5.9 The mathematics of cost drag · M05 Intraday and execution

Motivation

M01 computed the cost of one trade. Here we compute what costs do to a strategy over a year, and why the same cost per trade is far more damaging intraday than in swing trading (holding for days to weeks; M04).

5.9.1Net expectancy and Sharpe drag#

Setup

A strategy makes $N$ non-overlapping trades a year. Trade $i$ returns $r_i = e - c + \epsilon_i$ , where $e$ is the gross edge (expected gross return per trade), $c$ the round-trip cost (charges plus spread plus impact, as a fraction of notional), and $\epsilon_i$ i.i.d. with mean 0 and standard deviation $\sigma_t$ . Position size is constant.

Proposition 5.7net expectancy and Sharpe drag

The annual expected return, standard deviation and Sharpe ratio are

\mu_A = N(e - c), \qquad \sigma_A = \sqrt N\,\sigma_t, \qquad \text{SR}_A = \sqrt N\,\frac{e - c}{\sigma_t} = \text{SR}_A^{\text{gross}} - \sqrt N\,\frac{c}{\sigma_t}. \tag{5.25}

If the holding period is $h$ (in days), $N = H/h$ trades fit in $H$ trading days, $\sigma_t = \sigma_D\sqrt h$ for a random-walk price, and the edge scales as $e = a h$ (a drift-like edge of $a$ per day), then

\text{SR}_A(h) = \frac{\sqrt H}{\sigma_D}\Big(a - \frac{c}{h}\Big). \tag{5.26}

Proof

Sums of $N$ i.i.d. trade returns have mean $N(e - c)$ and variance $N\sigma_t^2$ ; divide. For (5.26), substitute $N = H/h$ , $\sigma_t = \sigma_D\sqrt h$ and $e = ah$ : $\sqrt{H/h}\,(ah - c)/(\sigma_D\sqrt h) = \sqrt H(ah - c)/(\sigma_D h)$ .∎

The cost term $\sqrt N c/\sigma_t$ is the Sharpe drag. Equation (5.26) shows that for a drift-like edge, the cost penalty is $c/h$ per day: it falls in inverse proportion to the holding period. Short holding periods are viable only if the edge per unit time is much larger at short horizons, which is what intraday traders implicitly claim.

5.9.2Intraday versus swing#

Worked example: intraday versus swing on a large-cap with $\sigma_D = 1.8\%$

	Intraday	Swing
Holding period	~1 hour of a 6.25-hour session	10 trading days
$\sigma_t$	$1.8\% \times \sqrt{1/6.25} = 0.72\%$	$1.8\% \times \sqrt{10} = 5.69\%$
Cost $c$ per round trip	10 bp (8.3 bp charges on ₹1 lakh, plus ~2 bp spread)	25 bp (24 bp delivery charges incl. DP, plus ~1 bp spread)
$c/\sigma_t$ (per-trade Sharpe needed to break even)	0.139	0.044
Trades a year $N$	500 (2 a day)	25
Sharpe drag $\sqrt N c/\sigma_t$	3.1	0.22
Annual cost $Nc$ (on notional)	50%	6.25%

(Volatility is assumed uniform across the session; the U-shape makes the opening hour more volatile and the midday hour less.)

An intraday strategy trading twice a day must have a gross annualised Sharpe above 3.1 just to break even; at 250 trades a year the hurdle is 2.2. The swing strategy's hurdle is 0.22. The intraday trader's cost per trade is 40% of the swing trader's, but relative to the risk taken per trade it is three times larger, and it is paid 20 times as often.

5.9.3Statistical power of frequency#

The counterpoint: intraday edges are faster to verify The number of years needed for a strategy with net per-trade Sharpe $s_t = (e-c)/\sigma_t$ to reach a t-statistic of 2 is $T_2 = 4/(N s_t^2)$ . An intraday strategy with a net edge of 5 bp per trade and $\sigma_t = 0.72\%$ at 500 trades a year needs 1.7 years; a swing strategy with a net edge of 50 bp and $\sigma_t = 5.69\%$ at 25 trades a year needs 21 years. High frequency gives statistical power. It does not give edge.

Practitioner notes

Costs are known in advance and certain; edges are estimated and uncertain. A backtest that clears costs by a small margin is a strategy that probably loses.
Leverage scales $e$ , $c$ and $\sigma_t$ together and does not change $c/\sigma_t$ . It cannot rescue a strategy with negative net Sharpe.
Annual cost $Nc$ is on notional, not on capital. At 5× intraday notional per unit of capital, 50% of notional is 250% of capital a year.