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Work out the best no-trade region to use when rebalancing in the presence of transaction costs.

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

The idea

With transaction costs there is a range in which doing nothing is best. Let λ be the manager's risk aversion and cov the covariance of the asset with the active portfolio. For each asset, marginal value added is alpha less a marginal risk penalty, so MCVA = α − 2 × λ × cov. Buy if MCVA exceeds the purchase cost PC. Sell if MCVA is below minus the sale cost SC. Otherwise hold. The no-trade region therefore runs from −SC to +PC in MCVA, and its width is PC + SC. Costs must be amortized to the same annual basis as alpha, so a shorter holding horizon widens the region. With zero costs the region shrinks to a point at zero. The usual mistakes are using alpha instead of MCVA, forgetting amortization, and swapping the buy and sell sides.

Formulas

Marginal contribution to value added

MCVA = α − 2 × λ × cov(i, A)

α
the asset's alpha
λ
the active risk aversion
cov(i, A)
the covariance of the asset with the active portfolio

Buy if MCVA exceeds the amortised purchase cost, sell if it is below minus the amortised sale cost, otherwise hold. Compare MCVA, not raw alpha, with the cost.

A practice question

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

What is the no-trade region in portfolio rebalancing?

  1. The range of alpha values between zero and the manager's active risk, inside which no trade is made.Why not: Mixes alpha with the risk target; the region is defined on marginal value added against costs.
  2. Range of MCVA where the gain from trading is below the cost, so the best action is to hold. - correct
  3. The range of asset weights within 5% of the benchmark weights, set by an active weight limit.Why not: This describes an active weight limit, not a no-trade region derived from costs.
  4. The set of assets whose alpha is exactly zero, since these cannot add any value after costs.Why not: Having a zero alpha does not by itself mean no trade; MCVA, not alpha alone, drives the decision.

If marginal value added lies between minus the sale cost and plus the purchase cost, a trade costs more than it earns, so the manager holds.

A harder one

For a stock, alpha is 3.0% and the marginal risk penalty (2 x lambda x covariance with the active portfolio) is 2.2%. The purchase cost is 2.4% one-way and the stock would be held for 2 years. What is the correct action?

  1. Do not trade: MCVA of 0.8% is below the amortized purchase cost of 1.2%. - correct
  2. Buy: the alpha of 3.0% exceeds the amortized purchase cost of 1.2%.Why not: Uses alpha instead of MCVA, which ignores the 2.2% risk penalty.
  3. Buy: MCVA of 0.8% is positive.Why not: Ignores that MCVA must exceed the amortized purchase cost, not just be positive.
  4. Sell: MCVA is below the purchase cost.Why not: Treats failure to clear the purchase cost as a sell signal; selling requires MCVA below minus the sale cost.

MCVA = 3.0 - 2.2 = 0.8%. Amortized purchase cost = 2.4 / 2 = 1.2%. Since 0.8% < 1.2% the stock is inside the no-trade region.

The hardest tier

At a 3-year horizon, the no-trade region is -0.4% to +0.6%. If the expected holding period is cut to 1.5 years, with the same underlying costs, the new region is:

  1. -0.2% to +0.3%Why not: Halves the region; a shorter horizon raises the amortized costs, so the region widens.
  2. -0.4% to +0.6%Why not: Leaves the region unchanged, ignoring that annual cost depends on the horizon.
  3. -1.2% to +1.8%Why not: Multiplies by 3, the old horizon, instead of the ratio of horizons (3 / 1.5 = 2).
  4. -0.8% to +1.2% - correct

Annual amortized cost = cost / years. Halving the horizon doubles it, so both limits double: -0.4 x 2 = -0.8% and 0.6 x 2 = +1.2%.

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