Which three numbers define any VaR?

"Our VaR is $2 million" is not information. It's a number missing its units. Every VaR figure carries three parameters, and changing any one of them changes the answer — sometimes by a factor of five. A practitioner never quotes VaR without all three.

Parameter 1 — the time horizon

Over what period is the loss measured? One day is standard for trading desks, where positions turn over fast. Ten days is the classic regulatory horizon for market risk. One month or a quarter suits asset managers who cannot reposition quickly.

Horizons are often scaled rather than recomputed, using the square root of time:

10-day VaR ≈ 1-day VaR × √10 ≈ 1-day VaR × 3.16

So a $200,000 one-day VaR implies roughly a $632,000 ten-day VaR. Convenient — and resting on an assumption you should hold lightly: that daily returns are independent and identically distributed. Real markets cluster: calm follows calm, chaos follows chaos. So the scaling inherits whichever regime the one-day figure was measured in. Scale up a one-day number taken from a calm stretch and the ten-day figure is too small, because the next ten days can leave the regime the estimate came from. Scale one taken mid-crisis and it is too large, because volatility reverts. The direction of the error is not fixed; what is fixed is that √10 is doing work the data has not verified.

Parameter 2 — the confidence level

How rare is the "bad day" you're asking about? 95% means one day in twenty. 99% means one day in a hundred — about 2.5 days a year. 99.9% means once in four years of trading days.

Under a normal-distribution assumption the multipliers are fixed:

• 95% → 1.645 standard deviations
• 97.5% → 1.960
• 99% → 2.326

So moving from 95% to 99% multiplies the number by 2.326 ÷ 1.645 ≈ 1.41. Anyone comparing two firms' VaR figures without checking the confidence level is comparing nothing.

Parameter 3 — the unit

Currency amount or percentage of portfolio value? "$2 million" tells a risk committee something immediate; "1.4% of assets" is what you need to compare a $140 million fund with a $14 billion one. Both are used; mixing them silently is a classic reporting error.

There is a third unit, and for an asset owner it is the one that matters. A pension fund or an insurer holds assets against liabilities, and the liabilities are valued with the discount rates that move its bonds. The position a VaR should describe is assets minus liabilities, the surplus. On an asset-only VaR a 20-year bond is the riskiest holding; on a surplus VaR it is the hedge, because its price and the liability move together. An asset-only figure reported to a pension board is precise and describes the wrong thing. The horizon follows: an owner that cannot reposition within a year has no use for a one-day figure scaled by the square root of anything.

The fourth parameter nobody labels

There is a hidden dial: the data window and the method. A VaR built on the last 250 days of a calm market and a VaR built on 2,000 days that include a crash will differ enormously on the same portfolio. So will a historical-percentile VaR and a normal-distribution VaR.

This dial is not usually printed on the report, which is exactly why it deserves your attention. When a firm's VaR falls, ask first whether the portfolio got safer or whether the window got calmer.

A worked comparison

One portfolio, one day, four quotes — all correct, all different:

Same risk. A 4.5× spread in the headline. This is why the full sentence matters.

Try it now

1. Take the 95% and 99% one-day figures you counted off the chart in the previous lesson — the same year is below if you want to redo them. Divide the 99% figure by the 95% figure.

2. Compare that ratio to the theoretical 1.41 you would get under a normal distribution. Bigger means the real tail is heavier than the bell curve implies — the subject of Unit 2. Change the third dial as well: switch the chart to Weekly and count again. A weekly VaR is not a daily VaR scaled by anything you can read off a page.
3. Scale your 1-day figure to 10 days with √10 and write down one reason that scaling might be too optimistic. No verdict on whether the level is acceptable — that judgement belongs to whoever owns the money.