diff --git a/alphalens/sharpe_test.py b/alphalens/sharpe_test.py new file mode 100644 index 00000000..1d399c48 --- /dev/null +++ b/alphalens/sharpe_test.py @@ -0,0 +1,141 @@ +import numpy as np +import scipy.integrate as integrate + + +def _compute_vhat(ret1, ret2): + nu = [np.mean(ret1), np.mean(ret2), np.mean(ret1 ** 2), np.mean(ret2 ** 2)] + nux = np.vstack([ret1 - nu[0], ret2 - nu[1], ret1 ** 2 - nu[2], ret2 ** 2 - nu[3]]).T + return nux + + +def _ar2(y, nlag, const=1): + # ols estimates for the AR(k) model + results = {} + n = y.shape[0] + results['y'] = y + results['negs'] = 1 + cols = [] + if const == 1: + cols.append(np.ones(n)) + for i in range(nlag): + cols.append(np.roll(y, i + 1)) + + x = np.vstack(cols).T + x = x[nlag:, :] + y = y[nlag:] + n_adj = len(y) + + b0 = np.linalg.lstsq(x.T.dot(x), x.T.dot(y), rcond=None)[0] + p = nlag + const + sige = ((y - x.dot(b0)).T.dot(y - x.dot(b0))) / (n - p + 1) + + results['meth'] = 'ar' + results['beta'] = b0 + results['sige'] = sige + results['yhat'] = x.dot(b0) + results['nobs'] = n + results['nadj'] = n_adj + results['nvar'] = nlag * const + results['x'] = x + return results + + +def _compute_alpha(v_hat): + t = v_hat.shape[0] + p = v_hat.shape[1] + numerator = 0 + denominator = 0 + for i in range(p): + results = _ar2(v_hat[:, i], 1) + rho_hat = results['beta'][1] + sig_hat = np.sqrt(results['sige']) + numerator = numerator + 4 * rho_hat ** 2 * sig_hat ** 4 / (1 - rho_hat) ** 8 + denominator = denominator + sig_hat ** 4 / (1 - rho_hat) ** 4 + + alpha_hat = numerator / denominator + return alpha_hat + + +def _gamma_hat(v_hat, j): + t = v_hat.shape[0] + p = v_hat.shape[1] + gamma_hat = np.zeros((p, p)) + if j >= t: + raise Exception('j must be smaller than the row dimension!') + else: + for i in range(j, t): + gamma_hat = gamma_hat + v_hat[i, :].reshape(-1, 1).dot(v_hat[i - j, :].reshape(1, -1)) + gamma_hat = gamma_hat / t + return gamma_hat + + +def _kernel_type(x, type): + if type == 'G': + if x < 0.5: + wt = 1 - 6 * x ** 2 + 6 * x ** 3 + elif x < 1: + wt = 2 * (1 - x) ** 3 + else: + wt = 0 + return wt + elif type == 'QS': + term = 6 * np.pi * x / 5 + wt = 25 * (np.sin(term) / term - np.cos(term)) / (12 * np.pi ** 2 * x ** 2) + return wt + else: + raise Exception('wrong type') + + +def _compute_psi(v_hat): + t = v_hat.shape[0] + alpha_hat = _compute_alpha(v_hat) + ss_star = 2.6614 * (alpha_hat * t) ** 0.2 + psi_hat = _gamma_hat(v_hat, 0) + j = 1 + while j < ss_star: + gamma = _gamma_hat(v_hat, j) + psi_hat = psi_hat + _kernel_type(j / ss_star, "G") * (gamma + gamma.T) + j += 1 + psi_hat = (t / (t - 4)) * psi_hat + return psi_hat + + +def _compute_se(ret1, ret2): + t = len(ret1) + mu1_hat = np.mean(ret1) + mu2_hat = np.mean(ret2) + gamma1_hat = np.mean(ret1 ** 2) + gamma2_hat = np.mean(ret2 ** 2) + gradient = np.zeros((4, 1)) + gradient[0] = gamma1_hat / (gamma1_hat - mu1_hat ** 2) ** 1.5 + gradient[1] = -gamma2_hat / (gamma2_hat - mu2_hat ** 2) ** 1.5 + gradient[2] = -0.5 * mu1_hat / (gamma1_hat - mu1_hat ** 2) ** 1.5 + gradient[3] = 0.5 * mu2_hat / (gamma2_hat - mu2_hat ** 2) ** 1.5 + v_hat = _compute_vhat(ret1, ret2) + psi_hat = _compute_psi(v_hat) + se = np.sqrt(gradient.T.dot(psi_hat).dot(gradient) / t) + return se + + +def sharpe_hac(ret1, ret2): + """ + This method performs a sharpe test on two return sequences, + returns: + diff: the difference in sharpe (sharpe1 - sharpe2) + pval: P-value under the assumption that H0 holds， H0: there is no difference between these two sharpes + + + For mathematical principles, please refer to the paper: http://www.ledoit.net/jef_2008pdf.pdf + """ + mu1_hat = np.mean(ret1) + mu2_hat = np.mean(ret2) + sig1_hat = np.std(ret1) + sig2_hat = np.std(ret2) + sr1_hat = mu1_hat / sig1_hat + sr2_hat = mu2_hat / sig2_hat + diff = sr1_hat - sr2_hat + se = _compute_se(ret1, ret2)[0, 0] + func = lambda x: (1 / np.sqrt(np.pi * 2)) * np.exp(-0.5 * x ** 2) + pval, err = integrate.quad(func, -1000, -abs(diff) / se) + pval = 2 * pval + return diff, pval diff --git a/alphalens/tests/test_sharpe_test.py b/alphalens/tests/test_sharpe_test.py new file mode 100644 index 00000000..9c8f0048 --- /dev/null +++ b/alphalens/tests/test_sharpe_test.py @@ -0,0 +1,407 @@ +from unittest import TestCase +import random +from ..sharpe_test import sharpe_hac +import numpy as np + + +class SharpeTestTestCase(TestCase): + + def choose_random(self, total_data, size): + res = [] + for i in range(size): + res.append(total_data[random.randint(0, len(total_data) - 1)]) + return res + + def test_false_negative(self): + """ + This method checks the probability of making false-negative errors. If the method is effective, + the probability of making a mistake should be close to 1-a in the case of a certain confidence level a. + The two return series used for testing are from SPY’s daily return data for the last 5 years, + with a sample size of 400 + + """ + spy_daily_rets = [-0.013919958838530699, 0.005853027750837958, 0.0044375278402715157, -0.0039632195264346359, + 0.012759817845710009, 0.0086145490161160243, -0.0021462779457527281, -0.01659340355533423, + 0.0052233854734780039, -0.012982218667322964, -0.0016490802552390216, -0.0034625449137485331, + -0.00041580988944023289, -0.025511715900802368, 0.001327399991813305, 0.018710800172858866, + 0.002396211921440905, 0.014731785428342636, 0.018102301674970178, -0.0035760152340462703, + 0.0082443528405238276, 0.0088712076964614361, 0.00089570498536284404, 0.0010913456291608004, + -0.0067916712089829412, -0.0046977729482915453, 0.01525711575998856, 0.0040515953835709162, + 0.00088710506244882303, -0.0013294782068258293, 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-0.020799767711140227, 0.0068382435369376271, + 0.025317501364825024, -0.017924637356247386, -0.030436302415500327, 0.022529366536312212, + -0.00032380273937104853, 0.014190752797302197, 0.014378992193044615, -0.0046667506261981462, + 0.025646883501219797, -0.0092798519610025609, -0.028290563701328653, 0.0053714830846338657, + 0.0088833970053499023, -0.0066749489376936033, 0.010835771116858073, 0.017363890238404123, + 0.00064930826339226755, -0.020304324539599183, -0.017234732998310909, 0.011999172716769646, + 0.0042424092938362357, 0.030497710566481206, -0.01010224823542627, 0.016953874729130014, + -0.0073429187002593821, 0.0021704464918497202, 0.012695282617934511, 0.01440614518950234, + -0.0017143802746979464, 0.0039617812433694333, 0.0046693326552784686, 0.008345357792013619, + 0.013240758849990319, -0.0028824933567535016, 0.025601233763300479, 0.012432325304123415, + -0.0075473925918101425, -0.0056104494621230794, -0.057756004181202414, 0.012256572704064661, + 0.0090712229645868447, 0.019219193677130653, -0.0043458863329258746, 0.00054480101868126241, + -0.0057801233350732728, 0.0067405535031435093, 0.0044099658790959229, -0.025318078389898302, + 0.010390293624437108, -0.02365843340167284, 0.014732351176588221, 0.013631585862567297, + 0.0063838750445575076, 0.0052163833075733468, 0.015407777564225844, -0.010378876305246254, + 0.0077462543831685604, -0.0054091671147944442, 0.01024107881178038, -0.0089409394282835564, + 0.012992376111816917, 0.0093449151745117476, -0.0032621865970733355, 0.0026650458200860871, + 0.0082847594621899834, 0.001942406117037887, 0.0057236052558695683, -0.011932809105651199, + -0.0064410243706065629, 0.0072931276297334424, -0.006188310281877496, 0.012173479871727011, + -0.0034450938172869705, 0.0078708562256928349, 0.006890637919946041, 0.0037715189488412015, + 0.006272347130476863, 0.0066849348068296344, 0.00071789656307030825, 0.0029890898221491824, + -0.0081060944717628036, 0.013820869512964684, -0.0017188750259312657, 5.9373608430890457e-05, + 0.0029685041707483428, 0.0021605943114215709, -0.0041051388068518158, 0.0031137866611310461, + 0.0035180039023237075, 0.010222418618353268, 0.0034410358101015781, 0.010055216506829279, + 0.0027477270111635477, 0.0058390600117643743, -0.0022250748823278155, 0.0080338508162507427, + 0.014606614101764226, -0.034020071003270647, -0.0088262530385461169] + + total_test_count = 5000 + reject_count = 0 + alpha = 0.05 + for i in range(total_test_count): + ret1 = np.array(self.choose_random(spy_daily_rets, 400)) + ret2 = np.array(self.choose_random(spy_daily_rets, 400)) + diff, pval = sharpe_hac(ret1, ret2) + if pval <= alpha: + reject_count += 1 + + percent = reject_count / total_test_count + print("the reject percent:" + str(percent)) + assert abs(percent - alpha) < 0.02 + + def test_definite_value(self): + """ + This sample value and result is taken from the paper: http://www.ledoit.net/jef_2008pdf.pdf + """ + ret1 = np.array( + [3.92670000, -2.3305000, -4.8583000, 1.96000000, -0.4397300, -2.7056000, 2.92570000, 3.87810000, -2.5249000, + 1.74590000, -4.0699000, 0.84013000, -0.5617100, 2.84340000, 3.11080000, 1.91650000, 1.24070000, 2.70060000, + 4.01650000, 1.19310000, 2.44220000, -1.6150000, 3.69480000, 2.07020000, 2.02090000, 0.83481000, 1.44590000, + 1.20520000, 1.51500000, 0.06133600, -4.8483000, 2.44980000, 4.55040000, 1.21710000, 5.40700000, -2.5487000, + 3.72540000, 0.54106000, -5.4384000, 5.18970000, 4.95600000, 4.63930000, 7.65870000, -5.7242000, 4.80640000, + -3.2255000, 3.90520000, 1.86220000, 0.10583000, 6.23280000, 4.70720000, 0.34273000, -1.0273000, 4.33750000, + 0.03111600, -16.625000, 3.97890000, 6.78550000, 5.69550000, 7.71210000, 3.20710000, -2.4622000, 3.89690000, + 1.76310000, -3.5363000, 5.06140000, -3.5352000, -1.9028000, -2.5466000, 3.85620000, 3.63410000, 9.75480000, + -4.8312000, 2.90380000, 3.82400000, -6.4514000, -5.0871000, 5.19070000, -2.5309000, 6.15040000, -5.8471000, + -0.8204800, -11.711000, 2.15650000, 2.76750000, -10.242000, -8.8416000, 12.0670000, 0.23470000, -0.6506600, + -5.3993000, -5.2209000, -10.378000, 2.62880000, 5.75680000, 2.07200000, -1.6054000, -2.5013000, 2.24750000, + -6.0060000, -0.4965000, -6.5602000, -8.6304000, -0.0425940, -10.261000, 8.24590000, 4.83680000, -5.7832000, + -3.3307000, -0.8245600, 1.73950000, 7.23180000, 3.48930000, 0.94715000, 2.39770000, 1.60400000, -1.8021000, + 5.55900000, 0.58602000, 5.52010000]) + ret2 = np.array([2.64760000000000, -0.816280000000000, -5.72400000000000, 0.0474760000000000, -4.24160000000000, + -7.59120000000000, 3.33910000000000, 6.75050000000000, -0.843120000000000, 4.28130000000000, + -4.41420000000000, 2.29520000000000, -3.21210000000000, 4.18450000000000, 3.66220000000000, + 2.78660000000000, 3.57730000000000, 10.0630000000000, 10.7530000000000, 0.906150000000000, + 1.90050000000000, -1.66430000000000, -1.68610000000000, -5.91620000000000, 0.996660000000000, + 5.41830000000000, -0.403410000000000, 5.81890000000000, 3.37120000000000, -4.26580000000000, + -10.7530000000000, 3.69750000000000, 7.75710000000000, -3.00710000000000, 4.93610000000000, + -3.78720000000000, 6.32270000000000, -5.64270000000000, -7.06980000000000, 1.74900000000000, + 8.63520000000000, 2.33160000000000, 9.29360000000000, -1.61820000000000, 5.63140000000000, + -8.20500000000000, 0.500030000000000, 1.09180000000000, -1.07180000000000, 8.75430000000000, + 5.11080000000000, 1.20070000000000, -4.22400000000000, 9.33900000000000, 0.463230000000000, + -21.9250000000000, 11.4950000000000, 2.99960000000000, 6.78450000000000, 12.2250000000000, + 10.2200000000000, -5.98690000000000, 12.7810000000000, 6.38120000000000, -4.78080000000000, + 8.73820000000000, -1.53230000000000, 5.41220000000000, -2.88540000000000, 10.4790000000000, + 10.8260000000000, 16.9360000000000, -0.699960000000000, 15.9100000000000, -2.98170000000000, + -17.0610000000000, -12.1860000000000, 16.9480000000000, -7.93020000000000, 14.7800000000000, + -10.2540000000000, -12.1910000000000, -26.5170000000000, 6.52520000000000, 1.67180000000000, + -22.6160000000000, -23.9400000000000, 19.0280000000000, -3.57650000000000, -11.1930000000000, + -13.3670000000000, -15.1100000000000, -23.9460000000000, 15.6860000000000, 10.0070000000000, + 0.0186130000000000, -7.06580000000000, -10.8050000000000, 4.75810000000000, -10.2730000000000, + -7.13690000000000, -16.3760000000000, -7.77480000000000, -3.71390000000000, -14.6970000000000, + 13.9630000000000, 13.4860000000000, -9.00860000000000, -0.636020000000000, -1.54890000000000, + 1.80770000000000, 4.97740000000000, 7.06600000000000, 1.26390000000000, 2.70540000000000, + 3.73190000000000, -1.55510000000000, 5.91610000000000, 2.27750000000000, 1.82370000000000]) + expected_p_val = 0.0646 + diff, pval = sharpe_hac(ret1, ret2) + assert abs(pval - expected_p_val) < 0.001