fic
CS
FIRSTBOSTON
April 26, 1995
FIXED INCOME RESEARCH BULLETIN
Implied Default Probabilities: A New Approach to Risky Debt
Clayton Perry
212/909-2682
Summary
Implied default probabilities are a useful mechanism for evaluating risky
debt. Rather than discounting promised cashflows at a high rate which reflects
the danger that they will not be received, we have more directly identified the
implied default probability in a set of Argentinian eurobonds by discounting
expected cashfiows at the risk-free rate. Furthermore, we have removed some
of the apparent inconsistencies inherent in assuming different probabilities
for similar bonds in the same time period. Thus we, have generated a set of
probabilities against which investors can measure their own opinions about
the riskiness of the securities to decide upon relative value.
This year has witnessed turmoil in emerging debt markets on a scale unseen since
the Brady restructuring. Following the Mexican devaluation in December, shock
waves spread as far afield as Poland and Thailand as investors fled for the safety of
Switzerland and Germany. In an environment characterized by this sort of "risk
reduction at any price" it is plausible that different risks may be priced
inconsistently, or indeed that equivalent risks offer widely differing returns.
Our approach to identifying such possibilities is to analyze the probability of
default which investors have priced into each of a set of similarly risky bonds. The
result is merely a different way of stating relative value, but has the advantage of
expression in terms of the specific risk that investors are concerned with - the
chances that they will get their money back. The statistic we calculate is the rate of
default on the bond which would give an expected return (net of probability
weighted credit losses) equal to the risk free rate of return. We have taken as our
sample the non-Brady dollar bonds issued by Argentina. These have been favored
for this exposition principally because there are five bonds of the same class with a
reasonably regular maturity schedule. Later bulletins will deal with Brady bonds
and other classes of risky debt.
This memorandum is for informative purposes only. Under no circumstances is it to be used or considered as an offer to sell, or a solicitation of an offer to buy any
security. Additional information is available upon request. Results based on hypothetical projections or past performance have certain inherent limitations. There is
no certainty that the parameters and assumptions used can be duplicated with actual trades. Any results shown do not reflect the impact of commissions and/or
fees, unless stated. Financial futures and options are not appropriate for all investors. Their relative merits should be carefully weighed. Where information has
been obtained from outside sources. It is believed to be reliable but is not represented to be accurate or complete. This document may not be reproduced in whole
or in part or otherwise made available without our written consent This firm may from time to time perform investment banking for, or solicit investment banking or
other business from any company mentioned. We or our employees may from time to time have a long or short position in any contract or security discussed.
Fixed Income Research
Methodology
Suppose you are due to receive a payment in one year's time. If you are 100% sure
you will collect, it is sensible to discount that payment at the relevant risk free rate,
the 1 -year U.S. Treasury bill yield. The less sure you are of receiving the payment
the higher will be the discount rate applied and the less will be the present value of
the payment. Now suppose you are 100% certain of getting either payment x or
payment y, where y is a smaller amount. If you discount both at the risk free rate
you get two net present values, one of which will be correct. To value this "either
or" payment the most logical thing to do is to assign probabilities to each of
payment x and payment y and take an average of the two NPVs weighted by the
assigned probabilities.
In our example, payment x represents the cashflow scheduled on a bond, while
payment y represents the recovery value - the amount the bond is worth in the
event of a default. The market price of the bond in its final coupon period will
represent the probability weighted average - or expected value - of these two
amounts discounted at the risk-free rate. This is represented algebraically as
follows:
c (1 - d) + Rd
P= _____________
1+r
where
P = dirty price of bond
c = cashflow
d = the probability of default in this period
R = recovery proceeds in event of default
r = risk free discount rate
For example, suppose a risky 1-year zero coupon bond is trading at a price of 83.33, giving a
yield of 20%. If the discount rate on the 1 -year U.S. Treasury bill is 5% and the assumed
recovery rate on the risky bond is 30%, then the equation becomes;
100 (1 - d) + 30d
83.33 = ___________
1.05
Solving for the default rate d gives a market implied default probability of 17.85%.
If the bond has two cashflows remaining, the price should equal the sum of the two discounted
expected values. However the calculation of the second expected value is slightly more
complicated because we must allow for the fact that the bond may default in the first coupon
period. Therefore our weights are now the default probability in the second period conditional on
making it through the first period. This conditionality is expressed by multiplying both terms by
the probability of no
2
Fixed Income Research
default in the first period (1-d). We can keep extending this for n scheduled cashflows in the
same manner. Note that each period's probabilities must be multiplied by the probability of no
default in all the previous periods.
If we assume that the probability of default in each of the individual coupon periods is the
same we can collapse this eq~ition to the following;
This is recognizable as an extension of the simple yield to maturity calculation which one
would normally solve for the discount rate, r, from the known price and cashflow schedule.
This discount rate or yield would usually reflect opinions of the riskiness of the cashflow
schedule. Our approach separates this discount factor from the risk weighting by using
the risk free discount rate from the U.S. Treasury spot curve and confining
expectations to the top line of the equation. If we assume a recovery value, we can back
out a value for d by iteration in the same manner as one would conventionally calculate yield
to maturity by iterating for r.
The methodology we have employed is set out diagrammatically below. The probability of
getting to a point on this binomial tree is the product of the probabilities identified on each
branch taken.
Exhibit 1. Binomial Tree of Default Probabilities
Cumulative
Default
Probability
The calculations we have described above can be solved for d to give the default
probability assumed for each single period. If a borrower has debt with many
maturities outstanding or has debt which capitalizes interest or amortizes, the per
period default rate can be especially useful. For many issuers however, of more
3
Fixed Income Research
interest is the cumulative default probability, the chance that a bond will not make it to a
particular point on our binomial tree. For instance, most investors will be interested in the
probability that their bond will not default before maturity. This is the probability that we
will get to the bottom right box on our binomial tree, which is just the product of all the
individual per-period probabilities of survival.
(1+d1)(1-d2)…(1-dn)
Again if we assume constant per-period probabilities, the probability of no default up to and
including period n is
(1-d) n
This means that the cumulative probability of default will be
1 - (1 - d) n
Assessing a
Recovery Value
The assessment of the recovery value of the bond in the event of default is both
difficult and crucial to the results. A corporate bond should have some value in
default from the claims that bond-holders have on the assets of the company.
Secured debt presents few problems, while unsecured debt will have a claim on
unencumbered assets which will presumably be sold at some discount to full value
in the case of liquidation. Assessing the price these assets would fetch in a
situation of distress, along with the costs of liquidating the company will present
some problems, but the degree of accuracy should be reasonable.
Sovereign debt, however, presents greater difficulty. Creditors are not in a position
to close down the country and sell off roads, bridges and dams. On the other hand,
a promise to pay a future series of cashflows and even past obligations in default
probably means more from a sovereign state than from a corporate entity.
Therefore it is difficult to imagine such debt ever trading at zero. Even the
defaulted paper of Liberia, one of the world's poorest countries, trades at around
2.5% of face value.
This is confirmed by historical analysis. Prior to the most recent Brady
restructuring, defaulted Latin American debt traded at prices between 28.88
(Brazil) and 46.50 (Argentina). Of course, these prices no doubt took account of
the high probability of restructuring, and therefore were somewhat more generous
than we should use here.
We approach this on the basis that defaulted paper would have a value relative to
the cashfiows still to be paid on the bond. Investors might assume a payment rate,
for example, of 30 cents on the dollar and take the present value of that percentage
of the remaining cashflows as the price of the defaulted paper. This is the amount
that would be realizable on the day of default, and accordingly constitutes our R.
The effect of this is that R is not constant but changes throughout the life of the
4
C S FIRSTBOSTON
Fixed Income Research
bond, which appears more theoretically consistent than a constant number. Of
course we do not know what discount rates will be prevailing at each coupon date,
and since we are valuing everything today we are interested in the discounted
value of each of the R's
where:
t
a = expected percentage payout in event of default.
Our example, Argentina, has a well-developed infrastructure and is well endowed with natural
resources. Outstanding debt is only 34% of GDP. This would lead us to believe that a figure of
around 30 cents on the dollar would be a reasonable investor expectation as to the post-default
payout on Argentina paper. Due to the arbitrary nature of such an estimate and the sensitivity of
our calculations to it, we have also derived the implied probabilities associated with payout rates
of 15% and 45%.
Some of these results deserve comment. Implied per-period default probabilities in general
increase with maturity, a fact we deal with further below. Note that the 1997, 1999 and 2000
bonds all have very similar rates with a big jump out to 2003. This is more apparent from the plot
below of the "implied default curve."
Fixed Income Research
The main feature of the table is that the numbers appear fairly small. However, when we look at
the cumulative default rate implied, the magnitude rises rapidly. This is the probability that a
bond will default before the given date, in this case its maturity. The results are set out below.
This gives a clearer view of where the market is placing its bets. Naturally the longer is a
bond's maturity, the greater the chance it will default before maturity. Note that even in the
case of the smallest recovery value investors believe there is nearly a half chance that the Dec
2003 bond will not pay its full schedule of cashflows.
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Fixed Income Research
Extending the
Model to NonConstant
Default Rates
The principal problem with this approach is one of inconsistency. Because we are
assuming constant per period default rates throughout the life of a bond, we get the
anomalous result that the probability of default by the same issuer in the same
period will be different when measured on different bonds. Of course this may be a
result of inconsistent market behavior, but equally it may show a maturity effect.
In the same way as yield-to-maturity discounts all cashflows at a uniform rate in
spite of a non-constant term structure, the assumption of constant per period
default rates over time may be unreasonable. Investors may be more certain of a
country's economic prospects in the short term than a few years from now, or a
scheduled election could increase default probability in its wake. As for corporate
issuers, one can envisage a new product launch or outcome to litigation which may
dramatically alter their credit quality.
Therefore we have extended the model to take account of the possibility of
variance in default rates over time. The methodology is merely to assume that the
shortest bond determines the implied default rate for its lifetime, and that the
cashflows on all other bonds during that period should be weighted by that
probability. The next longest bond then determines the probability until its
maturity and so on. Thus if a bond matures after 3 coupon periods we would work
out the constant per-period default probability for that security according to our
first method. This would give us d1, d2 and d3 for every bond on the curve. The
equation for the next bond would use those three rates as given, and we would then
be able to back out a constant value of d for the remaining periods in that bond.
This process can be repeated for every bond to give the "marginal" per-period
default rate for each; that is, the rate of d that applies for the final periods of that
bond's life. In this way we can "bootstrap" a credit curve in much the same way
that a series of coupon bonds can be used to generate a spot curve in yield terms.
The restriction is that we are only dealing with a handful of issues, and thus can
only vary the default probability a few times. Nevertheless this will give a
reasonable indication of the shape of the "credit curve" and may show up arbitrage
opportunities.
Mathematically this introduces non-constant default rates. The equation now
becomes more complicated but still manageable:
Fixed Income Research
Because we are (theoretically) varying the default rates every period, we lose the ability to
use exponential terms and must introduce product notation. This merely harks back to our
binomial tree and the notion of multiplying every term by all the probabilities that must be
satisfied to reach that point.
Thus for the 15% recovery rate, the implied probability of default in each of the semi-annual
periods up to August 1996 (d1,d2,d3) is 3.1 %. Thereafter the implied rate becomes 4.6%
until July 1997 (d4,d5) and so on as shown in the table below.
This gives a more accurate picture of the time at which the market is betting Argentina will
fail on payments. By once more multiplying through these per-period probabilities, we can
generate a more consistent picture of cumulative probabilities as follows.
There are caveats in the interpretation of these numbers, as discussed below, but they provide
the
market's best estimate of when Argentina is most likely to default on its obligations.
The overall message is that if investors believe that there is less than a 50% chance that
Argentina will default in the next eight years, they can earn more premium than they require
over a riskless asset.
8
Fixed Income Research
We stress that there are simplifying assumptions employed in our model. Firstly we have abstracted
from all liquidity and tax effects in the market, assuming instead that our universe of bonds is
homogeneous in respect of these attributes. This does not seem too restrictive an assumption when
dealing with eurobonds issued in reasonable size. More importantly we have assumed no correlation
between U.S. interest rates and rates of default. Incorporate bonds this may not be easily defendable, but
the size of Argentina's debt burden relative to its GDP - 34% - means that rates would have to rise much
further than is likely in the next couple of years for the increase in debt servicing to have a significant
impact on Argentina's finances. By making these assumptions we have been able to reappraise these
securities in terms that are more directly relevant to the investors who buy them.
I
CS FIRST BOSTON
June 15,1995
FIXED INCOME RESEARCH BULLETIN
Implied Default Probabilities Part 2:
Application to Brady Bonds
Clayton Perry
212/322-5998
Summary
Following our earlier analysis of default probabilities priced into risky euro
bonds, the purpose of this paper is to adapt the methodology to collateralized
Brady bonds. By calculating the default probability, we can more accurately
value the various component parts of a Brady bond and thus assess the true
return to country risk. We illustrate this by showing the considerable differences between conventionally calculated stripped yields and those worked out
using the implied default probability method.
This paper references and extends the analysis developed in our earlier bulletin
Implied Default Probabilities, A New Approach to Risky Debt, CS First Boston
Fixed Income Research, April 1995. The methodology employed here is explained
in detail in that publication and recapped only briefly here.
In the previous paper we discussed the concept of market implied rates of default
and looked at applying this to different issues by the same borrower to create a
basic term structure of default probability. The aim of this report is to amend the
methodology slightly to apply to Brady bonds and to examine the valuation
implications across different issuers. We begin by revisiting the analysis outlined
previously.
Recapping the
Methodology
Suppose you are due to receive a payment in one year's time. If you are 100% sure
you will collect, it is sensible to discount that payment at the relevant risk free rate,
the 1-Year U.S. Treasury bill yield. The less sure you are of receiving the payment.
the higher will be the discount rate applied and the less will be the present value of
the payment. Now suppose you are 100% certain of getting either payment x or
payment y, where y is a smaller amount. If you discount both at the risk free rate
you get two net present values, one of which will be correct. To value this "either
This memorandum is for informative purposes only. Under no circumstances is it to be used or considered as an offer to sell, or a solicitation of an offer to buy any
security. Additional information is available upon request. Results based on hypothetical projections or past performance have certain inherent limitations. There is
no certainty that the parameters and assumptions used can be duplicated with actual trades. Any results shown do not reflect the impact of commissions and/or
fees, unless stated. Financial futures and options are not appropriate for all investors. Their relative merits should be carefully weighed. Where information has
been obtained from outside sources. It is believed to be reliable but is not represented to be accurate or complete. This document may not be reproduced in whole
or in part or otherwise made available without our written consent This firm may from time to time perform investment banking for, or solicit investment banking or
other business from any company mentioned. We or our employees may from time to time have a long or short position in any contract or security discussed.
Fixed Income Research
or" payment the most logical thing to do is to assign probabilities to each of payment x and
payment y and take an average of the two NPVs weighted by the assigned probabilities.
In our example, payment x represents the cash flow scheduled on a bond, while payment y
represents the recovery value - the amount the bond is worth in the event of a default. The
market price of the bond in its final coupon period will represent the probability weighted
average - or expected value - of these two amounts discounted at the risk-free rate. This is
represented algebraically as follows:
P=
c (I - d) +Rd
I+r
where
P = price including accrued interest
c = cash flow
d = the probability of defaults in this period
R = recovery proceeds in event of default
r = risk free discount rate
For example, suppose a risky 1-year zero coupon bond is trading at a price of 83.33, giving a
yield of 20%. If the discount rate on the 1 -year U.S. Treasury bill is 5% and the assumed
recovery rate on the risky bond is 30%, then the equation becomes;
83.33 = 100 (1 - d) + 30d
1.05
Solving for the default rate d gives a market implied default probability of 17.85%.
If the bond has two cash flows remaining, the price should equal the sum of the two
discounted expected values. However, the calculation of the second expected value is slightly
more complicated because we must allow for the fact that the bond may default in the first
coupon period. Therefore, our weights are now the default probability in the second period
conditional on making it through the first period. This conditionality is expressed by
multiplying both terms by the probability of no default in the first period (1-d). We can keep
extending this for n scheduled cash flows in the same manner. Note that each period's
probabilities must be multiplied by the probability of no default in all the previous periods.
Fixed Income Research
If we assume that the probability of default in each of the individual coupon periods is
the same we can collapse this equation to the following;
The methodology we have employed is set out diagrammatically below. The probability
of getting to a point on this-binomial tree is the product of the probabilities identified on
each branch taken.
Exhibit 1. Binomial Tree of Default Probabilities
The Brady
Difference
Brady bonds are distinguished from Eurobonds mainly by collateralization of
some payments. Under the terms of the security, the issuer has defeased the
principal with U.S. Treasury STRIPS of the same maturity, and collateralized
several coupon payments on a rolling basis with a high quality security (usually
AA or better). The actual terms of the Brady issues vary by issuer; we refer readers
who are unfamiliar with the market to An Introduction to Emerging Countries
Fixed Income Instruments, CS First Boston May 1993.
The simple way to incorporate this collateralization of some cash flows into our
model is to adjust the calculation of a recovery value, R, to reflect the inclusion of
the collateralized coupons and principal at their full discounted value. Using
Mexico as the example, this means that the recovery value for default in the sixth
coupon period would be the discounted value of the sixth, seventh and eighth
coupons, and the discounted principal. In general terms this is the following;
Note that we have assumed no recovery over and above the collateral. This is a
very important and somewhat arbitrary assumption, as the bond in default would
Fixed Income Research
have some value attributable to the promised coupon flow outside of the collateral,
and any judgement may award creditors more than just the collateral value.
However we justify our assumption on the grounds that it will represent the base
case, and that, having defeased some of the payments, the issuer is less likely to
feel compelled to pay out more than they already have.
Armed with this information, we can now work through the equation on the previous
page to back out the implied default rates for a series of Brady bonds. Most issuers
have a Discount and a Par bond, a floating rate and fixed rate 30-year bond
respectively, which have reasonably uniform structures across issuers and thus lend
themselves to cross-market comparison. For the floating rate Discount bonds, we
assume coupons to be fixed according to LIBOR forward rates.
The following table summarizes our findings.
Applications:
Pricing
Components
The collateralization of some cash flows allows the division of the price among the
various components as follows.
P = Pg + Pc + Pp
where
P = price including accrued interest
Pg = price of coupon guarantee
Pc = price of risky coupon stream
Pp = price of principal redemption
Fixed Income Research
Conventionally this is split as shown below. The first part is the collateralized (and thus certain)
first three coupons, the second the risky stream of coupons following the first three, and the last
part is the principal. By putting a value to the first and last pieces we can work out what price is
charged for the risky coupon stream (Pc) and thus generate the yield on that piece of the bond.
To start with the easiest, the principal of the bond is approximately equivalent to off-the-run
U.S. Treasury STRIPS. We say approximately because although the STRIPS secure the
redemption payment, these securities are not obtainable before maturity and cannot therefore
be traded like a normal security. However, for current purposes the principal is treated as
Treasury STRIPS, and thus valued as follows.
Pp=
100
(1+rn) n
The split between the first part of the bond, the coupon collateral, and the risky coupon stream
is the most interesting. Conventionally the approach has been to take the collateralized
coupons as certain cash flows and discount them at the risk free rate. This value (along with
the principal value above) is subtracted from the price and the remainder is the cost of the rest
of the coupon stream.
The problem with this approach, depicted in exhibit 3, is that it gives no value to the rolling
collateral past the first three coupons. It implies that in eighteen months, if there is no default,
you have an uncollateralized coupon stream when in fact you will still have a partially
collateralized coupon stream. Thus the conventional method clearly oversimplifies the impact
of the collateral unless the bond defaults in the first period.
We believe that the correct way to value the guarantee is to price it as a set of contingent cash
flows. Look again at our binomial tree on the following page. Each of the boxes represents a
certain cash flow at a certain point in time, and thus has a known discounted value.
Furthermore, from our earlier calculation we know the values for d that the market implies,
and thus the probability of getting each of the flows. The product of the discounted value and
the probability of receiving it gives the expected value, which is what the risk neutral investor
would be willing to pay for a contingent cash flow.
Fixed Income Research
Therefore the fair value of the guarantee (Pg) is the total expected value of the top half
of the tree, leaving the expected value of the bottom half of the tree as the price of the
scheduled coupon stream (P,).
Mathematically, the fair price of the coupon guarantee can now be expressed as follows:
Remember for the purposes of this calculation we have already subtracted the value of the
principal and we assume zero recovery of non-collateralized flows, so the recovery is defined:
This gives a value for the coupon strip as follows:
Fixed income Research
Now we can substitute these values for different parts of the bond (Pc, Pg and Pp)
back into the original equation and collapse terms to give the following:
This is, of course, our original, risky bond price equation with the principal expressed separately.
Having thus closed the loop, we can look at the current pricing of Brady bonds and allocate value
to each of the three components, as shown in the following table.
Applications:
The True
Stripped Yield
The reallocation of value in the price of a Brady bond has important implications
for measures of relative value. One of the most frequently used indicators in the
Brady market is stripped yield. This is the internal rate of return on the risky cash
flows, and is thus comparable to an uncollateralized Eurobond issued by the same
borrower.
In order to calculate the stripped yield, we must be able to fairly assign the
proportion of the price of the bond that is attributable to the risky coupon stream as
only then can we measure the true return to country-specific risk.
The following table sets out the true stripped yields offered on Brady's and
contrasts them with the conventionally calculated measure. The true stripped yield
is generally higher by around 100 b.p., despite the fact that the coupon guarantee is
generally worth less (and thus the coupon stream more) under our method. The
reason for this is that the inclusion in the stripped coupon stream of a probability
weighted portion of the first three cash flows outweighs the higher price paid.
Fixed Income Research
Other
Considerations
Through our previously discussed default probability methodology, we have been
able to more accurately capture the price of the different risks inherent in a Brady
bond. This, in turn, has allowed us to calculate a true stripped yield and thus a
more consistent measure of valuation.
This analysis assumes risk neutrality on the part of investors such that they are
willing to pay $1 for each dollar of expected present value. Moreover we have
assumed for this paper that in the event of default, there would be no payment
other than the collateral. This assumption is open to debate, but we note that
relaxing it would change the numbers given but not the validity of the analysis.
Finally, we earlier mentioned the liquidity problem associated with the principal
collateral. Although the redemption payment is defeased with Treasury STRIPS,
there is no way of breaking that out of the bond before the scheduled maturity.
Therefore, any arbitrage would rely on swapping that amount back to its present
value using a 30-year zero-coupon swap. In fact any realization of the expected
value of coupon payments would also rely on swapping them, so for arbitrage
purposes all flows should be discounted from the swap curve.
0~
Conditional default risk refers to the prospect of bond default in a certain period, assuming no prior default.
Unlike cumulative default risk, conditional default risk need not rise looking forward. Conventionally, however,
emerging market participants presume it does.
The presumption is reasonable for strong credits, because their quality has so much more scope to decay than to
improve. For weak credits, however, the presumption should be reversed. Over time, low grade bonds that do not
default are likely to improve. These arguments can be refined using the theory of Markov chains.
Empirically, default term structures for U.S. corporate bonds are rising for A ratings but steeply inverted for
single-B. A Markov chain approach explains the evidence remarkably well, including the humped term
structures of Ba and Baa credits.
The discrepancy between market assessment and risk suggests an attractive arbitrage opportunity for longer-term
investors: buy calendar spreads in lower-grade bonds.
Term Structure of Default Risk
Introduction
As bonds age, their default risks change. Viewed
from the date of purchase, each successive coupon
is less likely to be paid, so the cumulative default
risk rises. But we might also ask: how likely is the
next coupon to be defaulted, assuming no prior
default? This is known as the conditional default
risk. Its term structure indicates whether that risk
rises or falls over time.
In emerging markets, the term structure of default
risk is typically priced as if it were flat or rising.
This paper marshals evidence suggesting, on the
contrary, that the term structure for weak credits
naturally declines.1
The term structure of default risk matters to
investors, because it ought to influence interest
rates. As an illustration, suppose the conditional
default risks are 10 basis points (bp) today and 6
bp tomorrow, that there is no post-default payout,
and that risk-free interest rates run 2 bp per day.
A risk-neutral investor (i.e., willing to accept any
bet expected to break even) will charge 12 bp
interest for a one-day loan, because his expected
payback per dollar is 99.90% times $1.0012 or
roughly the risk-free gross return of $1.0002.
Similarly, he will charge an average 10 bp per day
for a loan repayable in two days, because his
expected return per dollar is 99.90% times 99.94%
times $1.0010 squared, or roughly the risk-free
gross return of $1.0004. The daily interest rate
falls even though the cumulative default risk rises.
Note in the preceding example that for the first
day, the required interest premium over the riskfree rate is 10 bp, the same as the default risk. For
the second day of the two-day loan, the required
future interest rate is 8 bp, or a future premium of
6 bp, again the same as the conditional default risk.
This point has been emphasized by, among others,
J.S. Fons. "Using Default Rates to Model the Term
Structure of Credit Risk," Financial Analysts Journal.
September-October 1994. pp. 25-32; and L.T. Nielsen,
J. Saa-Requejo, and P. Santa-Clara, "Default Risk and
Interest Rate Risk: The Term Structure of Default
Spreads". INSEAD Working Paper. May 1993.
For longer time periods, the required premium can
be shown to equal d(1+r)/(1-d), where d is the
conditional default rate and r is the future risk-free
rate. Incorporating positive post-default payouts
and risk aversion further complicates the
relationship, but higher future default risks are still
associated with higher future premiums over riskfree interest rates.
Conventional View of Forward Default Risk
In emerging markets, default risk is typically
priced as if it were (a) rising for all types looking
forward, and (b) steeper, the lower the initial credit
rating.
For example, in April 1995, one-year, five-year,
and eight-year Argentine dollar Eurobonds traded
respectively at 600 bp, 700 bp, and 775 bp spreads
over U.S. Treasuries. Assuming a post-default
payout ratio of 15% of face value, the implied
conditional default risks escalate to 10.1% for
years five through eight from 6.9% for year one.2
For comparison, the yield differential between
triple-A U.S. corporate bonds and corresponding
U.S. Treasuries typically rises from 10-15 bp on
two-year paper to 40-50 bp on 30-year paper.
Intuitively, it makes sense for triple-A interest
premiums to widen looking forward. No credit has
ever defaulted in a year it was rated triple-A.
However, some triple-A credits have defaulted
after having decayed to lower credit ratings. In
probability terms, they have nowhere to go but
down.
It also makes sense for default risks to initially rise
more quickly for, say, single-A credits than for
triple-A credits. Having large servicing cushions,
triple-A credits decay in the first instance to other
A-levels, where immediate default risks remain
low. Single-A credits tend to decay sooner to sub
investment grade, where propensities to default are
much higher.
Clayton Perry, Implied Default Probabilities: A New
Approach to Risky Debt. Fixed Income Research
Bulletin, CS First Boston. 26 April 1995.
Term Structure of Default Risk
The chart above on the default history of U.S.
investment-grade corporate bonds, distinguished by
letter rating out to a ten-year horizon, lends
support to this view. Each term structure is
generally rising. Moreover, the lower the initial
credit quality, the steeper the term structure.3
Revised View of Forward Default Risk
On reflection, the default term structure cannot rise
for the weakest credits. The latter are more likely
to default in the near-term, but they also have more
upside. Moreover, the longer they go without
defaulting, the more likely they are to have
improved in quality. For example, if a weak
sovereign credit like Bulgaria can service its Brady
bonds punctually for the next 15 years, the chances
are creditors will no longer demand the 1500 bp
default spreads they carry today. Note that this
result does not hinge on our belief that Bulgaria is
ultimately "emerging" rather than "submerging".
It is simply the observation that 15 years of
successful debt servicing is more likely than not to
indicate improved credit quality.
Conditional default risks calculated from data in
Moody’s Investors Service. Corporate Bond Defaults
and Default Rates 1970-1994, January 1995.
To strengthen the intuition for this result, imagine a
blindfolded man wandering on an unfenced roof.
The starting position corresponds to an initial
rating, while falling off the roof corresponds to
default. For a man starting close to the edge,
disaster may seem imminent. But if he has
survived for a while-without failing, the chances
are he has worked part way back to the centre, and
hence is less likely to fall in the next instant.
Conversely, a man starting at the centre is bound to
come closer to the edge, so his conditional risks of
falling must increase.
Mathematically, we are predicting a forward
default risk conditional on an initial rating and no
interim default. Over time, the interim history
counts for more, relative to the initial assessment.
Hence, the differential default risk between the
strongest and weakest credits shrinks over time.
To consider a numeric example, suppose there are
two credit qualities: Strong (S) and Weak (W).
Current S credits never default, while current W
credits default 10% of the time. From one period
to the next, assuming no prior or current default. W
credits have a 20% chance of switching to S, while
S credits have a 5% chance of switching to W.
Term Structure of Default Risk
Looking forward, the conditional default risk for S
is 0.5% (i.e., 5% chance of switching to W, which
has 10% default risk) for the second period and
0.875% (5%*80% = 4% chance of path S-W-W
plus 90%*5% = 4.75% chance of path S-S-W) for
the third. The corresponding risk for W is 8% for
the second period and 6.5% for the third. Hence,
default risks rise looking forward for currently
Strong credits, and decline looking forward for
currently Weak credits.
The Appendix refines these themes, using the
mathematical theory of Markov chains. Basically,
the evolution of credit quality is a Markov chain to
the extent that the current rating supersedes all
prior rating information. The analysis confirms
that conditional default risks ultimately tend to
converge. In the short run, however, with more
than two credit qualities, forward default risks can
rise and later fall or vice-versa.
For evidence, we turn again to the default record of
U.S. corporate bonds. The facing charts
incorporate sub-investment-grade credits into the
comparisons and extend the horizon to 20 years.
Note that, contrary to the conventional view, but
compatible with our revised view:
Time ultimately narrows the risk gaps implied
by different credit ratings. The average
difference in conditional default risks between
investment and speculative grades shrinks from
4.2% in the first year to 1.3% in the tenth and
0.5% in the twentieth.
* The default term structure is steeply inverted
for single-B. Conditional default risk drops to
4.9% in the fourth year from 7.9% in the first.
* The default term structure for Ba credits is
humped. The conditional default risk rises to
2.5% in years two through five from an initial
1.7%, then declines gradually to 1.0%.
To further test the revised view, the Appendix
explores how well a Markov chain can replicate the
observed default profiles. The fit is excellent.
This is not to claim that foreign bonds, particularly
sovereign bonds, exhibit the same default profiles
as U.S. corporates. While the rating agencies
strive to ensure that assigned grades are
comparable across issuer types, there is no
guarantee that they succeed, or even that it is
possible. Even for U.S. corporates, future default
rates may depart from historic values. Our claim
is rather that the qualitative patterns predicted by
theory and confirmed in one important data set especially, the inverted default term structure for
weak credits - are likely to recur in others.
From an investment perspective, the main practical
implication is to recommend calendar spreads on
long-dated bonds for emerging markets. A followup paper elaborates this them&.
Term Structure of Default Risk
Appendix: Default as a Markov Chain
A Markov chain is a sequence of transitions from
one state to another, where the conditional
probabilities of transition depend only on the
transition itself, not on history or timing. The
evolution of credit quality can be viewed as a
Markov chain in which the states are the various
credit ratings, provided that current ratings make
past ratings irrelevant.
We will use the following terminology:
n
ptij
Pt
dt
Dt
Πj(i)
number of feasible states, labelled 1 to n
"transition" probability, given current state
i, of moving to state j in exactly t periods
nxn matrix with ij element ptij. Also the
t-times product of P with itself (where a
matrix product PQ has ij element pi1q1j +
... + pin qnj)
conditional default risk in period t, given
initial state i
column vector with n elements di
long-run probability of being in state j,
given initial state i
simplicity, we constrain our search in the following
ways:
- any credit classes lower than B ignored.
- credits allowed to move at most two classes up
or down per
- D1 assumed to exactly equal the observed
averages for the sample (which, among other
things, requires single-A credits to have 0.01%
lower one-period default rates than Aa)
These constraints mean that the P matrix selected
does not yield the best possible fit. Moreover,
because the estimating equations are highly nonlinear, we do not try to calculate significance
levels. Nevertheless, it is clear from the charts on
the facing page that the estimated term structures
track the observed values remarkably well.
To try to “replicate” the corporate default data 'm Chart 5
using a Markov chain, we search for a P and D1 such that the
equations Dt = Pt D1 approximately describe the observations. For