• Credit Risk

• How to price the credit risk?

– Credit risk and Default Probability
• Credit Ratings
– In the S&P/Fitch rating system, AAA is the best rating. After
that comes to AA, A, BBB, BB, B, CCC, CC, and C
– The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B,
Caa, Ca, and C
– Bonds with ratings of BBB (or Baa) and above are considered to
be “investment grade”; The rest are “junk” bonds
• Default Probabilities
– Predict default and estimate DP
» Altman’s Z-score (using accounting ratios)
» use historical data to estimate historical DP
» use credit spreads (credit risk priced by the market)
» use Merton’s model (modeling equity as a call option on the
assets of the firm)
» The importance of default correlation
– Altman’s Z-score (Manufacturing Companies)
• X1=Working Capital/Total Assets
• X2=Retained Earnings/Total Assets
• X3=EBIT/Total Assets
• X4=Market Value of Equity/Book Value of Liabilities
• X5=Sales/Total Assets

Z = 1.2X1+1.4X2+3.3X3+0.6X4+0.99X5
If the Z > 3.0 default is unlikely; if 2.7 < Z < 3.0 we should be on
alert. If 1.8 < Z < 2.7 there is a moderate chance of default; if Z < 1.8
there is a high chance of default

• 80-90% accuracy in predicting default within a year

• All these ratios are still relevant.
• ML techniques can be used to obtain a better predictor.
– Historical default probabilities
• Cumulative Average Default Rates (%) (1970-2015, Moody’s)

1 2 3 4 5 7 10
Aaa 0.000 0.011 0.011 0.031 0.087 0.198 0.396
Aa 0.022 0.061 0.112 0.196 0.305 0.540 0.807
A 0.056 0.170 0.357 0.555 0.794 1.345 2.313
Baa 0.185 0.480 0.831 1.252 1.668 2.525 4.033
Ba 0.959 2.587 4.501 6.538 8.442 11.788 16.455
B 3.632 8.529 13.515 17.999 22.071 29.028 36.298
Caa-C 10.671 18.857 25.639 31.075 35.638 41.812 47.843

The table shows the probability of default for companies starting with a
particular credit rating
A company with an initial credit rating of B has a probability of 8.529%
of defaulting by the end of the second year, 13.515% by the end of the
third year, and so on (PD during the third year = 13.515-8.529=4.99%)
• The probability of a B-rated bond defaulting during the third year
conditional on no earlier default is 0.0499/0.916 or 5.45%
– Hazard Rate
• The hazard rate (also called default density), λ(t), at time t, is
defined so that λ(t)∆t is the conditional default probability for a short
period between t and t+∆t
• If V(t) is the probability of a company surviving to time t
V(t+∆t) – V(t) = -𝜆𝜆(t)V(t) ∆t
dV(t)/dt = -𝜆𝜆(t)V(t)
𝑡𝑡
− ∫0 𝜆𝜆 𝜏𝜏 𝑑𝑑𝜏𝜏 �
This leads to 𝑉𝑉 𝑡𝑡 = 𝑒𝑒 = 𝑒𝑒 −𝜆𝜆𝑡𝑡
1 𝑡𝑡
𝜆𝜆̅ = 𝑡𝑡 ∫0 𝜆𝜆 𝜏𝜏 𝑑𝑑𝜏𝜏 is the average hazard rate between 0 and t.
• Q(t) is the probability of defaults by time t,
𝑄𝑄 𝑡𝑡 = 1 − 𝑉𝑉 𝑡𝑡 = 1 − 𝑒𝑒 −𝜆𝜆𝑡𝑡
– Recovery Rate
• The recovery rate for a bond is usually defined as the price of the
bond immediately after default as a percent of its face value
Recovery Rates; Moody’s: 1982 to 2015
Class Mean(%)

First Lien Bond 53.4

Second Lien Bond 49.7

Senior Unsecured 37.6

Senior Subordinated 31.1

Subordinated 31.9

Junior Subordinated 24.2

• Recovery rates tend to decrease as default rates

increase (both for mortgage default and corporate
bond default)
– Default Probability Using Credit Spreads
• Credit Default Swap
Like an insurance contract that pays off in case of a default
• If the reference entity (a country or company) defaults,
– Physical settlement: the buyer of the CDS has the right to sell to
the seller the bonds issued by the reference entity for their face
value
– Cash settlement: based on the difference between the face value
and the cheapest-to-deliver bond after the credit event (based on
the auction of the bonds within a certain maturity range)
– The spread payments stop after the default
• Example: Notional principal (e.g. $100 million) and maturity specified;
Protection buyer pays a fixed rate of 120 bps on the notional principal
(the CDS spread)
– Insurance premium 1.2 million per year
– If a default occurs and supposed the cheapest-to-deliver bond
issued by the reference entity is worth 40 cents per dollar, the
seller of CDS has to make a payment equal to $60 million (the
insurance is to bring the value back up to $100 million).
– Relating credit spread to the hazard rate
• Suppose s(T) is the credit spread for maturity T, s(T) should
compensate for the average loss rate.
• The average loss rate between time zero and time T is
approximately
𝜆𝜆̅ 𝑇𝑇 1 − 𝑅𝑅 = 𝑠𝑠 𝑇𝑇
where R is the recovery rate
Thus
𝜆𝜆̅ 𝑇𝑇 = 𝑠𝑠(𝑇𝑇)/ 1 − 𝑅𝑅
• This estimate is quite accurate in most situations
• Example: 1-year and 2-year bonds yield 150 and 180 more than the
risk-free rate, respectively. The recovery rate is estimated at 40%.
– Average 1 year hazard rate = 0.015/(1-0.4) = 2.5%
– Average 2 year hazard rate = 0.018/(1-0.4) = 3.0%
– Average 2nd year hazard rate = 3.5%
– Matching Bond Prices
• For more accuracy we can work forward in time choosing hazard
rates that match bond prices, using the bootstrap method
• Example:
– Given risk-free rate = 5% and
1-year, 2-year, 3-year bond yields: 6.5%, 6.8%, 6.95%
bond face value = 100, coupon rate 8% (semi-annual comp.)
– We can obtain
» bond prices (from the yields): $101.33, $101.99, $102.47
» bond prices (if they were risk-free, discounted using the
risk-free rate of 5%) would be: $102.83, $105.52, $108.08
– The present value of the expected default losses (the difference
between bond prices assuming they are risk-free and the actual
bond prices): $1.50, $3.53, and $5.61 for 1-year, 2-year, and 3-
year bonds
• Example (cont.)
– Consider a 1-year bond and assume that the recovery rate is
40% --- we want to relate the hazard rate λ1 to the expected
default loss of $1.50
– Assume that the defaults can happen only at the midpoints of
6-month intervals (default times are in 3 months and 9 months)
» This is a reasonable numerical approximation --- it would
be exact if the result linearly depends on the time of
default; so essentially it is the first-order linear
approximation of the default time dependence.
– The risk-free value of the bond at the 3-month point is
4e-0.05x0.25+104e-0.05x0.75 = $104.12
– The present value of the loss if there is a default at the 3-month
point is (104.12-40)e-0.05x0.25 = $63.33
– The risk-free value of the bond at the 9-month point is
104e-0.05x0.25 = $102.71
– The present value of the loss if there is a default at the 9-month
point is (102.71-40)e-0.05x0.75 = $60.40

x x
6 12
• Example (cont.)
– The probability of default in the first 6 months: 1 − 𝑒𝑒 −0.5𝜆𝜆1
– The probability of default during the following 6 months:
1 − 𝑒𝑒 −𝜆𝜆1 − 1 − 𝑒𝑒 −0.5𝜆𝜆1 = 𝑒𝑒 −0.5𝜆𝜆1 − 𝑒𝑒 −𝜆𝜆1
– The hazard rate λ1 must satisfy
1 − 𝑒𝑒 −0.5𝜆𝜆1 × 63.33 + (𝑒𝑒 −0.5𝜆𝜆1 −𝑒𝑒 −𝜆𝜆1 ) × 60.40 = 1.50
This gives λ1 =2.46%
– This is compared to 𝜆𝜆̅ 𝑇𝑇 = 𝑠𝑠(𝑇𝑇)/ 1 − 𝑅𝑅 = 1.5/(1-0.4)=2.5%
– With λ1 determined, we can obtain hazard rate λ2 …
» The hazard rate for the second year depends both on λ1
and λ2
– Risk-Free Rates
• The risk-free rate when credit spreads and default probabilities are
estimated is traditionally the LIBOR/swap zero rate. The risk-free
rate implied from the CDS (the difference between the risky bond
yield and the CDS spreads) is about 10 bps below the LIBOR/swap
rate and is close to the OIS rate.
– LIBOR is being phased out starting at the end of 2021
• Asset swaps (structured, for example, to swap the coupon payment
of a bond with LIBOR plus spread) provide a direct estimate of the
spread of bond yields over LIBOR.

– Real World vs Risk-Neutral Default Probabilities

• The default probabilities backed out of bond prices or credit default
swap spreads are risk-neutral default probabilities
• The default probabilities backed out of historical data are real-world
default probabilities
– Comparing two DP estimates

Cumulative 7-year default Average 7- year credit

Rating
probability(%): 1970-2015 spread (bp): 1996-2007
Aaa 0.198 35.74
Aa 0.54 43.67
A 1.345 68.68
Baa 2.525 127.53
Ba 11.788 280.28
B 29.028 481.04
Caa 41.812 1,103.70
Data from Moody’s and Merrill Lynch
– Comparing two DP estimates (cont..)

Rating Historical Hazard Rate Hazard Rate from bonds Ratio Difference
(% per annum)1 (% per annum)2
Aaa 0.028 0.596 21.0 0.568
Aa 0.077 0.728 9.4 0.651
A 0.193 1.145 5.9 0.952
Baa 0.365 2.126 5.8 1.761
Ba 1.792 4.671 2.6 2.879
B 4.898 8.017 1.6 3.119
Caa 7.736 18.395 2.4 10.659

Historical Hazard Rate: Calculated as −[ln(1-Q(7))]/7 where Q(7) is the

Moody’s 7-year default rate. For example, in the case of Aaa
companies, Q(7)=0.00198 and -ln(0.99802)/7=0.00028 or 2.8bps.
For investment-grade companies, the historical hazard rate is
approximately Q(7)/7.

Hazard Rate from bonds: Calculated as s/(1-R) where s is the bond

yield spread and R is the recovery rate (assumed to be 40%).
 – Expected Excess Return (extra risk premium) on Bonds
Rating Bond Yield Spread of risk-free Spread to Extra Risk
Spread over rate used by market compensate for Premium
Treasuries over Treasuries default rate in the (bps)
(bps) (bps)1 real world (bps)2
Aaa 78 42 2 34
Aa 86 42 5 39
A 111 42 12 57
Baa 169 42 22 105
Ba 322 42 108 172
B 523 42 294 187
Caa 1146 42 464 640

• Use the average spread (42bps) of our benchmark risk-free rate

over Treasuries.
• Spread to compensate for default rate in the real world: historical
hazard rate times (1-R) where R is the recovery rate. For example,
in the case of Baa, 22bps is 0.6 times 36.5bps.
– Possible Reasons for the Extra Risk Premium
• Corporate bonds are relatively illiquid
• The subjective default probabilities of bond traders may be much
higher than the estimates from Moody’s historical data
• Bonds do not default independently of each other. This leads to a
systematic risk that cannot be diversified away. As a result, bond
returns are highly skewed with limited upside. This may be priced in
by the market
– On an individual bond, there might be a 99.75% chance of a
7% return in a year (no default), and a 0.25% chance of a -60%
return in the year (default). The big tail risk can’t be diversified
away.
– Which DP Estimates Should We Use?
• Use risk-neutral estimates for valuing credit derivatives and
estimating the present value of the cost of default
• For risk management use real-world estimates for scenario analysis
– Using Equity Prices: Merton’s Model
• Merton’s model regards equity as an option on the assets of the firm
• In a simplified model, the equity value is
ET = max(VT −D, 0)
where VT is the value of the firm and D is the debt repayment required
• To use the Black-Scholes-Merton option pricing model for the value
of the firm’s equity E0 today. We need the value of its assets today,
V0, and the volatility of its assets, σV

E 0 = V0 N ( d 1 ) − De − rT N ( d 2 )
where
ln (V0 D) + ( r + σ V2 2) T
d1 = ; d 2 = d 1 − σV T
σV T
 • Volatilities are related by (Ito’s Lemma)
𝜕𝜕𝜕𝜕
dV = μVVdt + σVVdz, 𝑑𝑑𝑑𝑑 = … 𝑑𝑑𝑑𝑑 + 𝜕𝜕𝜕𝜕 𝜎𝜎𝑉𝑉 𝑉𝑉𝑉𝑉𝑉𝑉 = … 𝑑𝑑𝑑𝑑 + 𝜎𝜎𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸
∂E
σ E E0 = σV V0 = N ( d 1 ) σV V0
∂V
– Note the delta from BSM: N(d1)
• The two equations can be used to solve for V0 and σV, Given E0 and
𝜎𝜎𝐸𝐸 from the market
• The probability (risk-neutral) of default: N(-d2)
– An example
• A company’s equity is $3 million and the volatility of the equity is
80%
• The risk-free rate is 5%, the debt is $10 million and the time to debt
maturity is 1 year
• Solving the two equations (numerically) yields V0=12.40 and
σv=21.23%
• The probability of default is N(−d2) = 12.7%
• Credit Risk in Derivatives Transaction
– XVAs: CVA, DVA, FVA, MVA, KVA (adjustments in derivatives values)
– CVA
• Credit valuation adjustment (CVA) is an adjustment to the no-default
value of derivatives arising from the possibility of a counterparty
default
• CVA is calculated on a counterparty-by-counterparty basis.
– Agreements for bilaterally cleared transactions typically state
that outstanding transactions are netted in the event of a
default.
» If there are two outstanding transactions worth +10 and -6
with a counterparty, the potential loss will be 4.
– DVA
• Debit (or debt) valuation adjustment is an adjustment to a dealer’s
no-default value because the dealer itself might default.
• Like CVA, DVA must be calculated on a counterparty-by-
counterparty basis
– FVA and MVA: adjustments to the value of a derivatives portfolio for the
cost of funding derivatives positions
– KVA: capital valuation adjustment
– Calculation Issues
• The calculation of all the XVAs involves very time-consuming Monte
Carlo simulations for calculating expected future exposures, future
margin requirements, and future capital requirements
– Valuing Bilaterally Cleared Derivatives Portfolios
• Value after credit adjustments is (for the dealer):
No-default value − CVA + DVA
• Why does DVA increase the value of the portfolio of transactions
with the counterparty?
– It is a zero-sum game, what is worse for the counterparty is
better for the dealer.
 – CVA calculation

Time 0 t1 t2 t3 t4 tn=T

Counterparty q1 q2 q3 q4 ……………… qn
default probability

PV of dealer’s loss v1 v2 v3 v4 ……………… vn

given default

n
CVA = ∑ qi vi
i =1

• Default probabilities are calculated from credit spreads

 s (ti −1 )ti −1   s (ti )ti 

qi = exp −  − exp − 
 1− R   1− R 

The average hazard rate 𝜆𝜆�𝑖𝑖 = 𝑠𝑠(𝑡𝑡𝑖𝑖 )/(1 − 𝑅𝑅)

 – DVA calculation

Time 0 t1 t2 t3 t4 ……………… tn=T

Dealer default q1* q2* q3* q4* ……………… qn*

probability

PV of counterparty’s v1* v2* v3* v4* ……………… vn*

loss given default
n
DVA = ∑ qi ∗vi ∗
i =1
– Calculation of vi’s
• The vi is calculated by simulating the market variables underlying
the portfolio in a risk-neutral world
– If no collateral is posted, the loss on a particular simulation trial
during the ith interval is the PV of (1-R)max(Vi, 0) where Vi is
the value of the portfolio at the midpoint of the interval
– vi is the average of the values of the potential losses across all
simulation trials
• More details in actual scenarios:
– Collateral (posted by both the dealer and the counterparty),
collateral changes the risk exposures and potential loss
– Downgrade Triggers
» Collateral is required if the counterparty is downgraded
below a certain credit rating
» Examples: AIG was downgraded below AA on Sept 15,
2008, this triggered a collateral requirement that AIG was
not able to meet --- needs a government bailout.
– CVA Risks
• The CVA for a counterparty is regarded as a complex derivative
• Increasingly dealers are managing it like any other derivative
(measuring the risks by calculating the Greeks)
• Two sources of risk:
– Changes in counterparty spreads
– Changes in market variables underlying the portfolio
– Wrong-Way/Right-Way Risk
• So far we only consider that the probability of default qi is
independent of net exposure vi. But they are often not independent,
we need to pay attention to the wrong-way risk
• Wrong-way risk (potentially destabilizing risk) occurs when qi is
positively dependent on vi
• Right-way risk occurs when qi is negatively dependent on vi
– Wrong-Way Risk Examples:
• The counterparty uses CDS to sell protection to the dealer (AIG sold
lots of CDS in the 08’ financial crisis). The time when the credit
spread of the reference entity increases, the value of the protection
to the dealer increases (vi increases); because of default correlation
among companies, the counter party’s default probability qi also
increases.
• Counterparty is speculating by entering many similar (unhedged)
trades with one or more dealers could lead to wrong-way risk for the
dealers. If the trades move against the counterparty, the
counterparty’s probability of default is likely to move higher.

An alpha multiplier (as large as 1.4) is often used to increase the CVA
for wrong-way risk.
– 1st Simple Situation
• Suppose a portfolio with a counterparty consists of a single
uncollateralized derivative that always has a positive value to the
dealer and provides a payoff at time T (e.g. the dealer bought an
option from a counterparty)
– No payoff before maturity, the present value of the expected
exposure at time ti < T is f0, the present value of the derivative.
vi = (1-R)f0, 𝐶𝐶𝐶𝐶𝐶𝐶 = (1 − 𝑅𝑅)𝑓𝑓0 ∑𝑛𝑛𝑖𝑖=1 𝑞𝑞𝑖𝑖
The adjusted derivative value
𝑓𝑓0∗ = 𝑓𝑓0 − 𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑓𝑓0 1 − 1 − 𝑅𝑅 ∑𝑛𝑛𝑖𝑖=1 𝑞𝑞𝑖𝑖
– Now consider a zero-coupon bond that ranks equally with the
derivative in case of a default (same recovery rate), then
𝑓𝑓0∗ 𝐵𝐵0∗ 𝐵𝐵0∗
𝐵𝐵0∗ = 𝐵𝐵0 1 − 1 − 𝑅𝑅 ∑𝑛𝑛𝑖𝑖=1 𝑞𝑞𝑖𝑖 ; Thus = ; 𝑓𝑓 ∗ = 𝑓𝑓
𝑓𝑓0 𝐵𝐵0 0 𝐵𝐵0 0

𝐵𝐵0 = 𝑒𝑒 −𝛾𝛾𝑇𝑇 (the price of the zero-coupon bond assuming no

default)
∗ 𝑇𝑇
𝐵𝐵0 ∗ = 𝑒𝑒 −𝛾𝛾 (the price of the zero-coupon bond)
𝑓𝑓0 ∗ = 𝑒𝑒 −𝑠𝑠 𝑇𝑇 𝑇𝑇 𝑓𝑓
0 , where 𝑠𝑠 𝑇𝑇 = 𝛾𝛾 ∗ − 𝛾𝛾
– 1st Simple Situation (cont.)
• The CVA adjustment has the effect of multiplying the value of the
transaction by e-s(T)T, where s(T) is the counterparty credit spread
for maturity T.
• DVA = 0 (the position is always positive for the dealer)

• Example:
A 2-year uncollateralized option sold by a new counterparty to the
dealer has a Black-Scholes-Merton value of $3
Assume a 2-year zero coupon bond issued by the counterparty
has a yield of 1.5% greater than the risk-free rate
If there is no collateral and there are no other transactions
between the parties, the adjusted value of the option is
3e-0.015×2=2.91
– 2nd Simple Situation: Uncollateralized Long Forward with Counterparty
• For a long forward contract that matures at a time T
– The value of the contract at time t is
(Ft−K)e−r(T−t) [At time 0, the value is known (F0−K)e−rT ]
• The expected exposure at time t is
E[max((Ft-K)e-r(T-t),0)]=e-r(T-t)E[max(Ft-K,0)]
Assuming Geometric Brownian motion for Ft., the expected
exposure can be calculated as
w(t)=e-r(T-t) [F0N(d1(t))-KN(d2(t))], where
𝐹𝐹 𝐹𝐹
𝑙𝑙𝑙𝑙[ 𝐾𝐾0 ]+(𝜎𝜎2 /2)𝑡𝑡 𝑙𝑙𝑙𝑙 𝐾𝐾0 −(𝜎𝜎2 /2)𝑡𝑡
𝑑𝑑1 = 𝜎𝜎 𝑡𝑡
; 𝑑𝑑2 = 𝜎𝜎 𝑡𝑡

σ is the volatility of the forward price.

Thus we have
vi = e-rt(1-R)w(ti)=e-rT(1-R)[F0N(d1(ti))-KN(d2(ti))]
– 2nd Simple Situation (cont.)
Example：2 year Gold forward with the current forward price is $1,600
per ounce. K = 1,500, r = 5%
• Consider two one-year intervals, and as an approximation, assume
the defaults can occur only in the mid-year: t1 =0.5, t2=1.5
• Assume σ = 20% and R = 0.3
• We can calculate v1 = 92.67 and v2 = 130.65;
• Suppose we can further estimate q1 =0.02 and q2=0.03
CVA=q1v1+q2v2=0.02×92.67+0.03×130.65 = 5.77
Value the forward contract after CVA
(1600−1500)e-0.05×2 − 5.77 = 84.71
– Gaussian Copula Model：

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

V1 V2

- -

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

U1 U2

N(x)

F(V1) = N(U1) : V1  U1
• Examples:

V1 V2
• V1 Mapping to U1 and V2 Mapping to U2

V1 Percentile U1 V2 Percentile U2

0.2 20 -0.84 0.2 8 −1.41

0.4 55 0.13 0.4 32 −0.47
0.6 80 0.84 0.6 68 0.47
0.8 95 1.64 0.8 92 1.41
• Example of Calculation of Joint Cumulative Distribution
– The probability that V1 and V2 are both less than 0.2 is the
probability that U1 < −0.84 and U2 < −1.41
– Assume that the copula correlation is 0.5, this probability is
M( −0.84, −1.41, 0.5) = 0.043
where M is the cumulative distribution function for the bivariate
normal distribution
1 𝑈𝑈12 + 𝑈𝑈22 − 2𝜌𝜌𝑈𝑈1 𝑈𝑈2
exp −
2𝜋𝜋 1 − 𝜌𝜌 2 1 − 𝜌𝜌2
– 𝜌𝜌 is often estimated using the maximum-likelihood estimation
• 5000 Random Samples from the Bivariate Normal Dist. ρ=0.5.
5
4
3
2
1
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
-1
-2
-3
-4
-5
• Multivariate Gaussian Copula
– We can similarly define a correlation structure between
V1, V2,…Vn
– We transform each variable Vi to a new variable Ui that
has a standard normal distribution on a “percentile-to-
percentile” basis.
– The U’s are assumed to have a multivariate normal
distribution
• Factor Copula Model (an alternative way to define the
correlation structure)
– In a factor copula model, the correlation structure between the
U’s is generated by assuming one or more factors.
• Gaussian Copula Application: Credit Default Correlation
– The credit default correlation between two companies is
a measure of their tendency to default at about the same
time
– Default correlation is important in risk management when
analyzing the benefits of credit risk diversification
– It is important for determining the worst-case default rate
(tail risk of a loan portfolio).

• Model for Loan Portfolio Value (Vasicek)

– We map the time to default for company i, Ti, to a new variable
Ui and assume (ai to be the same)
Ui = aF + 1 − 𝑎𝑎 2 Zi, where F and the Zi have independent
standard normal distributions (𝐸𝐸 𝑍𝑍𝑖𝑖 𝑍𝑍𝑗𝑗 = 𝛿𝛿𝑖𝑖𝑖𝑖 ; 𝐸𝐸 𝑍𝑍𝑖𝑖 𝐹𝐹 = 0)
Note var(Ui) = 1 because E(F2) =1, E(Zi2) =1, E(FZi) = 0
– The copula correlation between Ui and Uj (i≠j), 𝐸𝐸 𝑈𝑈𝑖𝑖 𝑈𝑈𝑗𝑗 =
𝑎𝑎 2 𝐸𝐸 𝐹𝐹 2 = 𝑎𝑎2
• Analysis
– The aim
• To determine the probability distribution of the default rate, given the
average default probability PD and default correlation ρ
– Steps to derive the probability distribution
• Map the time to default Ti to Ui
• Assume Ui can be decomposed into a systematic component F and
an idiosyncratic component specific to the company i.
• Obtain the probability of default conditional on macro/systematic
variable F: Q(U|F), or Q(T|F)
– Default probabilities for different years are different simply
because the variable F is different. In our model, the lower the
F(this may indicate a bad economic condition) the higher the
probability.
• Given the probability distribution of F (here we assume a normal
distribution for F; this is part of the model assumption), we can then
derive the probability distribution of the default rate
– Finally, using the historical data of default rate and the method of
maximum likelihood, we can estimate PD and ρ
– Analysis
• To analyze the model, we
– assume the probability distribution of the default rate is the
same for all companies ({𝑎𝑎𝑖𝑖 } are the same)
– calculate the probability that, conditional on the value of F, Ui is
less than some value U
• The company defaults by time T (Ti is less than T) when 𝑈𝑈𝑖𝑖 < U, or
𝑈𝑈−𝑎𝑎𝐹𝐹
𝑍𝑍𝑖𝑖 < [Note: Ui = 𝑎𝑎F + 1 − 𝑎𝑎 2 Zi ]
1−𝑎𝑎2
» Ti and Ui are related by the percentile-to-percentile
mapping: N(Ui)=Q(Ti), where Q(T)=N(U)=PD (𝑈𝑈 =
𝑁𝑁 −1 (𝑃𝑃𝑃𝑃)) is the unconditional probability of default by time
T (if we consider 1-year default rate, T= 1 year).
• Conditional on F the probability of this is
𝑈𝑈−𝑎𝑎𝑎𝑎 𝑁𝑁−1 (𝑃𝑃𝑃𝑃)− 𝜌𝜌𝐹𝐹
𝑄𝑄 𝑇𝑇 𝐹𝐹 = 𝑁𝑁 = 𝑁𝑁 [𝑎𝑎 = 𝜌𝜌]
1−𝑎𝑎2 1−𝜌𝜌
– Low values of F give high default probabilities
– If 𝜌𝜌 = 0, Q(T|F)=Q(T)=PD
– Given a distribution of F, we have a distribution of the default rate
Cumulative Probability distribution of the default rate, G(y) = P(Q(T|F) < y)
𝑁𝑁−1 𝑃𝑃𝑃𝑃 − 𝜌𝜌𝐹𝐹 𝑁𝑁−1 𝑃𝑃𝑃𝑃 − 1−𝜌𝜌𝑁𝑁−1 (𝑦𝑦)
Q(T|F) < y  𝑁𝑁 < y  𝐹𝐹 >
1−𝜌𝜌 𝜌𝜌

Assuming F follows the standard normal distribution

𝑁𝑁−1 𝑃𝑃𝑃𝑃 − 1−𝜌𝜌𝑁𝑁−1 𝑦𝑦 1−𝜌𝜌𝑁𝑁−1 𝑦𝑦 −𝑁𝑁−1 𝑃𝑃𝑃𝑃
𝐺𝐺 𝑦𝑦 = 1 − 𝑁𝑁 = 𝑁𝑁
𝜌𝜌 𝜌𝜌

Note that 1 − 𝑁𝑁 𝑥𝑥 = 𝑁𝑁(−𝑥𝑥)

• In risk management, we are interested in the worst-case default
rate given a certain confidence level, X (the probability that the
default rate is less than WCDR)
G(WCDR) = X, or
1 − 𝜌𝜌𝑁𝑁 −1 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 − 𝑁𝑁 −1 𝑃𝑃𝑃𝑃
𝑁𝑁 = 𝐗𝐗
𝜌𝜌
𝑵𝑵−𝟏𝟏 𝑷𝑷𝑷𝑷 + 𝝆𝝆𝑵𝑵−𝟏𝟏 𝑿𝑿
WCDR(T,X)= 𝑵𝑵
𝟏𝟏−𝝆𝝆
– How to estimate PD and ρ?
• The probability density function for the default rate is (taking the
derivative of G(y))

2
1 − 𝜌𝜌 1 1 − 𝜌𝜌𝑁𝑁 −1 𝑦𝑦 − 𝑁𝑁 −1 (𝑃𝑃𝑃𝑃)
𝑔𝑔 𝑦𝑦 = 𝑒𝑒𝑒𝑒𝑒𝑒 (𝑁𝑁 −1 𝑦𝑦 )2 −
𝜌𝜌 2 𝜌𝜌

• Maximum likelihood estimate: Use the default rates ranging from

0.088% to 4.996% between 1970 and 2016 for all rated companies;

– Maximizing the sum of the logarithms (log-likelihood) of

∑𝑖𝑖 ln(𝑔𝑔 𝑦𝑦𝑖𝑖 ) for the data we get PD=1.32% and ρ = 0.098
• Probability Distribution for Default Rate

70

60

50

40

30

20

10

0
0 0.01 0.02 0.03 0.04 0.05 0.06
Default rate

• default correlation is important in determining the worst-case

default rate (WCDR), which describes the tail risk
With 𝞀𝞀=0.1, PD=1%  WCDR=7.7% (given 99.9% confidence
level)
Credit Derivatives
– Credit Default Swaps
• The buyer of the instrument acquires protection from the seller
against a default by a particular company or country (the reference
entity)
– Attractions of the CDS Market
• Allows credit risks to be traded in the same way as market risks
• Can be used to transfer credit risks to a third party
• Can be used to diversify credit risks
– Using a CDS to Hedge a Bond Position
• Example: Portfolio consisting of a 5-year par yield corporate bond
that provides a yield of 6% and a long position in a 5-year CDS
costing 100 basis points per year ~ a long position in a riskless
instrument paying 5% per year (before default the portfolio earns
5%, after default the portfolio can earn 5% by investing in risk-free
bonds).
• Bond yield spreads (measured relative to LIBOR) should be close to
CDS spreads
– CDS Valuation
• Example: Hazard rate for reference entity is 2% for a 5-year CDS,
survival probability e-0.02t

Survival
Time (years) Default Probability
Probability
1 0.9802 0.0198
2 0.9608 0.0194
3 0.9418 0.019
4 0.9231 0.0186
5 0.9048 0.0183
• Assume payments are made annually in arrears, that defaults
always happen halfway through a year, and that the expected
recovery rate is 40%, the risk-free rate 5%
 • Suppose the breakeven CDS rate is s per dollar of notional principal
Discount PV of Exp
Time (yrs) Survival Prob Expected Payment
Factor Pmt
1 0.9802 0.9802s 0.9512 0.9324s
2 0.9608 0.9608s 0.9048 0.8694s
3 0.9418 0.9418s 0.8607 0.8106s
4 0.9231 0.9231s 0.8187 0.7558s
5 0.9048 0.9048s 0.7788 0.7047s
Total 4.0728s
• Present value of expected payoff (DP*(1-R))
Expected Discount PV of Exp.
Time (yrs) Default Probab. Rec. Rate
Payoff Factor Payoff
0.5 0.0198 0.4 0.0119 0.9753 0.0116
1.5 0.0194 0.4 0.0116 0.9277 0.0108
2.5 0.019 0.4 0.0114 0.8825 0.0101
3.5 0.0186 0.4 0.0112 0.8395 0.0094
4.5 0.0183 0.4 0.011 0.7985 0.0088
Total 0.0506
 • PV of Accrual Payment Made (0.5s) in Event of a Default

Time Default Prob Expected Accr Pmt Disc Factor PV of Pmt

0.5 0.0198 0.0099s 0.9753 0.0097s
1.5 0.0194 0.0097s 0.9277 0.0090s
2.5 0.019 0.0095s 0.8825 0.0084s
3.5 0.0186 0.0093s 0.8395 0.0078s
4.5 0.0183 0.0091s 0.7985 0.0073s
Total 0.0422s

• PV of expected payments is 4.0728s + 0.0422s = 4.1150s

• The breakeven CDS spread is given by
4.1150s = 0.0506 or s = 0.0123 (123 bps)
• If a swap was negotiated some time ago with a CDS spread of
150bps, the value of the swap to the seller would be
4.1150×0.0150−0.0506 = 0.0111
per dollar of the principal.
– Securitization and the Credit Crisis of 2007
• Securitization
– Traditionally banks have funded loans with deposits
– Securitization is a way that loans can increase much faster than
deposits
• Asset-Backed Security (Simplified)

Senior Tranche
Asset 1 Principal: $75 million
Asset 2 Return = LIBOR + 60bp
Asset 3

Mezzanine Tranche
 SPV Principal:$20 million
Return = LIBOR+ 250bp
Asset n

Principal: Equity Tranche

$100 million Principal: $5 million
Return =LIBOR+2,000bp
– The Waterfall

Asset
Cash
Flows

Senior
Tranche

Mezzanine Tranche
Equity Tranche
 – ABS CDOs or Mezz CDOs (Simplified; more tranches in reality)

The mezzanine tranche is

repackaged with other
ABSs mezzanine tranches
Assets Senior Tranche (75%)
AAA
ABS CDOs
Senior Tranche (75%)
AAA
Mezzanine Tranche (20%)
BBB

Mezzanine Tranche
(20%) BBB

Equity Tranche (5%)

Not Rated

Equity Tranche (5%)

 – Losses to AAA Tranche of ABS CDO

Losses on
Losses on Losses on
Losses on Equity Losses on
Mezzanine Mezzanine Senior
Subprime Tranche Senior Tranche
Tranche of Tranche of Tranche
portfolios of ABS of ABS CDO Senior
ABS ABS CDO (75%)
CDO AAA Tranche
(75%)
10% 25% 100% 100% 0% AAA
Mezzanine
15% 50% 100% 100% 33.30% Tranche
(20%) Mezzanine
20% 75% 100% 100% 66.70% BBB Tranche
(20%) BBB
25% 100% 100% 100% 100%
Equity
Tranche (5%)
Example: BBB Tranches Not Rated Equity
Tranche
BBB tranches of ABS CDOs were often quite thin (1% wide) (5%)

The BBB tranche of the Mezz ABS CDO in this simplified example:
20%x20%=4%
This means that they have a quite different loss distribution from BBB
bonds and should not be treated as equivalent to BBB bonds
They tend to be either safe or completely wiped out (cliff risk)
This type of risk is very hard to manage
 – A More Realistic Structure
High Grade ABS
CDO

Senior AAA 88%

Junior AAA 5%
AA 3%
A 2%
BBB 1%
ABS
AAA 81%
NR 1%

AA 11%
Subprime
Mortgages
A 4% Mezz ABS CDO CDO of CDO
BBB 3% Senior AAA 62% Senior AAA 60%
BB, NR 1% Junior AAA 14% Junior AAA 27%
AA 8% AA 4%
A 6% A 3%
BBB 6% BBB 3%
NR 4% NR 2%
• U.S. Real Estate Prices, 1987 to 2016: S&P/Case-Shiller Composite-
10 Index

250.00

200.00

150.00

100.00

50.00

0.00
– What went wrong?
1. Starting in 2000, mortgage originators in the US relaxed their
lending standards and created large numbers of subprime first
mortgages.
2. This, combined with very low-interest rates, increased the demand
for real estate and prices rose.
3. To continue to attract first-time buyers and keep prices increasing
they relaxed lending standards further

• Features of the market: 100% mortgages, ARMs (adjustable-rate),

teaser rates, NINJAs, liar loans, non-recourse borrowing (personally
not liable)
• Loan approvals are “standardized”, depending largely on the
applicant’s FICO score.
• Mortgages were packaged in financial products and sold to investors
• Banks found it profitable to invest in the AAA-rated tranches
because the promised return was significantly higher than the cost of
funds and capital requirements were low
– What went wrong? (cont..)
4. In 2007 the bubble burst. Some borrowers could not afford their
payments when the teaser rates ended. Others had negative
equity and recognized that it was optimal for them to exercise their
“put options”.
5. Foreclosures increased supply and caused U.S. real estate prices
to fall. Products, created from the mortgages, that were previously
thought to be safe began to be viewed as risky (The correlation
plays a role here)
6. There was a “flight to quality” and credit spreads increased to very
high levels
7. Many banks incurred huge losses
– What the market participants didn’t anticipate
• Default correlation goes up in stressed market conditions
• Recovery rates are less in stressed market conditions
• A tranche with a certain rating cannot be equated with a bond with
the same rating. For example, the BBB tranches used to create ABS
CDOs were typically about 1% wide and had “all or nothing” loss
distributions (WCDR=100%, quite different from the BBB bond)

– Regulatory Arbitrage
• The regulatory capital banks were required to keep for the tranches
created from mortgages was less than that for the mortgages
themselves
– Incentives
• The crisis highlighted what is referred to as agency costs
– Mortgage originators (Their prime interest was in originating
mortgages that could be securitized)
– Valuers (They were under pressure to provide high valuations
so that the loan-to-value ratios looked good)
– Traders (They were focused on the next end-of-year bonus and
not worried about any longer-term problems in the market)
– A huge amount of new regulations (Basel II.5, Basel III, Dodd-
Frank, etc). For example:
• Banks are required to hold more equity capital with the definition of
equity capital being tightened
• Banks required to satisfy liquidity ratios
• CCPs and SEFs (Swap execution facilities) for OTC derivatives
• Bonuses limited in Europe
• Bonuses spread over several years
• Proprietary trading is restricted (Volcker’s rule)
– Lessons from the Crisis
• Beware irrational exuberance
• Do not underestimate default correlations in stressed markets
• The recovery rate depends on the default rate
• Compensation structures did not create the right incentives
• If a deal seems too good to be true (eg, a AAA earning LIBOR plus
100 bp) it probably is
• Do not rely on ratings alone (the default correlation is more
important)
• Transparency is important in financial markets
– ABSs and ABS CDOs were complex inter-related products
– Once the AAA-rated tranches were perceived as risky they
became very difficult to trade because investors realized they
did not understand the risks
– Other credit-related products with simpler structures (eg, credit
default swaps) continued to trade during the crisis.
• Re-securitization was a badly flawed idea
Model of the Short Rate
– The zero curve
The process for the instantaneous short rate, r, in the traditional risk-
neutral world defines the process for the whole zero curve in this world
The price at time t of a zero-coupon bond maturing at time T is
𝑃𝑃 𝑡𝑡, 𝑇𝑇 = 𝐸𝐸 𝑒𝑒 𝑟𝑟̅ 𝑇𝑇−𝑡𝑡 = 𝑒𝑒 −𝑅𝑅(𝑟𝑟,𝑇𝑇)(𝑇𝑇−𝑡𝑡)
where r̅ is the average r between times t and T and the yield on the
bond is
1
𝑅𝑅 t, T = − 𝑙𝑙𝑙𝑙E er� T−t
𝑇𝑇−𝑡𝑡
• Term structure models attempt to describe the evolution of the
whole term structure
• An equilibrium model usually starts with assumptions about
economic variables and derives a process for the short rate
• A no-arbitrage model is designed to exactly match today’s term
structure
– Equilibrium Models (Risk Neutral World)
• Rendleman & Bartter:
dr = μ r dt + σ r dz
• Vasicek:
dr = a(b-r)dt + σ dz
• Cox, Ingersoll, and Ross (CIR):
𝑑𝑑𝑑𝑑 = 𝑎𝑎 𝑏𝑏 − 𝑟𝑟 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑟𝑟𝑑𝑑𝑑𝑑
• Both Vasicek and CIR incorporate mean-reversion, and can be
solved analytically
• Possible shapes of the term structure in the Vasicek and CIR
models.
Zero Rate Zero Rate

Maturity Maturity

Zero Rate

Maturity
– Real vs. Risk-Neutral Processes: Vasicek
• The risk-neutral world process is
dr = a(b-r)dt + σ dz
• If the market price of interest rate risk is λ (negative) the real-world
process is
dr = a(b*-r)dt + σ dz
where
a(b*-r) = a(b-r) + λ σ, or b* = b + λ σ/a
– Estimating Parameters: A Simple Approach
• The real-world parameters can be estimated from historical
movements in the three-month rate
• The market price of risk can then be estimated so that yields match
the current term structure as closely as possible
• No arbitrage models of the short rate
A no-arbitrage model is a model designed to fit today’s term structure
of interest rates
A model for r can be made to fit the initial term structure by including a
function of time in the drift
– Ho-Lee Model
dr = θ(t)dt + σdz
• Many analytic results for bond prices and option prices
• Interest rates are normally distributed
• One volatility parameter, σ
• All forward rates have the same standard deviation

Short r
Rate Forward Rate

r r Curve

r
Time
– Hull-White Model
dr = [θ(t)-ar]dt + σdz
• Many analytic results for bond prices and option prices
• Two volatility parameters, a and σ
• Interest rates are normally distributed
• The standard deviation of a forward rate is a declining function of its
maturity

Short
r
Rate

r Forward
Rate
r Curve

r Time
– Black-Derman-Toy Model
d lnr = [θ(t)-a ln r]dt + σ(t)dz; a(t) = - σ’(t)/ σ(t)
• No analytic properties
• If no relation between a(t) and σ(t), we have Black-Karasinski Model
• Interest rates are normally distributed
– Options on Bonds
• In Vasicek and Hull-White model, the price of the call maturing at T
on a zero-coupon bond lasting to s is
LP(0,s)N(h)−KP(0,T)N(h−σP)
• The price of the put is
KP(0,T)N(−h+σP)−LP(0,s)N(h)
where
1 − e −2 aT
h=
1
ln
LP (0, s ) σ P
σ P P(0, T ) K 2
+
σ
σP = 1− e
a
[ ]
− a ( s −T )

2a
L is the principal and K is the strike price.
For Ho - Lee σ P = σ( s − T ) T
• Options on Coupon Bonds can be evaluated as a portfolio of zero-
coupon bonds.
– Interest Rate Trees
• The variable at each node in an interest rate tree is the ∆t-period rate
• Interest rate trees work similarly to stock price trees except that the
discount rate used varies from node to node
• Convenient to use a trinomial rather than a binomial tree for interest
rate, as it provides an extra degree of freedom, 14%
• Example: 3
12%
1.11* 12%
1

10% 10% 10%

0.35** 0.23 0

8%
8% 0
0.00

Payoff after 2 years is MAX[100(r – 0.11), 0] 6%

0
*: (0.25×3 + 0.50×1 + 0.25×0)e–0.12×1
**: (0.25×1.11 + 0.50×0.23 +0.25×0)e–0.10×1