Evaluate credit risk
Describe what is meant by credit spread, and describe components of credit spread
Discuss different approaches to modeling credit risk
Consider the main ares of work for actuaries in respect of the analysis of credit risk
Modeling credit default probabilities for a company using different types of models
Modeling credit risk exposure in a diversified portfolio
Analysis of credit spreads
There are details on the different types of credit risk models (technical)
Should focus on understanding the broad application of the concepts
(e.g pros and cons, and difficulties involved)
Subjects such as Metron model is just for completeness and revision
Exam note:
Module 28 will discuss the applications of some credit derivative tools
Stuff excluded from Sweeting 14.5.3:
Quantitative credit models
Probit and logit models
Discriminant analysis
k nearest neighbor approach
Support vector machines
Hazard Rates
Rate that a defined event occurs at a specified time on members of a defined group given that the event hasn’t yet occured to the members of that group
\(\mu(x)\): Force of mortality, hazard rate
\(\mu(x) \lim \limits_{k \rightarrow 0^+} \dfrac{1}{k} \times \Pr[T \leq x +k \mid T> x]\)
Probability of surviving \(t\) years
\(_{t}P_x = \exp \left\{ -\int \limits_{s=0}^t \mu_{x+s} ds \right\}\)
Risk of loss due to contractual obligations not being met (in terms of quantity, quality, or timing) either in part or in full, whether due to inability of, or decision by, the counterparty
Credit risk components
1. Default risk
Depends on the credit standing of the counterparty
Default can be any of the following:
Payment is missed
Financial ratio dropped below certain level
Legal proceedings start against the credit issuer
PV of assets fall below the liabilities due to economic factors
2. Credit spread risk
Credit spread:
Measure of the difference between the yield on a risky and risk-free security
Credit spread risk:
Risk of changes in value of an assets due to changes in the credit spread
(Reflecting a change in actual or perceived creditworthiness)
Components of default risk of a single counterparty
Probability of default
Loss given default
Components of default risk of a credit portfolio
Probability of default of each counterparty
LGD in respect of each counterparty (function of exposure and recoveries)
Level and nature of interactions between various credit exposures and other (non-credit) risks in a portfolio
Exposure
Can be clear and straightforward (e.g. loan or invoice) but not always (e.g. derivatives)
For derivatives, the value fluctuate with market movement
So even though the exposure at an exact point in time may be known, it should not be treated as the full extent of the exposure to the counterparty
Might need to make an estimate of the average exposure or model the possible future exposure using Monte Carlo
Recovery
Can take a long time to obtain and usually require legal proceedings
Can be strengthened with collateral or 3rd party guarantees
Sweeting explicitly defines credit risk excluding credit spread risk
But even so market risk and credit risk are not indepdenent and the inter relationship needs to be considered in the risk assessment;
Examples:
Pension schemes are exposed to market risks and are dependent on the credit risk of the sponsor
Credit risk associated with long term loans are dependent on changes in the yield curve
General techniques to minimize credit risk
Diversify exposure across multiple counterparties
Monitor exposure regularly
Take immediate action when default occurs
(more in Module 28)
Availability of relevant and reliable information is a particular problem for assessing credit risk
Creditor has an information asymmetry about the risks they are creating
e.g. Mortgagee knows much better about their personal situation and ability to repay the mortgage than the bank
Source of information
Credit issuer
(e.g. credit rating agencies will carry out in depth interviews with the institutions they rate)
Counterparty
(e.g. banks will have standardized questionnaires for new borrowers)
Publicly available data
(e.g. information disclosed under Basel disclosure rules or stock exchange listing rules)
Proprietary database
(e.g. companies such as Experian hold vast amounts of data on individual’s credit histories)
Need to be aware of the trade off between cost of obtaining more information and the benefit of improved analysis
There are both qualitative and quantitative methods
Can generate subjective assessment of default and credit spread risk
Assessments are based on relevant factors:
Nature of the contractural obligation
(e.g. loan’s seniority)
Level and nature of any security
(e.g. parental guarantees, collateral)
Nature of the borrower
(e.g. company’s industry sector or individual’s employment status)
Economic indicators
(e.g. inflation rates)
Financial ratios
(e.g. company’s gearing)
Face to face meetings with the credit issuer and/or counterparty
Assessment should consider how the risk may change over time (e.g. over the life of the contract or over an economic cycle)
Recall example of S&P’s process from Module 7
Assessment will ultimately consider risk of default and perceived creditworthiness of the counterparty over the time-horizon of interest
Advantages
Disadvantages
Excessive subjectivity
Lack of consistency between ratings (between sector, analyst, etc)
Meaning of subjective ratings may change over the economic cycle or as a result of changes in the economic environment
Ratings may fail to respond to changes in the economic cycle or circumstances of the counterparty (particularly there is often a reluctance to change a credit rating)
i.e. A decision making bias (anchoring)
Rely on taking financial data and converting into some form of credit measure (e.g. PD)
Types of credit models:
Credit-scoring
Forecast the likelihood of a counterparty defaulting at a particular point in time given certain fundamental information about the counterparty
Empirical models:
Analyse the incidence of default in the past for companies with a certain level of gearing, cashflow, profits etc
Expert models:
Use the opinions of experts to assess the likelihood of default
Structural modesl(firm value)
Estimate the likelihood of default using market information such as the company’s share price and the volatility of its share price (rather than fundamental financial data)
e.g. Merton and KMV models
Reduced form models
Does not model the mechanism leading to default
Model it as a statistical process that typically depends upon economic variables
e.g. Includes all credit migration models that estimate how a counterpatyr’s credit rating might behave over time
Credit portfolio models
Use to estimate credit exposures across several counterparties and may allow for diversification effect of uncorrelated creditors (or the aggregate effect of correlated ones)
e.g. Multivariate structural and multivariate credit migration models
Credit exposure models
For estimating complex credit exposures that are not straightforward to calculate
e.g. Monte Carlo for estimtating the expected and maximum credit exposures
Common difficulties
Lack of publicly available data on default experience
Skweness of the distribution of credit loss
Correlation of defaults between different counterparties
Model combinations
Models above are typically combined to get the default risk (e.g. credit migration model for credit ratings forecast and structural models to determine the PD)
Model risk from model combinations can be significant
As the results can vary significantly depending on the mixing distribution model selected (esp. for the extreme tail)
Important to understand the significance of each of the modeling assumptions when testing the model risk
Recovery
Another source of risk is recovery percentages given default
Difficult to estimate but have a large effect on the results
Based on model proposed by Merton in 1974 and uses option pricing theory along with equity share price volatility
From the shareholder’s p.o.v. the equity of a company = call option on its total assets
If the total asset > debt (at the time the debt has to be repaid)
\(\hookrightarrow\) Shareholders will repay the debt and own the company’s total assets
If the total assets < debt
\(\hookrightarrow\) Shareholders will walk away
Similarly: Value of debt = Value of risk-free bond - value of a put option on its total asset
If assets > debt
\(\hookrightarrow\) Bondholders will get the same amount at maturity as the holder of a risk-free bond
If asset < debt
\(\hookrightarrow\) Bond holder will lose the difference between the value of the company’s total assets at redemption and the value of the debt
We can use the black scholes formula for the value of the equity shares:
\(S_0 = X_0 \Phi(d_1) - B e^{-rT} \Phi(d_2)\)
\(S_t\): Value of equity at time \(t\)
\(X_t\): Value of asset at time \(t\)
\(T\): Time of redemption
\(r\): Continuously compounded risk free rate
\(B\): Nominal value of the debt
\(d_1 = \dfrac{\ln(X_0/B) + (r+\sigma^2_X/2)T}{\sigma_X \sqrt{T}}\)
\(d_2 = d_1 - \sigma_X \sqrt{T}\)
\(\sigma_X\) volatility of the company’s total asset (over the period to time \(T\))
Note that past volatility is observable but we need an estimate of future volatility
Need volatility of assets rather than the equity shares
Can use historic volatility or implied volatility from options on the equity shares
Equation needs to be solve with both \(X_t\) and \(\sigma_X\) (which is normally impossible)
Use Ito’s Lemma to get a second equation with the same two unknown and solve simultaneously
\(\sigma_S S_0 = \Phi(d_1) \sigma_X X_0\)
Merton methods results formulas
Probability (at time 0) of default occurring at time \(T\)
\(P(X_t \leq B) = \Phi(-d_2) = 1- \Phi(d_2)\)
Value of debt (at time 0)
\(B_0 = B e^{-rT} - [Be^{-rT} \Phi(-d_2) - X_0 \Phi(-d_1)]\)
Implied T-year spot rate (\(b\))
\(B_0 = Be^{-bT}\)
Implied credit spread relative to risk free debt
\(b-r\)
Advantages
Limitations (due to assumptions)
Markets are frictionless with continuous trading
Risk free rate is deterministic
\(X_t\) follows a log-normal random walk with fixed rate of growth and fixed volatility
(i.e independent of the company’s financial structure like level of gearing, which is unrealistic)
\(X_t\) is an observable trading security
(rarely a correct assumption)
Bond is a ZCB with only one default opportunity
Default results in liquidation
(Not necessarily the case in real life)
Another limitation:
Based on the same concept as Merton where a company will default when \(X_t\) falls below \(B\)
(or \(\tilde{B}\) which is based on the term structure of all the company’s liabilities, e.g. often taken as the liabilities falling within one year)
Distance to default
Number of s.d. that the company’s assets have to fall in value before they breach \(\tilde{B}\)
\(DD_0 = \dfrac{X_0 - \tilde{B}}{\sigma_X X_0}\)
Using empirical data on company defaults and how these defaults link with the \(DD\)
The model is used to estimate the likelihood of default for any given company over the coming year
Advantages (over Merton)
Coupon paying bonds can be modeled
More complex liability structures can be accommodated as the system uses the average coupon and the overall gearing level (rather than assuming ZCB)
\(X_0\) is not assumed to be observable
Derived from the value of the company’s equity shares
Limitations
For longer term exposure (> 1 year) credit migration models estimate how the credit rating might change over time
Estimating the PD in each future year
3 steps modeling process
Historical data used to determine transition probability and recorded in a rating transition probability matrices
Apply the matrices to the counterparty’s current rating to estimate the likelihood of each possible rating in each future year
Using the PD for a company of a given rating to estimate the chance of default in each future year
Advantages
Not overly impacted by volatility in equity markets
Does not rely on publicly traded share information
Disadvantages
Mostly due to reliance on:
Credit migration process following a time homogeneous Markov chain
There being a credit rating that reflects the company’s default likelihood through the business cycle (rather than reflecting the default chance in the current economic environment)
Therefore disadvantages includes:
Time homogeneity assumption has been criticized using empirical evidence and appears unintuitive
(e.g. recently downgrade company is more likely to be downgraded again than company that has been at the rating for a long time)
Assumes default probabilities for each rating in each future year can be estimated
Assumes that the likelihood of default can be determined solely by the company’s credit rating
Low number of distinct credit ratings (compared to the number of rated orgs) results in low level of granularity in the default estimates
Rankings of organizations by the different credit rating agencies do not always coincide
Not all organizations have obtained a (costly) credit rating
Ratings are sometimes unavailable (withdrawn)
e.g. if data required for rating has not been made available
Simpler form of credit migration model assumes that credit migration follows a martingale process
CreditMetrics (Single bond)
Estimates the value of a bond in one year’s time for each of its possible future ratings and deduces the bond’s expected future value
Combining this information with the transition probabilities produces an estimate of the variance of the bond’s value in one year’s time
Models to estimate behavior of a credit portfolio
Key challenge: understand the relationships between the various credit exposures so as to model the appropriate behavior (e.g. jointly-fat tails)
Can use a multivariate version of Merton (or KMV)
Can use an explicit copula as well
Extending the credit migration models
Assume each org in the portfolio has asset values that behave independently and log-normally
We can derive a model of the number of organization that default in each year
Combine with exposure, recoveries we can derive a default distribution via simulation
CreditMetrics (portfolio)
country specific indices and independent firm specific volatilityMonte Carlo simulations of these indices and firm specific volatility are used to derive potential:
Movements in equity values
Corresponding changes in the overall value of each org’s assets
Associated changes in rating
Implied *8changes in the value of the bonds** in the portfolio
(incl. the default experience)
Risk measures can then be applied to the simulated valuations
Assumptions
Each credit rating has an associated PD
\(\Delta\) in rating is a function of \(\Delta\) in the value of an org's assets and the volatility of the value of those assets
Value of the assets of each org in the portfolio behaves log-normally
Correlation between the asset values (of different orgs) can be estimated from the correlation between the corresponding equity values
Equity returns can be modeled using country specific indices and (independent) firm-specific volatility
Econometric and Actuarial Models
Econometric models
Estimate the default occurrence using combinations of macro economic variables such as the interest rates, inflation etc
Actuarial models
Use average default rates and volatility for the portfolio together with a broad brush estimate of future losses (does not require simulation)
(e.g. CreditRisk+)
These two models do not model the asset value going forward but try to estimate default rates for firms using external (e.g economic) or empirical data
Common Shock Models
For a portfolio of bonds, the probability of no defaults can be modeled using copulas (e.g. Marshall Olkin copula)
The process can be simplified if we assume each bond’s default follows a Poisson process
For a portfolio with:
\(N\) bonds
\(M = 2^N-1\) distinct shocks
Shock: An event that could occur that would knock out a particular subset of bonds
Probability that all bonds survive to time \(T\)
\(\Pr(\text{no defaults}) = \exp \left\{ - \sum \limits_{m=1}^M \lambda_{\{m\}} T \right\}\)
Let \(\lambda_i\) be the probability of the shock that affects just the \(i\)th bond
Let \(\lambda_{ij}\) (for \(i \neq j\)) be the probability of the shock that affects just the \(i\)th and \(j\)th bonds
Etc.
e.g. for \(N =3\), \(M=7\) the values of \(\lambda_{\{m\}}\) for \(m=1\) to \(7\) form the set \(\{ \lambda_{1},\lambda_{2},\lambda_{3},\lambda_{12},\lambda_{23},\lambda_{13},\lambda_{123}\}\)
Probability that only one bonds defaults:
\(\Pr(\text{Exactly one default}) = \sum \limits_{n=1}^N \underbrace{\left[1- \exp\left\{ -\lambda_n T\right\}\right]}_{(1)} \underbrace{\exp\left\{ -\left[ \left(\sum \limits_{m=1}^M \lambda_{\{m\}} T\right) - \lambda_n T \right] \right\}}_{(2)}\)
Probability that bond \(n\) defaults by itself
Probability that none of the other shocks (i.e. default combinations) occur
Further expansion is possible but becomes computationally demanding
Model the incidence of defaults by using copulas to describe the relationship between the times of default of bonds in a portfolio
Default times
For a portfolio of bonds the survival CDF \(\bar{F}(t)\) can be described in terms of the hazard rate \(h(t)\):
\(\begin{align} \bar{F}(t) &= \exp\left\{ -\int \limits_{s=0}^t h(s) ds \right\} & \\ &= e^{-ht} & \text{for constant hazard rate }h\\ \end{align}\)
For constant hazard rate the default time PDF is an exponential with parameter \(h\)
\(\begin{align} f(t) &= \dfrac{\delta F(t)}{\delta t} = \dfrac{\delta [ 1- \bar{F}(t)]}{\delta t} \\ &= he^{-ht}\\ \end{align}\)
Hazard rate can be estimated in various ways (incl. Merton model, published credit ratings, default history)
These all enable calculation of an implied default probability \(\alpha\) over a particular time horizon
Setting \(\alpha\) to the default CDF \(F(t) = 1 - e^{-ht}\)
We can solve for \(h = \dfrac{-\ln(1-\alpha)}{t}\)
Linking Default Time
Having modeled the default time above we can link them using suitably parameterized copula functions
The combined model enables calculation of the aggregate default rate for a bond portfolio
Normal copula is a common choice but a higher tail dependencies (e.g. Gumbel) may be more appropriate
Two common measures of recovery
Price after default
Short term measure
Ultimate recovery
Typically much larger than price after default
Not usually known until after 1 or 2 years after default
Difference due to market over reacting to the collapse while the receiver takes time to extract as much value as possible from the company’s residual assets
Loss given default depends primarily on
Seniority of the debt
Affects how the debt ranks compared to other debt
The more senior the debt the higher call it has on any remaining assets and hence a higher recover rate
Availability of collateral
Lender can take possession and seek the asset in the event of default
The more liquid and marketable the collateral the more value it has to the lender
Secondary factors:
Nature of the industry
Point in the economic cycle
Legal jurisdiction
Rights and actions of the other creditors
Future recovery rates may be modeled based on historical recovery rates (and volatility)