# Earnings at Risk – Calculating EAR

Earnings-at-Risk (EAR) is computed in order to evaluate the impact of interest rate change on earnings. The approach used is a VaR based approach that takes into account non-parallel shifts in the term structure and its impact on the earnings portfolio of the bank. The balance sheet items to be included in the calculation are those which are interest rate sensitive and generate income or expense cash flows. For the purpose of calculation, the book value of cash flows will be taken into account.

### Step 1: Determine look back period

Determine the period over which the risk is to be evaluated. For illustration purposes let us assume a look back period from 1st January 2009 to 30th June 2009, inclusive.

### Step 2: Data collection

The data is interest rate data for all available buckets. In our example we use the PKRV rates (these are government treasury rates) at all available buckets for the determined look back period.

### Step 3: Calculated the return series

Calculate the returns series for all the interest rate buckets by taking the natural logarithm of the ratio of successive rates. This is determined as follows for the given illustration.

### Step 4: Identify assets, liabilities & off balance sheet (OBS) instruments to be included in the calculation

Identify interest rate sensitive, income/expense generating on and off balance sheet items that will be included in the calculation. In our illustration our rate sensitive portfolio (as at 30th June 2009), comprises of the following items:

### Step 5: Calculate expected cash flows and days to maturity (DTM)

For each instrument identified in Step 4 above strip off cash flows (at book value) expected for each future period from the revaluation date (which is our case is the balance sheet data given above, i.e. 30th June 2009) for the life of the instrument and calculate the days to maturity for each cash flow.

**Note**: For floating rate instruments only include the cash flows that fall within one re-pricing period of the revaluation date. For the final cash flow before the expiry of that re-pricing period also include the principal amount. This step is illustrated below for INCOME BOND 05 (given at serial number 17 in the table above):

### Step 6: Slot the cash flows according to DTM

Define buckets for the when cash flows are expected to be paid/ received. Further break down these buckets into those that are representative of the underlying interest rate term structure to be used in the analysis (e.g. PKRV). Slot the cash flows determined in step 5 above into these buckets based on their DTMs. This is illustrated in our example as follows for INCOME BOND 05 (Serial no. 17):

### Step 7: Compute weights of each sub bucket with respect to overall bucket

At the overall rate sensitive asset, liability and off-balance sheet level calculate for each term structure bucket its weight with respect to the defined bucket. Weights are based on cash flows. This is illustrated for “Up to 1 month” bucket of the asset portfolio as follows. Please note that the methodology may be extended to whatever buckets that may be defined. For example if smaller buckets are defined such as separate buckets for 0-7 days, 8-15 days and 16-30 days as these corresponding to the actual rate buckets the rate buckets ways will be 100% of the defined buckets:

### Step 8: Compute correlated return series based on the weights calculated in Step 7 above

For overall assets, liabilities and off-balance sheet items in each defined bucket calculate the weighted (correlated) return series. The inputs in this step are the return series calculated in Step 3 and the weights computed in step 7 above. This is illustrated in our example for assets in the defined bucket of “Up to 1 month” below:

### Step 9: Compute rate VaR

For overall assets, liabilities and off-balance sheet items in each defined bucket calculate the rate VaR for the correlated return series calculated in Step 8 above. This is done using the following formula:

Rate VaR = Volatility of the return series × inverse of the cumulative standard normal distribution at the desired confidence level × square root of the desired holding period.

In our example this is illustrated for a confidence level of 99% and a holding period of 10 -days as follows:

* [For a more detailed step-by-step procedure for calculating VaR you may like to review the course “**Calculating Value at Risk**”]. *

### Step 10: Compute Rate Shift/ Shock

In this step we calculate the rate shock that we will subject the underlying interest rate sensitive portfolio to. This is done in two stages separately for overall assets, liabilities and off-balance sheet items in each bucket. In the first stage we determine the weighted average base rate based on the interest rates as of the revaluation date (i.e. the base rates) and the weights calculated in Step 7 above. In our example, for the overall assets in the “Up to 1 month” defined bucket this is illustrated as follows:

The second stage is to calculate the rate shift/ shock using the weighted base rate calculated in stage 1 and the Rate VaR calculated in Step 9, specifically separately for overall assets, liabilities and off balance sheet items for each defined bucket this is calculated as follows:

Rate Shift for the Bucket = Weighted base rate × Rate VaR

In our example this results in a rate shift for assets for the bucket “Up to 1 month” of 5.8594% (=12.87%×45.52%).

### Step 11: Compute the change in rate sensitive assets/ liabilities/ off- balance sheet items

Once we have determined the rate shift for the bucket we determine the impact this rate shift has on the assets/ liabilities/ off- balance sheet items respectively. This is carried out once again in a two stage process. In the first stage we compute the weighted average days to maturity (DTM) of the defined buckets. (Note: The DTM calculated in step 5 is DTM calculated for each instrument whereas the DTM calculated here is the DTM for the defined bucket).This is illustrated for assets in the defined bucket “Up to 1 month” in our example below:

Next using this weighted DTM and the rate shift computed in Step 10 above, we calculate the change in rate sensitive assets, liabilities and off-balance sheet items respectively for each bucket. Specifically we use the formula:

Change in assets/liabilities/ off-balance sheet items = Rate shift × Weighted DTM × Total RS Assets/ Liabilities/ Off-balance sheet items

This is illustrated as follows for the rate sensitive assets in the defined bucket “Up to 1 month”:

Change in RS Asset in bucket “Up to 1 month” = 5.8594%×0.0077×225432248=101,709.

It may be noted that this process is calculated individually and separately for interest rate sensitive (overall) assets, liabilities and off-balance sheet items for each defined bucket.

### Step 12: Compute the on-balance sheet and cumulative gaps for each defined bucket

For each defined bucket we will calculate the on-balance sheets gap as well as the cumulative gap which includes the OBS gap in the interest sensitive portfolio arising out of rate shocks. Specifically:

- On-balance sheet Gap = Change in RS assets less Change in RS liabilities
- Cumulative Gap = On-balance sheet Gap + Change in Off-balance sheet (OBS) RS items

This is illustrated for the “Up to 1 month” defined bucket as follows:

### Step 13: Compute Earnings at Risk for the given confidence level

The total EAR for a given confidence level is the sum of the calculated gaps (from Step 12 above) across all defined buckets. This may be done for only on-balance sheet items or it may also include gaps in OBS items. This is formulated by:

In our example we have used the 99% confidence level. This is illustrated for on-balance sheet Total EAR as follows:

Also see * ALM Training for Board and ALCO Members. 7 posts, 60 minutes*.