Simon Benninga. Financial Modelling 3-rd edition

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718 Chapter 27

b. Use a polynomial regression to determine the relation between the YTM and

the bond maturity.

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DILLARDS INC

102.50 6.690 1-Aug-07 -19.559 BB

GENERAL MTRS ACCEP CORP

101.75 3.800 15-Aug-07 -9.400 BB

GAP INC DEL

102.31 6.900 15-Sep-07 -3.732 BB

GENERAL MTRS ACCEP CPSMARTNBE

101.95 5.600 15-Sep-07 -3.493 BB

SEARS ROEBUCK ACCEP CORP

102.14 6.700 18-Sep-07 -2.817 BB

DELUXE CORP

101.38 3.500 1-Oct-07 -1.783 BB

SONAT INC

102.50 6.750 1-Oct-07 -2.811 BB

CMS ENERGY CORP

103.50 9.875 15-Oct-07 -1.802 BB

GENERAL MTRS ACCEP CPSMARTNBE

101.94 5.600 15-Oct-07 -0.909 BB

CASE CR CORP

103.00 6.750 21-Oct-07 -2.693 BB

GENERAL MTRS ACCEP CPSMARTNBE

101.98 6.000 15-Nov-07 0.768 BB

NORTHWEST PIPELINE CORP

102.50 6.625 1-Dec-07 0.721 BB

EL PASO ENERGY CORP MTN BE

102.46 6.950 15-Dec-07 1.595 BB

BROWNING FERRIS INDS

102.75 6.375 15-Jan-08 1.313 BB

PANAMSAT CORP NEW

102.50 6.375 15-Jan-08 1.739 BB

SPIEKER PPTYS L P

102.70 6.750 15-Jan-08 1.748 BB

MIRAGE RESORTS INC

103.25 6.750 1-Feb-08 1.193 BB

CONSTELLATION BRANDS INC

104.00 8.000 15-Feb-08 1.567 BB

DILLARDS INC

103.50 6.300 15-Feb-08 0.701 BB

GENERAL MTRS ACCEP CPSMARTNBE

101.54 5.250 15-Feb-08 2.767 BB

PSEG ENERGY HLDGS INC

103.75 8.625 15-Feb-08 2.557 BB

SEARS ROEBUCK ACCEP CORP INTER

102.00 6.200 15-Feb-08 2.954 BB

SEARS ROEBUCK ACCEP CORP INTER

101.80 6.100 15-Feb-08 3.174 BB

SEARS ROEBUCK ACCEP CORP INTER

102.39 6.650 15-Feb-08 2.784 BB

SMITHFIELD FOODS INC

103.50 7.625 15-Feb-08 1.982 BB

EOP OPER LTD PARTNERSHIP EOP

100.65 4.400 15-Mar-08 3.464 BB

BB CORPORATE BONDS, NONCALLABLE, 20 JULY 2007

4. Clean up the data in exercise 3 by eliminating obviously problematic points. Repeat

the regression exercise.

5. Graph the cleaned-up data for the CCC and BB bonds on one set of axes.

6. Graph the spread between the CCC and BB bonds using the polynomial regressions

from exercises 2 and 4.

Calculating Default-Adjusted Expected Bond Returns

28.1 Overview

In this chapter we discuss the effects of default risk on the returns from

holding bonds to maturity. The expected return on a bond that may

possibly default is different from the bond’s promised return. The latter

is deﬁ ned as the bond’s yield to maturity, the internal rate of return cal-

culated from the bond’s current market price, and its promised coupon

payments and promised eventual return of principal in the future. The

bond’s expected return is less easily calculated: We need to take into

account both the bond’s probability of future default and the recovery

rate, the percentage of its principal that holders can expect to recover in

the case of default. To complicate matters still further, default can happen

in stages, through the gradual degradation of the issuing company’s

creditworthiness.

In this chapter we use a Markov model to solve for the expected return

on a risky bond. Our adjustment procedure takes into account all three

of the factors mentioned: the probability of default, the transition of the

issuer from one state of creditworthiness to another, and the percentage

recovery of face value when the bond defaults. In sections 28.3–28.5 we

ﬁ rst use Excel to solve a relatively small-scale problem. We then use

some publicly available statistics to program a fuller spreadsheet model.

Finally, we show that this model can be used to derive bond betas, the

CAPM’s risk measure for securities (discussed previously in Chapters

8–11).

28.1.1 Some Preliminaries

Before proceeding, we deﬁ ne a number of terms:

•

A bond is issued with a given amount of principal or face value.

When the bond matures, the bondholder is promised the return of this

principal. If the bond is issued at par, then it is sold for the principal

amount.

1. Besides default risk, bonds are also subject to term-structure risk: The prices of bonds

may show signiﬁ cant variations over time as a result of changing term structure. This

statement will be especially true for long-term bonds. In this chapter we abstract from

term-structure risk, conﬁ ning ourselves only to a discussion of the effects of default

risk on bond expected returns.

720 Chapter 28

•

A bond bears an interest rate called the coupon rate. The periodic

payment promised to the bondholders is the product of the coupon rate

times the bond’s face value.

•

At any given moment, a bond will be sold in the market for a market

price. This price may differ from the bond’s coupon rate.

•

The bond’s yield to maturity (YTM) is the internal rate of return of

the bond, assuming that it is held to maturity and that it does not

default.

American corporate bonds are rated by various agencies on the basis

of the bond issuer’s ability to make repayment on the bonds. The classi-

ﬁ cation scheme for two of the major rating agencies, Standard & Poor’s

(S&P) and Moody’s, is given in the following table:

Long-Term Senior Debt Ratings

Investment-Grade Ratings Speculative-Grade Ratings

S&P Moody’s Interpretation S&P Moody’s Interpretation

AAA Aaa Highest quality

BB+

BB−

Ba1

Ba2

Ba3

Likely to fulﬁ ll

obligations; ongoing

uncertainty

AA+

AA−

Aa1

Aa2

AA3

High quality

B+

B−

High-risk obligations

A+

A−

Strong payment

capacity

CCC+

CCC

CCC−

Caa Current vulnerability

to default

BBB+

BBB

BBB−

Baa1

Baa2

Baa3

Adequate payment

capacity

In bankruptcy or

default, or other

marked shortcomings

When a bond defaults, its holders will typically receive some payoff,

though less than the promised bond coupon rate and return of principal.

2. Just to complicate matters, in the United States the convention is to add to a bond’s

listed price the prorated coupon (the accrued interest) between the time of the last

coupon payment and the purchase date. The sum of these two is termed the invoice

price of the bond; the invoice price is the actual cost at any moment to a purchaser of

buying the bond. In our discussion in this chapter we use the term market price to

denote the invoice price. The computation of the accrued interest is illustrated in

section 28.6.

721 Calculating Default-Adjusted Expected Bond Returns

We refer to the percent of face value paid off in default as the recovery

percentage.

28.2 Calculating the Expected Bond Return in a One-Period Framework

The bond’s yield to maturity is not its expected return: It is clear that

both a bond’s rating and the anticipated payoff to bondholders in the

case of bond default should affect its expected return. All other things

being equal, we would expect that if two newly issued bonds have the

same term to maturity, then the lower-rated bond (having the higher

default probability) should have a higher coupon rate. Similarly, we

would expect that an issued and traded bond whose rating has been

lowered would experience a decrease in price. We might also expect that

the lower the anticipated payoff in the case of default, the lower will be

the bond’s expected return.

As a simple illustration, we calculate the expected return of a one-year

bond that can default at maturity. We use the following symbols:

F = face value of the bond

P = price of bond

Q = annual coupon rate of the bond

π = probability that the bond will not default at end of year

π = fraction of bond’s value that bondholders collect upon default

The bond’s expected end-of-year cash ﬂ ow is π

(1 + Q)

F + (1 −

π)

F, and its expected return is given by

One-year bond expected return

Expected year-end cash flow

nnitial bond price, P

QF F

−

∗+ ∗+−∗∗

−

ππλ()()

This calculation is illustrated in the following spreadsheet:

3. The bond’s recovery percentage is not, as you might think, the payoff to the bondhold-

ers in the ﬁ nal settlement of a bankruptcy. Instead it is usually computed as the price

of the bond in the period immediately following a ﬁ nancial distress event.

722 Chapter 28

28.3 Calculating the Expected Bond Return in a Multiperiod Framework

We now introduce multiple periods into the problem. In this section we

deﬁ ne a basic Markov model that uses a ratings transition matrix to

compute a bond’s expected return. The model is illustrated using a very

simple set of ratings, much simpler than the complex rating system illus-

trated in section 28.1. Section 28.6 will use more realistic data.

We suppose that at any date there are four possible bond “ratings”:

A, B, C Bond ratings of solvent bonds in decreasing order of

creditworthiness.

D The bond is in default for the ﬁ rst time and pays off

recovery rate λ of the face value.

E The bond was in default in the previous period; it

therefore pays off 0 in the current period and in any

future periods.

The transition probability matrix Π is given by

Π=

⎡

⎣

⎢

ππππ

AA AB AC AD

BA BB BC BD

CA CB CC CD

00001

⎢⎢

⎤

⎦

⎥

The probabilities π

indicate the probability that in one period the bond

will go from a rating of i to a rating of j. In the numerical examples in

this and the following two sections, we use the following Π:

AB C

Face value, F 100

Price, P 90

Annual coupon rate, Q 8%

Default probability 20%

Recovery percentage 40%

Expected period 1 cash flow 94.4 <

=B2*(1+B4)*(1-B5)+B2*B6*B5

Expected return 4.89% <

=B8/B3-1

EXPECTED RETURN ON A ONE-YEAR BOND

WITH AN ADJUSTMENT FOR DEFAULT

PROBABILITY

723 Calculating Default-Adjusted Expected Bond Returns

ABC D E F

AB C D E

0.9700 0.0200 0.0100 0.0000 0.0000

0.0500 0.8000 0.1500 0.0000 0.0000

0.0100 0.0200 0.7500 0.2200 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000

What does this matrix Π mean?

•

If a bond is rated A in the current period, there is a probability of 0.97

that it will still be rated A in the next period. There is a probability 0.02

that it will be rated B in the next period and a probability of 0.01 that it

will be rated C. It is impossible for the bond to be rated A today and D

or E in the subsequent period.

•

A bond that starts off with a rating of B can—in a subsequent period—

be rated A (with a probability of 0.05), be rated B (with a probability of

0.8) or be rated C (probability 0.15). Bonds rated B in the current period

do not default (rating D) in the next period. The transition probabilities

from state C to states A, B, C, and D are 0.01, 0.02, 0.75, and 0.22,

respectively.

•

While it is possible to go from rating A, B, or C to any of ratings A, B,

C, or D, it is not possible to go from A, B, or C to E. This statement is

true because E denotes that default took place in the previous period.

•

A bond that is currently in state D (i.e., ﬁ rst-time default), will neces-

sarily be in E in the next period. Thus the fourth row of our matrix Π

will always be [0 0 0 0 1].

•

Once the rating is in E, it remains there permanently. Therefore, the

ﬁ fth row of the matrix Π will also always be [0 0 0 0 1].

28.3.1 The Multiperiod Transition Matrix

The matrix Π deﬁ nes the transition probabilities over one period. The

two-period transition probabilities are given by the matrix product Π

Π.

The following spreadsheet uses the array function MMult.

It shows that

the product Π

Π is

4. See discussion of matrix products and array functions in Chapter 34.

724 Chapter 28

Two-period transition probability =∗=ΠΠ

0 9420 0 0356 0 0202 0....00022 0 0000

0 0900 0 6440 0 2330 0 0330 0 0000

0 0182 0 0312 0 565

.....

...66 0 1650 0 2200

0 0000 0 0000 0 0000 0 0000 1 0000

0 0000 0 0000 0

.....

...00000 0 0000 1 0000..

⎡

⎣

⎢

⎤

⎦

⎥

Thus if a bond is rated B today, there is a probability of 9 percent that

in two periods it will be rated A, a probability of 64.4 percent that in two

periods it will be rated B, a probability of 23.3 percent that in two periods

it will be rated C, and a probability of 3.3 percent that in two periods it

will default (and hence be rated D).

Here is the spreadsheet:

ABCDEF

One-period transition matrix

ABCDE

0.9700 0.0200 0.0100 0.0000 0.0000

0.0500 0.8000 0.1500 0.0000 0.0000

0.0100 0.0200 0.7500 0.2200 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000

Two-period transition matrix

ABCDE

0.9420 0.0356 0.0202 0.0022 0.0000

0.0900 0.6440 0.2330 0.0330 0.0000

0.0182 0.0312 0.5656 0.1650 0.2200

0.0000 0.0000 0.0000 0.0000 1.0000

Cells B12:F16 contain the array formula =MMULT(B4:F8,B4:F8)

Three-period transition matrix

ABCDE

0.9157 0.0477 0.0299 0.0044 0.0022

0.1218 0.5217 0.2723 0.0513 0.0330

0.0249 0.0366 0.4291 0.1244 0.3850

0.0000 0.0000 0.0000 0.0000 1.0000

Cells B21:F25 contain the array formula =MMULT(B4:F8,B12:F16)

USING THE MMULT FUNCTION

To Compute Multiperiod Transition Matrices

725 Calculating Default-Adjusted Expected Bond Returns

In general, the year-t transition matrix is given by the matrix power

. Calculating these matrix powers by the procedure that we have illus-

trated is cumbersome, so we deﬁ ne a VBA function MatrixPower to

compute powers of matrices:

Function matrixpower(matrix, n)

If n = 1 Then

matrixpower = matrix

Else: matrixpower = Application.MMult _

(matrixpower(matrix, n - 1), matrix)

End If

End Function

The use of this function is illustrated in the following spreadsheet. The

function Matrixpower allows a one-step computation of the power of

any transition matrix:

ABCDEF

One-period transition matrix

ABCDE

0.9700 0.0200 0.0100 0.0000 0.0000

0.0500 0.8000 0.1500 0.0000 0.0000

0.0100 0.0200 0.7500 0.2200 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000

t10

t-period transition matrix

ABCDE

0.7648 0.0799 0.0699 0.0148 0.0706

0.2123 0.1429 0.1747 0.0432 0.4269

0.0450 0.0250 0.0755 0.0208 0.8338

0.0000 0.0000 0.0000 0.0000 1.0000

Cells B14:F18 contain the array formula =matrixpower(B4:F8,B10)

USING THE FUNCTION MATRIXPOWER

To Compute Multiperiod Transition Matrices

726 Chapter 28

From this example it follows that if a bond started out with an A rating,

there is a probability of 1.48 percent that the bond will be in default at

the end of ten periods and a 7.06 percent probability that it will have

defaulted in a previous period (rating E).

28.3.2 The Bond Payoff Vector

Recall that Q denotes the bond’s coupon rate and λ denotes the recovery

percentage payoff of face value if the bond defaults. The payoff vector

of the bond depends on whether the bond is currently in its last period

N or whether t < N:

Payoff Payoff ( )(, ) ,tt N

tt N

⎧

⎨

⎪

⎩

⎪

⎧

⎨

⎪

λλ

⎪⎪

⎩

⎪

The ﬁ rst three elements of each vector denote the payoff in nondefaulted

states, the fourth element λ is the payoff if the rating is D, and the ﬁ fth

element 0 is the payoff if the bond rating is E. (Recall that E is the rating

for the period after the bond defaults—in our model the payoff in rating

E is always zero.) The distinction between the two vectors depends, of

course, on the repayment of principal in the terminal period.

Before we can deﬁ ne the expected payoffs, we need to deﬁ ne one

further vector, which will denote the initial state of the bond. This current

state vector is a vector with a 1 for the current rating of the bond and

zeros elsewhere. Thus, for example, if the bond has rating A at date 0,

then Initial = [1 0 0 0 0]; if it has date-0 rating of B, then Initial =

[0 1 0 0 0].

We can now deﬁ ne the expected bond payoff in period t:

Et t

[()] ()Payoff Initial Payoff =∗∗Π

28.4 A Numerical Example

We continue using the numerical Π from the previous section to price a

bond with the following characteristics:

727 Calculating Default-Adjusted Expected Bond Returns

•

The bond is currently rated B.

•

Its coupon rate Q = 7 percent.

•

The bond has ﬁ ve more years to maturity.

•

The bond’s current market price is 100 percent of its face value.

•

The bond’s recovery percentage λ = 50 percent.

The following spreadsheet shows the facts in the preceding list as well

as the payoff vectors of the bond at dates before maturity (in cells F3:

F7) and on the maturity date (cells I3:I7). The transition matrix is given

in cells C10:G14, and the initial vector is given in C16:G16.

The expected bond payoffs are given in cells B20:I20. Before we

explain how they were calculated, we note the important economic fact

that—if the expected payoffs are as given—then the bond’s expected

return is calculated by the Excel IRR function. As Cell B21 shows, this

expected return is 4.61 percent. The actual formula in cell B21 is IRR(B20:

AN20). This allows the calculation of the IRR of bonds of maturity up

to 40 years.

ABCDEFGH

Bond price 100.00% Payoff (t<N) Payoff (N)

Coupon rate, Q 7% Cells to right 7% Cells to right 107%

Recovery rate,

50% are called 7% are called 107%

Bond term, N 5 "payoff1" 7% "payoff2" 107%

Initial rating B in row 20 50% in row 20 50%

0% 0%

AB C D E

Transition matrix ? A 0.9700 0.0200 0.0100 0.0000 0.0000

B 0.0500 0.8000 0.1500 0.0000 0.0000

C 0.0100 0.0200 0.7500 0.2200 0.0000

D 0.0000 0.0000 0.0000 0.0000 1.0000

0.0000 0.0000 0.0000 0.0000 1.0000

Initial vector 01 0 0

Formula in cell C16: =IF(UPPER(B6)="A",1,0)

Year 0 1 2 3 4 5 6 7

Expected payoffs -1.0000 0.0700 0.0842 0.0897 0.0899 0.8802 0.0000 0.0000

Expected yield 4.61% <

=IRR(B20:AN20,0)

CALCULATING THE EXPECTED BOND RETURN

=IF(year>bondterm,0,

IF(year=bondterm,MMULT(initial,MMULT(matrixpower(transition,year),payoff2)),

MMULT(initial,MMULT(matrixpower(transition,year),payoff1))))

IRR of expected

payoffs