Turnover costs and strategy capacity

Definition Fix a common pretrade reference capital $A > 0$ and common reference prices, and let $w_i^{\mathrm{old}}$ and $w_i^{\mathrm{new}}$ be the holdings of asset $i$ just before and just after a rebalance, both expressed as fractions of $A$ at those prices. The turnover of the rebalance is

summed over the traded assets, without the money-market account that balances the trades. The traded notional is $N = A\tau$ currency units, since $A \lvert w_i^{\mathrm{new}} - w_i^{\mathrm{old}} \rvert$ is the value of the shares of asset $i$ bought or sold. This convention counts purchases and sales separately, so a sale of $100$ currency units and a purchase of $100$ currency units give $N = 200$. Some reports use the one-way turnover $\tau/2$ instead. Either convention works as long as every cost coefficient is quoted per unit of the same measure. The two sets of weights share one capital base, so neither is renormalized to the wealth left after costs, and market moves between rebalances, external cash flows and financing need their own treatment in a full simulation. As a sum of absolute values, the turnover is a convex function of the new weights.

Definition This lesson uses a constructed cost law. A rebalance with traded notional $N = A\tau$ costs

currency units, where $c \ge 0$ is a proportional cost per currency unit traded, such as a half-spread plus fees, $V > 0$ is the market volume in currency units over the same execution period, and $k \ge 0$ is a dimensionless impact coefficient. The first term is the proportional cost and the second is the impact cost of the next card. Both are in currency units, because the second has squared currency divided by currency, and a cost formula that fails this check has usually lost a factor of turnover. As a function of the trades $A(w_i^{\mathrm{new}} - w_i^{\mathrm{old}})$, the cost is convex. The proportional part is a nonnegative multiple of a sum of absolute values, which has no derivative at a zero trade. The subdifferential of $\lvert x \rvert$ is the interval $[-1, 1]$ at $x = 0$ and $\{\operatorname{sign}(x)\}$ elsewhere. Hence, for one asset and a concave differentiable objective $f$ of the trade $x$, the trade $x = 0$ maximizes $f(x) - c\lvert x \rvert$ exactly when $\lvert f'(0) \rvert \le c$. A quadratic impact term has derivative zero at $x = 0$ and does not change this no-trade condition.

Definition An impact curve gives the average impact cost per currency unit traded as a function of the participation rate $x = N/V$, the traded notional as a fraction of the market volume over the execution period. This lesson uses the constructed linear curve $I(x) = kx$ with $k \ge 0$, so a trade of notional $N$ pays the impact cost $N \, I(N/V) = kN^2/V$. With $N = A\tau$, the impact cost as a fraction of capital is

linear in capital and quadratic in turnover. The linear curve with one aggregate volume is a modeling assumption, not an empirical law. Actual impact differs across assets, trade schedules and market conditions, is estimated with error, and need not be linear in participation. A different curve, for example a concave power law $I(x) = kx^{1/2}$, leads to different capacity formulas.

Definition Let $g$ be the strategy’s expected gross return per unit of capital over the period in which $\tau$ and $V$ are measured, and suppose that $g$, $\tau$, $c$, $k$ and $V$ do not depend on the capital. This is the expectation of a gross strategy return in the sense of Signal evaluation and performance attribution, before any trading cost. For capital $A > 0$ the expected net profit is $Ag - C(A)$, and dividing by $A$ gives the expected net return

The proportional fraction $c\tau$ does not depend on capital, and the impact fraction rises linearly in $A$ when $k\tau > 0$. With $\tau = 0$ both declared costs vanish and the net return is $g$. Holding $g$ fixed assumes that more capital does not change the opportunity the signal offers before costs. All of these are expectations in the stated model, and the realized net return of a period also contains the deviation of the realized gross return from $g$.

Definition The capacity of a strategy is the largest capital at which a stated performance criterion still holds under a stated cost model. The break-even capacity uses the criterion of a nonnegative expected net return. In the model of the previous cards, suppose $\tau > 0$, $k > 0$, $V > 0$ and the margin after proportional costs $m = g - c\tau$ is positive. Setting the net return to zero gives

and the expected net return is positive for $0 < A < A^{*}$ and negative for $A > A^{*}$. If $m \le 0$, no positive capital has a positive expected net return. If $\tau = 0$ and $g > 0$, or $k = 0$ and $m > 0$, the declared costs never bring the net return to zero, and the model gives no finite capacity. These are limits of the chosen cost equations, and risk limits, position limits or a gross return that falls with scale can still bound a real strategy. A required net return $h$ with $0 \le h < m$ gives the capacity $V(m - h)/(k\tau^2)$ in the same way.

Definition The expected net currency profit at capital $A \ge 0$ is

with $m = g - c\tau$. Under $k > 0$, $\tau > 0$ and $V > 0$ it is a strictly concave quadratic, with $\Pi'(A) = m - 2kA\tau^2/V$ and $\Pi''(A) = -2k\tau^2/V < 0$. If $m > 0$, the unique maximizer is

the maximal expected profit is $Vm^2/(4k\tau^2)$, and the net return at that capital is $m/2$. If $m \le 0$ and zero capital is allowed, then $\Pi(A) < 0$ for every $A > 0$ and zero capital maximizes the profit. At $A_{\mathrm{profit}}$ one more unit of capital adds as much expected cost as expected gross profit, while the average net return is still positive, which is why this capital lies below the break-even point. This is the unconstrained currency-profit problem under a fixed gross return and a fixed turnover. Maximizing the net return in percent has no solution among positive capitals, because that return decreases in $A$, and a risk-adjusted objective under risk limits is a different problem with a different answer.