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Explain why mapping reflects market-wide as well as position-specific risk, and quantify each for a portfolio given the underlying primitive risk factors.

FRM® Part II · Market Risk Measurement and Management · VaR Mapping · FRM2.MR.5.b

The idea

Mapping shows two layers of risk. Market-wide (general) risk is the part of a position's variance driven by the common risk factors it is mapped to. For a stock mapped to an index, its systematic volatility is β times the index volatility. Position-specific risk is the residual, idiosyncratic part left once the factor effect is removed. Write σM for the index volatility and σR for the residual volatility. For one stock, the total variance is Variance = β² × σM² + σR². With z the normal quantile and V the position value, the market-wide part, general VaR, is GVaR = z × β × σM × V. Specific VaR uses only the residual volatility, with no beta exposure at all, and is SVaR = z × σR × V. Beta exposures add linearly across positions, but residuals are assumed uncorrelated and combine as a square root of summed squares, so specific risk diversifies away in large portfolios. Never add the two: they are assumed uncorrelated. Total VaR combines the general and specific parts as VaR = √(GVaR² + SVaR²). Exam point: hedging beta removes general risk but leaves specific risk.

Formulas

Total variance of a stock under an index mapping

Variance = β² × σM² + σR²

β
the stock's beta to the index
σM
the index volatility
σR
the residual (specific) volatility

Residuals uncorrelated with the index. Variances add, volatilities do not; using β instead of β² is a trap.

General (market-wide) VaR of a position

GVaR = z × β × σM × V

z
the standard normal quantile
β
the beta to the index
σM
the index volatility over the horizon
V
the position value

Beta exposures add linearly across positions, so a futures hedge subtracts directly from β × V.

Specific VaR of a position

SVaR = z × σR × V

z
the standard normal quantile
σR
the residual volatility over the horizon
V
the position value

No beta enters. Across positions residuals are uncorrelated, so specific VaRs combine as a square root of summed squares.

Total VaR from general and specific VaR

VaR = √(GVaR² + SVaR²)

GVaR
the general (market-wide) VaR
SVaR
the specific VaR

The two parts are assumed uncorrelated. Adding them directly overstates VaR.

A practice question

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

In a single-index mapping, position-specific risk is best described as:

  1. beta multiplied by the index volatility and the position value, which is the market-wide component.Why not: Describes general (market-wide) risk, not the residual left after the factor effect is removed.
  2. the risk that every position loses value together in a market crisis when correlations rise.Why not: Describes systemic correlation risk; specific risk is the idiosyncratic residual.
  3. the part of a position's variance not explained by its exposure to the index, which diversifies away across many positions. - correct
  4. the difference between the 99 percent and 95 percent VaR of the position.Why not: Confuses specific risk with tail risk from a higher confidence level; it is a source of variance, not a confidence-level gap.

Total variance of a position equals beta² x index variance plus residual variance. The residual part is specific risk; because residuals are uncorrelated across positions it diversifies, whereas the beta-driven part does not.

A harder one

A USD 20 million position has a beta of 1.2 to a market index whose annual volatility is 16 percent. The one-month 95 percent market-wide (general) VaR, scaling by the square root of 12, is closest to:

  1. USD 1.52 millionWhy not: Leaves out beta and treats the position as having index-like risk.
  2. USD 1.82 million - correct
  3. USD 6.32 millionWhy not: Uses annual volatility without scaling it to one month.
  4. USD 2.58 millionWhy not: Uses the 99 percent quantile 2.326 instead of the 95 percent quantile 1.645.

Monthly index volatility = 16% / sqrt(12) = 4.62%. General VaR = 1.645 x 1.2 x 4.62% x 20m = 1.82m. The beta scales the exposure to the market-wide factor.

The hardest tier

A USD 20 million portfolio holds USD 10 million of stock X (beta 0.8, residual volatility 20 percent a year) and USD 10 million of stock Y (beta 1.4, residual volatility 25 percent a year). The residuals are uncorrelated with each other and with the index, whose annual volatility is 15 percent. The one-month 95 percent VaR, combining general and specific risk correctly, is closest to:

  1. USD 1.57 millionWhy not: Counts only the market-wide component and omits the residual risk.
  2. USD 3.09 millionWhy not: Adds general and specific VaR directly instead of combining them as a square root of summed squares.
  3. USD 2.18 million - correct
  4. USD 2.65 millionWhy not: Adds the two residual risks linearly, assuming perfectly correlated residuals, before combining with general risk.

Portfolio beta exposure = 10 x 0.8 + 10 x 1.4 = 22m, so general VaR = 1.645 x 22 x 15%/sqrt(12) = 1.57m. Specific VaR = 1.645 x sqrt(2² + 2.5²)/sqrt(12) = 1.52m. Total = sqrt(1.57² + 1.52²) = 2.18m.

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