Default risk can be defined as the risk that the counterparty to a transaction does not honor its obligation.

Default could be both in terms on monetary and non-monetary terms, and it's a part of every transaction.

For a bank or any other financial institution, the default risk refers to the risk of default on payment obligations, such as loans, and other financial transactions.

While referring to payments default on loans and bonds, the default risk is called credit risk. This means that the counterparty does not make the payment when it is due.

<strong>Payment Default</strong>

Payment default could occur because of various reasons:
<ol>
 	<li>The counterparty may refuse to accept the payment claim - repudiation</li>
 	<li>The counterparty may stop issuing all payments for a period of time - Moratorium. This generally happend in case of countries.</li>
 	<li>The counterparty may default on its loan obligation - credit default.</li>
</ol>
<strong>Insolvency:</strong> Insolvency is the situation where the counterparty is unable to pay.

<strong>Bankruptcy:</strong> Bankruptcy refers to the situation where the counterparty has filed for bankruptcy to ensure fair treatment for all creditors.

What is Default Risk?

In order to understand default risk, we will analyze the its key components.

<strong>Default arrival: </strong>As we learned earlier, a default occurs when a counterparty defaults on its obligation. In general, there are two scenarios: default or no default. The value for default arrival is taken as one if the default has occurred. Otherwise, the value is 0.

<strong>Exposure at Default (EAD):</strong> This refers to the total value that the bank is exposed to at the time of default. The bank will calculate the exposure at default for each obligor at the given time. Banks will use their own internal methods, using the IRB approach to calculate the exposure at default.

<strong>Loss Given Default (LGD): </strong>The third factor is the loss given default. This factor determines the losses incurred by the bank when an obligor defaults. The loss given defaults differs from the exposure at default because the bank is unlikely to lose all the value that they are exposed to. There will be some recovery and therefore the actual losses will be lesser. So banks will compare actual losses to the potential exposure at the time of default to calculate LGD.

LGD is not generally calculated for individual loans. Instead, it is calculated for a portfolio of loans.

<strong>Default Loss
</strong>
Using the above three factors, the bank can calculate the default loss as given below:
<div class="formula1"><strong>Default loss: Default arrival x EAD x LGD</strong></div>
The above formula assumes only the two cases where either there is a default or there is no default. In reality, banks will have to consider the probability of default to calculate the expected loss.
<div class="formula1"><strong>Expected Loss (EL) = PD x EAD x LGD</strong></div>
&nbsp;

<strong>Example:</strong>

Assume that the PD is 0.25, EAD is $1 million, and LGD is 60%, then the expected loss will be calculated as follows:
<div class="formula1"><strong>EL = 0.25 x 1,000,000 x 0.60

EL = $150,000</strong></div>
The banks use various methods to calculate probability of defaults, exposures, and loss given defaults. We will look at these methods in other articles.

Exposure, Default and Recovery Rates

While the terms expected loss and unexpected loss are commonly used in risk management, it is important to have a clear understanding of what they actually mean.

In general, expected loss as the name suggests is the expected loss from a loan exposure. On the other hand unexpected loss is the loss that exceeds the expectations.

<strong>Expected Loss </strong>
In statistical terms, the expected loss is the average credit loss that we would expect from an exposure or a portfolio over a given period of time.

The expected loss is measured using the following formula:

<strong>Portfolio Expected Loss:</strong> The total expected loss of a portfolio will simply be the summation of expected losses of individual assets. This is because the mean of the sum is the same as the sum of the mean.

Since the expected loss is what a business expects to lose in a year, the business will generally have budget for it and the losses can be borne as a part of the normal operating cash flows.

<strong>Unexpected Loss</strong>
The unexpected loss is the average total loss over and above the mean loss. It is calculated as a standard deviation from the mean at a certain confidence level at a certain confidence level. It is also referred to as Credit VaR.

A business will safeguard itself from unexpected losses by allocating capital.

Unlike expected loss, the expected loss of a portfolio is not calculated by adding the unexpected loss of individual assets. This is because unless there is perfect correlation, the standard deviation of sum will not be the same as the sum of standard deviation.

The unexpected loss of a portfolio at a 99% confidence level will be expressed as follows:
<div class="formula">UL99% = D99% - EL</div>
Where D<sub>99%</sub> represents the 99% Var Quantile.

If the 99% VaR level is $200m and the expected portfolio loss is $50, then the unexpected loss will be $150m.

The unexpected loss of a portfolio will be expressed as follows:
<div class="formula">[latex]UL (Portfolio) = \sum UL_i \rho_i[/latex]</div>
Apart from the expected and unexpected losses, there are also catastrophic losses, which are covered by insurance.

<strong>The Credit Loss Distribution
</strong>
The following diagram shows the credit loss distribution.

&nbsp;

<img class="aligncenter size-full wp-image-1991" title="loss_dislt" src="https://financetrain.com/wp-content/uploads/2011/05/loss_dislt.gif" alt="" width="400" height="304" />

&nbsp;

<strong>Characteristics of Credit Loss Distribution</strong>

The credit losses distribution has three key characteristics:
<ol>
	<li>It is      not symmetrical. There is a limited upside because the best scenario is      when there is no loss. However, there is extremely large downside, that      is, the losses can be huge.</li>
	<li>It is      highly skewed. The distribution is more concentrated toward small losses,      with very few chances of large losses.</li>
	<li> The distribution has heavy tail, i.e.,      the probability of large losses reduces very slowly.</li>
</ol>

Expected Loss, Unexpected Loss, and Loss Distribution

The recovery rates are a crucial element for calculating credit risk.

The loss given default of an asset or a portfolio is calculated as 1 minus the recovery rate.
<div class="formula">LGD = 1 - R</div>
The problem with recovery rates is that they are not easy to estimate and the data available for recovery rate is very fragmented and inconsistent.

The recovery rates are closely linked to the bankruptcy procedures which happens after an obligor has defaulted. Based on the procedures, all the creditors of the defaulter are paid a part of their claims based on the priority. For example, the secured creditors will have first claim on the collateral and other assets of the defaulting firm. The unsecured creditors will come after them, and the shareholders will get the last preference.

Another important issue is whether the payment received as recovery should be discounted to the time of default or not. In case of large obligors the payments will generally be discounted back.

The recovery rates can be generally defined in two ways:
<ol>
	<li><strong>Market      Value Recovery:</strong> This is the market value per unit of legal claim      amount, short time (1 to 3m) after the default</li>
	<li><strong>Settlement      Value Recovery:</strong> This refers to the value of the default settlement per      unit of legal claim amount, discounted back to the default date and after      subtracting legal costs</li>
</ol>
&nbsp;

Understanding Recovery Rates

It is practically impossible to accurately predict the recovery rates. There are various factors that will affect the recovery rates of a defaulted loan:
<ol>
	<li><strong>Collateral:</strong> The higher the value of collateral, the higher will be the recovery rate,      as the sale of collateral will be able to payoff more creditors.</li>
	<li><strong>Priority      class of the claim:</strong> The recovery rates are affected by the priority      class of the claim. This is because most bankruptcy procedures settle the      claims based on the seniority of the class. The highest recovery rates are      observed in the senior secured class and the lowest rates are observed in      the junior subordinated debt.</li>
	<li><strong>The      bankruptcy procedure used:</strong> The backruptcy procedure will also affect      the recovery rates. The UK bankruptcy procedures tend to be more      favourable towards creditors, while the procedures in USA and France are      more friendly towards the obligor. Accordingly, the UK will observe higher      recovery rates compared to US and France.</li>
</ol>
<h3>Other Empirical Factors</h3>
There are other less observed factors that affect the recovery rates:
<ol>
	<li><strong>Industry:</strong> The types of industry will affect the recovery rates. For example,      financial institutions tend of have higher recovery rates than other      industries. The more capital intensive the business is, the higher will be      the recovery rates.</li>
	<li><strong>The      obligor’s rating prior to default: </strong>If the obligor was having a poor      rating prior to default, then the recovery rates will be less because he      may have very few assets to liquidate.</li>
	<li><strong>Business      cycle &amp; average rating in the industry: </strong>When the economy is in      recession, the recoveries will be low. Similarly, when the business is in      the downswing of it’s business cycle, the recoveries will be low.</li>
</ol>
Given all the factors and the uncertainty of recovery rates, banks must be careful about the model they adopt for determining recovery rates.

Factors Affecting Recovery Rates

Most of the people use the beta distribution to model recovery rates.

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval (0, 1) parameterized by two positive shape parameters, typically denoted by α and β.

The function for the beta distribution is as follows:

[latex]f(x)=c.x^{a}(1-x)^{b}[/latex]

c is the normalization constant.

We can input different values of α and β to obtain different shapes of recovery rates.

The mean and standard deviation of the dataset can be used as the two parameters α and β of the beta distribution using the BETADIST() function in excel. The following graph shows four shapes of the curves based on the different values of mean and standard deviation.

<img class="aligncenter size-full wp-image-2004" title="Beta Distribution" src="https://financetrain.com/wp-content/uploads/2011/05/beta-dist.gif" alt="Beta Distribution" width="551" height="349" />

This method is considered quite flexible because it only requires the mean and variance for calibration.

<strong>Beta Distribution Spreadsheet</strong>

Download the following spreadsheet to see how the distribution is arrived at.

<a href="https://financetrain.com/wp-content/uploads/Beta-Distribution-for-Recovery-Rates.xls">Estimating Recovery Rates Using Beta Distribution</a> - This spreadsheet shows the estimation of recovery rates using beta distribution.

Using Beta Distribution for Estimating Recovery Rates

Foundations of Credit Risk Modelling

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Foundations of Credit Risk Modelling

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