The calendar vs threshold rebalancing section compares calendar against threshold rebalancing and prices the trades. It leaves the harder question open: given that you have chosen a threshold policy, what sets the threshold? A house rule of plus or minus five points applied to every sleeve is a default, not an answer, and it is wrong in a specific and predictable direction for at least two sleeves in any real book.
How is the optimal band width derived?
Derive it rather than memorise it. A corridor trades one cost against another. Letting the portfolio drift a half-width d away from policy in sleeve i means holding an unintended active position: long d of sleeve i and short d of everything else, funded from the rest of the book. That active position's variance is not d²σᵢ². It is
Var(active) = d² × (σᵢ² + σ_c² − 2·ρ_ic·σᵢ·σ_c) = d² × σ_diff²where σ_c is the volatility of the rest of the portfolio and ρ_ic the correlation between them. The whole of the corridor question runs through that one quantity, the volatility of the difference between the sleeve and the book it sits in. The cost of drifting is roughly ½·λ·d²·σ_diff² per unit of time, with λ the investor's risk aversion; the cost of not drifting is the transaction cost c paid every time the band is touched. Balancing the two, in the impulse-control form Constantinides derived for a transaction-cost problem in 1986 and which practitioner rules such as Masters's approximate, gives a half-width proportional to
d* ∝ ( c / (λ · σ_diff²) )^(1/3)Advertisement
What drives the width of a rebalancing band?
Every comparative static falls out of that expression, and each one is a sentence rather than a rule to memorise.
Transaction cost. d* rises with c. An expensive sleeve should be allowed to drift further, because each corrective trade buys the same risk reduction at a higher price. Multiply the cost by eight and the band widens by a factor of two.
Risk aversion. d* falls as λ rises. A more risk-averse investor pays more attention to the unintended active position and less to the trading bill, so the band tightens. The same sleeve legitimately carries different bands for two clients.
Sleeve volatility. d* falls as σᵢ rises, because a higher σᵢ raises σ_diff. The same drift in percentage points is a bigger risk deviation in a volatile sleeve than in a quiet one.
Correlation with the rest of the book. d* rises as ρ_ic rises, because a higher correlation lowers σ_diff. A sleeve that moves with everything else can drift a long way without changing the portfolio's risk much. This is the driver most house rules ignore, and it is the one that most often reverses their answer.
Worked example: mid-cap equity and gold
A worked comparison, two sleeves that end up on opposite sides for opposite reasons. Take a book whose non-sleeve remainder has a volatility of 11%.
| Sleeve | σᵢ | ρ with the rest | Round-trip cost | σ_diff | What each driver argues for |
|---|---|---|---|---|---|
| Mid-cap equity | 26% | 0.85 | 25 bp | 0.176295 | Volatility says narrow; correlation says wide; cost says narrow |
| Gold | 15% | 0.05 | 90 bp | 0.181521 | Volatility says wide; correlation says narrow; cost says wide |
The two σ_diff figures are nearly identical, 0.1763 against 0.1815, which is the first surprise: mid-cap equity is 11 points more volatile than gold and yet contributes almost exactly the same deviation risk per point of drift, because 0.85 of its movement is movement the rest of the book was making anyway. With σ_diff almost neutral, the cost term decides. The ratio of optimal half-widths is
d*_gold / d*_equity = ( (90/25) × (0.031080/0.032950) )^(1/3) = (3.6000 × 0.943247)^(1/3) = 1.5031so on a ±3.0 point band for mid-cap equity, gold's band should be about ±4.51 points. Isolating each driver from the equity baseline shows how much work each one does: the cost difference alone would give ±4.60, the volatility difference alone ±5.04, and the correlation difference alone ±2.22. The correlation term is the only one pulling the other way, and it removes roughly a point of width from what the other two would have granted.
The practical rule that survives this is short. Set bands from σ_diff and the trading cost, not from the sleeve's headline volatility, and expect the diversifying, expensive sleeves to end up with the widest bands and the cheap, highly correlated equity sleeves with the narrowest. Two honest limits go with it. The cube-root form assumes proportional costs, a constant opportunity set and continuous monitoring, none of which is exactly true, so treat the ratios as the usable output and the absolute widths as calibration. And a sleeve with a hard mandate limit has a corridor set by the mandate, not by this arithmetic; the optimisation applies inside the limit, never through it.
The funded-ratio and duration-gap arithmetic is the allocation-level version. AA1.05 takes it into the institutional client's world, where the same ratio drives a de-risking glide path, a currency-duration hedge ratio and a mandate-feasibility test on an escalating rather than a stated-present-value liability. Read it after this node if you work with pensions, insurers or endowments; nothing here is repeated there.