Transcript Credit Risk, Chapter 23

8.1
Credit Risk
Lecture n.8
8.2
Credit Ratings
• In the S&P rating system AAA is the best
rating. After that comes AA, A, BBB, BB, B,
and CCC
• The corresponding Moody’s ratings are Aaa,
Aa, A, Baa, Ba, B, and Caa
• Bonds with ratings of BBB (or Baa) and
above are considered to be “investment
grade”
• Another provider of ratings is Fitch
8.3
Some basic questions
• What is rating?
It is a system to provide an easy to understand
signal about the probability of default of a
certain financial instrument and the loss caused
by this default
• What is default?
– tricky question, it depends on the contract
• debt restructuring, failure to pay, bankruptcy
• What is loss given default?
8.4
Average Cumulative Default Rates (%)
(S&P Credit Week, April 15, 1996, Table 23.2, page 627)
Yrs
1
2
3
4
5
7
10
AAA
0.00
0.00
0.07
0.15
0.24
0.66
1.40
AA
0.00
0.02
0.12
0.25
0.43
0.89
1.29
A
0.06
0.16
0.27
0.44
0.67
1.12
2.17
BBB
0.18
0.44
0.72
1.27
1.78
2.99
4.34
BB
1.06
3.48
6.12
8.68
10.97 14.46
17.73
B
5.20
11.00
15.95 19.40
21.88 25.14
29.02
19.79 26.92
31.63 35.97
40.15 42.64
45.10
CCC
8.5
from default statistics to bond pricing
• we need to know, what is the default definition:
• changes according to the contract
• ISDA effort has not yet produced a standard
• changes according to the nature of the issuer (sovereign vs
private, corporate vs bank, etc)
• changes according to the legislations
• we need to know what is the loss caused by a default
• changes according to financial instruments
• changes according to covenants
•changes according to legislations
• changes according to sectors
• changes according to economic cycles
8.6
Example:
let’s start from these 2 ZCB curves
Maturity Risk-free Corporate
bond yield
yield
(years)
5.25%
5%
1
2
5%
5.50%
3
5%
5.70%
4
5%
5.85%
5
5%
5.95%
8.7
implied losses from defaults
• One-year Treasury bond (principal=$1) sells for
0.051
e
 0.951229
• One-year corporate bond (principal=$1) sells
for
0.05251
e
 0.948854
or at a 0.2497% discount
• This indicates that the holder of a corporate
bond expects to lose 0.2497% from defaults in
the first year
Notation
h(T1,T2): Expected proportional loss between
times T1 and T2 as seen at time zero
y (T ): yield on a zero-coupon corporate bond
maturing at time T
y *(T ) yield on a zero-coupon Treasury bond
maturing at time T
P (T ): price of a zero-coupon corporate bond
paying $1 at time T
P*(T ): price of a zero-coupon Treasury bond
paying $1 at time T
8.8
8.9
Estimating Default Statistics
from Bond Prices
P * (T )  P(T )
h(0, T ) 
P * (T )
Because P(T )  e
 y(T )T
and P (T )  e
*
[ y ( T )  y * ( T )]T
h(0, T )  1  e
Because h(T1 , T2 )  h(0, T2 )  h(0, T1 )
h(T1 , T2 )  e
[ y ( T1 )  y * ( T1 )]T1
e
[ y ( T2 )  y * ( T2 )]T2
 y* ( T ) T
8.10
implied forward losses
• Similarly the holder of a A-rated bond
expects to lose
e0.05502  e0.052
 0.009950
 0.05 2
e
or 0.9950% in the first two years
• Between years one and two the
expected loss is 0.7453%
another example
8.11
Suppose that the spreads over Treasuries for
the yields on 5 - and 10 - year BBB - rated zero
coupon bonds are 130 and 170 basis points.
In this case:
h( 0,5)  1  e0.0135  0.0629
h( 0,10)  1  e0.01710  01563
.
so that h(5,10)  01563
.
 0.0629  0.0934.
The expected loss on a BBB bond between years
5 and 10 is 9.34% of the no - default value
8.12
Bond Prices vs Historical
Default Experience
• The estimates of the probability of
default calculated from bond prices are
much higher than those from historical
data
• Consider for example A-rated bonds
• These typically yield at least 50 bps
more than Treasuries
8.13
Bond Prices vs Historical
Default Experience
This means that we expect to lose at least
1  e0.0055  0.0247
or 2.47% of the bond' s value over a 5 - year period.
Assume a low recovery rate of 30%.
The probability of default is then 2.47 0.7  353%
.
This is over five times greater than the 0.67%
historical probability
8.14
Possible Reasons for These
Results
• The liquidity of corporate bonds is less
than that of Treasury bonds
• Bonds traders may be factoring into
their pricing depression scenarios much
worse than anything seen in the last 20
years
8.15
A Key Theoretical Reason
• The default probabilities estimated from
bond prices are risk-neutral default
probabilities
• The default probabilities estimated from
historical data are real-world default
probabilities
• In the real world we earn an extra 40
bps per year
8.16
Real World vs Risk Neutral
World
• When we infer default probabilities or
expected losses from security prices they are
“risk-neutral”. When we infer them from
historical data they are “real-world”
8.17
Quantifying the Cost of Default
The cost of default has to be estimated
not only for traditional assets such as
bonds, but also for Derivatives:
• Those that are always assets to one
party and liabilities to the other (e.g.,
options)
• Those that can become assets or
liabilities (e.g., swaps, forward
contracts)
8.18
Notation
f *: value of derivative assuming defaults
are impossible
f : actual value of derivative
8.19
Independence Assumption
• The independence assumption states
that the variables affecting the price of a
derivative are independent of the
variables determining defaults
• This assumption (although not perfect)
makes pricing for default risk possible
8.20
Contracts that are Assets
• Consider a contract that promises a
payoff at time T
• The contract is assumed to rank equally
with an unsecured bond in the event of
a default
8.21
Contracts that are Assets
continued
f  f
 h( 0, T )
*
f
*
* [ y ( T ) y * ( T )]T
f  f e
8.22
A Simple Interpretation
• Use the “risky” discount rate rather than
the risk-free discount rate when
discounting cash flows in a risk-neutral
world
• Note that this does not mean we simply
increase the interest rate in option
pricing formulas
Credit Exposure for Contracts 8.23
That Can be Assets or Liabilities
Exposure
Contract value
8.24
Netting
• Most derivative contracts state that, if a
company defaults on one contract with
a counterparty, it must default on all
contracts with that counterparty
• This means that the total exposure is an
option on a portfolio rather than a
portfolio of options
8.25
Incremental Effect of a New
Contract
• The incremental effect of a new contract
on credit risk is found by calculating
expected credit losses with and without
the contract
• The incremental effect can be negative
8.26
Reducing Exposure to Credit
Risk
• Set credit limits
• Ask counterparty to post collateral
• Design contract to reduce credit risk (eg
margins)
• Include a downgrade trigger in contract
• use credit derivatives
8.27
Credit Derivatives
Examples:
• Credit default swap
• Total return swap
• Credit spread option
• CDO
• ABS
8.28
Credit Default Swap
• Company A has the right to sell a
reference bond for its face value to
company B in the event there is a
default on the bond
• In return, A makes periodic payments to
B
• The reference bond is issued by a third
party, C
8.29
Total Return Swap
• The total return from one asset or group
of assets is swapped for the total return
from another asset or group of assets
• If the deal is fair, the assets have the
same market value at the beginning of
the life of a total return swap
8.30
Uses of Total Return Swap
• Consider a bank in Texas lending
primarily to oil companies and a bank in
Michigan lending primarily to auto
manufacturers and their suppliers
• A total return swap allows them to
achieve credit risk diversification without
buying or selling assets
8.31
Credit Spread Option
• This is an option on the spread between
the yields earned on two assets.
• The option provides a payoff whenever
the spread exceeds some level
8.32
Attraction of Credit Derivatives
• Allows credit risks to be exchanged
without the underlying assets being
exchanged
• Allows credit risks to be managed
8.33
Does Credit risk mean only default?
• Credit risk is not a one-dimensional
problem of default
• It is also a problem of credit
deterioration, i.e. the worsening of the
rating (and then increased def.prob.)
• An example is constituted by mutual
funds investing in Investment Grade
bond. Their problem is downgrading to
sub-investment grades
One-Year Transition Matrix
Year End
Rating
Init
Rat
AAA
AA
A
BBB
BB
B
CCC
Def
90.81
8.33
0.68
0.06
0.12
0.00
0.00
0.00
AA
0.70
90.65
7.79
0.64
0.06
0.14
0.02
0.00
A
0.09
2.27
91.05
5.52
0.74
0.26
0.01
0.06
BBB
0.02
0.33
5.95
86.93
5.30
1.17
0.12
0.18
BB
0.03
0.14
0.67
7.73
80.53
8.84
1.00
1.06
B
0.00
0.11
0.24
0.43
6.48
83.46
4.07
5.20
CCC
0.22
0.00
0.22
1.30
2.38
11.24
64.86 19.79
Def
0.00
0.00
0.00
0.00
0.00
0.00
AAA
0.00
100
8.34
8.35
Default probability obtained via
option pricing: Merton’s Model
• Merton’s model regards the equity as an
option on the assets of the firm
• In a simple situation the equity value is
max(VT -D, 0)
where VT is the value of the firm and D
is the debt repayment required
8.36
Equity vs Assets
An option pricing model enables the
value of the firm’s equity today, E0, to be
related to the value of its assets today,
V0, and the volatility of its assets, sV
E0  V0 N (d1 )  De  rT N (d 2 )
where
lnV0 / D  (r  sV2 2)T
d1 
; d 2  d1  sV T
sV T
8.37
Volatilities
V0 E
sE 
sV
E 0 V
This equation together with the option pricing
relationship enables V and sV to be
determined from E and sE