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Show how transaction costs bear on the way a portfolio is constructed.

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

The idea

Transaction costs enter the optimization as a penalty on trades, charged on purchases and sales and usually different for each. Alpha is quoted per year while a cost is paid once, so the cost is amortized over the expected holding period, as below. Cost = round-trip cost ÷ holding years. A trade is worthwhile only if the improvement in alpha less risk exceeds that amortized cost. Higher costs therefore lower turnover and keep the portfolio nearer its current holdings, and a held asset is kept longer than an equivalent unheld asset is bought. Market impact usually rises with trade size, so splitting a large trade can cost less. The common mistake is comparing an annual alpha gain with a full, unamortized cost.

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

Alpha is annual and the cost is paid once, so it must be spread over the holding period. Comparing the annual gain with the full cost is the common mistake.

A practice question

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

How are transaction costs normally included in portfolio construction?

  1. A penalty subtracted from the objective on buys and sells, spread over the expected holding period. - correct
  2. As an addition to the alphas of the assets that are already held in the current portfolio.Why not: Costs reduce the attractiveness of trading; they do not add to the alpha of held assets.
  3. As a hard constraint that sets turnover to zero, so no trade is made unless costs are zero.Why not: A zero-turnover constraint would never allow any trade; costs are traded off against benefits.
  4. They are left out of construction and deducted only when performance is reported to clients.Why not: Ignoring costs in construction leads to excessive turnover; they belong in the objective.

Costs enter the optimization as a charge on trading, with buys and sells given their own cost, and because alpha is annual the cost is amortized over the holding period.

A harder one

A trade has a round-trip cost of 2.4% and the new position is expected to be held for 1.5 years. The annual alpha improvement from the trade is 2.0%. The net annual benefit after amortized costs is closest to:

  1. -0.4%Why not: Subtracts the full round-trip cost (2.0 - 2.4) without amortizing over the holding period.
  2. -1.6%Why not: Multiplies the cost by the holding period (2.4 x 1.5 = 3.6) rather than dividing by it.
  3. +0.4% - correct
  4. +0.8%Why not: Halves the cost (2.4 / 2 = 1.2) as though only one side were paid, without amortizing.

Amortized cost = 2.4% / 1.5 years = 1.6% a year. Net benefit = 2.0% - 1.6% = +0.4% a year, so the trade is worthwhile.

The hardest tier

Market impact cost for a trade is 0.2% of value for each USD 1 million traded, so a USD 10 million trade costs 2.0% of its value. If the order is split into two USD 5 million trades on different days, and the impact of the first has fully faded before the second, the total impact cost is:

  1. USD 200,000, the same as a single trade.Why not: Assumes cost is proportional to the amount traded and ignores that impact rises with size.
  2. USD 50,000, the cost of one half-size trade.Why not: Counts only one of the two trades.
  3. USD 400,000, because two trades double the cost.Why not: Doubles the cost for the number of trades, ignoring that each trade is half the size.
  4. USD 100,000, half the cost of a single trade. - correct

Single trade: 2.0% x 10m = USD 200,000. Each 5m trade: 1.0% x 5m = USD 50,000, so two cost USD 100,000. With convex impact, splitting lowers cost but delays the trade and risks losing alpha.

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

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