Credit Risk: Expected and Unexpected Losses (FRM Part 1)

1. Context

In this swatch, we look at a very simple credit risk formulation that models default as a one-step Bernoulli trial. Using properties of a Bernoulli trial, we aim to establish simple relations the expected and unexpected losses for a loan / credit, both of which are key ingredients in computing economic capital for credit risk. The details of the reading in which this topic appears are given below:

Area	Valuation and Risk Models
Reading	Capital Structure in Banks
Reference	Schroeck, Gerhard, Ch5. Capital Structure in Banks. In Risk Management and Value Creation in Financial Institutions, 1st Edition, New York, John Wiley & Sons, 2002.

2. Terminology

We will use the following symbols in the sections below:

$D$	Bernoulli variable to model default
$L$	Loss that depends on $D$ (default, no default)
$T$	Horizon, typically 1 year
$EA$	Exposure Amount at $T$
$PD$	Probability of Default (for period till $T$)
$LR$	Loss Rate (a random variable)
$\overline{LR}$	The average or expected value of $LR$
$EL$	Expected Loss
$UL$	Unexpected Loss

3. Key Ingredients

3.1 Exposure Amount ($EA$)

Dollar amount of exposure to a customer / credit counterparty at time of default. This includes all outstanding payments (including interest) at that time, which can be very different from the amount at the initiation of the credit (more true for derivatives like swaps). It can also be stated in % of notional amount.

3.2 Probability of Default ($PD$)

This is the borrower specific probability or likelihood that a borrower will default before the end of a predetermined period of time (horizon of 1 year) or prior to the maturity of the loan. It is linked to the borrower’s risk rating. $PD$ can and does change over time, an effect captured by transition matrices.

3.3 Loss Rate or Loss Given Default ($LR$)

The fraction of the exposure amount that is lost in case of default i.e. the amount that is not recovered after collateral is sold. $LR=1-RR$, where, $RR$ is the fractional recovery rate.

4. Default Event as a Bernoulli Variable

Default is modeled as a Bernoulli variable $D$ that can take values 1(if default happens) and 0(if default does not happen). Alongside, we define another random variable (dependent on $D$ that quantifies the loss in the two states (default, no default). $L$ and $D$ are illustrated below:

5. Assumptions

To get to expressions for expected and unexpected loss, we require two assumptions. Firstly, we assume exposure $EA$ is not random, or if it is, we are dealing here with it’s expected value. The exposure for loans and bonds can be more accurately determined for a given horizon. The same is not true for derivatives like swaps. Secondly, we assume that $D$ (the incidence of default) is independent of $LR$ (it’s severity). This assumption may not be true in reality and we revisit it as part of FRM Part II curriculum.

6. Expected Loss (EL)

From the figure, we know that $L=D \cdot EA \cdot LR$. The expected loss or expected value of $L$ ($EL=E(L)$ can be easily derived, as below. Here, we use $E(D)=PD$ and $E(LR)=\overline{LR}$): $$\begin{align} E(L) &=E(D \cdot EA \cdot LR) = EA \cdot E(D) \cdot E(LR) \\ &=EA \cdot PD \cdot \overline{LR}\\ \end{align}$$

7. Unexpected Loss (UL)

In this reading, Unexpected Loss is defined as the standard deviation of $L$. We already know that $L=EA\cdot D \cdot LR $ and $E(L)=EA \cdot PD \cdot \overline{LR}$. To compute the standard deviation (via variance), we first compute: $$\begin{align} E(L^{2})&=E(EA^2 \cdot D^{2} \cdot LR^{2})\\ &=EA^2 \cdot E(D^{2}) \cdot E(LR^{2})\\ &=EA^2 \cdot PD\cdot E(LR^{2}) \end{align}$$ Combining $E(L^{2})$ and $E(L)$, we arrive at: $$\begin{align} \mbox{var}(L)&=E(L^{2})-E(L)^2\\ &=EA^2 \cdot PD \cdot E(LR^2) − EA^2 \cdot PD^2 \cdot \overline{LR}^2\\ &=EA^2 \left[ PD \cdot E(LR^2) − PD \cdot \overline{LR}^2 + PD \cdot \overline{LR}^2 – PD^2 \cdot \overline{LR}^2 \right]\\ &=EA^2 \left[ PD \cdot \sigma^{2}_{LR} + \overline{LR}^2 \cdot \sigma^{2}_{D} \right] \\ \end{align}$$ where, we have used the definitions $\sigma^{2}_{LR}=E(LR^2)-\overline{LR}^2$ and $\sigma^{2}_{D}=PD\cdot(1-PD)$. We finally arrive at: $$UL=\sqrt{\mbox{var}(L)} = EA\cdot\sqrt {PD \cdot \sigma ^{2}_{LR}+\overline{LR}^{2}\cdot\sigma ^{2}_{PD}}$$