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Outline the inputs feeding portfolio construction and explain the difficulties that arise in working with them.

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

The idea

Portfolio construction takes four main inputs: alpha forecasts for each asset, a risk model that supplies the covariance matrix of returns, estimates of transaction costs, and an active risk aversion that sets how much risk is accepted for each unit of alpha. Each is hard to get right. Alphas are noisy and tend to be overstated. Covariance estimates from short samples are unstable. With more assets than observations the sample matrix is singular, so a factor model is normally used. Costs depend on trade size and market conditions and are known only after trading. The optimizer magnifies every error, because it takes large positions wherever alpha is overstated or risk understated. Exam point: garbage in, amplified garbage out.

Formulas

Information coefficient implied by alpha dispersion

implied IC = alpha dispersion ÷ residual volatility

alpha dispersion
the cross-sectional standard deviation of the alphas
residual volatility
the typical residual volatility of the stocks

Compare with the historical IC: an implied IC well above it means the alphas are overstated and should be scaled down before optimising.

A practice question

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

Which list best describes the inputs that a portfolio optimizer needs?

  1. Alpha forecasts, trailing returns and a target return, with no risk estimate because alpha already reflects risk.Why not: Treats alpha as self-contained; the optimizer still needs a covariance forecast to price the risk of each position.
  2. A covariance matrix, benchmark weights and transaction costs, with the alphas produced by the optimizer.Why not: Confuses the optimizer with the forecasting process; alphas are an input to the optimizer, not an output.
  3. Alpha forecasts, a covariance matrix from a risk model, transaction cost estimates and an active risk aversion. - correct
  4. Alpha forecasts and a covariance matrix only, with costs and risk aversion applied after the trades.Why not: Costs and risk aversion are part of the objective; applying them afterwards gives different trades.

A mean-variance construction maximizes alpha less a risk penalty and trading costs, so it needs alphas, a covariance forecast, cost estimates and the aversion that sets the alpha-versus-risk trade-off.

A harder one

Which statement about transaction cost inputs is most accurate?

  1. Costs depend on trade size, liquidity and timing and are known only after trading, so forecasts are uncertain. - correct
  2. Costs equal the commission plus half the quoted spread, so they are known exactly before any trade is made.Why not: Ignores market impact, which depends on size and conditions and is usually the larger part of cost.
  3. Costs are linear in trade size, so they are negligible for large trades and can be dropped from the objective.Why not: Costs are not linear in practice, and omitting them leads the optimizer to overtrade.
  4. Costs are the only input that can be measured exactly after the fact, so they do not need to be forecast.Why not: Ex post measurement does not help decisions made before trading; costs must still be forecast for the optimization.

Market impact rises with trade size and depends on liquidity and urgency, and it is measured only after execution. The optimizer therefore works with estimates that can be wrong, usually in the direction of too little cost.

The hardest tier

An alpha model has residual volatility of 25% and a historical IC of 0.05. The cross-sectional standard deviation of its alphas is 6% a year. What does this indicate?

  1. The implied IC is 0.024, so the alphas are too conservative and the whole set should be scaled up before use.Why not: Slips a decimal place when converting 6% and 25%; the ratio is 0.06 / 0.25 = 0.24, not 0.024.
  2. Implied IC is 0.24, about five times the historical 0.05, so alphas are overstated and should be scaled down. - correct
  3. The implied IC is 4.17, so the alphas are far too large and the whole alpha model should be discarded as invalid.Why not: Inverts the ratio (25 / 6) instead of dividing alpha dispersion by volatility.
  4. The implied IC is 0.24, which is consistent with a skilled manager, so no adjustment is needed.Why not: Accepts an IC far above the historical 0.05 without question; a skilled manager does not have an IC of 0.24.

Alpha dispersion should be about volatility x IC = 0.25 x 0.05 = 1.25%. The model shows 6%, which implies IC = 0.06 / 0.25 = 0.24, nearly five times the evidence. An optimizer would act on this overconfidence.

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