Read a Value at Risk figure and explain why expected shortfall tells you more about the bad days.
Marcus manages his own portfolio of about S$202,000, an example figure. He wants something more useful than "my portfolio is volatile": a figure he can plan around, such as "in a bad month I could lose about S$13,000."
Value at Risk (VaR) gives you that kind of figure. Banks report it to their regulators every day, and fund risk reports quote it too. It turns a vague statement about volatility into a money amount someone can plan around. It also has a major blind spot, one that many people suffered from in 2008, so you need to know both how to calculate it and what it leaves out.
VaR is the loss you would not expect to exceed on a given share of days or months, over a given period. A one-month 95% VaR of S$13,000 means that in 95 months out of 100 you'd expect to lose less than S$13,000 or make a gain. In the other 5 months you'd expect to lose more than S$13,000.
Three choices define any VaR figure: the confidence level (usually 95% or 99%), the period (such as one day or one month) and the method used to calculate it. Check all three before comparing two VaR figures.
For a personal account, the simplest method is the historical method. You take your actual past returns, sort them from worst to best, and read off the point where the worst 5% begins (for 95% VaR).
Marcus takes 60 monthly returns, which covers five years. 5% of 60 is 3, so his 95% VaR is the third-worst month. His five worst months, sorted, are -11.2%, -8.0%, -6.4%, -5.1% and -4.8%. The third-worst is -6.4%, so his one-month 95% VaR is 6.4%. On about S$202,000, 6.4% is about S$12,900, which is where his rough S$13,000 came from.
The steps for your own portfolio are:
Collect at least 60 monthly returns. Sort them from worst to best. Work out 5% of the number of months. Read off that month as your VaR percentage. Multiply by your portfolio value to get a money figure.
Spreadsheet percentile functions such as PERCENTILE interpolate between neighbouring months, so they can give a somewhat different figure from counting. Neither approach is wrong. Pick one, write down which you used, and keep using it so your figures stay comparable over time.
VaR tells you where the bad months begin. It says nothing about how bad they get beyond that line. Two portfolios can have identical VaR even if one's worst 5% of months all lose about 7% and the other's worst 5% include a month of minus 30%. The losses that do the most damage sit beyond the VaR line, in the region VaR ignores. Before the 2008 crisis, many banks reported modest VaR figures based on calm years, and the losses they then took were many times larger.
Expected shortfall, also called conditional VaR, measures what happens past the line. It tells you how bad a month is on average once it crosses the VaR line. You calculate it by averaging the losses at or beyond the VaR line, then multiplying by your portfolio value. The Basel Committee's market risk rules for banks, known as the Fundamental Review of the Trading Book, replaced VaR with expected shortfall as the main measure for market risk capital.
Marcus averages his three worst months, -11.2%, -8.0% and -6.4%, and gets about 8.5%. On his portfolio that is about S$17,300. He can now sum up his risk in one sentence: "In about one month in twenty, I should expect to lose at least S$12,900, and when that happens, the average loss is about S$17,300."
A quicker approach is the parametric method. It assumes returns follow a bell curve (a normal distribution) and estimates 95% VaR as the average monthly return minus 1.645 standard deviations, since 1.645 standard deviations is the cut-off for the worst 5% of a normal distribution. You then multiply by portfolio value.
Marcus's average monthly return is 0.5% and his standard deviation is 3.3%. 0.5 minus 1.645 times 3.3 gives about -4.9%, or about S$10,000, lower than his historical VaR of 6.4%. Under a bell curve, his worst month of -11.2% would be a once-in-many-decades event. Real returns have fat tails, meaning more extreme outcomes than a bell curve predicts, as Lesson 2.1, "Volatility: what standard deviation captures and what it misses", showed. A bell curve model understates risk exactly where it matters most, which is why the historical method is worth the effort.
Historical VaR and expected shortfall are built entirely from the months in the window you choose. Five calm years produce small numbers. The next crisis is not in the data yet, and the method gives no warning of this.
The first habit is to use the longest history available, ideally including at least one serious fall such as 2008 or early 2020. For holdings you haven't owned that long, an index that tracks them is a reasonable stand-in.
The second habit is to treat VaR and expected shortfall as a floor for what a bad month looks like, never a ceiling. The complementary step is stress testing, where you deliberately run your portfolio through scenarios worse than anything in your data. Lesson 11.6, "Stress test against past crises and made-up shocks", covers that other half of the approach.
For the activity, take at least 60 monthly returns for your portfolio, from your returns tab or from index data for your holdings, sort them, and find the third-worst month and the average of the worst three.
Using at least five years of data, calculate your portfolio's monthly historical Value at Risk and expected shortfall at 95%.
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