Final answer:
The portfolio's value-at-risk (VaR) at the 97.5% confidence level is 1m, while the expected shortfall (ES) is 19m, as it accounts for losses beyond the VaR threshold.
Step-by-step explanation:
To determine the value-at-risk (VaR) and expected shortfall (ES) for a portfolio with a 97.5% confidence level, we first need to understand the loss probabilities. The portfolio has a 9% chance of a 1m loss and a 1% chance of a 10m loss.
At the 97.5% confidence level, we are interested in losses that will not exceed a certain amount 97.5% of the time.
Our risks are as follows: with 90% certainty, there's no loss; with the next 9%, there is a 1m loss, and with the last 1%, there is a 10m loss.
Since our confidence level excludes the worst 2.5% outcomes, we look at the last percentile which is at the 1m loss (since 90% + 9% = 99%, which is above 97.5%).
Therefore, our VaR at the 97.5% confidence level is 1m.
To calculate the ES, we need to take the average of losses exceeding the VaR threshold. Here, that includes the full possible 10m loss since we only have two outcomes.
The ES is then calculated as (1m * 9% + 10m * 1%) / (9% + 1%) = (0.09m + 0.1m) / 0.1
= 1.9m / 0.1
= 19m.
This is because ES considers both the magnitude beyond VaR and the likelihood of occurrence. So the expected shortfall at the 97.5% confidence level for this portfolio is 19m.