Credit Risk

1
Credit Risk

• Yiling Lai
• 2008/10/3

2
Outline

• Introduction
• Credit Ratings
• Historical Default Probabilities
• Recovery Rates
• Estimating Default Probability from Bond Prices
• Comparison of Default Probability Estimates

3
Introduction

• Credit risk raises from the possibility that
  borrowers and counterparties in derivatives
  transactions may default.

4
Credit Rating

• Credit Rating assesses the creditworthiness of
  corporate bonds.

Default Risk SP Moodys ????
Low AAA Aaa twAAA
AA , AA, AA- Aa1, Aa2, Aa3 twAA, twAA, twAA-
A, A , A- A1, A2, A3 twA, twA, twA-
BBB, BBB, BBB- Baa1, Baa2, Baa3 twBBB, twBBB, twBBB-
BB, BB, BB- Ba1, Ba2, Ba3 twBB, twBB, twBB-
B, B, B- B1, B2, B3 twB, twB, twB-
High CCC Caa twCCC
Investment grade bond
Non-investment grade bond (high yield bond,
speculative grade bond or junk bond)
5
Historical Default Probabilities

• Average cumulative default rates(), 1970-2006.
  Source Moodys
• From this table, we can calculate unconditional
  default probability and conditional default
  probability.

Term 1 2 3 4 5 7 10 15 20
Aaa 0.000 0.000 0.000 0.026 0.099 0.251 0.521 0.992 1.191
Aa 0.008 0.019 0.042 0.106 0.177 0.343 0.522 1.111 1.929
A 0.021 0.095 0.220 0.344 0.472 0.759 1.287 2.364 4.238
Baa 0.181 0.506 0.930 1.434 1.938 3.959 4.637 8.244 11.362
Ba 1.205 3.219 5.568 7.985 10.215 14.005 19.118 23.380 35.093
B 5.236 11.296 17.043 22.054 26.794 34.771 43.343 52.175 54.421
Caa-C 19.476 30.494 39.717 46.904 52.622 59.938 69.178 70.870 70.870
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Historical Default Probabilities

• Unconditional default probability
• The probability of a bond defaulting during a
  particular year as seen at time 0.

Term 1 2 3 4 5 7
Aaa 0 0 0 0.026 0.073
Aa 0.008 0.011 0.023 0.064 0.071
A 0.021 0.074 0.125 0.124 0.128
Baa 0.181 0.325 0.424 0.504 0.504
Ba 1.205 2.014 2.349 2.417 2.23
B 5.236 6.06 5.747 5.011 4.74
Caa-C 19.476 11.018 9.223 7.187 5.718
Increasing
0.352 0.506-0.181
Decreasing
7
Historical Default Probabilities

• Conditional default probability (default
  intensity or hazard rate)
• The probability that the bond will default
  during a particular year conditional on no
  earlier default.

Average cumulative default rates Average cumulative default rates Average cumulative default rates Average cumulative default rates Average cumulative default rates
Term 1 2 3 4
Caa-C 19.476 30.494 39.717
Unconditional default probability Unconditional default probability Unconditional default probability Unconditional default probability Unconditional default probability
Term 1 2 3 4
Caa-C 19.476 11.018 9.223
The probability that the bond will survive until
the end of year 2 is 100-30.94969.506
Conditional default probability is 9.223/69.506
13.27 (for 1-year time period)
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Default Intensities
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(No Transcript)
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Default Intensities

• Q(t) the probability of default by time t.

11
Recovery Rates

• The recovery rate for a bond is normally defined
  as the bonds market value immediately after a
  default, as a percent of its face value.

Recovery rates on corporate bonds as a percentage of face value. 1982-2003 (Source Moodys) Recovery rates on corporate bonds as a percentage of face value. 1982-2003 (Source Moodys)
Class Average recovery rate ()
Senior secured 54.44
Senior unsecured 38.39
Senior Subordinated 32.85
Subordinated 31.61
Junior subordinated 24.47
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Recovery Rates

• Recovery rates are significantly negatively
  correlated with default rates.
• Recovery rate 59.1 - 8.356 x Default rate
• The recovery rate the average recovery rate on
  senior unsecured bonds in a year measured as
• The default rate the corporate default rate in
  the year measured as
• A bad year for the default rate is usually double
  bad because it is accompanied by a low recovery
  rate.

Default rate 0.1 gt Recovery rate 58.3
Default rate 3 gt Recovery rate 34.0
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Estimating Default Probability from Bond Prices

• Assumption The only reason a corporate bond
  sells for less than a similar risk-free bond is
  the possibility default.
• An approximate calculation

S 200 base points and R 40 gt
h0.02/(1-0.4)3.33
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A More Exact Calculation

• The corporate bond
• Period 5 years
• Coupon 6 per annum (paid semiannually)
• Yield 7 per annum (with continuous compounding)
• Price 95.34
• A similar risk-free bond
• Yield 5 per annum (with continuous compounding)
• Price 104.09
• The expected loss from default over the 5-year
  life of the bond is 104.9-95.348.75

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A More Exact Calculation

• Assumption defaults can happen at times 0.5,
  1.5, 2.5, 3.5, and 4.5 years (immediately before
  coupon payment dates).
• Notional Principal100
• Risk-free rates 5 (with continuous compounding)
• Recovery rate 40 gt 1004040

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A More Exact Calculation

• Consider the 3.5 years
• The expected value of the risk-free bond at time
  3.5 years
• The loss given default is 104.34-4064.34
• The PV of this loss is

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A More Exact Calculation

• Suppose that the probability of default per year
  (assumed to be the same each year) is Q.
• Calculating of loss from default on a bond in
  terms of the default probabilities per year, Q.

Time (years) Default Probabilities Recovery Amount () Risk-free Value() Loss given Default() Discount factor PV of expected loss ()
0.5 Q 40 106.73 66.73 0.9753 65.08Q
1.5 Q 40 105.97 65.97 0.9277 61.20Q
2.5 Q 40 105.17 65.17 0.8825 57.52Q
3.5 Q 40 104.34 64.34 0.8395 54.01Q
4.5 Q 40 103.46 63.46 0.7985 50.67Q
Total 288.48Q
288.48Q8.75 gt Q 3.03
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A More Exact Calculation

• We can extend this calculations by changing some
  assumptions.
• Example
• Defaults can take place more frequently
• A constant default intensity
• A particular pattern for the variation of default
  probabilities with time
• With several bonds we can estimate several
  parameters describing the term structure of
  default probabilities.

19
The Risk-Free Rate

• The benchmark risk-free rate that is usually used
  in quoting corporate bond yields is the yield on
  similar Treasury bonds.
• In fact, those derivative traders usually use
  LIBOR rates as short-term risk-free rates.
• They regard LIBOR as their opportunity cost of
  capital.
• Treasury rates are too low to be used as
  risk-free rates.

20
Asset Swaps

• Asset swaps asset interest rate swap or
• Asset swaps asset currency swap
• An asset swap transforms the character of an end
  users assets.
• Repacking an issue paying fixed rates into
  floating rates (or vice versa)
• Converting cash flow stated in one currency to
  another.
• It does not eliminate the asset from an
  investors portfolio.

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Fixed Rates to Floating Rates

• The investor transforms the yield on its fixed
  rate asset into a floating rate asset.

22
Currency Transformation

• The investor converts with the swap counterparty
  the FFr500mm principal cost of the asset and
  100mm, the investors base currency.

23
Asset Swaps Spread

• In practice, traders often use asset swap spreads
  as a way of extracting default probabilities from
  bond prices.
• Asset swap spreads provide a direct estimate of
  the spread of bond yields over the LIBOR/swap
  curve.
• Example
• An asset swap spread for a particular bond 150
  bps
• The LIBOR/swap zero curve is flat at 5
• The amount by which the value of the risk-free
  bond exceeds the value of the corporate is the
  present value of 150bps per year for 5years.
• Assuming semiannual payments

24
Asset Swaps Spread

• The sum of PV is 6.55 per 100 of principal.
• 288.48Q 6.55
• gt Q2.27

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Comparison of Default Probability Estimates (part
I)

• From historical data
• Based on
• Consider t7
• Consider an A-rated company in table 20.1
• Q(7) 0.00759

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Comparison of Default Probability Estimates (part
I)

• From bond prices
• Based on
• Recovery rate (R) 40
• Risk-free rate the 7-year swap rate minus 10
  bps.
• For A-rated bond, the average Merrill Lynch yield
  was 5.993
• The average swap rate was 5.398
• gt risk-free rate 5.398-0.15.298

s 0.05993-0.05298 0.00695
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Comparison of Default Probability Estimates (part
I)
Seven-year average default intensities ( per annum) Seven-year average default intensities ( per annum) Seven-year average default intensities ( per annum) Seven-year average default intensities ( per annum) Seven-year average default intensities ( per annum)
Rating Historical default intensity Default intensity from bonds Ratio Difference
Aaa 0.04 0.60 16.7 0.56
Aa 0.05 0.74 14.6 0.68
A 0.11 1.16 10.5 1.04
Baa 0.43 2.13 5.0 1.71
Ba 2.16 4.67 2.2 2.54
B 6.10 7.97 1.3 1.98
Caa and lower 13.07 18.16 1.4 5.50
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Comparison of Default Probability Estimates (part
II)

• Another way of looking at these results excess
  return over the risk-free rate

Expected excess return on bonds (bps) (A)-(B)-(C) Expected excess return on bonds (bps) (A)-(B)-(C) Expected excess return on bonds (bps) (A)-(B)-(C) Expected excess return on bonds (bps) (A)-(B)-(C) Expected excess return on bonds (bps) (A)-(B)-(C)
Rating Bond yield spread over Treasuries (A) Spread of risk-free rate over Treasuries (B) Spread for historical defaults (C) Excess return
Aaa 78 42 2 34
Aa 87 42 3 42
A 112 42 7 63
Baa 170 42 26 102
Ba 323 42 129 151
B 521 42 366 112
Caa 1132 42 784 305
Historical default intensity x (1-Recovery rate)
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Real-World vs. Risk-Neutral Probabilities

• The default probabilities implied from bond
  yields are risk-neutral probabilities of default.
• The risk-neutral valuation principle states that
  the price of a derivative is given by the
  expectation of the discounted terminal payoff
  under the risk neutral measure.
• This means that the default
• probability Q must be a risk-neutral
  probability.
• By contrast, the default probabilities implied
  from historical data are real-world probabilities
  of default.

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Why do we see such big differences between
real-world and risk-neutral default probabilities?

• Corporate bonds are relatively illiquid.
• The subjective default probabilities of bond
  traders may be much higher than those given in
  Tables 20.1.
• Bonds do not default independently of each other
  because of systematic risk.
• Unsystematic risk It is much more difficult to
  diversify risks in a bond portfolio than in an
  equity portfolio.

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Which one is better?

• The answer depends on the purpose of the
  analysis.
• Risk-neutral default probabilities
• To value credit derivatives
• To estimate the impact of default risk
• Real-world default probabilities
• To calculate potential future losses from default
  when carrying out scenario analyses