Basics of Credit Risk - Universität Hohenheim

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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

◮ 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

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 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

Chair for Banking and Finance Winter term 2009 Slide 5


Investment Banking and Capital Markets – Universität Hohenheim

Investment Banking and Capital Markets

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)


Investment Banking and Capital Markets – Universität Hohenheim

Investment Banking and Capital Markets

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

Chair for Banking and Finance Winter term 2009 Slide 7


Investment Banking and Capital Markets – Universität Hohenheim

Investment Banking and Capital Markets

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

Chair for Banking and Finance Winter term 2009 Slide 8


Investment Banking and Capital Markets – Universität Hohenheim

Investment Banking and Capital Markets

A Multi-Step 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 Multi-Step 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 Multi-Step 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 Multi-Step 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 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

Chair for Banking and Finance Winter term 2009 Slide 13


Investment Banking and Capital Markets – Universität Hohenheim

Investment Banking and Capital Markets

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

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 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)

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 market-making 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 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

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 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)

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

◮ 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

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

: 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)

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 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

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. 449-470

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

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

Chair for Banking and Finance Winter term 2009 Slide 29

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