Davinchu blog

Sharpe ratio, Z-score, AHPR and GHPR explained

Formulas and examples for the Sharpe ratio, Z-score (runs test), AHPR and GHPR in trading account statistics: what each metric means and how to read them together.

In short. The Sharpe ratio shows how steadily a strategy earns relative to the spread of its results. Z-score shows whether neighbouring trades depend on each other (streaks or alternation). AHPR and GHPR are the average return per trade: arithmetic and geometric, i.e. with compounding. All four are in the MetaTrader 5 strategy tester report and in monitoring services, but they are calculated differently β€” only compare values from the same method.

Sharpe ratio

The classic Sharpe ratio is excess return (above the risk-free rate) divided by the standard deviation of returns, usually annualised. In trading account statistics it is often calculated more simply β€” per trade: average trade result / standard deviation of results.

Example: average result +$12.5, standard deviation $110 β€” Sharpe β‰ˆ 0.11 per trade. The higher the value, the less the result depends on lucky outliers. A per-trade Sharpe and a fund's annual Sharpe are different numbers and cannot be compared.

Z-score

A runs test: how far the sequence of wins and losses is from random.

Z = (N Γ— (R βˆ’ 0.5) βˆ’ P) / √(P Γ— (P βˆ’ N) / (N βˆ’ 1)), where N is the number of trades, R the number of runs (consecutive trades of the same sign), P = 2 Γ— W Γ— L, W and L the number of winners and losers.

  • Z below βˆ’2 β€” fewer runs than random: wins tend to follow wins, losses follow losses.
  • Z above 2 β€” more runs: results tend to alternate.
  • Between βˆ’2 and 2 β€” no significant dependency. The Β±2 threshold corresponds to about 95% confidence.

Below 30 trades Z-score says almost nothing. A dependency is a reason to test changing the size after a win or a loss β€” but only on a long history.

AHPR and GHPR

A trade's holding period return HPR = 1 + result / balance before the trade. AHPR is the arithmetic mean of HPRs, GHPR the geometric mean: (HPR₁ Γ— HPRβ‚‚ Γ— … Γ— HPRβ‚™)^(1/n). They are usually shown in percent: (mean βˆ’ 1) Γ— 100.

Example: two trades, +10% and βˆ’10%. AHPR = 0% β€” Β«break-even on averageΒ». GHPR = √(1.1 Γ— 0.9) βˆ’ 1 β‰ˆ βˆ’0.5% per trade, and the account really lost 1%: 1.1 Γ— 0.9 = 0.99. GHPR is never above AHPR, and the bigger the spread of results, the bigger the gap. The account grows at the GHPR rate.

90%95%100%105%110%012345678910Balance vs the startTrade numberβˆ’1%βˆ’4.9%Balance: +10% and βˆ’10% in turnAHPR: 0%GHPR: βˆ’0.5%
The arithmetic mean (AHPR) promises zero, while the balance follows the GHPR line down: every pair of trades is 1.1 Γ— 0.9 = 0.99, a 1% loss. The account grows and shrinks at the GHPR rate.

Reading them together

MetricAnswers the question
Sharpehow stable the result is relative to its spread
Z-scorewhether trades depend on each other
AHPRaverage return per trade without compounding
GHPRhow fast the account actually grows or shrinks

None of them replaces profit factor, expectancy and drawdown β€” they are extra quality checks.

How Davinchu calculates them

In Davinchu these metrics are on the Β«TradesΒ» tab of the account page, across every closed trade of the selected period and from net results, including commission and swap:

  • Sharpe Ratio β€” the average trade result divided by the standard deviation of results, with no risk-free rate and not annualised;
  • Z-Score β€” by the runs test with the formula above, together with a probability;
  • AHPR / GHPR β€” from the balance before each trade, adjusted for deposits and withdrawals; on exchange futures accounts Funding for the period is shown in their place.

That makes the values comparable across your accounts and periods: the method is the same. On the Free plan two accounts connect at no cost.