Portfolio weights, drift and rebalancing

What you will learn

Derive capital-weight drift and an internal target transfer, then prove a common-path rebalancing contrast with reversal and continuation witnesses.

Prerequisites: Linked returns and external cash flows · Expected value as a weighted average · Inventory, cash and marked wealth

Separate simultaneous sleeves from consecutive periods

A portfolio return combines capital exposed at the same time. A linked return combines exposure through time. These operations answer different questions. Suppose a fictional 160-chip account holds two fully invested sleeves, A and B. Its initial fraction in A is w and its fraction in B is 1−w. Both sleeves experience the first period simultaneously. Their positive gross factors are a and b. Ending sleeve values are A=160wa and B=160(1−w)b. Total wealth is V=A+B. Dividing by 160 gives portfolio factor wa+(1−w)b, a capital-weighted sum rather than the product ab.

Assume no owner contributions or withdrawals, fees, taxes, debt, short positions, distributions outside the account or price impact. All capital is valued on the same clock. Fractional holdings are allowed and the boundary exchange can execute immediately at those valuations. The factor path is stipulated independently of our trades. Sleeve labels identify hypothetical exposures, not securities or investment recommendations. There are no probabilities in this experiment, so a favorable path cannot establish an expected advantage, risk reduction or suitability. Positive factors keep the denominator positive even at a zero initial sleeve weight.

Derive weight drift from holdings

With w=1/2, a=3/2 and b=1/2, both sleeves start with 80 chips. A ends with 120 and B with 40, so V=160. First-period return is zero: half of a 50 percent gain offsets half of a 50 percent loss. The ending fraction in A is 120/160=3/4. Unchanged share counts do not imply unchanged capital weights. A now receives three quarters of any subsequent wealth exposure even though it received only half initially.

In general the drifted A fraction is w′=wa/[wa+(1−w)b]. Subtract w, put both terms over the same denominator and factor: w′−w=w(1−w)(a−b)/[wa+(1−w)b]. For an interior weight, positive relative performance of A increases its weight. If a=b, there is no drift. If w is zero or one, there is no other sleeve to drift against. These are algebraic boundary checks, not empirical regularities inferred from a simulation.

The weights used for the first-period return are beginning weights. Substituting the ending 3/4 into that earlier return would give an invented 25 percent gain. It awards the larger weight to the sleeve after seeing it win. Ending weights are appropriate for the next period under unchanged holdings, not for rewriting exposure that already occurred. This timing error can survive correct individual sleeve arithmetic.

Construct a self-financing target restoration

Fix the target A weight at the original w. Let t denote chips transferred from A to B at the boundary; negative t means transferring from B to A. Post-trade holdings are A−t and B+t. Their sum remains V. The target equation is A−t=wV, so t=A−wV=160w(1−w)(a−b). This identity is a certificate: it proves both the target weight and internal financing. Long-only feasibility additionally requires −B≤t≤A. Our target lies between zero and one, so its exact solution always satisfies those bounds.

In the worked half-and-half case, transfer 120−(1/2)160=40 chips from A to B. The resulting holdings are 80 and 80, still totaling 160. The boundary trade earns no instantaneous gain. It changes the exposure of existing wealth. Buying 40 chips of B without selling A would instead require outside capital or borrowing, violating the experiment. Selling 40 from each sleeve would shrink the account and fail the conservation certificate.

Restoring weights is also different from restoring initial chip amounts. With w=1/4, the first boundary holdings are 60 and 60, totaling 120. The correct targets are 30 and 90, so t=30. Restoring the original A amount of 40 would leave A at 1/3 of the current total. A target is a fraction of current wealth, not a promise to maintain an old balance. Investor.gov describes allocation drift and internal or contribution-funded restoration; our equations isolate only the internal, costless case.

Compare policies on one common future path

Let the second-period gross factors be c for A and d for B. Holding unchanged ends with H=Ac+Bd. Restoring the target ends with R=(A−t)c+(B+t)d. Subtraction cancels the common holdings and gives R−H=t(d−c). Combining with the transfer identity yields R−H=160w(1−w)(a−b)(d−c). A restoration changes relative outcomes only through the transferred capital and its subsequent relative growth. It does not create a free return at the trade boundary.

For the half-and-half first period, t=40. Under reversal c=1/2,d=3/2, holding finishes at 120*(1/2)+40*(3/2)=120. Restoration finishes at 80*(1/2)+80*(3/2)=160. The advantage is 40, agreeing independently with 40*(3/2−1/2). Under continuation c=3/2,d=1/2, holding instead finishes at 200 and restoration at 160: the advantage is negative 40. The identical boundary decision can help or hurt on different stipulated paths.

Even when restoration wins, neither strategy necessarily makes money over the whole experiment. Compare final wealth with the original 160 as well as with the other policy. A relative advantage and an absolute gain have different baselines. With a flat second period, both finish at the boundary wealth regardless of the transfer. With equal next factors other than one, both grow by that common factor and still tie. No path comparison here supplies a probability of reversal or a claim that assets must compensate for earlier movement.

Build, challenge and audit an exposure certificate

In the workshop choose the initial A weight and a first-period pattern. Inspect the two boundary sleeve values, then move ten-chip packets in either direction. Every move displays both new holdings, their conserved sum and the resulting A weight. Construct the exact target restoration before entering the first return, the pre-trade drifted weight and the final policy difference. The target transfer is deliberately not supplied as a button. You must solve the exposure equation; the numerical checks alone cannot complete the activity.

Start with the half-and-half reversal case and predict a positive 40-chip difference. Change only the second path to continuation, rebuild the target and explain why the difference changes sign while the transfer does not. Change the initial A weight to one quarter to challenge the tempting original-balance rule. Test equal first-period growth, zero initial A and full initial A. These cases require zero transfer, although their returns and final wealth can differ. Model changes clear the constructed transfer, numerical answers and earlier success feedback; the draft is local to the visit and earns no account credit.

The audit separates a missed target from a wrong return clock, a wrong weight denominator and a reversed policy baseline. Hints explain the relevant equation rather than supplying a generic instruction to try again. In real execution, charges, indivisible units or unavailable liquidity can prevent this exact certificate. Costs can change both total wealth and the target equation. This lesson establishes the frictionless accounting benchmark; it does not silently apply that benchmark to an actual account or certify a trading strategy.

Hands-on: Construct a self-financing mix

Move conserved chips to restore a target weight, then challenge the policy comparison on reversal and continuation paths.

Common mistakes

• Multiplying simultaneous asset factors as though all capital experienced both sequentially.
• Using ending weights to recalculate an earlier return.
• Restoring old chip balances instead of fractions of current wealth.
• Buying the underweight sleeve without financing the purchase.
• Confusing relative policy advantage with absolute profit or expected performance.

Keep these ideas

• Portfolio factor is wa+(1−w)b; pre-trade weight is wa/[wa+(1−w)b].
• The self-financing transfer is t=A−wV=160w(1−w)(a−b).
• Restored minus held final wealth is t(d−c), on the same second-period path.
• Equal relative growth or a single occupied sleeve eliminates drift. Rebalancing is an exposure operation, not a guaranteed gain.

Further reading

• Investor.gov: Beginners’ Guide, Rebalancing 101 ↗

  Read Rebalancing 101 and When to Consider Rebalancing (HTML lines 147–164): differential growth, restoring allocation, internal transfers versus new contributions, and cost considerations. Supports conceptual definitions only. Original two-sleeve equations, chip paths, target certificate and checks are independently derived, not source examples. No probability, optimality, professional credential or video-review claim. Checked 2026-10-25.