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



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 




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 

◮ Reduced-form or intensity-based 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 price-driving factor 

◮ spreads for rating classes vary significantly 




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 




A Single-Step, Two-Stage 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 




A Single-Step, Two-Stage 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)



A Single-Step, Two-Stage 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 well-known expression on credit risk 




A Multi-Step 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 zero-coupon-bond 

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 





◮ 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 + 





◮ 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) 


i=1




◮ 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 


(27)




◮ 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 




A Multi-Step Model 

◮ calculating with zero-coupon 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 

+ 

m-1 

cf 

m-1 

1 - p m 

0 1 2 

m-1 

m 

◮ note the cash-flows in each period (coupon payments) 

... 

p m 

V cf 

m 

Default 

Vm 




A Multi-Step 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 credit-risky instruments 

with multiple cash-flows, 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 




Default Probability in a Continuous-Time 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 + ∆τ|τ ∗ > τ) 

| {z } 

probability for τ ∗ ∈[τ,τ+∆τ] 

St(τ) − St(τ + ∆τ) = St(τ) · P(τ ∗ ≤ τ + ∆τ|τ ∗ > τ) (30) 




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) 




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 market-making credit books 




Credit Default Swaps – Basic Structure 




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 




Credit Default Swaps – Credit Event Definition 

◮ Standardised products, easy to trade 

◮ Before the crisis: pure over-the-counter (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 




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 




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 cash-flows, 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) 




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 


(31) 

(32) 

(33)




◮ Discrete-time 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 


(34)




◮ 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 




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 

: mid-point 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) 





◮ We arrive at the formula for the valuation of a plain-vanilla CDS-contract: 

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 





to be continued. 




Literature 

◮ Merton, R. (1974): On the Pricing of Corporate Debt: The Risk Structure 

of Interest Rates, The Journal of Finance, 29, pp. 449-470 

◮ Felsenheimer, J., and Gisdakis P., Zaiser, M. (2006) Active Credit 

Portfolio Management, Wiley-VCH, Weinheim, ch 7, 10

Basics of Credit Risk - Universität Hohenheim

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