Holding Period and Turnover Optimization

A fisherman who reels in the moment he feels a bite catches more small fish but wears out his arm; one who waits too long lets fish wriggle off the hook. Every trading signal faces the same tension. A signal's predictive power fades with time — trade too slowly and you're capturing a stale, weakened version of it. But every trade crosses the bid-ask spread and pushes the price against you — trade too fast and the cost of chasing freshness eats the very edge you're chasing. Holding-period optimization is finding the point in between where an extra trade's expected benefit from fresher signal just equals its expected cost.

Setting benefit against cost

Model a signal's information coefficient — its correlation with future returns — as decaying exponentially with a half-life $h$:

$IC(t) = IC_0 \, e^{-t/h}$

In words: how predictive the signal still is after $t$ days is its starting predictiveness $IC_0$, shrunk by a decay factor that halves roughly every $h$ days. A fast-decaying signal (small $h$) wants to be rebalanced often to keep trading on a fresh, high-$IC$ version; a slow-decaying signal (large $h$) can be traded rarely without losing much.

Set that against a simple transaction cost model where cost per unit traded is $c$ (in basis points, covering spread and impact) and turnover per rebalance is $\tau$. Trading $n$ times over a period costs roughly $n \times \tau \times c$, while expected gross profit scales with the average $IC$ captured, which rises as $n$ increases (fresher signal each time) but with diminishing returns since $IC(t)$ is already close to $IC_0$ for small $t$. The optimum sits where the marginal profit from one more rebalance equals its marginal cost.

Worked example 1. Suppose a signal has $IC_0 = 0.05$ and half-life $h = 10$ trading days, and rebalancing weekly (every 5 days) versus daily changes how much decayed signal is captured. Trading weekly, the average IC captured over the week is roughly $IC(2.5) = 0.05 \times e^{-2.5/10} \approx 0.05 \times 0.78 = 0.039$. Trading daily instead captures close to the full $IC_0 = 0.05$ on average, a gain of about $0.011$ in average IC, roughly 28% more signal captured. But daily rebalancing multiplies the number of trades by 5x. If weekly turnover costs 0.6% of the portfolio in spread and impact ($\tau \times c$), daily rebalancing costs roughly $5 \times 0.6\% = 3\%$ over the same week — the cost grew far faster than the 28% signal gain, so weekly (or even less frequent) trading wins for this particular half-life and cost level.

A second example: the crossover point

Worked example 2. Take a slower signal: $IC_0 = 0.03$, half-life $h = 60$ days — a value-style signal that barely moves week to week. Trading it monthly (every 21 days) captures average $IC(10.5) = 0.03 \times e^{-10.5/60} \approx 0.03 \times 0.84 = 0.025$. Trading it weekly instead captures roughly $IC(2.5) = 0.03 \times e^{-2.5/60} \approx 0.03 \times 0.96 = 0.0288$ — a gain of only about 15% in average IC for going from monthly to weekly, but a roughly 4x increase in trade count and cost. For this slow-decaying signal, the extra cost of weekly trading swamps the tiny signal gain, and the optimizer should push the holding period out, not in — the opposite conclusion from the fast-decaying signal in example 1, even though the math is the identical formula. Half-life alone, not the strategy's "style," determines which side of the trade-off you're on.

What this means in practice

Grinold and Kahn's fundamental law of active management (see The Grinold Alpha Formula) shows that information ratio scales with the square root of the number of independent bets per year — which tempts researchers to trade as often as possible to rack up more bets. Turnover optimization is the necessary counterweight: more frequent rebalancing only adds independent bets if the signal has actually refreshed meaningfully, and every extra rebalance also adds a cost that scales roughly linearly (see The Square-Root Impact Law for how cost actually scales with trade size). A desk that ignores this and rebalances daily "to be safe" on a monthly-half-life signal is paying for trades that add turnover without adding information.