Most risk managers are intimately familiar with the expected loss for credit and interest rate risk. However, fewer risk managers are familiar with the concept of unexpected loss. For commercial banks, it is the unexpected loss that is more important for lending decisions and long-term profitability. We will outline how unexpected loss manifests itself in lending decisions and what commercial lenders must know to safeguard against unexpected loss for credit and interest rate risk.

**Expected Versus Unexpected Loss**

Expected loss (EL) is the average loss that we expect to experience from a loan or portfolio over a given period. The unexpected loss (UL) is the total loss over and above the expected loss. The UL is the variation over the EL that can be measured as a standard deviation from the mean at confidence levels. The UL concept is commonly expressed as either the capital-at-risk or the value-at-risk (VAR) and can be applied to both credit, interest rate, operational, and other forms of risk. Banks create reserves to offset EL and maintain capital to survive the impact of UL.

**Credit Example**

EL is calculated as the probability of default (PD) times loss given default (LGD) times the exposure at default. For example, on a $1mm senior secured loan, let us assume that the probability of default is 1% per annum (a common long-term value for better credits). If a default occurs, let us assume that the collateral can be liquidated to cover 70% of the loan or 30% LGD (an average LGD for senior secured loans). Therefore, in our example, the EL for the year is 1% times 30% times $1mm, or $3,000. A small amount of reserve would be required for this credit.

However, EL is just a segment, and a very small segment, of the economic risk to the bank. The bank will allocate ALLL for the loan to offset the EL loss. A basic loan pricing model will align the expected future loss for the loan with the bank’s ALLL methodology and new CECL accounting standards.

The much greater risk to the bank is the loss that has not been reserved – the UL. The UL can dwarf the exposure for the unexpected loss. Let’s consider some real numbers. In the example of the loan above with 1% PD and 30% LGD, the variation around our assumptions can be quite large depending on our confidence level and amount of skew in the distribution. The one standard deviation around the PD levels can be 1, 2, or even 3 times the PD percentage. Therefore, in our example above we expect that out of 100 loans in any year only one of them will default, in times of stress with a standard deviation of PD at 3%, it is possible at 67% confidence level that 3% of the loans will default, and at 95% confidence level that 6% of the loans will default and at 99.5% confidence that 9% of the loans will default.

The same distribution concept can also be determined for LGD, where historically the one standard deviation for LGD for senior secured loans is about 9%. Therefore, while we expect that our recovery is 70%, in times of stress with a standard deviation of LGD at 9%, it is possible at 67% confidence level that our recovery is only 61%, and at 95% confidence level that our recovery is 52%, and at 99.5% confidence that recovery is only 43%.

All of a sudden, while the EL per annum for a loan is quantified as $3,000, the UL for that same loan can be $46,800 at 99.5% confidence level. It is this UL figure that requires banks to hold capital, and it is this risk that bankers must quantify and manage. The expected losses on loans are offset through a decrease in revenue through ALLL, and the unexpected losses are the extreme events that are offset through the bank’s capital.

The graph below shows an example of credit VAR distribution. The UL can be many times (10 or even 100 times) the economic exposure of the EL.

**Interest Rate Example**

The same concept of UL can be applied to interest rate risk. A banker recently asked why banks should be concerned about interest rate risk since the yield curve is so flat. The yield curve is the equivalent of the expected path of interest rates and the EL from interest rate risk for banks is currently next to zero – the market doesn’t expect interest rates to move much for the next one to two years. However, it is the unexpected possibility of interest rates moving that possess the risk to banks. The variability around the expectation of the market represents the UL from interest rate movement.

Again, UL from interest rates is not in the expectation but variability around that expectation. The graph below shows where the market expects short term interest rates to be for the next three years in the white line. The green lines above and below the white line are the distribution in decile around the expectation. It is this distribution that represents the risk to the bank - interest rates rise, and a fixed loan is funded with short-term deposits. If the future always reflected the market’s expectation, there would be no risk to the bank or borrower regardless of the shape of the yield curve.

We can measure the UL from interest rate risk using the VAR concept. The market prices the value of each of the rate paths of the green lines shown in the graph above. We can then run a $1mm credit under various interest rate paths and calculate the UL to the bank based on current interest rate paths and various terms. The UL from interest rate movement for a one-year loan of $1mm is $10,800 (or 1.1% of the loan amount). The UL increases to $18,300 for a three-year loan, and $27,200 for a five-year loan, and $63,800 for a ten-year loan. The EL from interest rate movement on this loan is about zero.

The flat yield curve has not eliminated the interest rate risk for lenders. The UL from interest rate risk remains regardless of the shape of the yield curve. In fact, because of the current flatness of the curve, borrowers are pressuring banks to stretch on fixed rate duration, exposing banks to more UL from interest rate risk.

**Conclusion**

Bankers should pay particular attention to UL because this risk has substantially more consequences, thank EL. The EL to a bank is easily offset through revenue adjustment and reserves, but the UL is more difficult to quantify and has substantially more influence on long-term return on equity.

Submitted by Chris Nichols on May 22, 2019