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Estimate portfolio volatility by applying key rate analysis together with principal components analysis.

FRM® Part I · Valuation and Risk Models · Modeling Non-Parallel Term Structure Shifts and Hedging · FRM1.VR.13.e

The idea

Key rate analysis and PCA combine to give portfolio volatility. First express each principal component as the basis point move it causes at every key rate for a one standard deviation move in that factor. The portfolio's exposure to a factor is the sum over key rates of KR01 × that factor's move. Because the factors are uncorrelated, no covariance terms are needed. Portfolio variance equals the sum of the squared exposures, and volatility is the square root of that sum. If the factor loadings are given as unit length eigenvectors, multiply each exposure by that factor's own standard deviation before squaring. Multiply the daily volatility by a z value and the square root of the horizon in days to turn it into VaR. A portfolio with zero DV01 is not riskless, since it can carry large slope or curvature exposure. Using only the level factor understates risk.

Formulas

Portfolio exposure to a principal component

Eᵢ = Σ KR01 × mᵢ

KR01
the portfolio's KR01 at each key rate, the gain from a one basis point fall
mᵢ
the basis point move factor i causes at that key rate for a one standard deviation move

Summed across the key rates. If the loadings are unit eigenvectors rather than one standard deviation moves, multiply the result by the factor's standard deviation.

Portfolio volatility from principal component exposures

σP = √(Σ Eᵢ²)

Eᵢ
the exposure to factor i, scaled to a one standard deviation move

Works because the factors are uncorrelated. Adding absolute exposures assumes perfect correlation and overstates risk; reporting the sum of squares without the root gives the variance.

A practice question

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

A portfolio has KR01s (gain from a one basis point fall, USD thousand) of 5, 10 and -8 at the 2-, 5- and 10-year key rates. A one standard deviation rise in the level factor moves these key rates by +7.5, +8.0 and +7.0 basis points. What is the portfolio's change in value, in USD thousand?

  1. +61.5Why not: Treats KR01 as the gain from a rise in rates rather than a fall, so the sign is wrong.
  2. -61.5 - correct
  3. -52.5Why not: Multiplies the 2-year move by the net KR01 of 7 (5 + 10 - 8) instead of using each key rate's own move.
  4. -172.5Why not: Adds the absolute values of the KR01s (23) and applies the 2-year move to the sum.

Value change = -(5 x 7.5 + 10 x 8.0 + (-8) x 7.0) = -(37.5 + 80.0 - 56.0) = -61.5. KR01 is the gain per basis point fall, so a rise gives a loss, and each KR01 is multiplied by its own key rate move.

A harder one

Using the factor table of key rate moves in basis points for a one standard deviation move in each factor (key rates 2, 5 and 10 years; Level: +7.5, +8.0, +7.0; slope: -3.0, +0.5, +3.0; curvature: +1.5, -2.0, +1.0), what is the volatility of a portfolio with KR01s of -4, 12 and 3 (USD thousand per basis point fall)? Assume the factors are uncorrelated with unit variance in this scaling.

  1. USD 141.0 thousandWhy not: Sums the absolute factor exposures, which assumes the factors are perfectly correlated.
  2. USD 9,027 thousandWhy not: Reports the variance (sum of squared exposures) without taking the square root.
  3. USD 95.0 thousand - correct
  4. USD 87.0 thousandWhy not: Uses only the level factor, which understates volatility because slope and curvature exposures also add to variance.

Exposures: level = -4 x 7.5 + 12 x 8.0 + 3 x 7.0 = 87.0; slope = -4 x -3.0 + 12 x 0.5 + 3 x 3.0 = 27.0; curvature = -4 x 1.5 + 12 x -2.0 + 3 x 1.0 = -27.0. Volatility = sqrt(87² + 27² + -27²) = 95.0.

The hardest tier

The covariance matrix of daily changes in the 2-, 5- and 10-year key rates has three eigenvectors (loadings across the three key rates): level (+0.55, +0.60, +0.58), slope (-0.71, -0.03, +0.71) and curvature (+0.44, -0.80, +0.41). The daily standard deviations of the three factors are 6.0, 2.5 and 1.0 basis points. A portfolio has KR01s of 12, -5 and 9 (USD thousand per basis point fall) at the three key rates. What is the portfolio's daily volatility, in USD thousand?

  1. 15.8Why not: Omits the factor standard deviations, treating each factor as a one basis point move.
  2. 25.4Why not: Multiplies squared exposures by the standard deviations rather than by the variances.
  3. 54.7 - correct
  4. 70.8Why not: Adds the absolute factor risks, assuming the factors are perfectly correlated instead of uncorrelated.

Exposures per unit of factor: 8.82, -1.98, 12.97. Multiply by factor standard deviations: 52.92, 4.95, 12.97. Volatility = square root of the sum of squares = 54.7. Variances, not standard deviations, add.

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