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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…
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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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