Credit Risk Modeling A.Y. 2018/19 M.Sc. in Finance Introduction and model-independent formulas

A.Y. 2018/19 M.Sc. in Finance

Introduction and model-independent formulas

Luca Regis¹

1DEPS, University of Siena

Agenda

Classes are held on: Wed 18-19.30; Thu 10-12 with the following calendar: 13/3, 14/3,20/3,21/3, 15/5, 16/5, 22/5, 23/5, 29/5, 30/5.

Office Hours: on Wednesdays, 14-16 (check my website for updates!)

Personal webpage: docenti-deps.unisi.it/lucaregis Agenda:

1 Introduction: measuring and managing credit risk.

2 Default modeling: intensity-based approach.

3 Default modeling: firm-value or structural approach.

4 Pricing defaultable assets (also credit derivatives).

Main references

Duffie, D. and K. J. Singleton, Credit Risk. Pricing, measurement and Management, 2003, Princeton University Press.

Brigo, D. and F. Mercurio, Interest Rate Models - Theory and Practice with Smile, Inflation and Credit, 2006, Springer Finance.

Brigo, D. and Morini, M. and A. Pallavicini, Counterparty Credit Risk, Collateral and Funding, 2013, Wiley Finance.

Section 1

Introduction

What is credit risk?

Definition

Credit risk is the risk associated to the default or to changes in the market value of a credit or of an asset held in a portfolio following a credit-linked event happening to issuers or counterparties.

Why modeling credit risk?

Measure the risks inherent a portfolio, for internal risk management needs or for capital requirements;

Price instruments that can be affected by such events.

Ingredients:

1 Default arrival;

2 Losses: what happens if default occurs? Exposure.

3 Effects on the valuation of:

1 Corporate bonds;

2 Government bonds;

3 Credit derivatives;

4 Any Over-the-Counter derivative.

This course will be focusing on the risk associated to a credit-linked event.

Default or credit-linked event: failure to fulfill some important obligation.

Payment associated to credit derivatives, f.i., can be triggered by these credit-linked events:

1 Bankruptcy: a court declares it;

2 Failure to pay: missed payment (coupons, bills, ...);

3 Restructuring: debt of a company is redefined (amount, maturity,...);

4 Repudiation/moratorium: bankruptcy/restructuring of sovereign debt;

5 Obligation and acceleration default: violation of covenants in the debt contract.

Peculiarities in the Economics of Credit Risk

Asymmetric information between the borrower, that knows more about its own quality, and the lender. As a consequence:

Adverse selection: if the interest charged to lenders is too high, none or only the poor quality ones will demand credit. Possible solution: limit the access to credit, to incentivize the selection of good quality borrowers.

Moral hazard: the larger the borrowing, the higher the incentive for borrowers to undertake risky behaviors.

Main issues

Credit risk is a hot topic: credit crisis was at the basis of the crisis started in 2007/8;

Credit derivatives market: boomed before the crisis, now it is still alive but declining in size (around 10 trillion gross market value).

Now: focus shifted from pricing derivatives to risk effects recognition and regulatory compliance.

Lots of technical issues:

1 Modeling the default event;

2 Assessing exposures, i.e. losses in case of default;

3 Adjusting prices and fair values to account for counterparty default risk: Credit Value Adjustment (CVA);

4 Computing credit risk measures (Credit VaR);

5 Simulating, forecasting portfolio losses under real-world measure P; pricing under a risk-neutral measure Q;

6 Dependence with other risks and among defaults.

Regis Credit Risk Modeling A.Y. 2018/19 M.Sc. in Finance Introduction and model-independent formulas8/45

Size of the derivatives market (Source: BIS)

Traditional approaches to credit risk: Credit Ratings

Rating agencies (Moody’s, Standard & Poor’s, Fitch) are specialized companies that provide assessments of the credit quality of issuers of debt instruments.

Credit ratings measure the worthiness of debtors, using accounting data, historical data about default frequencies, expert judgement... they assign a “class” to an issuer: AAA, AA, A, (S & P) ....Aaa (Moody’s)..

Main distinction: investment (higher rating) vs. speculative grade (lower ratings, sometimes issues are referred to as junk bonds).

Ratings are public and were extensively used as inputs by companies for their risk calculations.

Default Rates

The use of ratings to measure default probabilities

Ratings changes may have effects on portfolio values and compositions, may trigger contingent payments.

Can be used to assess the likelihood of default of an issuer or a reference entity: use the average frequency at which similar entities, i.e. subjects with same credit rating, have defaulted.

Be careful:

1 They provide a “naive” assessment of the probability of defaults, are intended to provide measures of credit quality that are not affected by the business cycle BUT...

2 Evidence of momentum: a downgrading is more likely to be followed by another downgrading.

3 historical default probabilities, used to simulate default arrivals and actual portfolio losses, NOT for pricing!

Default rates per rating class

Multi-year solvency probabilities

The table below provides the cumulative average default rates by credit rating....

... BUT an issuer can change rating over time!

One year vs. multi-year default rates

Consider an A-rated bond, where the average default rate of A-rated firms is qA.

The survival probability over one year is thus 1 − qA.

The default probability over two years, for a firm rated A at time 0, assuming q_A stays constant, is NOT

1 − (1 − qA)²,

because the firm can change rating at the end of the first year.

It is necessary to model rates of migration across ratings:

transition matrices allow to do this, modeling the

probabilities that firms rated x at the beginning of one year will have rating y at the end of that year.

One-year transition matrix: example

Rating transitions

Average annual transition rates can be interpreted as a matrix π of 1-year transition probabilities, πij.

Given the assumptions that transition probabilities are constant over time and that ratings are the sole determinant of issuer’s credit risk, ratings can be seen as a Markov chain.

Hence, the multi-year transition matrix is simply πⁿ. Vintage effects: duration in a certain rating class affects the probability of a change.

Accounting for business cycles

To account for the effect of business cycle states on transition probabilities we can extend the setting.

Π_ij(x) denotes the transition probability from i to j when the business cycle is in state x (doubly stochastic transition model).

Assuming that X1, X2, ..., Xt is a Markov process, conditionally on the path of X, πij = Πij(X1)Πij(X2) · · · Πij(Xt).

In practice, we can treat the pair (Xt, Ct), where Ct denotes the rating, as a Markov Chain.

Traditional Approaches (2): Altman’s Z-score

Introduced by Ed Altman (1968), the Z-score is an accounting-based statistical model.

Z = 0.012X₁+ 0.014X₂+ 0.033X₃+ 0.006X₄+ 0.999X₅ X1: Net Working Capital/Total Assets; X2 Retained

Earnings/Total Assets; X₃ EBIT/Total Assets; X₄ Market Value of Equity/Book Value of Debt; X5 Sales/Total Assets.

The higher the Z-score, the lower the probability of bankruptcy.

Find thresholds that discriminate between bankrupt vs.

non-bankrupt firms to predict early defaults (2 years is ok!).

Credit spreads/1

Consider:

1 A default-free zero-coupon bond: pays 100 at time 1 always;

2 A defaultable zero-coupon bond: it pays

if there is no default between current time 0 and 1: 100 if there is default between current time 0 and 1:

Rec = 100(1 − L)

Rec is the recovery, the complement to 100 of L, the (Percentage) Loss Given Default;

Assume default occurring with probability q.

Assume a risk-free rate r.

The fair promised interest rate of the defaultable bond y must

be: 100

1 + y = [(1 − q)100 + q100(1 − L)]

1 + r

Credit Spreads/2

It follows that:

1 + r = (1 + y) [1 − q + q − qL]

1 + r = (1 + y) [1 − qL]

y = r

1 − qL

| {z }

>r

+ qL

1 − qL

| {z }

>0

> r.

y − r is the credit spread, the extra-remuneration due to the presence of default.

y − r is somewhat close to Lq, the expected loss rate.

If L = 0 or q = 0: y = r.

If L = 1: y = ^r+q_1−q, y − r is close to q.

Default probabilities and credit spreads

Since a typical recovery rate Rec/100 is 50%, a typical credit spread should be close to half the annual default probability, but indeed in reality spreads are much higher, due to risk premia, cyclicality, liquidity,....

Source: Huang and Huang (2002), published in 2012 on The Review of Asset Pricing Studies.

Regis Credit Risk Modeling A.Y. 2018/19 M.Sc. in Finance Introduction and model-independent formulas22/45

Risk premium

We said that

y = r

1 − qL + qL 1 − qL.

Q: what is the probability measure under which the q in the above formula is computed?

A: It is the risk-neutral probability Q obtained adjusting the real-world or historical probability P by the price of default risk, i.e. for the presence of a risk premium associated to the default risk.

Indeed, abstracting from recovery, i.e. setting L = 0, the price of a 1-year Ba-rated bond (see table in slide before) will not be (1 − 1.29%)/(1 + r), but will be equal to 1 − q^∗ where q^∗ is the risk neutral default probability.

Risk-neutral vs. actual probabilities

Let us take the opposite perspective: we observe the market quote of a defaultable coupon bond, traded at par and paying 108 euros in 1 year. Historical default rates suggest p = 0.99.

How can we recover p^∗, the risk-neutral survival probability?

Assume L = 50% of the par value, r = 6%. Then, p^∗ is obtained solving

100 = 1

1.06[p^∗108 + 50%(1 − p^∗)108] , p^∗ = 0.965.

The difference between p and p^∗ can be rationalized in terms of the presence of a default risk premium arising from agents’ risk aversion. q^∗ is typically larger than q!

Cyclicality

There is documented evidence (H.Chen, JF, 2010) that both default rates and recovery rates are cyclical, i.e. are respectively higher and lower in recessions. This, for instance, lowers the value of corporate debt, amplifying spreads.

Section 2

Pricing

A more technical view on the pricing of a defaultable zero coupon bond

When there is default risk, the time-t value of a bond paying 1 at a future date T , ¯P (t, T ) is the risk neutral expectation of the discounted payoff 1:

1_{{τ >t}}P (t, T ) = E¯ D(t, T )1_{{τ >T }}+ RecD(t, TR)1_{{τ ≤T }}|G_t , where:

E is an expectation under the risk-neutral measure;

1_{A} is an indicator function;

τ is the time of default;

Gtis the filtration at time t comprehensive of the “market”

information Ftand of the information on whether the default has happened, σ({τ < u}, u ≤ t) ;

D(t, s), s ≥ t, the discount factor, can be either deterministic or stochastic; Rec can be stochastic; TR= τ or T .

A more technical view on the pricing of a defaultable zero coupon bond/2

Notice that, denoting with P (t, T ) the price of a bond with maturity T when there is no default risk, i.e. the time-t value of a bond paying 1 at a future date T ,

P (t, T ) = E [D(t, T )1|Gt] = Eh

e^−RtTr(s)ds|G_ti , and assuming that the Recovery is paid at maturity T_R= T , the price of the defaultable bond can be seen as:

1_{{τ >t}}P (t, T )¯ = ED(t, T )(1 − 1_{{τ ≤T }}) + (1 − L)D(t, T )1_{{τ ≤T }}|G_t =

= P (t, T )

| {z }

Default-free bond

− EL × D(t, T )1_{{τ ≤T }}|G_t

| {z }

Disc. Risk-neutral expected Loss

.

Defaultable coupon bonds

The Discounted Risk-neutral Expected Loss in the previous formula is the so-called Credit Value Adjustment (CVA).

Similarly, the discounted payoff of a coupon bond issued by a company, paying coupons c_i at payment dates T_i, with i = 1, ...n is

n

X

i=1

ciD(t, Ti)1{τ >T_i}+ D(t, Tn)1{τ >T_n}+ RecD(t, TR)1{τ ≤T }.

TR is the time at which the recovery Rec is paid.

Its time-t price can be evaluated discounting the payoff under the risk-neutral measure as for the zero-coupon bond.

Credit derivatives / Single-name

Payoffs of Single-name products are linked to the credit events of one reference entity.

Credit Default Options: put option that gives the right to the buyer to be reimbursed for losses if a credit-event occurs before maturity;

Credit Default Swaps (CDS): periodic premium vs. protection;

Credit-linked notes, step-up bonds: bonds whose payments are linked to credit events associated to a reference entity.

Floating rate notes: structured as interest rate swaps (IRS) in which the fixed payer holds a bond paying coupons, that is paid to the payer of the floating rate. The floating payer pays

LIBOR rate+ a spread.

Types of credit derivatives / Multi-Name

Payoffs of Multi-Name products are linked to credit events of more than one reference entity.

Basket default swap: first-to default, k-th to default, last-to-default. Like CDS, but a set of reference entities is defined and protection is paid when the first, k-th or last reference entity experiences default. More risk than CDS:

leverage for protection seller; lower rate, partial hedging of multiple defaults by buyer.

Multi-name products/2

Index CDS’s (iTraxx, CDX): protection from default of many entities, each with equal notional; protection leg pays loss increment occurring at each default in exchange for a periodic premium.

Bespoke Tranche Opportunities (BTOs): collection of CDS’s with same maturity on different names; total loss is tranched into attachment points (e.g. A and B) and the incremental difference between B and total loss is paid, when this exceeds A. Notice that the rate paid is reduced when a default occurs.

Why using credit derivatives?

Who are the...

Protection Sellers? investment banks, insurers, asset management companies and reinsurers;

Buyers? mostly banks and asset management companies.

Purpose:

Modify or customize credit exposures (e.g. targeted to a selected maturity);

Modify credit concentrations (by industry, geographical area,...);

Go short on defaultable bonds (easier or cheaper than on bonds market);

Credit Default Swaps

Credit Default Swaps (CDS) are contracts in which 2 parties, the protection buyer (B) and the protection seller (S), agree that:

If a reference entity (E) defaults at a certain time T_a < τ < T_b, S pays B a certain deterministic cash amount LGD (protection leg). In exchange, B pays S a rate R at regular intervals (premium leg ), T_a+1, ..., until T_b, or until default τ .

LGD is the protection provided to B against default, and it is the notional of the contract.

CDS’s are the basic derivative used to mitigate default risk, e.g.

to protect B when it holds a bond issued by E.

From a modeling point of view, two types of needs:

1 models that can price correctly CDS’s;

2 models that can be calibrated using CDS’s.

Price of a CDS/1

Given the annual rate R, with α_i the distance between two successive payments (typical: 3 months), the price of a CDS at t < T_a is given by

PCDS (t, R, LGD) = E^t





D(t, τ )(τ − Tβ(τ )−1)R1{T_a<τ <T_b}

| {z }

Accrued rate at default

+

+

b

X

i=a+1

D(t, Ti)αiR1{τ >T_i}

| {z }

CDS Payments if no default

− 1{T_a≤τ ≤T_b}D(t, τ )LGD

| {z }

Protection payment at default











 .

Common assumption: payment of last R is not at default, but postponed to Ti and interest accrual period equal to either zero or accrued for the whole period; T_{β(τ )} is the first payment time T_i after default.

Price of a CDS/2

Pricing a CDS amounts to establishing the rate R, i.e. the payments of the fixed leg.

R is set at contract start t = Ta= 0 s.t. the contract is fair:

P V (Premium Leg) = P V (Protection Leg).

Leaving aside accrued interest:

Et

" _b X

i=1

D(0, T_i)α_iR1_{{τ >Ti}}

#

= EtLGD × D(0, τ )1_{{0<τ ≤Tb}} .

Hence, if R is constant, when Ta= 0 and t = 0:

R = R_0,b(0) = EtLGD × D(0, τ )1_{{0<τ ≤Tb}} α_iEt

hP_b

i=1D(0, T_i)1_{{τ >Ti}}i .

Modeling prices of credit derivatives

Prices depend on:

1 Estimated exposures: losses in the event of default, that define the protection (fixed LGD);

2 Dynamics of interest rates;

3 The way default time(s) is (are) modeled;

4 Dependence (or lack of) between default time τ and interest rates.

However, we can start by providing some model-free valuation formulas, under independence between default arrival and discount factors.

Model-independent Price of CDS/ Premium Leg

Assume D(s, t) independent of τ for any 0 ≤ s ≤ t. Leaving accrual aside, the time-0 value of the premium leg is

Et

" _b X

i=a+1

D(0, Ti)αi1{τ >T_i}

#

=

=

b

X

i=a+1

αiR Et[D(0, Ti)]

| {z }

P (0,Ti)

Et[1{τ ≥T_i}]

| {z }

Q(τ ≥Ti)

=

= R

b

X

i=a+1

αiP (0, Ti)Q[τ ≥ Ti]

| {z }

“DV01” or unit premium leg

=

= P remLeg_0,b(P (0, ·), Q(τ ≥ ·)

Model-independent Price of CDS/Protection Leg

Now, assuming also a constant LGD, the value of the Protection Leg P rotLeg_a,b(LGD; P (0, ·), Q(τ ≥ ·) is

EtLGD × D(0, τ )1_{{Ta≤τ ≤Tb}} =

= LGDEt

Z +∞

0

1_{{Ta≤t≤Tb}}D(0, t)1{τ ∈[t,t+dt)}



=

= LGD

Z Tb

Ta

ED(0, t)1{τ ∈[t,t+dt)} =

= LGD

Z Tb

Ta

E [D(0, t)] E1{τ ∈[t,t+dt)} =

= LGD

Z Tb

Ta

P (0, t)Q(τ ∈ [t, t + dt)) =

= −LGD

Z Tb

Ta

P (0, t)d_tQ(τ ≥ t).

Model-independent Survival Probabilities from CDS Prices

The above formulas are model-independent, i.e. they are valid for all the modeling choices (interest rate dynamics, default arrival modeling,...). They involve only the initial default-free zero-coupon curve and the survival probabilities at time 0.

When Ta= 0, t = 0 the market quotes the fair price R^mkt_0,b (0) for different maturities T_b.

Solving for the different Tb, starting from the shortest one, the equation

P rotLeg_0,b(LGD; P (0, ·), Q(τ > ·)) =

= P remLeg0,b(R^mkt_0,b (0); P (0, ·), Q(τ > ·)) for Q(τ > ·), it is possible to find the model-independent market implied survival curve : “stripping”.

Counterparty credit risk in derivatives transactions

Consider a transaction for an OTC derivative product, where a default-free party enters a transaction with a counterparty (C).

Such derivative offers a discounted payoff Π(s, t), with t final maturity, when C is default-free as well.

If we instead allow for the possibility that C defaults:

Π(s, t) = 1¯ _{τC>t}Π(s, t) + 1_{τC<t}[Π(s, τC)+

+ RecD(s, τC)(EτCΠ(τC, t))⁺− D(s, τ_C)(−EτCΠ(τC, t))⁺ . If no default: original payoff.

if default: payments before default + recovery of residual NPV if positive - residual NPV if negative.

Counterparty credit risk in derivatives transactions/2

It can be proved that, in general the payoff is equivalent to:

Π(s, t) = Π(s, t) − LGD1¯ {τ_C≤t}D(s, τC)(EτCΠ(τC, t))⁺.

and as a consequence:

Es[ ¯Π(s, t)] = Es[Π(s, t)] − Es[LGD1{τ_C<t}(EτCΠ(τC, t))⁺]

| {z }

Unilateral Counterparty Value Adjustment

.

Original payoff - LGD * a call option with zero strike on the residual NPV at default if default occurs prior to maturity.

Example: counterparty risk on equity options

Suppose the buyer (A) of an equity option on a firm B is default free. When he enters the transaction, he buys the option from a counterparty C. Assuming no counterparty risk, the payoff of the option is

Π(0, T ) = D(0, T )(S_T^B− K)⁺. If instead C can default, the payoff is modified:

Π(0, T )¯ = D(0, T )(S_T^B− K)⁺

− LGD1_{τ

C<T }D(0, τ_C)(EτCD(τ_C, T )(S_T^B− K)⁺).

It can be proved, through iterated expectations, that this payoff has same time-0 value of:

D(0, T )1 − LGD1_{{τC<T }} (S^B_T − K)⁺) =

= D(0, T )Rec1_{{τ <T }}+ 1_{{τ >T }} (S_T^B− K)⁺.

Two approaches to default arrival modeling

We will deal now with the modelling of default arrival and pricing accordingly.

Two main approaches to credit risk modeling, that differ, from a technical point of view, in the way default arrival, i.e. the time τ at which the default of a company occurs is modeled:

1 Reduced-form approach: main idea: disaggregate the value of cash flows in a default-free versus a cum-default environment;

the intensity at which default occurs has a given assumed form;

default is an exogenous event.

2 Structural approach: main idea: default occurs because the firm experiences financial deficiencies. Default arrival is connected to the changes in the fundamentals of the firm, i.e.

when firm value hits some boundary.

Markov Chains (Billingsley)

Take S to be a finite or countable set (state space, contains the states of the system).

Suppose for each pair i and j in S there is an assigned nonnegative number pij s.t. P

j∈Spij = 1, for i ∈ S.

Let X0, X1, ..., Xn be a sequence of random variables whose ranges are contained in S (history of the system).

The sequence is a Markov chain if for every n and every sequence i₀, ..., i_n for which P [X₀= i₀, ..., X_n= i_n] > 0,

P [X_n+1 = j|X₀= i₀, ..., X_n= i_n] = P [X_n+1= j|X_n= i_n] = p_inj. p_ij’s are the transition probabilities.