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FRM Part 2 · Worked study notes

Risk Management for Investment Portfolios

Portfolio risk depends on weights, factor exposures and co-movement. For an active manager, compare portfolio weights and returns with the benchmark, then measure tracking error and concentration. Small capital weights can still contribute substantial risk.

The method at a glance

Correlation changes diversification

Illustrative comparison · % volatility · bars start at zero

  • Correlation 014.14
  • Correlation 0.517.32
  • Correlation 120

Equal weights; both assets have 20% annual volatility. Variance = 0.02 + 0.02 × correlation.

  • 01Active weights

    How it works: Portfolio weight minus benchmark weight identifies the relative position.

    Watch for: Calling a benchmark underweight an absolute short position.

  • 02Tracking error

    How it works: Standard deviation of portfolio-minus-benchmark returns.

    Watch for: Subtracting benchmark volatility from portfolio volatility.

  • 03Risk contribution

    How it works: Examine covariance with the portfolio, leverage and factor concentration.

    Watch for: Assuming risk contribution equals capital weight.

Original worked example

Equal weights do not fix portfolio volatility

Two assets have 20% annual volatility and equal weights. Compare correlations of 0, 0.5 and 1 using the two-asset variance formula.

  1. 1

    Variance = 0.02 + 0.02 × correlation.

  2. 2

    At correlation 0: volatility = √0.02 = 14.14%.

  3. 3

    At 0.5: √0.03 = 17.32%; at 1: √0.04 = 20%.

The same capital allocation has different risk as co-movement changes. Stress correlation assumptions alongside weights.

Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.

Apply the method and check its limits

Before you start: Weights, covariance matrices and benchmark-relative returns.

Use active weights for tracking error

a = portfolio weights - benchmark weights; TE² = aᵀΣa

A portfolio weighted 60/40 against a 50/50 benchmark has active weights +0.10 and -0.10. With both asset volatilities 20% and zero correlation, active variance is 0.0008 and tracking error is 2.83%. The portfolio’s total volatility is about 14.42%. Different weight vectors produce different risk measures from the same covariance matrix.

Attribute risk, not only invested capital

Marginal volatility contributionᵢ = (Σw)ᵢ / σp; component contributionᵢ = wᵢ × marginal contributionᵢ

For nonzero portfolio volatility, component contributions sum to total volatility. An asset’s contribution depends on covariance with the portfolio, so equal capital weights do not guarantee equal risk contributions. A hedge can contribute negatively. Check the risk definition before interpreting a pie chart of portfolio weights as a chart of risk.

Benchmark exposure changes the question

A portfolio holding 8% in an asset against a 12% benchmark weight is 4 percentage points underweight, despite still owning the asset. Its active return depends on relative performance. For a two-position active allocation of +4% and -4%, a 10-point relative return difference contributes 0.04 × 0.10 = 0.4 percentage points before costs.

Work through the topic

Calculate benchmark-relative risk, distinguish it from total portfolio risk, and add drawdown and liquidity analysis before recommending a portfolio response.

Build active returns before calculating tracking error

Active return is portfolio return less the specified benchmark return for the same period. Tracking error is the standard deviation of that active-return series or a model-based forecast of it, depending on the context. It is not the average shortfall from the benchmark and is not calculated by subtracting two standalone volatility figures.

For an original four-month example, portfolio returns are 3%, -1%, 2% and 0%, while benchmark returns are 1%, 0%, 1% and 2%. Active returns are +2%, -1%, +1% and -2%, averaging zero. Using the sample standard deviation, squared deviations sum to 10 squared percentage points; divide by 3 and take the square root to obtain monthly tracking error of approximately 1.826%.

Multiplying by √12 gives approximately 6.325% annualized tracking error under assumptions supporting square-root-of-time scaling, including an appropriate absence of serial dependence. Four observations are deliberately used to make the arithmetic visible, not to provide a credible production estimate. A longer relevant sample or a forward-looking risk model is needed for a practical assessment.

Original monthly returns and benchmark differences

1
Portfolio and benchmark: 3% and 1%
Active return: +2 percentage points
2
Portfolio and benchmark: -1% and 0%
Active return: -1 percentage point
3
Portfolio and benchmark: 2% and 1%
Active return: +1 percentage point
4
Portfolio and benchmark: 0% and 2%
Active return: -2 percentage points

Trace active risk to portfolio decisions

A benchmark-relative portfolio can be analyzed using active weights: portfolio weights minus benchmark weights. Those differences can create sector, duration, currency, style or security-specific exposures. A small net active weight can conceal large offsetting positions in different securities. Review the economic factors behind the active positions, not only the total amount invested.

Suppose a manager is 10 percentage points overweight a long-duration bond sector and 10 points underweight a short-duration sector. Total bond weight may match the benchmark while rate sensitivity differs materially. A parallel rate move and a curve twist can produce different active results. Use the mapped tenor exposures to explain the risk rather than assuming that equal asset-class weights imply benchmark neutrality.

Risk contributions depend on covariance and the chosen model. A position that diversifies one portfolio can add concentration to another. Compare the estimated contribution with scenario outcomes and the investment rationale. A portfolio change should be evaluated after transaction costs, liquidity effects and constraints, not approved solely because a model reports lower tracking error.

Add path-dependent losses and recovery needs

Volatility and tracking error summarize variation, but investors also experience the path of wealth. For an original portfolio-value path of 100, 120, 90 and 108, the maximum peak-to-trough drawdown in the observed path is (120 - 90)/120 = 25%. The ending value is above the initial 100, yet the investor experienced a substantial decline from the intervening peak.

Recovering from 90 to the prior peak of 120 requires a gain of 30/90 = 33.33%, not 25%. The denominator changes after the loss. At the final value of 108, the portfolio remains 10% below the earlier peak. Specify whether values include contributions, withdrawals and reinvested distributions before interpreting a wealth-path measure.

A risk response must also consider liquidity. Selling positions to reduce measured volatility can realize losses or increase market impact, while maintaining a hedge may require cash collateral. Combine the benchmark-relative analysis with the investor’s obligations and ability to hold through stress. A lower tracking-error estimate alone cannot determine whether the resulting portfolio better serves the mandate.

Can average active return be zero while tracking error is positive?

Yes. Positive and negative active returns can average to zero while varying substantially. In the four-month example, the mean is zero and the sample standard deviation is about 1.826% per month.

Why is a 25% loss followed by a 25% gain insufficient to recover?

The gain applies to a smaller base. Starting at 100, a 25% loss leaves 75, and a 25% gain then produces 93.75. Returning from 75 to 100 requires 33.33%. Always identify the denominator.

Try a changed assumption

If portfolio and benchmark weights match exactly, what is tracking error in the stated covariance model?

Show answer and reasoning

Active weights are all zero, so aᵀΣa = 0. Total portfolio volatility can still be positive because it uses portfolio weights rather than active weights.

Original learning check. No email or account needed.

Is tracking error always undesirable?

No. Active management deliberately takes benchmark-relative risk. The question is whether that risk is understood, permitted and compensated after costs.

Sources and scope

Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.

Prepared by MiloPrep editorial team. MiloPrep is independent of GARP. References do not imply endorsement or independent review of these notes.