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Outline practical matters in building portfolios, such as choosing a suitable level of risk aversion, aversion to particular risks, and making sure alphas cover the whole universe properly.

FRM® Part II · Risk Management and Investment Management · Portfolio Construction · FRM2.IM.4.e

The idea

Three practical choices matter. First, risk aversion should be set from the target, not guessed. Let IR be the information ratio, λ the active risk aversion, and write risk in decimals. With unconstrained optimization, the active risk follows the first relation below. Risk = IR ÷ (2 × λ). Turning it round gives the aversion that delivers a target. Lambda = IR ÷ (2 × target risk). Scaling all alphas by a factor has the same effect on positions as dividing λ by that factor. Second, a manager can be more averse to particular risks, such as a sector or common factor, by adding a separate penalty on that risk alone, rather than raising overall aversion and shrinking every bet. Third, alphas should cover the whole universe. Assets with no forecast must be given a neutral, benchmark-consistent alpha, since an arbitrary zero can create an unintended bet between covered and uncovered assets.

Formulas

Active risk aversion from a risk target

λ = IR ÷ (2 × ψ)

IR
the expected information ratio
ψ
the target active risk, in decimals

Holds for an unconstrained optimisation. Scaling all alphas up by a factor needs λ raised by the same factor to keep active risk unchanged.

Total active risk from uncorrelated parts

active risk = √(factor risk² + residual risk²)

factor risk
the active risk from common factors
residual risk
the active residual (specific) risk

The two must be uncorrelated; risks add in variance, not in standard deviation.

A practice question

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

Which formula relates active risk aversion to the manager's target in an unconstrained optimization?

  1. Lambda = target active risk / (2 x IR)Why not: Inverts the relationship; higher IR with a given target means higher, not lower, aversion.
  2. Lambda = IR x target active risk / 2Why not: Multiplies instead of dividing; aversion falls as the target active risk rises.
  3. Lambda = IR / (2 x target active risk) - correct
  4. Lambda = 2 x IR / target active riskWhy not: Puts the 2 in the numerator, doubling aversion instead of halving the ratio.

The optimal active risk is IR / (2 x lambda), so lambda = IR / (2 x target active risk). A higher target means less risk aversion.

A harder one

A manager is comfortable with total active risk but wants less exposure to one industry factor. What is the most appropriate response?

  1. Raise the overall risk aversion, which cuts every active position.Why not: Reducing all bets lowers all risk and cuts alpha everywhere; the problem is one risk source.
  2. Reduce all alphas by the same fraction.Why not: Equivalent to raising overall risk aversion and shrinks every bet, not just the factor exposure.
  3. Double the weight on residual risk, which cuts the factor exposure indirectly.Why not: Residual risk is unrelated to the factor exposure; penalizing it targets the wrong risk.
  4. Add a larger penalty on that factor's risk alone, so only that exposure is cut. - correct

A separate aversion to a specific risk, such as a sector or common factor, lets the manager control it without changing the rest of the portfolio.

The hardest tier

At lambda = 8 the optimizer produces an active risk of 5%. All alphas are then multiplied by 1.25. To keep active risk at 5%, with no costs or constraints, lambda should become:

  1. 6.4Why not: Divides lambda by 1.25 instead of multiplying; larger alphas need more risk aversion.
  2. 12.5Why not: Multiplies by 1.25² (1.5625), treating the effect as variance.
  3. 10.0 - correct
  4. 8.0Why not: Leaves lambda unchanged; larger alphas at the same lambda would raise risk to 6.25%.

Positions are proportional to alpha / (2 x lambda). Alphas up 25% need lambda up 25% to leave positions unchanged: 8 x 1.25 = 10.

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

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