Value at risk: the idea · RiskAtoms Academy

Math Atoms

By the end you will be able to

1State what a VaR figure means, with its confidence level, horizon and currency.
2Compute parametric VaR as z × σ × value, and explain where each part comes from.
3Scale VaR to a longer horizon with the square-root-of-time rule.
4Say what VaR does not tell you, and when the normal version misleads.

Why it matters

“Our one-day 99% value at risk is $148,044.” That sentence fits a whole portfolio, every asset and every correlation, into one number a board can read. It is the most quoted figure in risk management, and one of the most misread.

Core explanation

One number for a bad day

Volatility (L2.5.01: Measuring volatility) tells you the size of a typical day. Scenarios (L2.5.05: Scenario analysis) tell you what happens if one particular thing goes wrong. Value at risk sits in between. It answers: how bad is a bad day, where “bad” is defined by probability?

What a VaR figure says

“The fund's one-day 99% VaR is $148,044” means: over the next trading day, the loss should exceed $148,044 with probability 1%, about one day in a hundred. Every VaR figure needs three labels: a confidence level (99%), a horizon (one day) and a currency (US dollars). In this course VaR is always quoted as a positive number: the size of the loss.

Picture the distribution of the fund's possible P&L tomorrow (L2.5.02: Return distributions). Cut off the worst 1% of it. The cut point is the 1% quantile of P&L (M-C09: Quantiles). Change its sign and you have the 99% VaR. That is all VaR is. The hard part is knowing the distribution. There are three main ways to get it:

Method	Where the distribution comes from	In this course
Parametric	Assume a normal distribution with the portfolio's volatility	This lesson
Historical	Replay the P&L the current positions would have made on past days	L2.5.07: Historical VaR
Monte Carlo	Simulate thousands of possible days from a model	M5.2: Market risk modelling

The parametric version. If daily P&L is normal with mean zero and standard deviation σ × value, then the 99% VaR is 2.326 standard deviations: the 99% point of the standard normal (M-C06: The normal distribution). The fund's daily dollar volatility is $63,638 (L2.5.03: Correlation and diversification), so its 99% VaR is 2.326 × $63,638 = $148,044. Raise the confidence and the VaR grows; it never reaches a limit, because the normal tail never ends:

Confidence	z	1-day VaR	10-day VaR	Exceeded about
90.0%	1.2816	$81,555	$257,901	1 day in 10
95.0%	1.6449	$104,675	$331,012	1 day in 20
97.5%	1.9600	$124,728	$394,426	1 day in 40
99.0%	2.3263	$148,044	$468,157	1 day in 100
99.5%	2.5758	$163,921	$518,363	1 day in 200
99.9%	3.0902	$196,656	$621,882	1 day in 1,000

Your callRead the VaR statement

0 of 3

“The most the fund can lose tomorrow is $148,044.”
“On roughly 1 trading day in 100 we should expect to lose more than $148,044.”
“When things go badly, on the worst 1% of days, we lose $148,044 on average.”

What VaR leaves out. It says nothing about how big the loss is on the bad days beyond the cut. Two portfolios can share a VaR while one of them hides a catastrophic tail. And the parametric version trusts the normal curve, which, as L2.5.02: Return distributions showed, is too thin in the tails of real returns.

The formal version

For a P&L over horizon $h$ , the VaR at confidence $\alpha$ is the loss that is exceeded with probability $1-\alpha$ :

P\big(\text{P\&L}_h < -\text{VaR}_{\alpha,h}\big) = 1 - \alpha \quad\Longleftrightarrow\quad \text{VaR}_{\alpha,h} = -\,q_{1-\alpha}\big(\text{P\&L}_h\big)

where $q_{1-\alpha}$ is the $(1-\alpha)$ quantile. Under the normal assumption with zero mean this becomes:

\text{VaR}_{\alpha,h} = z_{\alpha} \times \sigma \times \sqrt{h} \times V

with $z_{\alpha}$ the standard normal quantile (1.645 at 95%, 2.326 at 99%), $\sigma$ the daily volatility of returns and $V$ the portfolio value.

The mean. Over a short horizon the expected return is small next to the volatility, so this course sets it to zero, which gives a slightly larger (more prudent) VaR. Including it gives $\text{VaR} = (z_{\alpha}\sigma\sqrt{h} - \mu h)\,V$ . For the fund, the average daily return over the 500 days is 0.0395%, or $3,953 a day: the 99% one-day VaR would fall from $148,044 to $144,091. Over 10 days the mean term grows to $39,530, because the mean scales with $h$ and the volatility only with $\sqrt h$ . Whichever convention is used, it must be stated and used consistently.

The square-root-of-time rule, and when it fails

Multiplying one-day VaR by

\sqrt{h}

(L2.5.01: Measuring volatility) assumes the days are independent, drawn from the same distribution, with no trend and with the positions held unchanged. It fails when volatility clusters (a calm week followed by a storm), when returns trend or reverse, and when the portfolio is traded during the horizon. Fat tails also fade over longer horizons, so √h scaling of a fat-tailed one-day figure is not exact either. It is a convention, not a law.

Try it

The explorer opens on the fund. Move the confidence from 95% to 99.9% and watch the shaded tail shrink while the VaR grows. Stretch the horizon to 10 days: the VaR grows by √10, not by 10. Then change the volatility and the value, and check that VaR is proportional to both.

Try itParametric VaR explorer

Confidence level99.0%

Horizon1 day

Portfolio value$10m

Daily volatility0.64%

99.0% 1-day VaR

$148,886

z2.3263

σ × √h × value$64,000

√h1.000

Exceeded about1 in 100

On about one day in 100, the loss should be larger than this, if the normal model is right.

Worked example

The fund's parametric VaR

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.

Try each step before revealing it.

From volatility to VaR

Step 1 of 6

1
Daily dollar volatility
From L2.5.03: Correlation and diversification, the fund's daily volatility is 0.6364%. In dollars: $0.006364 \times \$10{,}000{,}000 = \$63{,}638$ .

Regulatory lens

VaR entered bank regulation with the Basel Committee's 1996 Market Risk Amendment. Banks allowed to use their own models computed a 99%, 10-day VaR (scaling up a one-day figure with √10 was allowed) on at least one year of data. The capital charge was the larger of yesterday's VaR and the average of the last 60 days' VaR times a multiplier of at least 3. Backtesting set the multiplier: over 250 days, 0 to 4 exceptions at 99% was the green zone, 5 to 9 yellow (a higher multiplier) and 10 or more red. The Fundamental Review of the Trading Book (FRTB, BCBS January 2016, revised January 2019) replaces 99% VaR with 97.5% expected shortfall (L2.5.08: Expected shortfall), though VaR remains in use for backtesting. The detail is in M4.3: Market risk capital (FRTB) and CVA, part of Level 4: Regulation, Capital & Accounting.

Sparks

SparkThe 4:15 report

In the late 1980s Dennis Weatherstone, chairman of J.P. Morgan, asked for a single page every day, soon after the market closed, saying how much the bank could lose over the next day. It became known as the “4:15 report”, after the time it landed. In October 1994 J.P. Morgan published the method and its data free of charge as RiskMetrics, and VaR spread across the industry.

SparkThe 1997 debate

In 1997 the magazine Derivatives Strategy ran a debate between Philippe Jorion, author of a standard text on VaR, and Nassim Nicholas Taleb, then an options trader. Taleb argued that VaR gives a false sense of security, because it is blind to the rare events that matter most. Jorion replied that a measured risk, with its limits understood, is better than none. Both sides still have followers.

Key takeaways

VaR is the loss exceeded with probability 1 − α over horizon h: a quantile of the P&L distribution, quoted as a positive amount.
Every VaR needs a confidence level, a horizon and a currency.
Parametric VaR = z × σ × √h × value, with z = 1.645 at 95% and 2.326 at 99% (mean set to zero here).
VaR of a portfolio is less than the sum of the parts when assets are not perfectly correlated.
VaR says nothing about how big losses are beyond it, and the normal version understates fat tails.

Reaction

Check what stuck

Question 1 · objective 2

A $2 million position has a daily volatility of 1.5%. Assuming normal returns with zero mean, what is its one-day 99% VaR, in dollars? (z at 99% is 2.3263.)

Question 2 · objective 3

A desk's one-day 99% VaR is $50,000. Using the square-root-of-time rule, what is its 10-day 99% VaR, in dollars?

Question 3 · objective 2

A €5 million portfolio has a daily volatility of 1.2%. What is its one-day 95% VaR, in euros? (z at 95% is 1.6449.)

€

Question 4 · objective 1

The fund's one-day 99% VaR is $148,044. Which statement is right?

AThe fund cannot lose more than $148,044 tomorrowBOn about one day in 100, the fund should lose more than $148,044COn its worst 1% of days the fund loses $148,044 on averageDThe fund will lose $148,044 on 99% of days

Question 5 · objective 4

When does the square-root-of-time rule give a poor 10-day VaR?

AWhen returns are independent from day to dayBWhen volatility clusters, returns trend, or the positions change during the 10 daysCWhen the confidence level is 99%DWhen the portfolio holds only bonds

Glossary

Value at risk (VaR): A loss level that is exceeded only with a small, chosen probability over a chosen horizon. Quoted as a positive amount of money.
Confidence level: The probability, such as 95% or 99%, that the loss stays at or below the VaR.
Horizon: The period over which the loss is measured, such as one day or ten days.
Quantile: The value below which a given share of a distribution lies. The 1% quantile of P&L is the VaR at 99%.
Parametric VaR: VaR computed from a formula for the distribution, usually the normal: z × σ × value. Also called variance-covariance or delta-normal VaR.
Exception: A day on which the actual loss is larger than the VaR. Also called a breach or exceedance.
Square-root-of-time rule: Scaling a one-day risk figure to h days by multiplying by √h. Valid only if days are independent and alike.

Deep Dives & Origins

Deep Dives

P. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, 3rd edition, 2007. Intermediate.
J.P. Morgan/Reuters, RiskMetrics: Technical Document, 4th edition, 1996. Advanced.

Origins

Basel Committee on Banking Supervision, Amendment to the capital accord to incorporate market risks, January 1996.
Basel Committee on Banking Supervision, Supervisory framework for the use of “backtesting” in conjunction with the internal models approach to market risk capital requirements, January 1996.
Basel Committee on Banking Supervision, Minimum capital requirements for market risk, January 2016, revised January 2019.
P. Jorion and N. N. Taleb, “The Jorion-Taleb debate”, Derivatives Strategy, April 1997.

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Scenario analysis

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Historical VaR