Value at Risk

While the Greek letters view portfolio risk from multiple angles and generate numerous risk measures, the VaR (Value at Risk) metric is an attempt to express portfolio risk in a single parameter. It answers a question roughly like this:

With 99% confidence, what is the minimum loss the portfolio could incur over the next 10 days?

If we assume that the portfolio’s expected return is normally distributed, there is a threshold on the left tail of the distribution beyond which the worst X% of outcomes occur. VaR shows the loss at precisely that threshold.

However, because events beyond that threshold may unfold differently across portfolios, another parameter was introduced: Expected Shortfall (ES). Put simply, if VaR tells us how bad things could get, ES tells us the expected loss if things do get that bad.

In the diagrams below, two portfolios are shown with the same VaR but different expected losses (one has a fatter tail).

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Historical Simulation of Risk Measures

Risk measures can be derived either through historical analysis or through modeling. Below is an example of the historical simulation approach:

Let’s assume we have a portfolio worth approximately $200,000.

We take three years of historical stock price data, which gives us more than 700 historical scenarios.

Accordingly, we can calculate the portfolio’s daily returns, which produces a frequency distribution of returns.

If we then sort these scenarios, we can identify the threshold corresponding to our confidence level of interest.

In the upper right part of the table, the 1-day risk is shown. The worst 1% of cases intersect at $6,279, while the average loss in those worst cases is $8,402.

In the lower right part, the 1-day risk is transformed into a 10-day risk. As time increases, risk increases as well, using the following formula:

N-day VaR = 1-day VaR × √N

It is interesting to ask:
$26,000 / $200,000 = 13% — how risky is that parameter?
It depends on the type of portfolio:

Copilot:

Portfolio	Typical 10-day VaR Range	Is 13% High?
Equity index portfolio	3–8%	High
Single-name equities / concentrated book	8–20%	Normal
Options / volatility trading	15–40%	Low–moderate
Crypto	20–60%	Low
Fixed income	1–4%	Very high

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P.S.
VaR – Historical Simulation – Excel

Adapted from:
Options, Futures & Other Derivatives, John C. Hull

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