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Characterise dispersion, account for what causes it, and set out ways to keep its different forms under control.

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

The idea

Dispersion is the difference in returns among portfolios that follow the same strategy, for example separate client accounts run to one model. It is not tracking error against a benchmark. Causes include different starting holdings and inception dates, different cash flows, client-specific restrictions, different trade timing and execution, and separate optimizations that react differently to small input changes. It is controlled by managing every account toward a common model portfolio, trading accounts together, rebalancing more often, and standardizing guidelines. Each step has a cost. More trading costs more, and restrictions cannot always be removed. Suppose two accounts have independent tracking errors of s against the model. Then Dispersion = s × √2. That is about 1.41 times s, so the two accounts differ by more than either does from the model.

Formulas

Dispersion between two accounts

dispersion = s × √2

s
each account's tracking error against the model portfolio

Assumes the two accounts' deviations from the model are independent and equal in size; dispersion is not tracking error against a benchmark.

A practice question

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

How is dispersion defined?

  1. The difference in returns among portfolios managed under the same strategy or model. - correct
  2. The difference between a portfolio's return and its benchmark's return.Why not: This is the active return or tracking error against a benchmark.
  3. The volatility of an individual stock's return.Why not: This is stock-level volatility, not a difference across portfolios.
  4. The difference between forecast alpha and realized alpha.Why not: This describes forecast error, not variation across accounts.

Dispersion is about portfolios that should look alike but do not, for instance separate accounts following one model.

A harder one

Two accounts follow the same model. Each has an independent tracking error of 0.6% against the model portfolio. The standard deviation of the return difference between the two accounts is closest to:

  1. 0.60%Why not: Takes one account's tracking error and ignores that the difference involves both accounts.
  2. 1.20%Why not: Adds the two standard deviations; independent risks add in variance.
  3. 0.85% - correct
  4. 0.42%Why not: Divides by the square root of 2 instead of multiplying.

Variance of the difference = 0.6² + 0.6² = 0.72, so the standard deviation is 0.85% (0.6 x square root of 2).

The hardest tier

Each account's independent tracking error against the model falls from 1.0% to 0.5% after more frequent rebalancing. What happens to dispersion between two accounts?

  1. It falls by 0.50 percentage points, from 1.00% to 0.50%.Why not: Omits the square root of 2 and treats dispersion as one account's tracking error.
  2. It falls by about 0.71 percentage points, from 1.41% to 0.71%. - correct
  3. It falls by about 0.41 percentage points, from 1.41% to 1.00%.Why not: Halves the variance rather than the tracking error.
  4. It is unchanged, because the square root of 2 factor applies to both cases.Why not: The factor is the same but it multiplies a smaller number, so the level falls.

Dispersion = tracking error x square root of 2: 1.0 x 1.414 = 1.41% before and 0.5 x 1.414 = 0.71% after. It halves, so it falls by 0.71 points. The trade-off is the extra trading cost.

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