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Chapter 23: Default-Adjusted Expected Bond Returns

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 coupon

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 (tbondterm,0,
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 (tbondterm,0,
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 (tbondterm,0,
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 (tbondterm,0,