How it works
Illustrative dataThe Module 2.5 data set
500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.
Problem 1A 10-day limit for a bank share
L2.5.01 · L2.5.06A market risk analyst at an asset manager
A portfolio manager wants to hold $3,000,000 of the bank share from the module's data set. Over the 500 days its daily returns have a sample standard deviation (n − 1) of 1.9427%. The firm's limit is set on 10-day 99% VaR, computed with the normal model, mean zero and the square-root-of-time rule.
What is the 10-day 99% VaR of the holding, in $?
Problem 2Rebalancing to 50/50
L2.5.03 · L2.5.06A portfolio manager
The fund is $10,000,000, now 60% in the equity index fund and 40% in the bond fund, with a one-day 99% normal VaR of $148,044. The board proposes 50/50. From the 500 days: equity daily volatility 1.0987%, bond daily volatility 0.3229%, correlation −0.2717.
What would the fund's one-day 99% VaR be at 50/50, in $?
Problem 3A bad open
L2.5.04 · L2.5.05A trader on the fund's desk
The fund holds its $6,000,000 equity index fund, its $4,000,000 bond fund (DV01 $2,400.82) and the 12 index put contracts, whose dollar delta is −$1,403,449. Overnight news: the index will open 5% lower, yields 25 bp higher, and implied volatility up 3 points.
Using sensitivities only (beta 1, DV01, dollar delta), what is the estimated P&L of the fund with the puts, in $? (A loss is negative.)
Problem 4Last year's VaR
L2.5.07A market risk analyst
The regulator asks for historical VaR on the most recent 250 days only (days 251 to 500). The ten worst fund losses in that window are:
| Rank | Day | Loss |
|---|---|---|
| 1 | 388 | $412,600 |
| 2 | 394 | $319,600 |
| 3 | 385 | $232,064 |
| 4 | 386 | $184,550 |
| 5 | 371 | $111,226 |
| 6 | 433 | $110,276 |
| 7 | 275 | $109,114 |
| 8 | 404 | $108,596 |
| 9 | 334 | $101,420 |
| 10 | 253 | $99,514 |
Use the convention of this module: the k-th worst loss, k = n(1 − α) rounded up.
What is the one-day 99% historical VaR on the last 250 days, in $?
Problem 5The tail behind the VaR
L2.5.07 · L2.5.08The same analyst, a day later
Using the same ten worst losses from the last 250 days, the risk committee now wants the one-day 97.5% expected shortfall, the measure the FRTB uses.
What is the 97.5% historical ES on the last 250 days, in $?
Problem 6Where is the breaking point?
L2.5.04 · L2.5.09A risk manager preparing a board paper
Without the puts, the board says a loss of $1,500,000 would force the fund to close. Your scenario team expects that in a sell-off driven by inflation, bond yields would rise 50 bp for every 10% the index falls. Use the equity fund's beta of 1 and the bond fund's DV01 of $2,400.82.
How far must the index fall, in %, to reach the breaking point?
Problem 7What should the board hear?
L2.5.02 · L2.5.06 · L2.5.08 · L2.5.09The chief risk officer
You have the module's figures for the $10,000,000 fund: daily returns with skewness −0.62 and excess kurtosis 5.68; normal one-day 99% VaR $148,044; historical 99% ES $279,443; historical 95% VaR $85,204 against normal $104,675; and a 30% index crash scenario.
Which statement is right?
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