WCAS - Economics of Banking and Insurance
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WCAS - Economics of Banking and Insurance
Assignment: Risk Measurement
These questions are due by the end of next week (you have 2 weeks for this project!). As usual, upload your answers to Canvas. Do not hesitate emailing me if you have any questions about the assignment.
1 Operational Risk
The data set provided in the ile oprisk_data contains operational risk events classiied by event type and business line with the corresponding dollar loss, settlement date, and industry. There are three versions of the ile, completely identical: one for Excel, one for Stata, and one for Matlab. Read irst all the problem before you start working on it.
1. By event type, plot an histogram of the losses (for all industries). What is wrong with the igures? Can you say something about the distribution of losses?
2. Transform the data by taking natural logarithms and repeat the exercise. What can you infer from the igures now? Explain in words what the transformation has done to the data. Overlay a normal distribution density on top of the histogram. You can do this in Stata by using the normal option, or by using the ksdensity function in Matlab.
3. Construct a table with two columns: in one column, show the average annual frequency of events in each event type (rows); in the second column, show the average annual severity of losses for each event type, using data for all industries.
4. Curve Fitting. Now that we have seen how a normal distribution may approximate the logarithm of the loss data we will estimate the 99.9 percentile of the loss for each event type.
o Estimate the parameters of the normal distribution of the log-losses. Don’t think too hard, this is easier than what you are thinking: u is the mean and a is the standard deviation.
o With the estimated parameters for each event type, compute the maximum loss that could occur, by event type, with a probability of at most, 0.1%.
o Display the results in a table with event types in rows and a column with the estimated u, a, and 99.9th percentile.
o Describe the results in terms of which units of measure are posing higher risks to the irm.
5. Repeat the exercise (1-4) just for the“Finance and Insurance”sector (code 52). How do the loss distributions difer from those plotted and estimated for all the sectors?
Note: to ilter by business line or event type in Matlab (in case you decide to use the data in Matlab format), you might ind useful the command strcmp. For example, if you want to know which observations correspond to CPBP, you’d deine an index variable with =find(strcmp(BISEventTypeLevel1,’Clients Products and Business Practices’)).
For the percentiles, you will want to use the function quantile. In Stata, it will be useful to become familiar with the histogram function and the diferent options that can be used (in addition to the the summarize and its options).
2 Credit Risk
The ile fund_ratings .dta contains 10 years of quarterly data for publicly traded irms. The data contains the name of the irm and identiier, the value of equity (market value of the irm), value of debt (total liabilities), and rating according to S&P. For your convenience, I also added the next quarter’s rating for a given irm and the change in market value.
1. Compute the transition matrix that we have seen in class. The matrix should have ratings at time t in the rows and ratings at t + 1 in the columns. Each cell should contain the fraction of irms that, over the sample, transitioned from the rating given by the row, to a rating the following quarter on the columns.
2. BONUS: Compute the 1-year (T = 1) probability of default according to the Merton model for 3 diferent irms. Remember that to compute the probability of default in the Merton model you need to use the value of equity, the volatility of assets (aV ), the book value of debt (D), the average growth of market value of equity (uv ), and the properties of the normal distribution. Assume that the value of debt is well approximated by the liabilities of the irm and that the value of assets (V) is equal to the value of debt (D) plus the value of equity (slide 15 of week 5’s notes).
P(VT ≤ D) = 
( 1 2ln(D/V)—(uV—2aV)T
aV^T)
Volatility of equity can be deined for our purposes as the standard deviation of the changes in market value of equity, for a given irm. How does the ranking among the three irms you have chosen compare with the ranking based on S&P’s long term rating?
3 Value-at-Risk
In this problem, we will learn how to compute the Value-at-Risk from the empirical distribution and also from a itted distribution. Suppose that you work in risk management and you are in charge to set limits for an equity trader. Your task today is to set the limits on how much the trader can invest in a particular stock, say General Motors (NYSE:GM). The irm-wide limits are such that the traders cannot lose more than $1 million in a given day in a given position with 99% conidence level.
1. Download 5 years of daily stock prices for GM from your favorite stock price source. Hint: the easiest one, as we’ve seen in class, would be Google Finance. Compute the daily re- turns as 
; for simplicity, let’s ignore dividend payments, ERM only cares about price luctuations. Plot a histogram of the distribution of returns.
2. Given the history of returns, what is the worst return that the trader is exposed at the 99% conidence level? This is the same as asking what is the Value-at-Risk at the 99% conidence level. What is the Value-at-Risk at the 5% conidence level? Report the 1st percentile, 5th percentile, 10th percentile, and 50th percentile (median) of the distribution of stock returns.
3. How much money, in dollars, can the trader invest in GM and stay within the risk limits of $1 million of daily maximum loss with 99% conidence?
Now suppose that you think the returns distribution is well approximated by a normal distribu- tion. The normal distribution is a two-parameter distribution: mean and variance, therefore, you can estimate those parameters without any complex econometric approach.
1. Plot the histogram again, but this time add a normal density over the histogram to assess how good of an assumption normality is. (Hint: Stata’s histogram command has an option to do this automatically.)
2. Compute the Value-at-Risk at the 99% conidence level based on the estimated normal dis- tribution of returns. Remember that the VaR is the maximum loss (or minimum returns in this case) that could occur with probability 1 — α, i.e., 1% of the cases.
3. How does the investment limit that you would establish for the trader, in dollars, compare to the limit estimated based on the empirical distribution above?
2023-02-17
Risk Measurement