Quantitative note

The Idealized Sharpe Discounts

Hit Rate To Sharpe covers the idealized Sharpe. Every discount to that takes the same form: state the trade as a random variable, apply \(SR_{\text{trade}} = \mathbb E[X]\big/\sqrt{\operatorname{Var}(X)}\), and compute the two moments. Symbols follow its table 1.

01 The discounts

D1 — Magnitude dispersion

Each trade is a sign times a magnitude, the two independent:

\[ X = S\cdot M, \qquad S=\begin{cases}+1 & \text{w.p. } p\\ -1 & \text{w.p. } 1-p\end{cases}, \qquad M>0,\quad \mathbb E[M]=m,\quad \operatorname{sd}(M)=s_M,\quad c=\frac{s_M}{m} \] \[ SR_{\text{trade}} \;=\; \frac{\mathbb E[X]}{\sqrt{\operatorname{Var}(X)}} \] \[ \mathbb E[X] \;=\; \mathbb E[S]\,\mathbb E[M] \;=\; (2p-1)\,m \tag{D1.1} \] \[ \mathbb E[X^{2}] \;=\; \underbrace{\mathbb E[S^{2}]}_{=\,1}\,\mathbb E[M^{2}] \;=\; m^{2}+s_M^{2} \;=\; m^{2}\left(1+c^{2}\right) \tag{D1.2} \] \[ \operatorname{Var}(X) \;=\; \mathbb E[X^{2}]-\big(\mathbb E[X]\big)^{2} \;=\; m^{2}\left[1+c^{2}-(2p-1)^{2}\right] \tag{D1.3} \]

Discount 1 — per-trade Sharpe with dispersed magnitudes

\[ SR_{\text{trade}} \;=\; \frac{(2p-1)\,m}{m\sqrt{1+c^{2}-(2p-1)^{2}}} \;=\; \frac{2p-1}{\sqrt{1+c^{2}-(2p-1)^{2}}} \tag{D1.4} \]

Dispersion reaches \(SR_{\text{trade}}\) only through the variance.

Limits

\(c=0\) returns the idealized form of §01.
\(m\) cancels in (D1.4) — Sharpe is scale-invariant.
Near \(p=0.5\), \((2p-1)^{2}\) is negligible against \(1+c^{2}\), leaving \(SR_{\text{trade}}\approx 2e\big/\sqrt{1+c^{2}}\). Table 1.
As \(c\to\infty\), \(SR_{\text{trade}}\to 0\), while \(p\) and \(\mathbb E[X]\) are unchanged throughout.

D2 — Payoff asymmetry

Wins and losses keep fixed sizes, but no longer the same one:

\[ X = \begin{cases}+W & \text{w.p. } p\\[2pt] -L & \text{w.p. } 1-p\end{cases}, \qquad W,L>0, \qquad b=\frac{W}{L} \] \[ SR_{\text{trade}} \;=\; \frac{\mathbb E[X]}{\sqrt{\operatorname{Var}(X)}} \] \[ \mathbb E[X] \;=\; pW-(1-p)L \;=\; L\big[pb-(1-p)\big] \tag{D2.1} \] \[ \mathbb E[X^{2}] \;=\; pW^{2}+(1-p)L^{2} \;=\; L^{2}\big[pb^{2}+(1-p)\big] \tag{D2.2} \] \[ \operatorname{Var}(X) \;=\; L^{2}\Big\{pb^{2}+(1-p)-\big[pb-(1-p)\big]^{2}\Big\} \;=\; L^{2}\,p(1-p)\,(b+1)^{2} \tag{D2.3} \]

Discount 2 — per-trade Sharpe with asymmetric payoffs

\[ SR_{\text{trade}} \;=\; \frac{L\big[pb-(1-p)\big]}{L\,(b+1)\sqrt{p(1-p)}} \;=\; \frac{pb-(1-p)}{(b+1)\sqrt{p(1-p)}} \tag{D2.4} \]

Unlike the other discounts, (D2.4) is not a multiplier on \(2e\) — it replaces it. Setting the numerator to zero gives the break-even ratio \(b^{\ast}=(1-p)/p\). It is also the fair odds. So \(p\) and \(b\) trade off against each other. The larger the payoff, the smaller the hit rate needed.

Limits

\(b=1\) returns the idealized form of §01.
\(L\) cancels in (D2.4) — Sharpe is scale-invariant.
The numerator vanishes at \(b^{\ast}=(1-p)/p\): 1.500 at \(p=0.4\), 0.667 at \(p=0.6\).
As \(b\to\infty\), \(SR_{\text{trade}}\to\sqrt{p/(1-p)}\) — a finite ceiling, where D1 had no floor. Table 3.

D3 — Costs

Take the D1 trade and subtract a round-trip cost \(C>0\), paid on every trade whether it wins or loses. Express it as a fraction of the typical move, \(\kappa = C/m\):

\[ X = S\cdot M - C \] \[ \mathbb E[X] \;=\; (2p-1)m - C \;=\; m\big[(2p-1)-\kappa\big] \tag{D3.1} \] \[ \operatorname{Var}(X) \;=\; \operatorname{Var}(S\cdot M) \;=\; m^{2}\left[1+c^{2}-(2p-1)^{2}\right] \tag{D3.2} \]

Discount 3 — per-trade Sharpe net of costs

\[ SR_{\text{trade}} \;=\; \frac{(2p-1)-\kappa}{\sqrt{1+c^{2}-(2p-1)^{2}}} \tag{D3.3} \]

\(C\) is deterministic, so (D3.2) is (D1.3) unchanged: cost moves the numerator and nothing else. Every other discount is a multiplier on the denominator; this one is a subtraction from the edge, which is why it alone drives \(SR_{\text{trade}}\) through zero at a finite parameter value.

Limits

\(\kappa=0\) returns D1.
The numerator vanishes at the break-even hit rate \(p^{\ast}=0.5+\kappa/2\).
Against gross, the multiplier is \(1-\kappa/(2p-1)\): linear in \(\kappa\), zero at \(\kappa=2p-1\). Table 4.
A random \(C\) adds to (D3.2) as well, so variable slippage is worse than its own average.

D4 — Correlation across simultaneous positions

\[ k_{\text{eff}} = \dfrac{k}{1+(k-1)\rho}, \qquad \lim_{k\to\infty}k_{\text{eff}} = 1/\rho \]

\(k\) positions held at once, equally sized, pairwise correlation \(\rho\). Tabulated in §02, table 5.

Derivation — effective breadth and the \(1/\rho\) ceiling

Hold \(k\) positions at once, equally sized, each with return \(X_i\) of mean \(\mu\) and standard deviation \(\sigma\), and average pairwise correlation \(\operatorname{Corr}(X_i,X_j) = \rho\) for \(i\neq j\). The portfolio return is \(P = \sum_{i=1}^{k} X_i\).

Step 1 — the mean is linear in \(k\). Expectation is linear whatever the dependence:

\[ \mathbb E[P] = \sum_{i=1}^{k}\mathbb E[X_i] = k\mu \tag{D4.1} \]

Step 2 — the variance is not. There are \(k\) diagonal terms and \(k(k-1)\) off-diagonal ones, each \(\operatorname{Cov}(X_i,X_j) = \rho\sigma^2\):

\[ \operatorname{Var}(P) = \sum_i \operatorname{Var}(X_i) + \sum_{i\neq j}\operatorname{Cov}(X_i,X_j) = k\sigma^2 + k(k-1)\rho\sigma^2 = k\sigma^2\big[1+(k-1)\rho\big] \tag{D4.2} \]

This is the whole result in one line: the mean collects \(k\) terms and the variance collects \(k^2\) of them. Independence is the special case that kills the second sum.

Step 3 — divide. Taking (D4.1) over the square root of (D4.2) and pulling \(\mu/\sigma = SR_{\text{trade}}\) out:

\[ \begin{aligned} SR_{\text{portfolio}} &= \frac{k\mu}{\sigma\sqrt{k[1+(k-1)\rho]}} = \frac{\mu}{\sigma}\sqrt{\frac{k}{1+(k-1)\rho}} = SR_{\text{trade}}\sqrt{k_{\text{eff}}} \\[6pt] k_{\text{eff}} &\equiv \frac{k}{1+(k-1)\rho} \end{aligned} \tag{D4.3} \]

Step 4 — the limit. Divide numerator and denominator of \(k_{\text{eff}}\) by \(k\):

\[ k_{\text{eff}} = \frac{1}{1/k + \rho - \rho/k} \;\xrightarrow[k\to\infty]{}\; \frac{1}{\rho} \tag{D4.4} \]

and \(k_{\text{eff}}\) is increasing in \(k\), so the limit is a supremum — the ceiling is approached from below and never exceeded. The half-way point follows by setting \(k_{\text{eff}} = 1/2\rho\) in (D4.3):

\[ \frac{k}{1+(k-1)\rho} = \frac{1}{2\rho} \;\Longrightarrow\; 2k\rho = 1+(k-1)\rho \;\Longrightarrow\; k\rho = 1-\rho \;\Longrightarrow\; k = \frac{1}{\rho}-1 \tag{D4.5} \]

Limits

\(\rho = 0\): (D4.3) gives \(k_{\text{eff}} = k\) and the naive \(\sqrt N\) of From hit rate to Sharpe, §02 is recovered exactly.
\(\rho = 1\): \(k_{\text{eff}} = k/k = 1\) — \(k\) copies of the same bet are one bet, as they must be.

Annualizing, the effective trade count is \(N_{\text{eff}} = 252\,k_{\text{eff}}\) and \(SR_{\text{ann}} = SR_{\text{trade}}\sqrt{N_{\text{eff}}}\), which is how D4 enters the cascade. Note that (D4.3) is the same statement as Grinold's fundamental law with breadth read as \(k_{\text{eff}}\) rather than as a headcount.

02 Tables

Table 1 — dispersion drag

At p = 0.54; undiscounted baseline SR_trade = 0.0803, SR_ann = 2.85
c	SR_trade	Exact multiplier	1/√(1+c²)	SR_ann at N = 1,260
0 (equal)	0.0803	1.000	1.000	2.85
0.25	0.0778	0.970	0.970	2.76
0.50	0.0717	0.894	0.894	2.55
0.75	0.0641	0.799	0.800	2.28
1.00	0.0567	0.706	0.707	2.01
1.50	0.0444	0.553	0.555	1.58
2.00	0.0358	0.446	0.447	1.27

From D1, equation (D1.4). The exact multiplier and the \(1/\sqrt{1+c^{2}}\) approximation agree to three decimals throughout — the \((2p-1)^{2}\) term contributes almost nothing at a 54% hit rate.

Table 2 — coefficient of variation by magnitude law

Assume a law for the magnitude M; its first two moments give c = s_M/m
Distribution of \|move\|	\(m=\mathbb E[M]\)	\(s_M=\operatorname{sd}(M)\)	\(c=s_M/m\)	Multiplier	Sharpe lost
All moves equal	\(m\)	\(0\)	0	1.000	0%
Uniform on \([0,a]\)	\(a/2\)	\(a/2\sqrt3\)	0.577	0.866	13%
Half-normal \(\|N(0,\sigma)\|\)	\(\sigma\sqrt{2/\pi}\)	\(\sigma\sqrt{1-2/\pi}\)	0.756	0.798	20%
Exponential\((\lambda)\)	\(1/\lambda\)	\(1/\lambda\)	1.000	0.707	29%
Lognormal\((\mu,\sigma)\), \(\sigma=1\)	\(e^{\mu+\sigma^{2}/2}\)	\(m\sqrt{e^{\sigma^{2}}-1}\)	1.311	0.607	39%
Pareto\((x_m,\alpha)\), \(\alpha=2.5\)	\(\dfrac{\alpha x_m}{\alpha-1}\)	\(\dfrac{x_m\sqrt\alpha}{(\alpha-1)\sqrt{\alpha-2}}\)	0.894	0.745	25%

Each row assumes a distribution for \(|move|\), reads \(m\) and \(s_M\) off it, and divides to get \(c\) — which then fixes the drag through (D1.4). The scale parameter cancels in every row — \(a\), \(\sigma\), \(\lambda\) and \(x_m\) all drop out of \(s_M/m\) — so \(c\) is a property of the distribution's shape alone, not a free parameter and not something bet sizing can change. Half-normal already costs 20%; real trade-magnitude distributions are typically near-exponential or lognormal, putting \(c\approx1\) and a ~30% loss as the realistic default rather than a pessimistic case, which is why the cascade in §04 uses \(c=1\). The Pareto row is the one to read carefully: \(c=1/\sqrt{\alpha(\alpha-2)}\) is acutely sensitive to the tail index, running 0.577 at \(\alpha=3\) to 2.18 at \(\alpha=2.1\), and the variance does not exist at all for \(\alpha\le2\). Multipliers here use \(1/\sqrt{1+c^{2}}\); the exact form carrying \(-(2p-1)^{2}\) is table 1.

Table 3 — payoff ratio and break-even

Per-trade Sharpe by hit rate and win/loss ratio b = W/L
Hit rate p	b* = (1−p)/p	b = 0.5	b = 1	b = 2	b = 3	b → ∞
40%	1.500	−0.544	−0.204	+0.136	+0.306	0.816
45%	1.222	−0.436	−0.101	+0.235	+0.402	0.905
50%	1.000	−0.333	+0.000	+0.333	+0.500	1.000
54%	0.852	−0.254	+0.080	+0.415	+0.582	1.083
60%	0.667	−0.136	+0.204	+0.544	+0.714	1.225

From D2, equation (D2.4). The b = 1 column is §01. A 40% hit rate is profitable from \(b>1.5\) onward, and at \(b=3\) beats a 54% strategy paying even money — hit rate alone ranks these backwards. The last column is the \(b\to\infty\) ceiling \(\sqrt{p/(1-p)}\): unlimited payoff does not buy unlimited Sharpe.

Table 4 — cost drag

At p = 0.54 (2p−1 = 0.08), c = 1
κ	Net edge (2p−1)−κ	SR_trade	Multiplier 1−κ/(2p−1)	Break-even p*
0	0.080	0.0567	1.000	50.0%
0.01	0.070	0.0496	0.875	50.5%
0.02	0.060	0.0425	0.750	51.0%
0.04	0.040	0.0283	0.500	52.0%
0.06	0.020	0.0142	0.250	53.0%
0.08	0.000	0.0000	0.000	54.0%

From (D3.3). The multiplier falls linearly and reaches zero at \(\kappa=2p-1\); every other discount decays asymptotically and never crosses. Read the last column as the accuracy the costs demand: at \(\kappa=0.08\) a 54% hit rate is exactly break-even — right more often than not, and earning nothing.

Table 5 — effective breadth by position count and correlation

k_eff = k / [1 + (k−1)ρ]
Positions k	ρ = 0	ρ = 0.02	ρ = 0.05	ρ = 0.10	ρ = 0.20	ρ = 0.50
1	1.00	1.00	1.00	1.00	1.00	1.00
2	2.00	1.96	1.90	1.82	1.67	1.33
5	5.00	4.63	4.17	3.57	2.78	1.67
10	10.00	8.47	6.90	5.26	3.57	1.82
20	20.00	14.49	10.26	6.90	4.17	1.90
50	50.00	25.25	14.49	8.47	4.63	1.96
100	100.00	33.56	16.81	9.17	4.81	1.98
∞ (ceiling)	∞	50.00	20.00	10.00	5.00	2.00

From D4. Read down a correlated column: at \(\rho = 0.2\), going from 5 names to 100 — a twentyfold rise in nominal breadth — moves \(k_{\text{eff}}\) from 2.78 to 4.81, and it can never pass 5. Since Sharpe scales as \(\sqrt{k_{\text{eff}}}\), that entire expansion is worth a factor of 1.32.

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