Simon Benninga. Financial Modelling 3-rd edition

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728 Chapter 28

Note the use of the IF statement in translating the bond’s initial rating

(cell B6) to the initial vector given in row 16. To avoid confusion, we

write this as IF(Upper(B6)=“A”,1,0), etc. This method guarantees that

even if the bond’s rating is entered as a lowercase letter, the initial vector

will come out correctly.

28.4.1 How to Calculate the Expected Bond Payoffs

As indicated in the previous section, the period-t expected bond payoff

is given by the following formula E[Payoff(t)] = Initial

Payoff(t).

The formula in row 21 uses two IF statements to implement this

formula:

=IF(year>bondterm,0,

IF(year=bondterm,MMUL(initial,MMULT(matrixpower(transition,

year),payoff2)),

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

)))

Here’s what these statements mean:

•

First IF: If the current year is greater than the bond term N (in our

example N = 5), then the payoff on the bond is 0.

•

Second IF: If the current year is equal to the bond term N, then the

expected payoff on the bond is MMULT(initial,MMULT(matrixpower

(transition,year),payoff2)). Here transition is the name for the transition

matrix in cells C10:G14 and payoff2 is the name for cells I3:I7.

•

If the current year n is less than the bond term, then the expected

payoff on the bond is MMULT(initial,MMULT(matrixpower(transition,

C18),payoff1), where payoff1 is the name for cells F3:F7.

Copying this formula gives the whole vector of expected bond

payoffs.

28.5 Experimenting with the Example

We can gain some insight into the relation between a bond’s expected

return, its coupon rate, and its yield to maturity (YTM) by constructing

some data tables. In the following data table we compute the bond’s

expected return as a function of its recovery percentage λ:

729 Calculating Default-Adjusted Expected Bond Returns

We conclude that a bond selling at par will, for every recovery percent-

age λ, have expected return less than the coupon rate. If the bond’s initial

rating is lower, then the expected return is less for every λ:

ABCDEFGH

Recovery percentage, l

4.61% <

=B21 , Tableheader

0% 2.43% 7%

10% 2.87% 7%

20% 3.31% 7%

30% 3.74% 7%

40% 4.18% 7%

50% 4.61% 7%

60% 5.03% 7%

70% 5.46% 7%

80% 5.88% 7%

90% 6.30% 7%

100% 6.71% 7%

Note: The data table has a series with the

coupon rate appended so that in the graph we

can see the convergence of the bond expected

return to the coupon rate (cells C30:C40)

Data Table: Recovery Percentage and Expected Yield

Bond Expected Return and Recovery Rate

Bond price = 100%, Bond Rating = B, Coupon = 7.00%

10090807060

Recovery rate (%)

Expected return (%)

5040302010

ABCDEFGH

Recovery percentage,

-4.69% <

=B21 , Tableheader

0% -14.72% 7%

10% -12.73% 7%

20% -10.74% 7%

30% -8.73% 7%

40% -6.71% 7%

50% -4.69% 7%

60% -2.66% 7%

70% -0.62% 7%

80% 1.43% 7%

90% 3.49% 7%

100% 5.55% 7%

Note: The data table has a series with the

coupon rate appended so that in the graph we

can see the convergence of the bond expected

return to the coupon rate (cells C30:C40)

Data Table: Recovery Percentage and Expected Yield

Bond Expected Return and Recovery Rate

Bond price = 100%, Bond Rating = C, Coupon = 7.00%

-20

-15

-10

-5

Recovery rate (%)

Expected return (%)

0 100908070605040302010

As shown in the next example, when the bond price is below par

(meaning that the bond is sold at less than 100 percent of its face value),

the bond’s expected return can be both below and above its coupon

rate.

730 Chapter 28

28.6 Computing the Bond Expected Return for an Actual Bond

In this section we illustrate the computation of a bond expected return

for an actual bond. Although the principles used are the same as those

discussed earlier, we introduce three innovations:

ABCDEFGHI

Bond price 88.00%

Payoff (t<N) Payoff (N)

Coupon rate, Q 7% Cells F6:F6 7% Cells I3:I6 107%

Recovery rate, λ

50% are called 7% are called 107%

Bond term, N 8 "payoff1" 7% "payoff2" 107%

Initial rating B in row 19 50% in row 19 50%

0% 0%

AB C D E

Transition matrix

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

Initial vector

0 1 0 0 0.0000

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

Year 0 1 2 3 4 5 6 7 8

Expected payoffs -0.8800 0.0700 0.0842 0.0897 0.0899 0.0867 0.0818 0.0761 0.6914

Expected yield 6.30%

Recover percentage, l

6.30% <

Table header

0% 3.33% 7%

10% 3.94% 7%

20% 4.54% 7%

30% 5.13% 7%

40% 5.72% 7%

50% 6.30% 7%

60% 6.87% 7%

70% 7.43% 7%

80% 8.00% 7%

90% 8.55% 7%

100% 9.10% 7%

Note: The data table has a series with the

coupon rate appended so that in the graph we

can see the convergence of the bond expected

return to the coupon rate (cells C30:C40)

Data table: Recovery percentage

and expected yield

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

Bond Expected Return and Recovery Rate

Bond price = 88%, Bond Rating = B, Coupon = 7.00%,

YTM = 9.183%

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

1009080706050

Recovery rate (%)

Expected return (%)

40302010

731 Calculating Default-Adjusted Expected Bond Returns

1. We compute the bond’s actual price using its quoted price and the

accrued interest. The accrued interest is jargon for the unpaid part of

the bond coupon since the last interest payment. In U.S. bond markets, the

accrued interest is added to the quoted bond price to compute the amount

actually paid for the bond. In most European bond markets, the quoted

bond price is the actual price paid for the bond and there is no separate

accrued interest calculation. The accrued interest is deﬁ ned as follows:

Accrued interest

Current date Last interest date

Next inter

−

eest date Last interest date

Periodic interest

−

∗

2. We use actual payment dates for the bond and use the XIRR function

to compute the bond’s expected yield.

3. We use an actual transition matrix for bond ratings.

The bond we analyze is a CCC-rated bond issued by AMR (the parent

company of American Airlines). Originally issued on 15 May 1991 and

maturing on 12 March 2021, the AMR bond has a coupon of 10.55

percent payable semiannually on 15 May and 15 November.

When we looked up the bond on Yahoo on 20 July 2005, its price was

76.75 percent of par:

To the quoted price of 76.75 percent we must add the bond’s accrued

interest:

732 Chapter 28

Accrued interest calculation

Last payment date 15-Mar-05

Next payment date 15-Sep-05

Current date 20-Jul-05

Percentage of period 0.69

Accrued interest 0.0364

It follows that the actual price paid for the bond is 76.75 percent + 3.64

percent = 80.39 percent.

In the following spreadsheet we compute the AMR expected return:

ABCDEFGHIJ

Bond price 76.75%

Payoff (t<N) Payoff (N)

Coupon rate, Q 10.55% Vector to

AAA

5.28% Vector to 105.28%

Actual price (includes accrued) 80.39% <

=B2+N8 right

5.28% right 105.28%

Recovery rate,

50.00% called

5.28% called 105.28%

Maturity date 12-Mar-21 "payoff1"

BBB

5.28% "payoff2" 105.28%

Current date 20-Jul-05

5.28% 105.28%

Initial rating CCC

5.28% 105.28%

Bond YTM 14.78%

CCC

5.28% 105.28%

Default

50.00% 50.00%

0% 0%

Transition matrix Æ

AAA AA A BBB BB B CCC Default E

AAA 0.9366 0.0583 0.0040 0.0008 0.0003 0.0000 0.0000 0.0000

0.0000

AA 0.0066 0.9172 0.0694 0.0049 0.0006 0.0009 0.0002 0.0001

0.0000

A 0.0007 0.0225 0.9176 0.0519 0.0049 0.0020 0.0001 0.0004

0.0000

BBB 0.0003 0.0025 0.0483 0.8926 0.0444 0.0081 0.0016 0.0022

0.0000

BB 0.0003 0.0007 0.0044 0.0667 0.8331 0.0747 0.0105 0.0098

0.0000

B 0.0000 0.0010 0.0033 0.0046 0.0577 0.8419 0.0387 0.0530

0.0000

CCC 0.0016 0.0000 0.0031 0.0093 0.0200 0.1074 0.6396 0.2194

0.0000

Default 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000

AAA AA A BBB BB B CCC Default E

Initial vector

000 0 0 0100

Computing the AMR bond expected return

Date 20-Jul-05 15-Nov-05 15-Mar-06 15-Sep-06 15-Mar-07 15-Sep-07 15-Mar-08 15-Sep-08 15-Mar-09 15-Sep-09

Expected payoffs -0.8039 0.1509 0.1066 0.0781 0.0594 0.0471 0.0387 0.0329 0.0287 0.0257

Expected bond return 2.47% <

=XIRR(B32:AP32,B31:AP31)

CALCULATING THE EXPECTED BOND RETURN

This version computes the expected bond return for AMR taking into account actual dates

Uses annual transition matrix

Original

rating

Probability of migrating to rating by year end (%)

=IF(E31>$B$6,0,

IF(E31=$B$6,MMULT(initial,MMULT(matrixpower(transition,E30),payoff2)),

MMULT(initial,MMULT(matrixpower(transition,E30),payoff1))))

Annualized IRR of expected

payoffs computed using actual

dates with

XIRR

Coupon paid

semiannually

Assuming a recovery rate of 50 percent, the bond has an expected

return of 2.47 percent.

733 Calculating Default-Adjusted Expected Bond Returns

28.6.1 What Are the Recovery Rates?

The recovery rate is obviously a critical factor in computing the bond’s

expected return. Considerable information exists on the recovery rates

in bankruptcy from various industries. A table from an article by Edward

Altman and Velore M. Kishore follows; from this table we can see that

the average recovery rate from a variety of industries was 41 percent.

Recovery Rates by Industry: Defaulted Bonds by Three-Digit SIC Code, 1971–1995

Recovery Rate

Industry SIC Code

Number of

Observations Average

Weighted

Observation

Median

Average

Standard

Deviation

Weighted

Public utilities 490 56 70.47 65.48 79.07 19.46

Chemicals, petroleum,

rubber and plastic products

280,290,300 35 62.73 80.39 71.88 27.10

Machinery, instruments, and

related products

350,360,380 36 48.74 44.75 47.50 20.13

Services–business and

personal

470,632,720,730 14 46.23 50.01 41.50 25.03

Food and kindred products 200 18 45.28 37.40 41.50 21.67

Wholesale and retail trade 500,510,520 12 44.00 48.90 37.32 22.14

Diversiﬁ ed manufacturing 390,998 20 42.29 29.49 33.88 24.98

Casino, hotel, and recreation 770,790 21 40.15 39.74 28.00 25.66

Building materials, metals,

and fabricated products

320,330,340 68 38.76 29.64 37.75 22.86

Transportation and

transportation equipment

370,410,420,450 52 38.42 41.12 37.13 27.98

Communication,

broadcasting, movies,

printing, publishing

270,480,780 65 37.08 39.34 34.50 20.79

Financial institutions 600,610,620,630,

670

66 35.69 35.44 32.15 25.72

Construction and real estate 150,650 35 35.27 28.58 24.00 28.69

General merchandise stores 530,540,560,570,

580,000

89 33.16 29.35 30.00 20.47

Mining and petroleum

drilling

100,103 45 33.02 31.83 32.00 18.01

Textile and apparel products 220,230 31 31.66 33.72 31.13 15.24

Wood, paper, and leather

products

240,250,260,310 11 29.77 24.30 18.25 24.38

Lodging, hospitals, and

nursing facilities

700 through 890 22 26.49 19.61 16.00 22.65

Total 696 41.00 39.11 36.25 25.56

Source: E. Altman and V. M. Kishore, “Almost Everything You Wanted to Know about Recoveries on Defaulted

Bonds,” Table 3, Financial Analysts Journal, November/December 1996, pp. 57–64.

734 Chapter 28

Using the Altman and Kishore numbers, the average recovery percent-

age for transportation companies is 38.42 percent, with a standard devia-

tion of 27.98 percent. Taking one standard deviation on either side of the

average, we can conclude that the recovery percentage for a transporta-

tion company is somewhere around (38.42 percent − 27.98 percent, 38.42

percent + 27.98 percent) = (∼10 percent, ∼66 percent).

In the next spreadsheet we have “backward engineered” a plausible

set of recovery ratios, from 55 to 65 percent, for the AMR bond. These

“guesstimates” for the AMR recovery are based on two assumptions:

•

The AMR bond should not have an expected return signiﬁ cantly more

than the riskless rate of return, which at the time of our calculations was

around 4 percent.

•

The AMR bond expected return should be signiﬁ cantly less than its

YTM of ∼15 percent. The YTM is based on promised payments, and we

ﬁ nd it implausible that these should correspond to the expected

returns.

This calculation gives an expected bond yield of between 3.85 and 6.86

percent for the AMR bond (highlighted area):

ABCDEFGH

Recovery percentage,

YTM

0.0% -7.93% 14.78%

10.0% -6.22% 14.78%

20.0% -4.36% 14.78%

30.0% -2.32% 14.78%

40.0% -0.05% 14.78%

55.0% 3.85% 14.78%

60.0% 5.31% 14.78%

65.0% 6.86% 14.78%

80.0% 12.15% 14.78%

90.0% 16.27% 14.78%

100.0% 20.93% 14.78%

Cell B41 contains the data table header =B33

Data Table: Recovery Percentage and Expected AMR Bond Return

-10

-5

0.0

Recovery percentage l (%)

Expected return (%)

100.080.060.040.020.0

28.7 Semiannual Transition Matrices

The analysis of the AMR bond in the previous section assumes that the

annual transition probabilities are also valid for bonds paying semian-

nual coupons. We could reﬁ ne this assumption by computing a semian-

nual transition matrix from the S&P data. Such a matrix would be the

square root of the Π matrix. This calculation cannot be easily done in

735 Calculating Default-Adjusted Expected Bond Returns

Excel. In the following spreadsheet we have used a computation from

Mathematica to ﬁ nd the semiannual transition matrix.

AAA AA A BBB BB B CCC Default E

AAA 0.9366 0.0583 0.0040 0.0008 0.0003 0.0000 0.0000 0.0000

0.0000

AA 0.0066 0.9172 0.0694 0.0049 0.0006 0.0009 0.0002 0.0001

0.0000

A 0.0007 0.0225 0.9176 0.0519 0.0049 0.0020 0.0001 0.0004

0.0000

BBB 0.0003 0.0025 0.0483 0.8926 0.0444 0.0081 0.0016 0.0022

0.0000

BB 0.0003 0.0007 0.0044 0.0667 0.8331 0.0747 0.0105 0.0098

0.0000

B 0.0000 0.0010 0.0033 0.0046 0.0577 0.8419 0.0387 0.0530

0.0000

CCC 0.0016 0.0000 0.0031 0.0093 0.0200 0.1074 0.6396 0.2194

0.0000

Default 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000

AAA AA A BBB BB B CCC Default E

AAA

0.9677 0.0303 0.0015 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000

0.0034 0.9574 0.0362 0.0021 0.0002 0.0004 0.0001 0.0000 0.0000

0.0003 0.0117 0.9573 0.0272 0.0023 0.0010 0.0000 0.0003 -0.0001

BBB

0.0001 0.0012 0.0254 0.9439 0.0238 0.0038 0.0008 0.0017 -0.0007

0.0002 0.0003 0.0018 0.0359 0.9115 0.0406 0.0056 0.0068 -0.0026

0.0000 0.0005 0.0017 0.0018 0.0314 0.9161 0.0225 0.0510 -0.0248

CCC

0.0009 -0.0001 0.0016 0.0050 0.0105 0.0624 0.7988 0.2706 -0.1495

Default

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

AAA AA A BBB BB B CCC Default E

AAA

0.9677 0.0303 0.0015 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000

0.0034 0.9574 0.0362 0.0021 0.0002 0.0004 0.0001 0.0001 0.0000

0.0003 0.0117 0.9573 0.0272 0.0023 0.0010 0.0000 0.0001 0.0000

BBB

0.0001 0.0012 0.0254 0.9439 0.0238 0.0038 0.0008 0.0010 0.0000

0.0002 0.0003 0.0018 0.0359 0.9115 0.0406 0.0056 0.0041 0.0000

0.0000 0.0005 0.0017 0.0018 0.0314 0.9161 0.0225 0.0261 0.0000

CCC

0.0009 0.0000 0.0016 0.0050 0.0105 0.0624 0.7988 0.1208 0.0000

Default

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

COMPUTING THE SQUARE ROOT OF THE TRANSITION MATRIX

The square root of the one-year transition matrix as computed by

Mathematica

Note the negative entries

The one-year transition matrix

Original

rating

Probability of migrating to rating by year end (%)

The semiannual transition matrix. To eliminate the negative entries in the

Mathematica

matrix:

a. We assume that a transition from AAA, .

. . , CCC

> E is impossible (i.e., last

column is zero except for last two entries).

b. We set all other negative entries to zero.

c. We set default probability so that each row sums to 1.

Original

rating

Probability of migrating to rating by year end (%)

Original

rating

Probability of migrating to rating by year end (%)

5. Mathematica is a high-powered computational program. See http://www.wolfram.com.

736 Chapter 28

If we use the semiannual transition matrix to compute the expected

bond returns, we get the following results:

ABCDEFGHIJ

Bond price 76.75%

Payoff (t<N) Payoff (N)

Coupon rate, Q 10.55% Vector to

AAA

5.28% Vector to 105.28%

Actual price (includes accrued) 80.39% <

=B2+N8 right

5.28% right 105.28%

Recovery rate,

50.00% called

5.28% called 105.28%

Maturity date 12-Mar-21 "payoff1"

BBB

5.28% "payoff2" 105.28%

Current date 20-Jul-05

5.28% 105.28%

Maturity (years) 15.65

5.28% 105.28%

Number of semiannual payments 30.00

CCC

5.28% 105.28%

Initial rating CCC

Default

50.00% 50.00%

Bond YTM 14.81%

0% 0%

Transition matrix Æ

AAA A

AA BBBBBBCCCDefaultE

AAA

0.9677 0.0303 0.0015 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000

0.0034 0.9574 0.0362 0.0021 0.0002 0.0004 0.0001 0.0001 0.0000

0.0003 0.0117 0.9573 0.0272 0.0023 0.0010 0.0000 0.0001 0.0000

BBB

0.0001 0.0012 0.0254 0.9439 0.0238 0.0038 0.0008 0.0010 0.0000

0.0002 0.0003 0.0018 0.0359 0.9115 0.0406 0.0056 0.0041 0.0000

0.0000 0.0005 0.0017 0.0018 0.0314 0.9161 0.0225 0.0261 0.0000

CCC

0.0009 -0.0001 0.0016 0.0050 0.0105 0.0624 0.7988 0.1208 0.0000

Default

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

AAA AA

A BBB BB B CCC Default E

Initial vector

000 0 0 0100

Period 012 3 4 56789

Date 20-Jul-05 15-Sep-05 15-Mar-06 15-Sep-06 15-Mar-07 15-Sep-07 15-Mar-08 15-Sep-08 15-Mar-09 15-Sep-09

Expected payoffs -0.8039 0.1068 0.0903 0.0771 0.0664 0.0578 0.0508 0.0451 0.0405 0.0367

Expected yield 8.07% <

=XIRR(B31:AP31,B30:AP30)

CALCULATING THE EXPECTED BOND RETURN

This version computes the expected bond return for AMR taking into account actual dates

Uses semiannual transition matrix

Original

rating

Probability of migrating to rating by year end (%)

Coupon paid

semiannually

Recovery

percentage,

Annual

transition

matrix

Semiannual

transition

matrix

0% -3.05% 0.93%

10% -1.75% 2.14%

20% -0.30% 3.45%

30% 1.32% 4.86%

40% 3.15% 6.40%

55% 5.25% 8.07%

60% 7.68% 9.90%

65% 10.51% 11.89%

80% 13.83% 14.07%

90% 17.73% 16.45%

100% 22.33% 19.04%

Expected bond return

COMPARING THE EXPECTED RETURNS WITH

SEMIANNUAL VS. ANNUAL TRANSITION MATRICES

Using actual dates and XIRR

Expected Bond Returns with Annual and

Semiannual Transition Matrices

-5

Recovery percentage l (%)

Expected return (%)

Annual

transition matrix

Semiannual

transition matrix

100908065605540302010

The semiannual transition matrix gives expected bond returns that are,

in general, higher than those given by the annual transition matrix:

737 Calculating Default-Adjusted Expected Bond Returns

28.8 Computing Bond Beta

A vexatious problem in corporate ﬁ nance is the computation of bond

betas. The model presented in this chapter can easily be used to compute

the beta of a bond. Recall from Chapter 2 that the capital asset pricing

model’s security market line (SML) is given by

Er r Er r

dfdmf

() [( ) ]=+ −β

where E(r

) is the expected return on debt, r

is the return on riskless

debt, and E(r

) is the return on equity market portfolio.

If we know the expected return on debt, we can calculate the debt β,

provided we know the risk-free rate r

and the expected rate of return

on the market E(r

). Suppose, for example, that the market risk premium

E(r

) − r

= 8.4 percent and that r

= 7 percent. Then a bond having an

expected return of 8 percent will have a β of 0.119:

CBA

Market risk premium, E(r

) - r

8.40%

Expected bond return 8.00%

Implied bond beta 0.119 <

=(B4-B3)/B2

CALCULATING A BOND'S BETA

CBA

Tax-adjusted SML (see section 2.6)

Market risk premium, E(r

) - r

8.40%

Corporate tax rate, T

40%

Expected bond return 8.00%

Implied bond beta 0.089 <

=(B11-B9)/(B8+B9*B10)

If we use the tax-adjusted version of the SML (see section 2.6), then

the bond SML becomes r

= Cost of debt = r

+ β

Debt

[E(r

) −

(1 − T

)].

This formula gives the bond beta as

Using our data for AMR, we get the following results for the classical

SML model: