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Make the accounting identity your first test

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Start with the idea

Signals are intentions; positions and fills create financial results. A good simulator can explain every dollar through an accounting identity before it reports a Sharpe ratio.

Symbols, units & horizon

wₜ₋₁: capital weights held before current returns
Rₜ: aligned asset return vector
T: transpose for the weighted sum
R_net: portfolio return after all listed costs
c_trade,c_borrow,c_funding: costs as fractions of NAV for the same period
w_target: intended posttrade weights
w_pretrade: weights after price drift and before trading
Δw: target minus pretrade weight
cᵢ: cost per unit weight traded
| |: absolute value, making sales and purchases count positively

When and why to use this

Use the identity to reconcile trade-level and portfolio-level results and to inspect a suspected timing bug. Turnover connects signal instability to actual economic drag.

A strategy must translate forecasts into positions and then into cash flows. Use beginning-of-period positions for the returns earned during that period. Costs follow actual trades, including exits and changes in size, rather than just the existence of a signal.

R_{t}^{net} = w_{t - 1}^{𝖳} R_{t} - c_{t}^{trade} - c_{t}^{borrow} - c_{t}^{funding}

Algebra and arithmetic

Express the cash ledger as a return

Beginning positions $q_{i, t - 1}$ earn $q_{i, t - 1} Δ P_{i, t}$ , including correctly accounted distributions. Divide by prior NAV and recognise $\frac{q P}{N A V} = w$ to obtain $w^{T} R$ .
Divide each dollar expense by the same NAV, then subtract trading, borrow and funding costs. Financing on cash and derivatives needs a consistent ledger convention.

Work it by hand

A .5 weight in an asset returning 2% contributes 1% of NAV. Costs of .08%+.02%+.01% leave .89% net.

{turnover}_{t} = \sum_{i} | w_{i, t}^{target} - w_{i, t}^{pretrade} |, c_{t}^{trade} = \sum_{i} c_{i, t} | Δ w_{i, t} |

Algebra and arithmetic

Translate traded weight into costs

Actual rebalancing change is target weight minus drifted pretrade weight. Use its magnitude for traded notional/NAV: $| Δ w_{i} |$ .
Sum magnitudes for turnover. Multiply each by its one-way cost rate and sum for portfolio cost. A move from +.2 to −.1 trades .3 of NAV.

Work it by hand

Trades of .3 and .1 of NAV at rates 10 bp and 20 bp cost .3×10+.1×20=5 bp of NAV; total absolute turnover=.4.

Weights are relative to NAV. Pretrade weights drift with prices; comparing targets to yesterday’s targets misses this drift. This turnover definition is total absolute traded weight; some reports divide it by two. Quote the convention. The cost rate is per traded dollar and includes the appropriate spread, commission, and impact estimates.

Test a constant-price path: the only loss should be explicitly modelled costs.
Test a single position by hand, then confirm the portfolio agrees with the cash and position ledgers.
Add the initial capital observation when measuring drawdown; otherwise an immediate first-period loss can disappear.
Stress costs, delayed fills, unavailable borrow, stale prices, and halved liquidity before accepting the model.

Research sources, review dates and limitations

Extend the research question

Keep an experiment ledger containing all trials, preprocessing choices, source versions and failed hypotheses. Reproduce one small result before escalating complexity.

Continue with the connected research module →

Connect the ideas: Cash flows and accounting

Retrieve: Track units, signed cash movements and ownership at each event.

Check the change: Instrument obligations, financing and external capital flows change the ledger you need.

Math & notation → Markets & returns → Trading different assets → Execution & microstructure → Fund operations & capstone → Putting it all together

Explain it yourself: Why can an account balance rise without an investment profit?

Self-assessed. Write your explanation before opening this comparison.

A deposit raises the balance without being investment P&L. Reconcile external flows separately from fills, fees and marked holdings.

Connect the ideas: Information and decision time

Retrieve: Use only information available when the decision is made.

Check the change: Observation dates, release delays, revisions and label maturity require different availability checks.

Statistics → Point-in-time data → Time series → Research & robust tuning → Financial machine learning → Putting it all together

Explain it yourself: Does shifting a feature by one row guarantee that it was available?

Self-assessed. Write your explanation before opening this comparison.

No. A revised value or delayed release may still contain unavailable information. Audit actual availability timestamps and fit preprocessing inside each training window.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def portfolio_net_return(weights, asset_returns, trade_cost, borrow_cost, funding_cost):
    if len(weights) != len(asset_returns):
        raise ValueError("Asset arrays must align")
    gross = sum(w*r for w,r in zip(weights,asset_returns))
    return gross-trade_cost-borrow_cost-funding_cost

def turnover_cost(target, pretrade, unit_costs):
    if not len(target) == len(pretrade) == len(unit_costs):
        raise ValueError("Asset arrays must align")
    changes = [abs(a-b) for a,b in zip(target,pretrade)]
    return sum(changes), sum(c*dw for c,dw in zip(unit_costs,changes))

print(turnover_cost([.6,.4], [.5,.5], [.001,.001]))

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