability that in one year’s time it will still be rated
A? Suppose that it is 91.305%. Now what is the probability
that it will be rated AA or even AAA, or in default? We can
represent these probabilities over the one year time horizon by
a transition matrix. An example is shown in Table 40.3.
This table is read as follows. Today the company is rated A. The probability that in one
year’s time it will be at another rating can be seen by reading across the A row in the table.
Thus the probability of being rated AAA is 0.092%, AA 2.42%, A 91.305% etc. The highest
probability is of no migration. By reading down the rows, this table can be interpreted as
either a representation of the probabilities of migration of all companies from one grade to
another, or of company XYZ had it started out at other than A. Whatever the grade today, the
company must have some rating at the end of the year even if that rating is default. Therefore
the probabilities reading across each row must sum to one. And once a company is in default,
it cannot leave that state, therefore the bottom row must be all zeros except for the last number
which represents the probability of going from default to default, i.e. 1.
This table or matrix represents probabilities over a ﬁnite horizon. But during that time a
bond may have gone from A to BBB to BB; how can we model this sequence of migrations?
This is done by introducing a transition matrix over an inﬁnitesimal time period. We can model
continuous-time transitions between states via Markov chains.
We will model migrations over the short time period from t to t + dt. Since this time period
is very short, the chance of any migration at all is small. The most likely event is that there is
no migration. I am going to scale the probability of a change of state with the size of the time
step dt; any other scaling will lead to a meaningless or trivial model. If the transition matrix
over the time step is Pdt then I can write
Pdt = I + dt Q,
for some matrix Q and where I is the identity matrix. The sum of the entries in each row
of Q must sum to zero, and the bottom row must only contain zeros since default is an
absorbing state. I will use P(t, t ) to denote the transition matrix over a ﬁnite time interval from
t until t .
Table 40.3
AAA
AA
A
BBB
BB
B
CCC
Default
An example of a transition matrix
AAA
AA
A
BBB
BB
B
CCC
Default
0.90829
0.00665
0.00092
0.00042
0.00039
0.00044
0.00127
0
0.08272
0.90890
0.02420
0.00320
0.00126
0.00211
0.00122
0
0.00736
0.07692
0.91305
0.05878
0.00644
0.00361
0.00423
0
0.00065
0.00583
0.05228
0.87459
0.07710
0.00718
0.01195
0
0.00066
0.00064
0.00678
0.04964
0.81159
0.07961
0.02690
0
0.00014
0.00066
0.00227
0.01078
0.08397
0.80767
0.11711
0
0.00006
0.00029
0.00009
0.00110
0.00970
0.04992
0.64479
0
0.00012
0.00011
0.00041
0.00149
0.00955
0.04946
0.19253
1
665
666
Part Four credit risk
40.13.1
The Forward Equation
By considering how one can change from state to state during the time step dt and the relevant
probabilities we ﬁnd that the relationship between P(t, t ) and Pdt is simply
P(t, t + dt) = P(t, t )Pdt .
In terms of Q this is
P(t, t + dt) = P(t, t )(I + dt Q).
Subtracting P(t, t ) from both sides and dividing by dt we get
∂P(t, t )
= P(t, t )Q.
∂t
This ordinary differential equation is the forward equation and must be solved with
P(t, t) = I.
The solution of this matrix equation for constant Q is
P(t, t ) = e(t −t)Q .
(40.7)
The exponential of a matrix is deﬁned via an inﬁnite sum so that
e(t −t)Q =
∞
i=0
1
(t − t)i Qi .
i!
We can use Equation (40.7) in several ways. First, suppose that at time t = 0 company
XYZ is rated A. Supposing that we know Q, how can we ﬁnd the probability of being in any
particular state at the future time T ? This is simple. We just need to ﬁnd the third row down
in the matrix P(0, T ), with ei to denote the row vector with zeros everywhere except in the ith
column, corresponding to the initial state. In our case i = 3. The answer to the question is
ei P(0, T ) = ei eT Q .
Another way to use the solution of the forward equation is to deduce the matrix Q from the
transition matrix over a ﬁnite time horizon. In other words, we can solve
eT Q = P(0, T )
for Q. Why might we want to do this? One reason is that some rating agencies, and other ﬁrms,
publish the transition matrix for a time horizon of one year, for example Table 40.3. If you
want to know what might happen for shorter timescales than that (and you believe the one-year
matrix) then you should ﬁnd Q.
Suppose that we can diagonalize the matrix Q in the form
Q = MDM−1 ,
credit risk Chapter 40
where D is a diagonal matrix. If we can do this then the entries of D are the eigenvalues of Q.
We can then write
∞
P(0, T ) = eT Q =
i=0
∞
=
i=0
∞
=
i=0
1 i i
T Q
i!
1 i
T (MDM−1 )i
i!
1 i
T (MDM−1 ) . . . (MDM−1 )
i!
i
∞
=M
i=0
1 i i −1
T DM .
i!
But since D is diagonal, when it is raised to the ith power the
with each diagonal element raised to the ith power:
i d i 0
d1 0 0 0 0
1
0 di
0 d2 0 0 0
2
Di = 0 0 d3 0 0 =
0 0
0 0 0 d
0 0
0
4
0 0 0 0 d5
0 0
result is another diagonal matrix
0
0
d3i
0
0
0
0
0
d4i
0
0
0
0
.
0
d5i
From this it follows that
P(0, T ) = MeT D M−1 ,
where eT D is the matrix with diagonal elements eT di . The eigenvalues of the two matrices
P(0, T ) and Q are closely related. The strategy for ﬁnding Q is to ﬁrst diagonalize P(0, T ) to
ﬁnd M eT D , from which it is a simple matter to determine the matrix Q.
40.13.2
The Backward Equation
The backward equation, which has a similar meaning to the backward equation for diffusion
problems, can be derived in a similar manner. The equation is
∂P(t, t )
= −QP(t, t ).
∂t
(40.8)
40.14 THE PRICING EQUATION
Having built up a model for rating migration, let’s look at how to price risky bonds. We will
concentrate on zero-coupon bonds. In the previous section we derived forward and backward
equations for the transition matrix. The link between the backward equation and contract prices
in the Brownian motion world is retained in the Markov chain world, so I will skip most of
the details.
667
668
Part Four credit risk
40.14.1
Constant Interest Rates
The price of the risky bond depends on the credit rating of the company. We will therefore
need one value per rating. The column vector V will have as its entries the bond value for each
of the credit states. Assuming for the moment that interest rates are constant, this vector will
be a function of t only. In the same way that the value of an option is related to the backward
equation for the transition density function, we now have the following equation for the bond
value:
dV
+ (Q − rI)V = 0,
dt
This is just the backward equation (40.8) with an extra discounting term. The ﬁnal condition
for this equation is
V(T ) = 1,
where 1 is the column vector consisting of 1s in all the rows, except the last where there is
a zero.
How does the equation change if there is a recovery on default?
Where would the market price of risk appear if you were able to hedge against change
of rating?
40.14.2
Stochastic Interest Rates
The extension to stochastic interest rate is quite straightforward. The governing equation is
∂V
∂V 1 2 ∂ 2 V
+ (α − λβ)
+ 2β
+ (Q − rI)V = 0,
2
∂t
∂r
∂r
40.15 CREDIT RISK IN CBS
The risk of default can be very important for the convertible bond (CB), discussed in some
depth in Chapter 33 but with little reference to credit issues. The CB is like a bond in that it
pays its owner coupons during its life and a principal at maturity. However, the holder may
convert the bond at speciﬁed times into a number of the underlying stock. This feature makes
the CB very like an option. The reader is referred to Chapter 33 for all the details and the
notation.
We can combine the ideas in the present chapter with those from Chapter 33 to derive a
model for CBs with risk of default priced in. I will present two possible approaches.
credit risk Chapter 40
40.15.1
Bankruptcy when Stock Reaches a Critical Level
We can model the default of the issuing company by saying that should its stock fall to a level
Sb then it will default. Such a model is similar in spirit to that described in Section 39.2. We
only need to add the condition
V (Sb , t) = 0,
to our favorite (no credit risk) CB model.
40.15.2
Incorporating the Instantaneous Risk of Default
Another possibility, in line with the instantaneous risk of default model described
above, is to have an exogenous default triggered by a Poisson process, as in
Section 40.4. In a two-factor CB setting we arrive at
∂V
∂ 2V
∂ 2V
∂ 2V
+ 12 σ 2 S 2 2 + ρσ Sw
+ 12 w2 2
∂t
∂S
∂S∂r
∂r
∂V
∂V
+ (r + p)S
+ (u − λw)
− (r + p)V = 0.
∂S
∂r
In this model we have stochastic r and stochastic S. We can have p as constant, a
known function of time, or even a known function of r and S.
This approach has the advantage that it reduces to common market pricing practice in the
absence of any stock dependence. This allows us to price instruments in a consistent fashion.
When the stock price is very small the above model will yield a CB price that is close to
the price of a non-convertible bond. It is market experience, however, that in such a situation
the price of the CB falls dramatically. The market sees the low stock price as an indicator of
a very sick company. This can be modeled by having the instantaneous risk of default being
dependent on the stock price, p(S). If p goes to inﬁnity sufﬁciently rapidly as S → 0 we ﬁnd
that the CB value goes to zero. Note that there is no more effort involved, computationally, in
solving such a problem since we must anyway solve for the CB value as a function of S. In
Figure 40.11 is shown the value of a CB with and without credit risk taken into account in a
one-factor model, with deterministic interest rates.
In Figure 40.12 is shown the output of a two-factor model for the CB with risk of default.
The interest rate model was Vasicek, ﬁtted to a ﬂat 7% yield curve.
40.16 MODELING LIQUIDITY RISK
Liquidity risk affects the price of a bond through the yield spread. The yield spread is the
difference between the bid and ask price expressed as a percentage of the mid-price. For
example, if the bid price is $0.97, and the ask price is $1.03 so that the mid-price is $1.00 then
the yield spread is 6%. Measuring the yield spread in percentage terms rather than dollar terms
ensures consistency between bonds with different prices.
A bond is liquid if it is traded in large volume. There are many buyers and sellers and so the
dealer can easily match both sides of the transaction. The uncertainty faced by dealers when
matching buyers to sellers is small, leading to a relatively small bid/ask spread.
669