14 5.00 5.85 3.34285 5.00 5.95 4.6390
with Treasury yields in the way indicated in Figure 26.1. However, there are sometimes factorscausing Treasury bonds to have artificially low yields. For example, in the United States the intereston Treasury bonds is not taxed at the state level whereas the interest on most other bonds(including corporate bonds) is taxed at both the federal and state level. This has led most analyststo use the LIBOR zero curve as the risk-free zero curve.
We will use the data in Table 26.1 to illustrate how bond yields can be related to default losses.The table shows that zero-coupon risk-free interest rates are assumed to be 5% for all maturities. Thassumes that a corporation issues zero coupon bonds with maturities between one and five yearsand that the yields on the bonds range from 5.25% for a one-year maturity to 5.95% for a five-yearmaturity.
The value of a one-year risk-free bond with a principal of 100 is 100e7°*, or 95.1229. The value
of a similar corporate bond is 100e~°°> = 94.8854. The present value of the loss from defaultson the corporate bond is therefore 95.1229 — 94.8854 — 0.2375. This means that we expect0.2375/95.1229 = 0.2497% of the no-default value of the corporate bond to be lost from defaults.
Consider next a two-year bond. The value of a two-year risk-free bond with a principal of 100 is100e~°°*? = 90.4837. The value of a similar corporate bond is 100e~°°*** = 89.5834. The presentvalue of the expected loss from defaults on the two-year corporate bond is 90.4837 — 89.5834 =0.9003. We therefore expect 0.9003/90.4937 = 0.9950% of the no-default value of the two-yearcorporate bond to be lost from defaults.
These results and those for other bond maturities are shown in the final column of Table 26.1.Probability of Default Assuming No Recovery
Define:
y(T): Yield on a T-year corporate zero-coupon bondy*(T): Yield on a 7-year risk-free zero-coupon bondQ(T): Probability that corporation will default between time zero and time TThe value of a T-year risk-free zero-coupon bond with a principal of 100 is 100e” while thevalue of a similar corporate bond is 100e~*”, The expected loss from default is therefore
Looe" MT _ e-MNT)
If we assume that there is no recovery in the event of default, the calculation of Q(T) is relativelyeasy. There is a probability Q(T) that the corporate bond will be worth zero at maturity and aCredit Risk 613
Table 26.2 Probabilities of default in the example in Table 26.1assuming no recovery in the event of default
 
 
Year Cumulative default Default probabilityprobability (%) in year (%)I 0.2497 0.24972 0.9950 0.74533 2.0781 1.08314 3.3428 1.26475 4.6390 1.2962
probability 1 — Q(T) that it will be worth 100. The value of the bond is therefore®{O(T) x 0+ [1 — Q(T) x 100}e°" = 100[1 — Q(T"The yield on the bond is y(T), so that
100e7™? = 100[1 — O(TyJe" P*This means thater OT _ g-XnTO(T) = rororQ(T) =1- eb OMIT (26.1)
Assuming no recovery, Table 26.2 shows the cumulative probability of default and the prob-ability of default in each year for the example in Table 26.1. The cumulative probability of defaultis the same as the expected percentage loss from default in the final column of Table 26.1. Theprobability of default in each year is obtained by subtracting consecutive cumulative defaults. Forexample, the probability of default in year 3 is 3.3428 — 2.0781 = 1.2647%.
Example 26.1 Suppose that the spreads over the risk-free rate for 5-year and a 10-year BBB-ratedzero-coupon bonds are 130 and 170 basis points, respectively, and there is no recovery in the event of
default. From equation (26.1),
Q(5) = 1 — e885 = 0.9629
O(10) = 1 — e 7*!9 — 0.1563It follows that the probability of default between five years and ten years is0.1563 — 0.0629 = 0.0934
Hazard Rates
At this stage it is appropriate to mention that there are two ways of quantifying probabilities ofdefault. The first is in terms of what are known as hazard rates. The second is in terms of thedefault probability density.
> Note that we discount the expected payoff from the bond at the risk-free rate. As we will see later, this means thatthe probability of default, O(7), that we are estimating is a risk-neutral probability.614 CHAPTER 26
The hazard rate, A(f), at time ris defined so that A(t) St is the probability of default between time tand t + dt conditional on no earlier default (i.c., h(t) dt is the probability of default between time tand time t + st conditional on no default between time zero and time t). The default probabilitydensity, g(t), is defined so that g(¢) 6¢ is the unconditional probability of default between times tand t + dt as seen at time zero.
Either A(t) or g(t) can be used to describe the default probabilities. They provide equivalentinformation and the relationship between h(t) and q(t) is
gt) = hie” hoe
Most of our analysis in this chapter and the next will be in terms of the default probability densityrather than in terms of hazard rates. Indeed we have already started to use the default probabilitydensity. The probabilities of default listed for each year in Table 26.2 are the unconditionalprobabilities of default as seen at time zero. Consider year 5. Table 26.2 lists the probability ofdefault as 1.2962%. The corresponding hazard rate is the probability of default in year 5conditional on no default up to year 4. The probability of no default prior to year 4 is1 — 0.033428 = 0.966572%. The hazard rate for year 4 is therefore 0.012962/0.966572 = 1.3410%.
Recovery Rates
The analysis presented so far assumes no recovery on bonds in the event of a default. This is notrealistic. When a company goes bankrupt, entities that are owed money by the company file claimsagainst the assets of the company. The assets are sold by the liquidator and the proceeds are used tomeet the claims as far as possible. Some claims typically have priorities over others and are met morefully. Table 26.3 provides historical data on the amounts recovered by different categories of claimsin the United States. It shows that senior secured debtholders received an average of 52.31 cents perdollar of par value while junior subordinated debtholders received an average of only 19.69 cents perdollar of par value.*
We define the recovery rate as the proportion of the “claimed amount”’ received in the event of adefault. If we make the assumption that the claimed amount for a bond is the no-default value ofthe bond, the calculation of the probability of default is simplified. We use the same notation asbefore and suppose the expected recovery rate is R. In the event of a default the bondholderreceives a proportion R of the bond’s no-default value. If there is no default, the bondholder
Table 26.3 Amounts recovered on corporate bonds as a percentof par value from Moody’s Investor’s Service (January 2000)
 
Mean Standard deviationClass (%) (%)Senior secured 52.31 25.15Senior unsecured 48.84 25.01Senior subordinated 39.46 24.59Subordinated 33.17 20.78Junior subordinated 19.69 13.85
° In these statistics the amount recovered is estimated as the market value of the bond one month after default.Credit Risk 615
receives 100. The bond’s no-default value is 100e7”""" and the probability of a default is Q(T).The value of the bond is therefore
[1 — O(T)100e7" + Qcry100Re PO”
so that100e" = [1 — O(T00e7" "4 Q(T)100Re” OFThis giveseV OT _ g-XDTO) = Tarorpag bO- Oror) = (26.2)
More Realistic Assumptions
Equation (26,2) is often used to provide a quick estimate of the probability of default. It is basedon the assumption that the amount claimed in the event of a default equals the no-default value ofthe bond, This assumption is analytically convenient, but does not correspond to the way bank-ruptcy laws work in most countries. A more realistic assumption is that the claim made in the eventof default equals the bond’s face value plus accrued interest. Equation (26.2) also assumes thatzero-coupon corporate bond prices are either observable or calculable. In reality probabilities ofdefault must usually be calculated from coupon-bearing corporate bond prices, not zero couponbond prices. We now present a way of extracting default probabilities from coupon-bearing bondsfor any assumptions about the claimed amount.
Assume that we have chosen a set of N coupon-bearing bonds that are either issued by thecorporation being considered or by another corporation that has roughly the same probability ofdefault as this corporation at all future times. We assume that defaults can happen on any of thebond maturity dates. (To be precise, defaults can occur immediately before each bond maturitydate. Later we generalize the analysis to allow defaults to occur at any time.) Suppose that thematurity of the ith bond is t; with t) < tf <--- < ty. Define:
;: Price of the jth bond today
j: Price of the jth bond today if there were no probability of default (ic., the price of arisk-free bond promising the same cash flows as the jth bond)
F(t): Forward price of the jth bond for a forward contract maturing at time f (t <¢;)assuming the bond is default-free. (Note that F(t) is the forward value of G, not B;.)
v(t): Present value of $1 received at time ¢ with certaintyC,(t): Claim made by holders of the jth bond if there is a default at time t (t < t,)R,(t): Recovery rate for holders of the jth bond in the event of a default at time t (t < t;)@;;: Present value of the loss from a default on the jth bond at time t;p;: The probability of default at time ¢;
ease of exnosition we a mo at intasact +.For case of exposition we assume that interest r
and claim amounts are known with certainty. It can .n be shown that our results are still true if thisassumption does not hold provided that (a) default events, (b) risk-free interest rates, and(c) recovery rates are mutually independent. (The amount claimed in the event of default can be
deterministic and that both recovery ratesaeterministic ana that ootn recovery rates616 CHAPTER 26
either the no-default value of the bond or the face value plus accrued interest. The recovery rate isset equal to its expected value in a risk-neutral world.)
The price at time t of the no-default value of the jth bond is F,(¢). If there is a default at time t,the bondholder makes a recovery at rate R(t) on a claim of C,,(t). It follows that
oj; = ot F(t) — Rt )C (tI
There is a probability p; of the loss a,;; being incurred. The total present value of the losses on thejth bond is therefore given by
iGj — By = > pio; (26.3)i=l
This equation allows the p’s to be determined inductively. The first probability, p,, is (G, — B))/ou.The remaining probabilities are given by
j-1_ G,—- B,- ye, Pi;
jj
P; (26.4)
another company that is considered to have the same risk of default as the reference entity. Thismeans that the p; should be the same for all bonds. The recovery rates can in theory vary accordingto the bond and the default time. From now onward we make the simplification that the expectedrecovery rate is the same for all bonds issued by a particular company and is independent of time. Wewill denote this expected recovery rate by R. This means that the above equation for a; ; becomes
As mentioned earlier, the N bonds used in the analysis are issued either by the reference entity Th
aj = v(t, )[Fy(t;) — RC\(t;)) (26.5)
An Iflustration
Table 26.4 provides hypothetical data on six bonds issued by a particular corporation. The bondshave maturities ranging from one to ten years. The coupons are assumed to be paid semiannually,the risk-free zero curve is assumed to be flat at 5% (semiannually compounded), and the expectedrecovery rate is assumed to be 30%. Table 26.5 shows the default probabilities for the twoalternative assumptions we have mentioned about the claim amount (see Problem 26.6 for thecalculation of the first two rows of this table). The first assumption is that the claim amount equalsthe no-default value of the bond; the second is that it equals the bond’s face value plus accrued
Table 26.4 Data on bonds issued by a corporation
Bond life Coupon Bond yield
 
(years) (%) (%)1 7.0 6.60
2 7.0 6.70
3 70 6.80
4 70 6.90
5 7.0 7.00
10 7.0 7.20Credit Risk
Table26.5 Implied probabilities of default for data in Table 26.4assuming defaults can happen only at bond maturity dates
 
 
Default probabilityTime Claim = Claim =(years) No-default value Face value + Accrued interest1 0.0224 0.02242 0.0249 0.02473 0.0273 0.02694 0.0297 0.02915 0.0320 0.031210 0.1717 0.1657
617
interest, It can be seen that the two assumptions give similar results. This is usually the case. Forthe probabilitics to be markedly different, it would be necessary for the coupons on the bonds to bemuch, much greater or much, much less than the risk-free rate.
When Defaults Can Happen at Any Time
The analysis used to derive equation (26.4) assumes that defaults can take place only on bondmaturity dates. It can be extended to allow defaults at any time. As earlier, we define q(t) as thedefault probability density. (This means that g(t) ét is the probability of default between times tand t + dt as seen at time zero.)
One approach is to assume that g(t) is constant and equal to gq; for t;_, < t < t;. Setting
Bij = [ v(t)[F(t) — RC (tat
i-1
a similar analysis to that used in deriving equation (26.4) gives
_ Gj By Di) aby
4,; Bij
Table 26.6 Implied default probability densities for data inTable 26.4 assuming defaults can happen at any time
Default probability density
 
Time Claim = Claim =(years) No-default value Face value + Accrued interest0-1 0.0220 0.02191-2 0.0245 0.02422-3 0.0269 0.02643-4 0.0292 0.02854-5 0.0315 0.0305
5-10 0.0295 0.0279
(26.6)
(26.7)618 CHAPTER 26
The parameters 8;; can be estimated using standard procedures, such as Simpson’s rule, forevaluating definite integrals. Table 26.6 shows the values of q; calculated for the example inTable 26.4.
Claim Amounts and Value Additivity
When we make the assumption that the claim amount is the no-default value of the bond,C,(t) = F(t), it can be shown (see Problem 26.12) that the value of the coupon-bearing bondB; is the sum of the values of the underlying zero-coupon bonds. This property is referred to asvalue additivity. It implies that it is theoretically correct to calculate zero curves for different ratingcategories from actively traded bonds and use them for pricing less actively traded bonds.
When we make the more realistic assumption that C,(t) equals the face value of bond j plusaccrued interest at time t, value additivity does not apply (except in the special case where therecovery rate is zero). This means that there is in theory no zero-coupon yield curve that can beused to price corporate bonds exactly for a given set of assumptions about default probabilitiesand expected recovery rates.
Asset Swaps
In practice, traders often use asset swap quotes as a way of extracting default probabilities frombond prices on the assumption that the LIBOR curve is the risk-free curve.
As explained in Chapter 6, an interest rate swap can be used to exchange a fixed return for afloating return. Suppose that an investor owns a fixed-rate five-year corporate bond that iscurrently worth par and pays a coupon of 6% per year. Suppose further that the LIBOR zerocurve is flat at 4.5% with semiannual compounding. In a five-year plain vanilla interest rate swap,4.5% would he exchanged for LIROR. Equivalently 6% would be exchanged for LTROR plus150 basis points in a such a swap. This means that an investor can use a five-year interest rate swapto exchange the bond’s coupons for LIBOR plus 150 basis points. The investor still bears the creditrisk that the bond might default. The asset swap would be referred to as swapping the bond’sreturn for a 150-basis-point spread over LIBOR.
The swap transaction shows that the investor is compensated for credit risk by receiving an extra150 basis points per year for five years. The value of 150 basis points per year (paid semiannually)using the LIBOR curve for discounting is $6.65 per $100 of face value. It follows that, if thecorporate bond had been risk-free, its value would increase $6.65 per $100 of face value. In theterminology of this section, B; = 100 and G; = 106.65 for this bond and equations (26.4) or (26.7)can be used for estimating default probabilities.
Suppose next that the bond is worth only $95 per $100 of par value and pays a coupon of 5%per year. We continue with our assumption that the LIBOR curve is flat at 4.5% with semiannualcompounding. The asset swap would usually be structured so that the bondholder pays $5 per $100of par value up front and then exchanges coupons for LIBOR plus a spread. Because 4.5% isswapped for LIBOR in a regular five-year swap, the coupon of 5% is swapped for LIBOR plus50 basis points. An up-front payment of $5 is equivalent to $1.1279 per year for five years paidsemiannually. The investor therefore requires a further 112.79 basis points to compensate for theup-front payment. The total floating payment should therefore be LIBOR plus 162.79 basis points.The asset swap would be referred to as exchanging the bond yield for a 162.79-basis-point spreadover LIBOR. The value of 162.79 basis per year for five years is $7.22 per $100 of par value. Itfollows that, in this case, B; = 95.00 and G; = 95.00 + 7.22 = 102.22.
When asset swap spreads are known for bonds with a number of different maturities, we canCredit Risk
mn»awo
 
Table 26.7 Average cumulative default rates (%)
 
Term (years)
 
 
Rating I 2 3 4 5 7 10 15
AAA 0.00 0.00 0.04 0.07 0.12 0.32 0.67 0.67AA 0.01 0.04 0.10 0.18 0.29 0.62 0.96 1.39A 0.04 0.12 0.21 0.36 0.57 1.01 1.86 2.59BBB 0.24 0.55 0.89 1.55 2.23 3.60 5.20 6.85BB 1.08 3.48 6.65 9.71 12.57 18.09 23.86 27.09B 5.94 13.49 20.12 25.36 = 29.58 36.34 = 43.41 48.81
ccc 25.26 34.79 42.16 48.18 54.65 58.64 62.58 66.12
Source: Standard & Poor’s, January 2001.
use equations (26.4) or (26.7) to estimate default probabilities. The asset swap spread depends onthe credit rating of the bond and its maturity. In theory, it should also be slightly dependent onthe bond’s coupon. In practice, because bonds are not highly liquid, the credit spread is often
assumed to be the same for all counons. A auc ted asset swan spre dis then asgumed ts annly toassumcd to be tne same lor au coupons. A quotea asset swap spread is tncen assumed to appiy to
a corporate bond selling at par for the purposes on applying equations (26.4) or (26.7).
26.2 HISTORICAL DATA
 
 
We now consider how default probabilities can be estimated from historical data. Table 26.7 istypical of the information produced by rating agencies. It shows the default experience throughtime of\companies that started with a certain credit rating. For example, a bond issue with aninitial crédit rating of BBB has a 0.24% chance of defaulting by the end of the first year, a 0.55%chance of defaulting by the end of the second year, and so on. The probability of a bonddefaulting during a particular year can be calculated from the table. For example, the probabilityof a BBB-rated bond defaulting during the second year is 0.55 — 0.24 = 0.31%.
As indicated in Figure 26.1, for investment grade bonds the probability of default in a year tendsto be an increasing function of time. For bonds with a poor credit rating, the reverse is often true.In Table 26.7, the probabilities, as seen at time zero, of a AA bond defaulting during year 1, 2, 3, 4,and 5 are 0.01%, 0.03%, 0.06%, 0.08%, and 0.11%, respectively. The corresponding probabilitiesfor a CCC bond are 25.26%, 9.53%, 7.37%, 6.02%, and 6.47%.’ For a bond with a good creditrating, some time must usually elapse for the fortunes of the issuer to decline to such an extent thata default happens. For a bond with a poor credit rating, the next year or two may be critical. If theissuer survives this period, its probability of default per year can be expected to decline.
26.3. BOND PRICES vs. HISTORICAL DEFAULT EXPERIENCE
 
 
The default probabilities in Table 26.7 are significantly less than we would expect from an analysisof bond prices. Consider, for example, a five-year A-rated zero-coupon bond. We suppose that the
7 The hazard rates in the five years are 25.26%, 12.75%, 11.30%. 10.41%, and 12.49%, respectively.620 CHAPTER 26
 
yield on this bond is 50 basis points above the risk-free rate. Assuming a zero recovery rate,equation (26.1) gives an estimate for the probability of default during the five years of
1 — 7 0 5*5 = 0.0247
or 2.47%. By contrast, the historical probability of default over five years for an A-rated bond isfrom Table 26.7 only 0.57%.
The two key assumptions we have made here are: (i) A-rated bonds yield 50 basis points abovethe risk-free rate; and (ii) the recovery rate is zero. Both assumptions are conservative. In practice,the yield on five-year A-rated bonds is often more than 50 basis points above the risk-free rate andthe recovery rale is nonzero, As we increase either the yield of A-rated bonds or the assumedrecovery rate, the probability of default calculated from bond prices increases.®
Altman was one of the first researchers to comment on the discrepancy between bond prices andhistorical default data.? He showed that, even after taking account of the impact of defaults, aninvestor could expect significantly higher returns from investing in corporate bonds than frominvesting in risk-free bonds. As the credit rating of the corporate bonds declined, the extent of thehigher returns increased.
Why do bond prices give much higher probabilities of defaults than historical data? One réasonmay be that investors raise the expected returns they require on corporate bonds to compensat& fortheir relatively low liquidity. Another reason may be that bond traders are allowing in their pricingfor the possibility of “depression scenarios” much worse than anything seen during the timeperiod covered by the historical data. However, there is an important theoretical reason, which wewill now explain.
26.4 RISK-NEUTRAL vs. REAL-WORLD ESTIMATES
 
We return to the five-year A-rated bond example. If the yield is 50 basis points above the risk-freerate and there is no recovery, the probability of default during the five-year life of the bondestimated from the bond price is 2.47%. This is over four times the estimate of 0.57% fromhistorical data.
It turns out that 2.47% is an estimate of the probability of default in a risk-neutral world,whereas 0.57% is an estimate of the probability of default in the real world. To show this, we firstnote that the price of the corporate bond is 2.47% less than that of a risk-free bond. One way ofgetting this pricing difference is by assuming:
(i) The expected cash flow from the corporate bond at the end of five years is 2.47% less thanthe expected cash flow from a risk-free bond at the end of the five years; and(ii) The discount rates appropriate for the two cash flows are the same.
The second of these two assumptions is correct in a risk-neutral world (i.e., a world where theexpected return required by all investors on all investments is the risk-free interest rate). The first
5 For example, if the recovery rate is 30% and the yield above the risk-free rate is 70 basis points, bond prices predicta probability of default of about 5% during the five-year period.
° See, for example. E. I. Altman, “Measuring Corporate Bond Mortality and Performance,” Journal of Finance,44 (1989), 902-22.Credit Risk 621
assumption corresponds to a probability of default of 2.47%. We conclude that the prices ofcorporate bonds are consistent with 2.47% being the probability of default in a risk-neutral world.
Approximately the same differential between the corporate bond price and the risk-free bondprice is calculated if we assume:
(i) The expected cash flow from the corporate bond at the end of five years is 0.57% less thanthe expected cash flow from the risk-free bond at the end of the five years; and
(ii) The discount rate appropriate for the corporate bond’s expected cash flow is about 0.38%higher than the discount rate appropriate for the risk-free bond cash flow.
This is because a 0.38% increase in the discount rate leads to the corporate bond price beingreduced by approximately 5 x 0.38 = 1.9%. When combined with the corporate bond’s lowerexpected cash flow, we get a total difference between the corporate bond price and the risk-freebond price of approximately 1.9 + 0.57 = 2.47%, which is what is observed in the market.
The 0.57% estimate of the probability of default is therefore the correct estimate in the real worldif the correct discount rate to use for the bond’s cash flows in the real world is 0.38% higher than inthe risk-neutral world. Arguably a 0.38% increase in the discount rate when we move from the risk-neutral world to the real world is not unreasonable. A-rated bonds do have some systematic(Le., nondiversifiable) risk. (When the market does badly, they are more likely to default; when themarket does well, they are less likely to default.) The excess expected return of the market over therisk-free rate is about 5%. Using the capital asset pricing model, a 0.38% excess expected return forthe corporate bond is consistent with the corporate bond having a beta of about 0.38/5 = 0.076. Itis interesting that what seems to be a large discrepancy between default, probability estimatestranslates into a relatively small adjustment to expected returns for systematic risk.'°
This explanation of the difference between estimates calculated from bond prices and thosecalculated from historical data is consistent with the pattern of Altman’s results, mentioned earlier.As the credit rating of a bond declines, it becomes more similar to equity and its beta increases. As
result, the excess of the expected return required by investors over the risk-free rate also increases.
At this stage, it is natural the ask whether we should use real-world or risk-neutral defaultprobabilities in the analysis of credit risk. The answer depends on the purpose of the analysis.When valuing credit derivatives or estimating the impact of default risk on the pricing ofinstruments, we should use risk-neutral default probabilities. This is because we likely to beimplicitly or explicitly using risk-neutral valuation in our analysis.