Brownian Bridge™

1. A strategy earns a daily excess return series with mean $\bar{r} = 0.05\%$ and standard deviation $\hat{\sigma} = 1\%$ over $T = 1260$ trading days (5 years). Compute the annualised Sharpe ratio and the t-statistic for $H_0: \text{SR} = 0$ (assume Gaussian returns, $\kappa = 0$ ).

Annualised SR

= 0.79

; t-statistic

\approx 1.77

Annualised SR

= 0.79

; t-statistic

\approx 2.00

Annualised SR

= 0.79

; t-statistic

\approx 1.26

Annualised SR

= 1.26

; t-statistic

\approx 2.00

2. A researcher tests 100 trading strategies on the same historical dataset. 8 strategies have p-values below 0.05. Applying the Benjamini-Hochberg procedure at FDR = 5%, what is the BH threshold for the $k$ -th ranked p-value, and approximately how many strategies would BH expect to reject under the global null (all SR = 0)?

BH threshold for rank

k

k/(100 \times 0.05) = k/5

; expect 5 rejections under global nullBH threshold for rank

k

k \times 0.05/100 = k/2000

; under global null (all SR=0), BH controls FDR

\leq 5\%

, so expected false discoveries

\leq 0.4

BH threshold for rank

k

k \times 0.05/100

; under global null BH expects exactly 0 rejectionsBH threshold for rank

k

k/(100 \times 0.05)

; expect 100 rejections since all tests are run

3. A fund manager reports a 3-year backtest with annualised SR = 2.5, and discloses that 80 parameter combinations were tried before selecting the reported strategy. The average pairwise correlation of all 80 strategies is 0.6. Estimate the effective number of independent tests and assess whether SR = 2.5 is credible.

Effective tests

N_{\text{eff}} = 80

; SR = 2.5 is credible since it exceeds the expected max of 3.0Effective tests

N_{\text{eff}} = 80/(1+79\times0.6) \approx 1.6

; SR = 2.5 is credible since expected max SR with 1.6 independent tests is lowEffective tests

N_{\text{eff}} = 80/(1+79\times0.6) \approx 1.6

; but since

N_{\text{eff}}

is barely above 1, the 80 strategies were essentially all the same strategy with minor variations — SR = 2.5 still needs 3 years of data to establish significance at 5%Effective tests

N_{\text{eff}} = 80

; expected max SR

\approx \sqrt{2\ln 80} \approx 3.0

; SR = 2.5 < 3.0 so it is not significant even before adjusting for correlation

4. Using end-of-day closing prices to simulate a strategy that rebalances at the close is free of look-ahead bias, provided you use prices strictly from the current day's close and no later data.

truefalse

5. The Deflated Sharpe Ratio (Bailey & López de Prado 2014) requires three inputs beyond the observed Sharpe: $N$ (strategies tried), $\bar{\rho}$ (average pairwise correlation), and the return distribution moments (skewness, excess kurtosis). Why is $\bar{\rho}$ required — what does it capture that $N$ alone does not?

\bar{\rho}

controls for the fact that correlated strategies share training data, which inflates the backtest SR

\bar{\rho}

determines the effective number of independent tests: high

\bar{\rho}

means fewer independent bets on the true data, so the expected maximum SR over

N

correlated trials is lower than over

N

independent trials — adjusting for

\bar{\rho}

prevents the DSR from over-penalising searches over similar strategies

\bar{\rho}

measures the correlation of strategy returns with the market, controlling for beta

\bar{\rho}

is needed to compute the Shanken correction for errors-in-variables

6. A walk-forward backtest is conducted with training window 3 years, test window 1 year, rolling forward annually. The strategy's OOS Sharpe over 5 years of testing is 0.8. An in-sample Sharpe over the full training period is 1.5. What additional analysis would a rigorous quant conduct before concluding the strategy is viable?

None — OOS SR > 0 is sufficient; the strategy is viableCompute the t-statistic on the OOS SR (5 years × 252 = 1260 obs); test for autocorrelation in OOS returns; decompose OOS performance by sub-period to check for regime concentration; estimate net-of-cost SR; run the DSR using the number of strategies and parameter combinations tried during developmentOnly compute the t-statistic — OOS testing already controls for multiple hypothesis testingVerify that the IS SR is higher than OOS SR — this is expected and confirms no look-ahead bias

Quiz: Backtesting and Statistical Testing

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