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Chapter 23: Default-Adjusted Expected Bond Returns

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Yields are not expected returns

Stock markets work in expected returns: CAPM/SML

Bond markets work in yields, but yields confuse promised payments with expected payments .

Stocks versus bonds – totally different approaches!

In stock markets:

Expected returns

Gordon growth model: Dividend0*(1+expected div. growth)t

CAPM: E(r) = rf + b*[E(rM) – rf]

Gordon growth : Risk identified with div. growth

CAPM: Risk identified with b

Bond markets: yields, yield curves, and ratings

Where’s the risk?

What’s the expected bond return?

What’s the beta?

Expected bond return = IRR of expected bond payments

IRR is the risk-adjusted expected bond return

Components of expected bond payments:

Return of principal

Recovery percentage (in case of default)

(Very simple) Example

One period bond

Sold $90, coupon = 8%

Default probability, 20%

Recovery percentage, 40%

Expected return: 4.89%

For the one-period case: notice that E(rBond) = IRR

(Very simple) Example

Multi-period case

Assume that default depends on ratings

Assume stable ratings transition (“migration”) matrix

Augment matrix to deal with period after default

In next slides we examine the expected bond return on 5-year bond

Ratings: A, B, C, D

D = default; bond payoff = recovery percentage, l

Augmented rating: E

Defined as period past D – bond payoff zero

Augmented ratings transition matrix P:

A, B, C are obvious ratings

D is default

E is period after default

Notice that E is a “sink”

Augmented transition matrix

Financial Modeling (Chapter 23) defines a function called MatrixPower.

Other data

Two payoff vectors:

For period before or at maturity N:

Payoff if rating = A 7% (t

Bonds sold below par can have expected returns > YTM

An example: 20 July 2005

Augmented transition matrix

Source: Standard & Poors, 1981 – 2000

Ignoring AMR partial period problem

Note computation of accrued interest (cell N7, used in B4)

Note: we’ve annualized the IRR in cell B31

The IRR function calculates semiannual yield, which needs to be annualized

Altman-Kishore give average recovery for “transportation” as 38.42%. This indicates that the cost of AMR’s bond is ~3%

AMR expected return vs. recovery rate – ignores partial period

Altman & Kishore: Recovery percentages

We’ve used the matrix of annual transition probabilities, even though the bond has semiannual coupons.

Shouldn’t we find the semiannual transition matrix?

Can’t be done in Excel?

Problem: Semi-annual versus annual transition matrix

Semi-annual transition matrix

Semi-annual transition matrix: expected returns are, in general, higher

Semi-annual vs annual transition matrix

Computations using actual dates

Previous calculations ignore the “partial period” problem

Next spreadsheet uses XIRR in Excel and actual dates to compute expected bond return

Comparing expected returns with…

Click to edit Master text styles

Second level

Third level

Fourth level

Fifth level

Last – what is the bond’s b ?

Note: AMR’s equity

What’s left?

Better transition matrices

Industry-specific transition matrices

Time-dependent transition matrices

More data on recovery ratios

Expected Bond Returns and the Credit Risk Premium, written by and

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2373845

Face value, F

Annual coupon rate, Q

Default probability

Recovery percentage

Expected period 1 cash flow

<-- =B2*(1+B4)*(1-B5)+B2*B6*B5
Expected return
<-- =B8/B3-1
EXPECTED RETURN ON A ONE-YEAR BOND
WITH AN ADJUSTMENT FOR DEFAULT
PROBABILITY
ABCDEFGHIJKLM
0.97000.02000.01000.00000.0000
0.05000.80000.15000.00000.0000
0.01000.02000.75000.22000.0000
<-- Transition matrix
; powers of
0.00000.00000.00000.00001.0000
0.00000.00000.00000.00001.0000
Period2 Period4
ABCDE ABCDE
0.94200.03560.02020.00220.0000
0.89090.05710.03870.00660.0066
0.09000.64400.23300.03300.0000
0.14700.42520.28370.05990.0843
0.01820.03120.56560.16500.2200
0.03020.03840.32750.09440.5094
0.00000.00000.00000.00001.0000
0.00000.00000.00000.00001.0000
0.00000.00000.00000.00001.0000
0.00000.00000.00000.00001.0000
Period3 Period5
ABCDE ABCDE
0.91570.04770.02990.00440.0022
0.86740.06430.04650.00850.0132
0.12180.52170.27230.05130.0330
0.16670.34880.27800.06240.1442
0.02490.03660.42910.12440.3850
0.03450.03790.25170.07210.6038
0.00000.00000.00000.00001.0000
0.00000.00000.00000.00001.0000
0.00000.00000.00000.00001.0000
0.00000.00000.00000.00001.0000
MULTIPERIOD TRANSITION MATRIX,
Uses VBA-defined Excel function MatrixPower(matrix,power)
(see Financial Modeling)
Bond price 100.00%
Payoff (t

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

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

IRR of expected

ABCDEFGHIJK

Recover percentage,

4.61%<-- Table header
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%,
Recovery = 50%
0%10%20%30%40%50%60%70%80%90%100%
ABCDEFGHIJ
Recover percentage,
7.23%<-- Table header
10%5.41%7%
20%5.87%7%
30%6.33%7%
40%6.78%7%
50%7.23%7%
60%7.68%7%
70%8.13%7%
80%8.57%7%
90%9.01%7%
100%9.44%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 = 90%, Bond Rating = B, Coupon = 7.00%,
Recovery = 50%
0%10%20%30%40%50%60%70%80%90%100%
Recover percentage,
10.90%YTM<-- Table header
0%1.26%24%
10%4.28%24%
20%7.50%24%
30%10.90%24%
40%14.47%24%
50%18.20%24%
60%22.05%24%
70%26.00%24%
80%30.04%24%
90%34.14%24%
100%38.30%24%
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 = 50%, Bond Rating = C, Coupon = 11.00%,
Recovery = 30%, YTM = 23.97%
0%10%20%30%40%50%60%70%80%90%100%
original rating
probability of migrating to rating by year end (%)
A BCDEFGHIJK
Bond price 76.75% 3
Payoff (t

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

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

Annualized IRR of

expected payoffs

Coupon paid

semiannually

Currentdate- =*periodicinterestpayment

Nextinterestdate-Lastinterestdate

Accrued interest calculation

Last payment date

Next payment date

Current date

Percent of period passed

Accrued interest

<-- =(N6-N4)/(N5-N4)*B3/2
A BCDEFGHIJK
Recovery percentage,
YTM<-- Table header
0% -2.78%14.08%
10% -1.54%14.08%
20% -0.18%14.08%
30% 1.33%14.08%
40% 3.01%14.08%
50% 4.90%14.08%
60% 7.03%14.08%
70% 9.43%14.08%
80% 12.14%14.08%
90% 15.19%14.08%
100% 18.61%14.08%
Note: The data table has a series with the
YTM appended so that in the graph we
can see the relation of the bond expected
return to the YTM.
Data table: Recovery percentage
and expected yield
AMR Bond Expected Return vs . Recovery Rate
Bond price = 77%, Bond Rating = CCC,
Coupon = 10.55%, YTM = 14.08%
0%20%40%60%80%100%
Recovery percentage
Recovery Rates by Industry: Defaufted Bonds by Three-Digit SIC Code, 1971-95
Altman & Kishore, "Almost Everything You Wanted to Know about
Recoveries on Defaulted Bonds," Table 3
Financial Analysts Journal, November/December 1996, pp. 57- 64
Recovery Rate
Observations
observations
Public utilities
Chemicals, petroleum, rubber and plastic products
280,290,300
Machinery, instruments, and related products
350,360,380
Services--business and personal
470,632,720,730
Food and kindred products
Wholesale and retail trade
500,510,520
Diversified manufacturing
Casino, hotel, and recreation
Building materials, metals, and fabricated products
320,330,340
Transportation and transportation equipment
370,410,420,450
Communication, broadcasting, movies, printing, publishing
270,480,780
Financial institutions
600,610,620,630,670
Construction and real estate
General merchandise stores
530,540,560,570,580,000
Mining and petroleum drilling
Textile and apparel products
Wood, paper, and leather products
240,250,260,310
Lodging, hospitals, and nursing facilities
700 through 890
To eliminate negative entries
We assumed that a transition from A,
> E was impossible

We set other negative entries equal to zero

We set default probability in each row so that row sum

THE MATRIX BELOW WAS COMPUTED WITH MATHEMATICA

A BCDEFGHIJK

Bond price 76.75%

Payoff (t

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

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

Annualized IRR of

expected payoffs

Coupon paid

semiannually

percentage,

transition

Semiannual

transition

0%-2.78%0.98%

10%-1.54%2.12%

20%-0.18%3.33%

30%1.33%4.63%

40%3.01%6.03%

50%4.90%7.52%

60%7.03%9.13%

70%9.43%10.85%

80%12.14%12.70%

90%15.19%14.67%

100%18.61%16.76%

Expected bond return

COMPARING THE EXPECTED RETURNS WITH

SEMIANNUAL VS. ANNUAL TRANSITION MATRICES

Expected Bond Returns with Annual and

Semiannual Transition Matrices

0%20%40%60%80%100%120%

Recovery percentage

transition matrix

Semiannual

transition matrix

ABCDEFGHIJK

Bond price 76.75%

Payoff (t

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