Charta is on Google Play: practise on the go. Get the app
Charta

Compute VaR with a parametric method, covering both normally distributed and lognormally distributed asset prices.

FRM® Part II · Market Risk Measurement and Management · Estimating Market Risk Measures: An Introduction and Overview · FRM2.MR.1.b

The idea

Parametric VaR assumes a distribution for profit and loss (P&L) and plugs in estimated parameters. The standard normal quantile z is 1.645 at 95 percent and 2.326 at 99 percent. If P&L is normal with mean μ and standard deviation σ, then VaR = −μ + z × σ. A positive mean therefore reduces VaR. To scale to a horizon of T days, assuming independent daily returns, use T-day values: the daily σ times the square root of T and the daily μ times T. If instead the price is lognormal, so geometric (log) returns over the horizon are normal with mean μ and standard deviation σ, then VaR = value × (1 − e^(μ − z × σ)). Lognormal VaR can never exceed the position value. Matched comparison only: with the same μ and σ and a positive move z × σ − μ, it is smaller than the normal approximation, value × (z × σ − μ), as the exponential falls short of the linear. Typical errors are the wrong sign on the mean and scaling volatility linearly with time. Exam point: match the formula to the distribution.

Formulas

Parametric VaR with normal P&L

VaR = −μ + z × σ

μ
the mean P&L over the horizon, in money
z
the standard normal quantile, 1.645 at 95% and 2.326 at 99%
σ
the standard deviation of P&L over the horizon, in money

P&L is normal. A positive mean lowers VaR; adding it instead of subtracting it is the usual sign error.

Normal VaR over a horizon of T days

VaR = −μ × T + z × σ × √T

μ
the mean daily P&L
T
the horizon in days
z
the standard normal quantile
σ
the standard deviation of daily P&L

Daily returns are independent, so the mean scales with T and the standard deviation with √T. Scaling σ linearly with time overstates VaR.

Parametric VaR with a lognormal price

VaR = V × (1 − e^(μ − z × σ))

V
the position value
μ
the mean log return over the horizon
z
the standard normal quantile
σ
the standard deviation of the log return over the horizon

Log returns are normal, so the loss can never exceed V. It is smaller than the normal approximation V × (z × σ − μ) whenever z × σ − μ is positive.

A practice question

From the question bank, with the reason each wrong option is wrong.

Daily P&L of a portfolio is normally distributed with mean USD 0.5 million and standard deviation USD 4 million. The one-day 99 percent VaR is closest to:

  1. USD 9.81 millionWhy not: Adds the mean instead of subtracting it; a positive expected profit reduces VaR.
  2. USD 9.31 millionWhy not: Ignores the mean P&L entirely and uses z x sigma only.
  3. USD 8.81 million - correct
  4. USD 6.08 millionWhy not: Uses the 95 percent quantile of 1.645 instead of the 99 percent quantile of 2.326.

For normal P&L, VaR = -mean + z x sigma = -0.5 + 2.326 x 4 = USD 8.81 million.

A harder one

A portfolio worth USD 100 million has monthly geometric returns that are normally distributed with mean 1.0 percent and standard deviation 6.0 percent, so the price is lognormal. The one-month 95 percent VaR is closest to:

  1. USD 8.49 million - correct
  2. USD 8.87 millionWhy not: Applies the normal P&L formula (z x sigma - mu) to the value, ignoring the exponential in the lognormal VaR formula.
  3. USD 9.40 millionWhy not: Drops the mean return from the exponent of the lognormal formula.
  4. USD 10.30 millionWhy not: Uses the wrong sign on the mean return inside the exponent, treating the drift as a loss.

Lognormal VaR = value x (1 - exp(mu - z sigma)) = 100 x (1 - exp(0.01 - 1.645 x 0.06)) = 100 x (1 - 0.9151) = USD 8.49 million.

The hardest tier

A portfolio holds USD 10 million of asset A and USD 6 million of asset B. Daily returns are jointly normal with zero means, daily volatilities of 1.5 percent for A and 2.0 percent for B, and correlation 0.3. The one-day 99 percent VaR of the portfolio is closest to:

  1. USD 0.63 millionWhy not: Adds the stand-alone VaRs, which assumes a correlation of one and ignores diversification.
  2. USD 0.45 millionWhy not: Sets the correlation to zero, omitting the covariance term in the portfolio variance.
  3. USD 0.51 million - correct
  4. USD 0.36 millionWhy not: Uses the 95 percent quantile of 1.645 where 99 percent requires 2.326.

Dollar volatilities are 0.15 and 0.12 million. Portfolio sigma = sqrt(0.0225 + 0.0144 + 2 x 0.3 x 0.15 x 0.12) = 0.2184; VaR = 2.326 x 0.2184 = USD 0.51 million.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Estimating Market Risk Measures: An Introduction and Overview