Basics of Credit Risk  Universität Hohenheim
Investment Banking and Capital Markets – Universität Hohenheim
Basics of Credit Risk
Investment Banking and Capital Markets
Winter 2009/10
Chair for Banking and Finance Winter term 2009 Slide 1
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Risk associated with Credit
◮ Default risk
◮ Market price risk
What drives the market’s view of credit risk?
◮ Rating changes
◮ Balance sheet information
◮ Implied equity volatility
◮ new issuance activity
◮ economic growth
⇒ no chance of using simple models
Chair for Banking and Finance Winter term 2009 Slide 2
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Risk Models
◮ Structural models
◮ Go back to a seminal work of Robert Merton
◮ default event if value of assets does not exceed the value of debt, equity’s
worth zero
◮ allows for the relative valuation of debt and equity → investment strategies
◮ Reducedform or intensitybased models
◮ Default is not clearly defined
◮ Error term which hits the company by accident
◮ Default probability drawn from the traded credit spread
◮ Transition matrix models
◮ try to derive the probability of default from rating migrations
◮ not very successful: rating transitions are just one pricedriving factor
◮ spreads for rating classes vary significantly
Chair for Banking and Finance Winter term 2009 Slide 3
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Moody’s Definition of Default Events
◮ A missed or delayed interest or principal payment, including payments
made within a grace period
◮ Filing for bankruptcy and related legal triggers that block the timely
payments of interest or principal
◮ Consummation of a distressed exchange
Chair for Banking and Finance Winter term 2009 Slide 4
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A SingleStep, TwoStage Model
V risky
◮ V risky : Value of the bond
◮ PD: Probability of default
◮ LGD: Loss given default
1  PD
PD
No Default
V
V Default
LGD
Chair for Banking and Finance Winter term 2009 Slide 5
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A SingleStep, TwoStage Model (continued)
◮ The expected loss is calculated by
◮ EL: Expected Loss
◮ Rec: Recovery rate
EL = PD × LGD = PD × (1 − Rec) (23)
◮ The value of the risky contract is therefore
V risky = (1 − PD) × V No default + PD × V Default
Chair for Banking and Finance Winter term 2009 Slide 6
(24)
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A SingleStep, TwoStage Model (continued)
Rearranging yields
V risky = V No default − PD × (V No default − V Default
= V No default × [1 − PD × (1 −
default
V
) ] (25)
V No default
 {z }
Rec
 {z }
LGD
 {z }
EL
Equation (25) is a wellknown expression on credit risk
Chair for Banking and Finance Winter term 2009 Slide 7
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A MultiStep Model for Zero Coupon Bonds
V risky
0
1  p 1
p 1
No Default
V1 V Default
1
= V risky
1
1  p 2
p 2
Cash Flow Structure zerocouponbond
No Default
V2 = V terminal
V Default
2
0 1 2
◮ V risky
i : Value of the risky claim at time i
◮ V Default
i
: Value of the claim in the case of a default
◮ pi: PD between i − 1 and i
Chair for Banking and Finance Winter term 2009 Slide 8
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A MultiStep Model for Zero Coupon Bonds
◮ Expectation for V risky
i
or
V risky
0 = (1 − p1) × V
V risky
1 = (1 − p2) × V
No default
1
No default
2
V risky
0 = (1 − p1)(1 − p2)
 {z }
survival probability for two time steps
+p1V default
1
+ p1 × V Default
1
+ p2 × V Default
2
+ (1 − p1)p2V default
2
No default
V2 +
Chair for Banking and Finance Winter term 2009 Slide 9
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A MultiStep Model for Zero Coupon Bonds
◮ Generalisation of the above
◮ equidistant points in time τi with i = 1, . . . , n
◮ pi is the PD between τi−1 and τi (if it has not defaulted before)
◮ V risky
i (m) indicates the value of a claim at time τi that runs until τm
◮ and the survival rate
S(m) =
◮ note that S(m) = S(m − 1)(1 − pm)
mY
(1 − pi)
Chair for Banking and Finance Winter term 2009 Slide 10
i=1
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A MultiStep Model for Zero Coupon Bonds
◮ The unconditional probability of default between i − 1 and i is therefore
◮ it follows that
S(i − 1) − S(i) = S(i − 1) − S(i − 1)(1 − pi)
V risky
0 (m) = S(m)V terminal +
= S(i − 1)(1 − (1 − pi))
= S(i − 1)pi (26)
mX
i=1
(S(i − 1) − S(i))V default
i
Chair for Banking and Finance Winter term 2009 Slide 11
(27)
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A MultiStep Model for Zero Coupon Bonds
◮ The cumulative default probability is the complement of the survival
probability
Fi(m) = 1 − Si(m)
◮ which is the cumulative default probability from t + τi to t + τm, and
obviously
Fi−1(i) = pi
Chair for Banking and Finance Winter term 2009 Slide 12
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A MultiStep Model
◮ calculating with zerocoupon bonds is nice, but coupon bonds and other
instruments (as CDS) have regular payments, thus
V risky
0
1  p 1
p 1
V risky
V
+
1
cf
1
V Default
1
1  p 2
p 2
V risky
V
+
2
cf
2 ...
V Default
2
Cash Flow Structure coupon bond
V risky
V
+
m1
cf
m1
1  p m
0 1 2
m1
m
◮ note the cashflows in each period (coupon payments)
...
p m
V cf
m
Default
Vm
Chair for Banking and Finance Winter term 2009 Slide 13
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
A MultiStep Model
◮ The value of the claim in this case is
◮ thus, we arrive at
V risky
i−1
V0(m) =
= (1 − pi)(V risky
i
mX
i=1
S(i)V cf
i +
mX
i=1
+ V cf
i ) + piV default
i
(S(i − 1) − S(i))V default
i
(28)
(29)
which is the price of a central pricing relation for creditrisky instruments
with multiple cashflows, you can value regular bonds, zero coupon bonds,
credit derivatives such as credit default swaps, and even portfolio
derivatives such as collateralised debt obligations.
◮ survival function and the recovery model necessary
◮ if V default is independent of the time, the second term of (29) will collapse
to (1 − S(m))V default
Chair for Banking and Finance Winter term 2009 Slide 14
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Default Probability in a ContinuousTime Approach
◮ τ ∗ : stochastic variable as the time until default
◮ Ft(τ) = P(τ ∗ ≤ τ) St(τ) = P(τ ∗ > τ) with
◮ Ft(0) = 0, lim Ft(τ) = 1, Ft(τ + ∆τ) ≥ Ft(τ) ∀ ∆τ > 0, and St(τ) =
τ→∞
1 − Ft(τ)
◮ The default probability of the claim in the time frame τ to τ + ∆τ is
Ft(τ + ∆τ) = P(τ ∗ ≤ τ + ∆τ)
which is equivalent to
= P(τ ∗ ≤ τ) + [1 − P(τ
 {z }
Ft (τ)
∗ ≤ τ)]
 {z }
St (τ)
· P(τ ∗ ≤ τ + ∆ττ ∗ > τ)
 {z }
probability for τ ∗ ∈[τ,τ+∆τ]
St(τ) − St(τ + ∆τ) = St(τ) · P(τ ∗ ≤ τ + ∆ττ ∗ > τ) (30)
Chair for Banking and Finance Winter term 2009 Slide 15
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Dealing with Credit Risk
◮ The default risk modeled before is tradable
◮ The main instruments are
◮ Credit Default Swaps (CDS),
◮ Credit Linked Notes (CLN),
◮ Indices on CDS, in Europe especially iTraxx and the very young SovX WE,
◮ Total Return Swaps (TRS), and
◮ Collateralised Debt Obligations (CDOs)
Chair for Banking and Finance Winter term 2009 Slide 16
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Derivatives – Market Participants
End Users Application
Asset Managers Diversify in credit risk
Portfolio balancing tools
Hedge funds Relative value trading
Gaining leveraged exposure
Proxy hedge against CDO tranches
Circumspect trading strategies
Credit correlation trading desks Proxy hedge against CDO tranches
Gaining leveraged exposure
Model trading
Bank proprietary desks Trading and marketmaking credit books
Chair for Banking and Finance Winter term 2009 Slide 17
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Basic Structure
Chair for Banking and Finance Winter term 2009 Slide 18
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Market
◮ Huge market, open position end of 2007: USD 60 trillion, today ca USD
24 trillion
70,000.00
60,000.00
50,000.00
40,000.00
30,000.00
20,000.00
10,000.00

Credit default swaps Outstanding, billions of USD
1H01
2H01
1H02
2H02
1H03
2H03
1H04
2H04
1H05
2H05
1H06
2H06
1H07
2H07
1H08
2H08
1H09
Chair for Banking and Finance Winter term 2009 Slide 19
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Credit Event Definition
◮ Standardised products, easy to trade
◮ Before the crisis: pure overthecounter (OTC) product, now more and
more clearing house usage
◮ ISDA (International Swaps and Derivatives Association) issued the
standardised “credit event” definition (1999/2003)
1. Failure to pay
2. Bankruptcy
3. Restructuring
4. Repudiation/moratorium (relevant to sovereign underlyings)
5. Obligation acceleration
6. Obligation default
Chair for Banking and Finance Winter term 2009 Slide 20
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Settlement in the case of a credit event
1. Physical settlement
◮ Protection seller pays face value
◮ Protection buyer hands over the reference obligation
◮ Widely used, problems if the total value of the CDS exceeds the total value
of the underlying entity (e.g. Delphi)
2. Cash settlement
◮ Protection seller pays
P = 1 − (Rec + accrued interest on reference obligation)
◮ Recovery rate is usually agreed upon in advance or estimated after the
default
◮ If recovery rate is fixed in advance: digital settlement
Chair for Banking and Finance Winter term 2009 Slide 21
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – An Example
◮ Bank A has granted loans to corporations from the semiconductor
industry, C1 and C2, USD 10mn each
◮ High default correlation, A buys protection on C1 to lower overall risk
◮ This costs 125bps per year on USD 10mn, or
0.0125 · USD 10 000 000 = USD 125 000
◮ To prevent negative cashflows, the bank sells protection on a corporation
with a low default correlation to the semiconductor industry. If the spread
of this CDS is 125bps as well, the bank has diversified its portfolio for a
cost of zero (+ fees)
Chair for Banking and Finance Winter term 2009 Slide 22
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Basic Valuation
◮ From the perspective of the protection seller:
V CDS
PS
= V PL − V DL
◮ the value consists therefore of a premium leg (PL) and a default leg (DL)
◮ From the perspective of a protection buyer, it is obviously
V CDS
PB = −V CDS
PS = V DL − V PL
◮ The default leg usually consists of two components: the default payment
and the accrued premium
V DL = V DP − V AP
Chair for Banking and Finance Winter term 2009 Slide 23
(31)
(32)
(33)
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Basic Valuation
◮ Discretetime CDS pricing algorithm (market standard): JP Morgan model
V (τm) =
mX
CFi · R(τi) · S(τi) +
i=1
 {z }
V PL
+
mX
CF default
i
· R(τi) · (S(τi−1) − S(τi))
i=1
 {z
−V
}
DL
◮ Note that this is a modification of (29), with a discount factor R
◮ Needed: structure the premium payments CFi and the payment
conditional on default, CF default
i , the survival probability function S(τ);
R(τi) can be derived by bootstrapping
Chair for Banking and Finance Winter term 2009 Slide 24
(34)
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Basic Valuation
◮ With CFi being simply the product of ∆τ = τi − τi−1 and the contractual
CDS rate c(τm)
V PL =
mX
c(τm) · ∆τ · R(τi) · S(τi) (35)
i=1
◮ standard maturity days for CDS according to ISDA (2003): March 20,
June 20, September 20, and December 20
◮ ∆τ is usually three months
Chair for Banking and Finance Winter term 2009 Slide 25
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Valuation
◮ To determine V DL , we have to formalise the default payment (1 − Rec)
and the accrued premium
◮ To keep things simple, we assume that defaults occur in the middle of τi−1
and τi, the accrual factor will be ∆τ
: midpoint approximation
2
◮ Thus:
V DP = (1 − Rec)
V AP =
mX
i=1
mX
R(τi) · (S(τi−1) − S(τi)) (36)
i=1
c(τm) · ∆τ
2 · R(τi) · (S(τi−1) − S(τi)) (37)
Chair for Banking and Finance Winter term 2009 Slide 26
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Valuation
◮ We arrive at the formula for the valuation of a plainvanilla CDScontract:
V CDS = V PL − V DL = V PL + V AP − V DP
=
mX
c(τm) · ∆τ · R(τi) · S(τi)
i=1
+
mX
i=1
−(1 − Rec)
c(τm) · ∆τ
2 · R(τi) · (S(τi−1) − S(τi))
mX
R(τi) · (S(τi−1) − S(τi)) (38)
i=1
Chair for Banking and Finance Winter term 2009 Slide 27
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Credit Default Swaps – Valuation
to be continued.
Chair for Banking and Finance Winter term 2009 Slide 28
Investment Banking and Capital Markets – Universität Hohenheim
Investment Banking and Capital Markets
Literature
◮ Merton, R. (1974): On the Pricing of Corporate Debt: The Risk Structure
of Interest Rates, The Journal of Finance, 29, pp. 449470
◮ Felsenheimer, J., and Gisdakis P., Zaiser, M. (2006) Active Credit
Portfolio Management, WileyVCH, Weinheim, ch 7, 10
Chair for Banking and Finance Winter term 2009 Slide 29