CS FIRSTBOSTON - NYU



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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

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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.

[pic]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;

[pic]

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

[pic]

Cumulative The calculations we have described above can be solved for d to give the default

Default probability assumed for each single period. If a borrower has debt with many

Probability 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

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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 The assessment of the recovery value of the bond in the event of default is both

Recovery Value 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

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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 t

[pic]

where: 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%.

[pic]

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

[pic]

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.

[pic]

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 The principal problem with this approach is one of inconsistency. Because we are

Model to Non- assuming constant per period default rates throughout the life of a bond, we get the

Constant anomalous result that the probability of default by the same issuer in the same

Default Rates 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:

[pic]

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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.

[pic]

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.

[pic]

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.

[pic]

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.

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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 differ-

ences 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 Suppose you are due to receive a payment in one year's time. If you are 100% sure

Methodology 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.

[pic]

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;

[pic]

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

[pic]

The Brady Brady bonds are distinguished from Eurobonds mainly by collateralization of

Difference 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;

[pic]

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.

[pic]

Applications: The collateralization of some cash flows allows the division of the price among the

Pricing various components as follows.

Components

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

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[pic]

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.

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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,).

[pic]

Mathematically, the fair price of the coupon guarantee can now be expressed as follows:

[pic]

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:

[pic]

This gives a value for the coupon strip as follows:

[pic]

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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:

[pic]

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.

[pic]

Applications: The reallocation of value in the price of a Brady bond has important implications

The True for measures of relative value. One of the most frequently used indicators in the

Stripped Yield 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

[pic]

Other Through our previously discussed default probability methodology, we have been

Considerations 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 ~

[pic]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 For longer time periods, the required premium can

be shown to equal d(1+r)/(1-d), where d is the

As bonds age, their default risks change. Viewed conditional default rate and r is the future risk-free

from the date of purchase, each successive coupon rate. Incorporating positive post-default payouts

is less likely to be paid, so the cumulative default and risk aversion further complicates the

risk rises. But we might also ask: how likely is the relationship, but higher future default risks are still

next coupon to be defaulted, assuming no prior associated with higher future premiums over risk-

default? This is known as the conditional default free interest rates.

risk. Its term structure indicates whether that risk

rises or falls over time. Conventional View of Forward Default Risk

In emerging markets, the term structure of default In emerging markets, default risk is typically

risk is typically priced as if it were flat or rising. priced as if it were (a) rising for all types looking

This paper marshals evidence suggesting, on the forward, and (b) steeper, the lower the initial credit

contrary, that the term structure for weak credits rating.

naturally declines.1

For example, in April 1995, one-year, five-year,

The term structure of default risk matters to and eight-year Argentine dollar Eurobonds traded

investors, because it ought to influence interest respectively at 600 bp, 700 bp, and 775 bp spreads

rates. As an illustration, suppose the conditional over U.S. Treasuries. Assuming a post-default

default risks are 10 basis points (bp) today and 6 payout ratio of 15% of face value, the implied

bp tomorrow, that there is no post-default payout, conditional default risks escalate to 10.1% for

and that risk-free interest rates run 2 bp per day. years five through eight from 6.9% for year one.2

A risk-neutral investor (i.e., willing to accept any For comparison, the yield differential between

bet expected to break even) will charge 12 bp triple-A U.S. corporate bonds and corresponding

interest for a one-day loan, because his expected U.S. Treasuries typically rises from 10-15 bp on

payback per dollar is 99.90% times $1.0012 or two-year paper to 40-50 bp on 30-year paper.

roughly the risk-free gross return of $1.0002.

Similarly, he will charge an average 10 bp per day Intuitively, it makes sense for triple-A interest

for a loan repayable in two days, because his premiums to widen looking forward. No credit has

expected return per dollar is 99.90% times 99.94% ever defaulted in a year it was rated triple-A.

times $1.0010 squared, or roughly the risk-free However, some triple-A credits have defaulted

gross return of $1.0004. The daily interest rate after having decayed to lower credit ratings. In

falls even though the cumulative default risk rises. probability terms, they have nowhere to go but

down.

Note in the preceding example that for the first

day, the required interest premium over the risk- It also makes sense for default risks to initially rise

free rate is 10 bp, the same as the default risk. For more quickly for, say, single-A credits than for

the second day of the two-day loan, the required triple-A credits. Having large servicing cushions,

future interest rate is 8 bp, or a future premium of triple-A credits decay in the first instance to other

6 bp, again the same as the conditional default risk. A-levels, where immediate default risks remain

low. Single-A credits tend to decay sooner to sub

This point has been emphasized by, among others, investment grade, where propensities to default are

J.S. Fons. "Using Default Rates to Model the Term much higher.

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 Clayton Perry, Implied Default Probabilities: A New

Interest Rate Risk: The Term Structure of Default Approach to Risky Debt. Fixed Income Research

Spreads". INSEAD Working Paper. May 1993. Bulletin, CS First Boston. 26 April 1995.

Term Structure of Default Risk

[pic]

The chart above on the default history of U.S. To strengthen the intuition for this result, imagine a

investment-grade corporate bonds, distinguished by blindfolded man wandering on an unfenced roof.

letter rating out to a ten-year horizon, lends The starting position corresponds to an initial

support to this view. Each term structure is rating, while falling off the roof corresponds to

generally rising. Moreover, the lower the initial default. For a man starting close to the edge,

credit quality, the steeper the term structure.3 disaster may seem imminent. But if he has

survived for a while-without failing, the chances

Revised View of Forward Default Risk are he has worked part way back to the centre, and

hence is less likely to fall in the next instant.

On reflection, the default term structure cannot rise Conversely, a man starting at the centre is bound to

for the weakest credits. The latter are more likely come closer to the edge, so his conditional risks of

to default in the near-term, but they also have more falling must increase.

upside. Moreover, the longer they go without

defaulting, the more likely they are to have Mathematically, we are predicting a forward

improved in quality. For example, if a weak default risk conditional on an initial rating and no

sovereign credit like Bulgaria can service its Brady interim default. Over time, the interim history

bonds punctually for the next 15 years, the chances counts for more, relative to the initial assessment.

are creditors will no longer demand the 1500 bp Hence, the differential default risk between the

default spreads they carry today. Note that this strongest and weakest credits shrinks over time.

result does not hinge on our belief that Bulgaria is

ultimately "emerging" rather than "submerging". To consider a numeric example, suppose there are

It is simply the observation that 15 years of two credit qualities: Strong (S) and Weak (W).

successful debt servicing is more likely than not to Current S credits never default, while current W

indicate improved credit quality. 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

Conditional default risks calculated from data in S credits have a 5% chance of switching to W.

Moody’s Investors Service. Corporate Bond Defaults

and Default Rates 1970-1994, January 1995.

Term Structure of Default Risk

[pic]

Looking forward, the conditional default risk for S 4.2% in the first year to 1.3% in the tenth and

is 0.5% (i.e., 5% chance of switching to W, which 0.5% in the twentieth.

has 10% default risk) for the second period and * The default term structure is steeply inverted

0.875% (5%*80% = 4% chance of path S-W-W for single-B. Conditional default risk drops to

plus 90%*5% = 4.75% chance of path S-S-W) for 4.9% in the fourth year from 7.9% in the first.

the third. The corresponding risk for W is 8% for * The default term structure for Ba credits is

the second period and 6.5% for the third. Hence, humped. The conditional default risk rises to

default risks rise looking forward for currently 2.5% in years two through five from an initial

Strong credits, and decline looking forward for 1.7%, then declines gradually to 1.0%.

currently Weak credits.

To further test the revised view, the Appendix

The Appendix refines these themes, using the explores how well a Markov chain can replicate the

mathematical theory of Markov chains. Basically, observed default profiles. The fit is excellent.

the evolution of credit quality is a Markov chain to

the extent that the current rating supersedes all This is not to claim that foreign bonds, particularly

prior rating information. The analysis confirms sovereign bonds, exhibit the same default profiles

that conditional default risks ultimately tend to as U.S. corporates. While the rating agencies

converge. In the short run, however, with more strive to ensure that assigned grades are

than two credit qualities, forward default risks can comparable across issuer types, there is no

rise and later fall or vice-versa. guarantee that they succeed, or even that it is

possible. Even for U.S. corporates, future default

For evidence, we turn again to the default record of rates may depart from historic values. Our claim

U.S. corporate bonds. The facing charts is rather that the qualitative patterns predicted by

incorporate sub-investment-grade credits into the theory and confirmed in one important data set -

comparisons and extend the horizon to 20 years. especially, the inverted default term structure for

Note that, contrary to the conventional view, but weak credits - are likely to recur in others.

compatible with our revised view:

From an investment perspective, the main practical

Time ultimately narrows the risk gaps implied implication is to recommend calendar spreads on

by different credit ratings. The average long-dated bonds for emerging markets. A follow-

difference in conditional default risks between up paper elaborates this them&.

investment and speculative grades shrinks from

[pic]

Term Structure of Default Risk

Appendix: Default as a Markov Chain simplicity, we constrain our search in the following

ways:

A Markov chain is a sequence of transitions from

one state to another, where the conditional - any credit classes lower than B ignored.

probabilities of transition depend only on the - credits allowed to move at most two classes up

transition itself, not on history or timing. The or down per

evolution of credit quality can be viewed as a - D1 assumed to exactly equal the observed

Markov chain in which the states are the various averages for the sample (which, among other

credit ratings, provided that current ratings make things, requires single-A credits to have 0.01%

past ratings irrelevant. lower one-period default rates than Aa)

We will use the following terminology: These constraints mean that the P matrix selected

does not yield the best possible fit. Moreover,

n number of feasible states, labelled 1 to n because the estimating equations are highly non-

ptij "transition" probability, given current state linear, we do not try to calculate significance

i, of moving to state j in exactly t periods levels. Nevertheless, it is clear from the charts on

Pt nxn matrix with ij element ptij. Also the the facing page that the estimated term structures

t-times product of P with itself (where a track the observed values remarkably well.

matrix product PQ has ij element pi1q1j +

... + pin qnj)

dt conditional default risk in period t, given

initial state i

Dt column vector with n elements di

Πj(i) long-run probability of being in state j,

given initial state i

[pic]

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

[pic]

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