**Background**

Risk management in the financial industry is often supported by quantitative analysis applying various economic, mathematical or statistical techniques in order to better understand the range of possible outcome and uncertainties linked to financial positions. In this case study, we will evaluate the level of risk in a portfolio containing two assets which had distinct variabilities as observed through their historical movements. The goal of this case study is to apply quantitative techniques to support the understanding of the corresponding risk and discussion on the limitations linked to the analysis.

The case study includes 2 assets which are represented by their respective daily returns (i.e. characterizing the change in daily asset prices).

Note that the asset A returns are reflected using lognormal returns:

The asset B return are captured using the relative returns convention:

In both cases, reflect the underlying asset Price at time in days as observed at the end of the trading day.

The portfolio data includes the total market value (in CHF) held in both stocks at the end of each day (i.e. and ). Note that the value of the portfolio is measure at the end of the trading day.Note that for this case study, the days are all considered consecutive with a unique ID.

**Task:**

- Assuming that the last 1249 observed daily returns (including the current day) are representative of a typical daily movement in the asset price, how much can we lose in each asset over a one-day holding period with a 99% confidence level at each observation day? Compute the amount that can be lost over a one-day holding period in each individual asset (in CHF) at a 99% confidence level for each observation day. Compute also the relative change in portfolio value (i.e. proportion of the lossesagainst the portfolio value) for each position for such confidence level.
**Hint:**Consider how each asset returns impact the asset price to build an empirical distribution in possible asset prices over a one-day movement.

- Questions for discussion:

- What is this measure representing and how can it be used to manage risk?
- Given 1249 observed historical movements, how is the 99% confidence level computed in practice, please explain.
- What can happen outside the 99% confidence level?

- Assuming that the last 1249 observed daily returns (including the current day) are representative of (i) a typical daily movement in the asset price and (ii) the co-movements in the two asset prices, compute the amount that can be lost over a one-day holding period in the overall portfolio (combining the 2 assets, in CHF) at a 99% confidence level for each observationday. Compute also the relative change in portfolio value (i.e. proportion of the losses against the portfolio value)for such confidence level.
**Hint:**Similar to task 1, consider how the portfolio value can move using each daily historical movements across the two assets to build an empirical distribution in possible portfolio value over a one-day movement.

- Questions for discussion:

- How does the results compare to the results under point 1?
- What can be noticed on some days?
- What would be the observation about the nature of the 2 assets and their covariance?
- How is the covariance impacting the result and how would the outcome change the covariance is higher or lower between in the two-asset portfolio?

- Compute the average and variance of the daily return over the 1249 observations relevant for the last portfolio observation day for the asset A and compute the 1 percentile return analytically assuming a normal distribution, then apply this obtained returns to the value of the portfolio position in asset A. Compare the obtained result to the previously computed result in step 1 for this asset, by plotting the cumulative probability distribution based on the empirical distribution (obtained in task 1) and the normal distribution (up to the 2 percentile of the normal distribution).

- Questions for discussion:

- How the analytically computed 1 percentile compare to the historical based result? What can be observed?
- In your opinion, what are the pros and cons between using a historical-based approach compared to a parametric/analytical approach? Which approach would you prefer and why?
- When a cumulative distribution is constructed based on discrete observations what is the implication on the curve?
- Assuming a normal distribution, what is the expectation regarding the number of times the value will fall outside the 99 confidence level during one year (i.e. 250 daily iterations)? Please elaborate your response

**Additional notes**

- The case study can be solved using any suitable tool (e.g. Excel, VBA, Python, other coding language)
- Multiple analytical techniques can be used to solve the case study, we are interested in the assumptions made and approaches taken
- The case study will be assessed both in terms of the quantitative outcome (i.e. results obtained), in terms of analytical techniques used (i.e. how the results were obtained) and in terms of qualitative factors (e.g. understanding of key assumptions made, limitations, interpretation of results)
- The case study is made-up and based on fictitious data prepared specifically to discuss the case study tasks