Explain what revising and rebalancing a portfolio involve, and examine how alpha, risk, trading costs and the investment horizon pull against one another.

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

The idea

Revising a portfolio means updating alphas and risk estimates when new information arrives. Rebalancing means trading the existing portfolio toward the new optimum, also to correct drift from price moves. Four forces pull against each other. Alpha argues for trading quickly, especially when the signal decays fast. Risk argues for trading back when drift has raised risk above target. Transaction costs argue for trading less. The investment horizon decides how many years of benefit repay a cost, so a longer horizon amortizes costs over more years and supports more trading. Trade only when the annual gain from the revision exceeds the amortized cost. Noisier alphas and higher costs call for lower turnover, not higher.

Formulas

Net annual benefit of a trade after amortised costs

net benefit = Δα − round-trip cost ÷ holding years

Δα: the annual improvement in alpha (value added) from the trade
round-trip cost: the cost of buying and later selling, paid once
holding years: the expected years the new position is held

A longer horizon spreads the cost over more years and supports more trading; trade only when the net benefit is positive.

Net alpha for a rebalancing frequency

net alpha = gross alpha − n × c

gross alpha: the alpha captured a year at that frequency
n: the number of rebalancings a year
c: the cost of each rebalancing

The most frequent rebalancing captures the most gross alpha but may not give the highest net alpha.

A practice question

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

What does rebalancing a portfolio involve?

Selling every holding and rebuying the new optimal portfolio in full.Why not: Rebalancing trades only as far as is worthwhile, not a full liquidation.
Trading toward the new optimum, including drift from price moves, only as far as the gain beats the cost. - correct
Rescaling the alpha forecasts so that they match the information coefficient.Why not: Scaling alphas is an input step, not a trading decision.
Changing the benchmark to reflect the portfolio's current holdings.Why not: The benchmark is fixed by mandate; rebalancing changes holdings, not the benchmark.

Revising updates the inputs; rebalancing is the trade that moves the portfolio toward the new optimum, balancing alpha and risk gains against transaction costs.

A harder one

Signal A has an alpha half-life of one month and signal B has a half-life of two years. Which statement about rebalancing is most accurate?

Signal A must be traded frequently to capture its alpha before it decays, so costs limit how much of it can be captured; signal B can be traded gradually. - correct
Signal A can be traded rarely, because a fast-decaying signal is mostly noise.Why not: Fast decay is about timing, not noise; slow trading lets the alpha decay before it is captured.
Signal B must be traded more often than signal A, because its alpha lasts longer.Why not: Slow signals give time to trade gradually and need less frequent trading.
Both signals call for the same trading frequency, because the cost of a trade is the same.Why not: The right frequency depends on how quickly alpha decays relative to the cost of trading.

A fast signal loses value quickly unless it is traded on, so costs decide how much of it can be captured. A slow signal gives time to trade gradually and amortize costs.

The hardest tier

Alpha forecasts become noisier (lower IC) while trading costs rise. Which adjustment is consistent with the trade-offs in rebalancing?

Rebalance more often to recapture the lost alpha.Why not: More trading raises costs and chases noise; lower IC lowers the gain per trade.
Raise the active risk target to compensate for the lower IC.Why not: A lower IR means a lower optimal active risk (IR / (2 x lambda)), not a higher one.
Rebalance less often and trade less, because the gain from each revision is smaller and the cost is larger. - correct
Shorten the assumed holding horizon so costs are amortized more aggressively.Why not: A shorter horizon raises the annual cost of a given trade and so increases the barrier.

Lower IC cuts the benefit of acting on forecast changes and higher costs raise the hurdle, so the optimal policy is lower turnover and a wider no-trade region.

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

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