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Walk through how a risk budget is put together, and compute the allocations of risk to each asset class and to each active manager.

FRM® Part II · Risk Management and Investment Management · VaR and Risk Budgeting in Investment Management · FRM2.IM.6.h

The idea

A risk budget splits total fund risk into components. Total risk is the fund's VaR at the chosen confidence level: VaR = z × σp × fund value. For asset class i, beta is measured against the fund itself: βi = cov(Ri, Rp) ÷ var(Rp). Asset class i's share of total fund risk is its weight times that beta: Risk share = wi × βi. The shares sum to 100 percent, so component VaRs, each share times total VaR, add back to the fund's VaR. Stand-alone VaR, z × σi × wi × fund value, ignores diversification and sums to more than the total unless every correlation is 1. For active managers with uncorrelated active returns, fund tracking error is the square root of the summed squares: TE = √[Σ (wi × TEi)²]. Manager i's share of active risk is its squared term divided by that sum: Manager share = (wi × TEi)² ÷ Σ (wi × TEi)². Common mistakes are to use weights as risk shares, to sum stand-alone figures, and to forget the square root. Exam point: compute the covariance with the portfolio, then multiply by weight.

Formulas

Total VaR of a fund

VaR = z × σp × fund value

z
the normal quantile for the confidence level
σp
the fund's return volatility
fund value
the market value of the fund

Normal returns at the chosen confidence level. Find σp from the full covariance matrix, not a weighted average of volatilities.

Beta of an asset class against the fund

βᵢ = cov(Rᵢ, Rp) ÷ var(Rp)

cov(Rᵢ, Rp)
the covariance of class i's return with the fund's return
var(Rp)
the variance of the fund's return

Measured against the fund itself, not a benchmark index. Compute the covariance with the portfolio first, then multiply by weight to get the risk share.

Asset class share of total fund risk

Risk share = wᵢ × βᵢ

wᵢ
the weight of class i in the fund
βᵢ
the beta of class i against the fund

The shares sum to 100 percent, so share × total VaR gives component VaRs that add up. Using the weight alone as the risk share is the classic mistake.

Fund tracking error from uncorrelated managers

TE = √[Σ (wᵢ × TEᵢ)²]

wᵢ
the share of the fund given to manager i
TEᵢ
manager i's tracking error against its benchmark

Only for uncorrelated active returns. Taking the weighted average of tracking errors, or forgetting the square root, overstates or misstates the result.

Manager's share of fund active risk

Manager share = (wᵢ × TEᵢ)² ÷ Σ (wᵢ × TEᵢ)²

wᵢ
the share of the fund given to manager i
TEᵢ
manager i's tracking error against its benchmark

Uncorrelated active returns. Using unsquared weighted tracking errors treats risks as additive and understates the share of a high tracking-error manager.

A practice question

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

Two managers have active returns that are uncorrelated. Their contributions to fund-level active risk, stated as standard deviations of the fund's active return, are 3.0 percent and 4.0 percent. The fund's tracking error is:

  1. 7.0 percentWhy not: Adds the tracking errors, which assumes perfect correlation.
  2. 3.5 percentWhy not: Averages the two figures, ignoring that risks accumulate.
  3. 5.0 percent - correct
  4. 25.0 percentWhy not: Adds the variances 9 + 16 and forgets to take the square root.

Uncorrelated risks add in variance: sqrt(3² + 4²) = sqrt(25) = 5.0 percent.

A harder one

A USD 1,000 million fund holds 60 percent equities with volatility of 16 percent and 40 percent bonds with volatility of 5 percent, with a correlation of 0.2. The equities' share of total fund risk (component VaR as a share of total VaR) is closest to:

  1. 92.4 percent - correct
  2. 88.8 percentWhy not: Uses only the equity variance term over total variance and omits its covariance with bonds.
  3. 82.8 percentWhy not: Uses the stand-alone volatility shares, w x sigma, which ignore correlation.
  4. 60.0 percentWhy not: Equates the risk share with the 60 percent capital weight.

Portfolio variance = 0.010384 (vol 10.19 percent); cov(equity, portfolio) = 0.016; share = 0.6 x 0.016 / 0.010384 = 92.4 percent. Bonds supply the remaining 7.6 percent.

The hardest tier

A fund allocates 40 percent, 40 percent and 20 percent to three active managers with tracking errors of 3.0, 2.0 and 6.0 percent against their benchmarks. Active returns are uncorrelated. The share of the fund's total active risk contributed by the third manager (its component tracking error divided by the fund tracking error) is closest to:

  1. 37.5 percentWhy not: Uses each manager's weighted tracking error as a share of the sum, which treats risks as additive.
  2. 54.5 percentWhy not: Uses the third manager's stand-alone tracking error as a share of the three added together, ignoring weights.
  3. 40.9 percent - correct
  4. 20.0 percentWhy not: Uses the capital weight of the third manager as its risk share.

With uncorrelated active returns, component TE_i = (w_i x TE_i)² / TE_fund, so its share is (0.2 x 6)² / [(1.2)² + (0.8)² + (1.2)²] = 1.44 / 3.52 = 40.9 percent.

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