Credit Risk Management
Introduction
Why Credit Risk Management is important?
To answer this question, we will construct a hypothetical example: Suppose a company is asking a bank for 5 billion Euros. Somewhere in the bank’s credit department a senior analyst has to decide if the loan will be approved or not? For this purpose he has to check the creditworthiness\(_1\) of the company, using the bank internal rating\(_2\). We will learn about these terms on the way.
\(^1\) How much the customer is worthy to receive a credit or loan
\(^2\) A rating is an indication of creditworthiness
Expected Loss
Background
In the previously given example, suppose the senior analyst approves the loan to the company. The company is unable to repayment within the certain period of time, i.e. there is a default on the Loan. In that case the bank will have a huge amount of loss. For this reason we have insurance. Moreover, history shows us that even good customers have potential to default on their financial obligation such that an insurance for not only the critical but all loans in the bank’s credit portfolio \(_3\).
The basic idea behind insurance remains the same. For example, in health insurance the cost of few sick customers are covered by the total sum of revenues from the fees paid to the insurance company by all customers.
Now we all know, the insurance premium paid by the customers are not the same. Suppose Mr.A & Mr.B both applies for a medical insurance for cancer. The insurance company asks both of them if they smokes or not. Mr.A is a smoker and Mr.B is not. In that case the insurance company will ask more premium from Mr.A than Mr.B because the risk of cancer for Mr.A is greater than Mr.B.
Similar thing will happen for loans also. Suppose 2 companies ABC & XYZ is asking bank for a loan of same amount. In that case, the management of bank will check the bank’s internal rating for those companies and then decide about giving out loans and risk premium on that basis.
But where does the Expected Loss play its part ?
The basic idea is very simple. The bank assigns every customer a Default Probability, a loss fraction called Loss given default and a Exposure at Default.
Default Probability is the probability that a loan will go to default.
Loss given Default or LGD is the fraction of the loan’s exposure expected to be lost in case of a default.
Exposure at Default or EAD is the percentage to be lost in a given time period.
The loss is defined by a Loss Variable, L defined as-
\[ \tilde L = EAD * SEV * L \] where \(L = 1_D\), \(P(D) = DP\), \(SEV=\) severity of loss, \(E(SEV) = LGD\)
Here \(D =\) The event that the customer defaults within a certain period of time.
The Expected Loss of any customer is the Expectation of the corresponding loss variable, \(\tilde L\). In this case we are assuming \(EAD\) & \(LGD\) are constants. But it is not necessarily the case under all circumstances. There are various situations in which, for example, the EAD has to be modeled as a random variable. \(EAD\), \(SEV\), \(D\) are independent of each other.
\[\begin{aligned} EL &= E(EAD . SEV . L) \\ &= EAD.LGD.E(L) \\ &= EAD.LGD.E(1_D) \\ &= EAD.LGD.DP \\ \end{aligned}\]
We will consider EAD a deterministic or non-random. But the severity of default (SEV) is considered as a random variable, with it’s expectation as LGD. i.e. \(E(SEV)=LGD\).
Suppose a bank sanctions a loan of 10 billion dollars to a customer. The customer has an Expected Loss of 2 Billion dollars. So the bank would like to have an insurance of 2 billion dollars.
The Default Probability
The task of assigning a default probability to every customer in the bank’s credit portfolio is far from being easy. There are essentially two approaches to default probabilities:
- Calibration of Default Probabilities from market data:
Most famous example is- Expected Default Frequencies (EDF) used by KMV Corporation.
- Calibration of Default Probabilities from ratings:
In this approach, DPs are associated with ratings, and ratings are assigned to each customer by external rating agencies (S&P, Mood’s investor services etc.) or bank’s internal rating methodologies.
Ratings
Rating is basically the creditworthiness of the customer. Various companies like Mood’s, S&P provide ratings for customers. The rating scaling are more or less same. The credits are- Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C with creditworthiness is highest for Aaa and lowest for C.
The process of assigning default probabilities to rating is called calibration. Now we will see very simple example of- calibration of default probabilities to Ratings. This is done by using historical corporate bond default or in simple words with the help of historical default cases. This is an example of Historic Corporate Bond default frequencies.
| Ratings | 1983 | 1984 | 1985 | 1986 | 1987 |
|---|---|---|---|---|---|
| Aaa | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
| Aa | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
| A | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
| Baa | 0.00% | 0.10% | 0.05% | 0.11% | 0.06% |
| Ba | 0.70% | 0.50% | 0.00% | 0.66% | 0.43% |
| B | 1.04% | 0.90% | 0.79% | 0.96% | 0.88% |
| Caa | 3.02% | 2.63% | 4.09% | 3.57% | 2.31% |
| Ca | 6.76% | 5.57% | 7.23% | 9.03% | 7.02% |
| C | 12.21% | 11.00% | 12.89% | 17.29% | 14.33% |
A quick solution is-
- Define \(h_i(R)\) as the historic default frequencies for year i and rating class R.
- Define the sample mean of DF for rating R, \(m(R)\) & sample variance \(s(R)\) as- \[ \begin{aligned} m(R) &= \frac{1}{n}\sum_{i=1}^{n}h_i(R) \\ s(R) &= \frac{1}{n-1}\sum_{i=1}^{n}(h_i(R)-m(R))^2 \end{aligned}\]
- If we plot the ratings in numeric i.e. Aaa as 1 and so on in the x-axis and logarithm of \(m(R)\) in the axis, we can see that- we can fit a regression line.
- There is strong evidence from various empirical default studies that default frequencies grow exponentially with decreasing creditworthiness. For this reason we have chosen an exponential fit (linear on logarithmic scale)
- Using the regression equation we can estimate the Default probabilities to each class.
The empirical formula of default frequency is- \[DF(creditworthiness) = \exp([1-creditworthiness].\alpha)\] where \(\alpha > 0\).As \[\begin{aligned} & creditworthiness \downarrow \\ & \implies (1-creditworthiness) \uparrow \\ & \implies \exp((1-creditworthiness).\alpha) \uparrow \\ & \implies Df(creditworthiness) \uparrow \end{aligned}\]
Exposure at Default or EAD
EAD is a quantity, specifying the bank have to it’s borrower. It has two parts- Outstandings & Commitments.
Outstandings is the portion of the exposure already drawn by the obligor.
Commitments is the amount the bank has promised to the obligor. Commitments can be divided into two portions- drawn & undrawn.
\[\begin{aligned} \hat {EAD} &= Outstandings + \hat {Drawn} \\ &= Outstandings + \frac {\hat {Drawn}}{Commitments}.Commitments \\ &= Outstandings + \gamma. Commitments \\ \end{aligned}\]
\(\hat{Drawn}\) is the estimate of the Drawn amount from the commitments.
Story time: Consider a hypothetical situation, where ABC company asks for 10 billion dollars from XYZ bank as a loan for a project and XYZ bank approves the loan. Initially ABC company takes 3 billion dollars but after sometimes the project goes south. The board of directors have realized that the company will be unable to pay the 3 billion dollars outstandings due to that project. But to try to turn around the project one last time, the company asks the bank for the rest of the 7 billion dollars. Now since the bank approved the loan, they have to pay the company 10 billion dollars when asked (if not the company ABC is bankrupted). So in that case there is a high chance that the project will not be successful & the whole 10 billion dollars will be defaulted. But if the bank recognized the company ABC’s situation earlier, they would not approve the rest of the 7 billion dollars because the loan was already going to default.
Moral of the story is- often the borrower has some informal advantage that the bank recognizes the financial distress of its borrower with some delay.
Covenants are legally binding clauses, and if breached will trigger compensatory or other legal action. The bank may force the borrower to provide more collateral or renegotiate the terms of loan.
If the bank picks up indications of financial distress early enough to react before the customer has drawn on her committed line (A committed credit line includes funds available to a company which cannot be rescinded by a financial institution without proper notice), that will save the bank from a huge amount of losses.
The problem with quick action of lending institution is specially critical for borrowers with good credit score (i.e. borrowers with high creditworthiness). Because the bank tend to focus on more critical cases than good credit score of the customer. In terms of hypothesis testing, we can write like this-
\[\begin{aligned} \mathbb{H_0}: \text{The customer is in financially good position}\qquad \text{vs}\qquad \mathbb{H_1}: \text{The customer is in financially bad position} \end{aligned}\]| \[\frac{True\ scenario}{Test\ result}\] | H0 | H1 |
| H0 | Good | Type-2 error |
| H1 | Type-1 error | Good |
In hypothesis testing, we always want tests having Power greater than Type-1 error i.e.-
\[\begin{aligned} P(H_0\ is\ NOT\ accepted|\ H_0\ is\ true) &< P(H_0\ is\ NOT\ accepted|\ H_1\ is\ true) \\ Type-1\ error &< Power\ of\ the\ test \end{aligned}\]
In this case the event,
\(\mathbb{H_0\ \text{is NOT accepted}\quad|\ H_0\ \text{is true}}\) = The borrower is in good financial condition but the bank did not accept it & didn’t gave the borrower Commitment part of the exposure.
\(\mathbb{H_0\ \text{is NOT accepted}\quad|\ H_1\ \text{is true}}\) = The borrower is in bad financial condition & the bank didn’t gave the borrower Commitment part of the exposure.
Since the quick action by lending institution focuses more on critical cases than good credit score, any statistical hypothesis testing following the above Power > Type-1 error condition will be useful.
Loss Given Default or LGD
LGD can be described as “the proportion of the Loss the bank will suffer in case of a default”. General form is-
\[\begin{aligned} LGD &= 1- Recovery\ rate \\ &= 1- \frac{Amount\ Recovered}{Amount\ Loaned} \end{aligned}\]
So, this sums up the brief overview of what is Credit risk management and its pertaining Expected losses. As the amount of loss in case of a loan default is not constant, We would further explore the distribution of losses using parametric and non-parametric (Empirical) approaches in my future posts.