Question: Calculate Jack Henry & Associates's Information Ratio from September 16, 2023 to January 30, 2024. (Annualized (US standard uses arithmetic mean excess return); 252 trading days per year) Using unadjusted prices, write down the indicator formula, collect the raw data, calculate the answer, and keep four decimal places. Please list the raw data and key calculation process.
Formula: IR = E[r - r_b] * 252 / TE; TE = \sigma(r - r_b) * \sqrt{252}; r_b is the benchmark daily return
Data:
date,close,benchmark_close
2023-09-15,154.49000549316406,4450.31982421875
2023-09-18,155.19000244140625,4453.52978515625
2023-09-19,152.69000244140625,4443.9501953125
2023-09-20,152.8800048828125,4402.2001953125
2023-09-21,148.61000061035156,4330.0
2023-09-22,147.88999938964844,4320.06005859375
2023-09-25,147.74000549316406,4337.43994140625
2023-09-26,147.5399932861328,4273.52978515625
2023-09-27,148.44000244140625,4274.509765625
2023-09-28,151.6699981689453,4299.7001953125
2023-09-29,151.13999938964844,4288.0498046875
2023-10-02,149.8699951171875,4288.39013671875
2023-10-03,146.63999938964844,4229.4501953125
2023-10-04,147.5,4263.75
2023-10-05,148.5800018310547,4258.18994140625
2023-10-06,148.97000122070312,4308.5
2023-10-09,147.8000030517578,4335.66015625
2023-10-10,148.82000732421875,4358.240234375
2023-10-11,147.17999267578125,4376.9501953125
2023-10-12,144.4600067138672,4349.60986328125
2023-10-13,145.42999267578125,4327.77978515625
2023-10-16,147.89999389648438,4373.6298828125
2023-10-17,149.02999877929688,4373.2001953125
2023-10-18,147.0800018310547,4314.60009765625
2023-10-19,145.61000061035156,4278.0
2023-10-20,143.1199951171875,4224.16015625
2023-10-23,143.10000610351562,4217.0400390625
2023-10-24,143.55999755859375,4247.68017578125
2023-10-25,141.9600067138672,4186.77001953125
2023-10-26,140.3300018310547,4137.22998046875
2023-10-27,138.10000610351562,4117.3701171875
2023-10-30,139.7899932861328,4166.81982421875
2023-10-31,140.99000549316406,4193.7998046875
2023-11-01,140.35000610351562,4237.85986328125
2023-11-02,141.92999267578125,4317.77978515625
2023-11-03,144.85000610351562,4358.33984375
2023-11-06,145.36000061035156,4365.97998046875
2023-11-07,143.42999267578125,4378.3798828125
2023-11-08,149.94000244140625,4382.77978515625
2023-11-09,149.2100067138672,4347.35009765625
2023-11-10,147.99000549316406,4415.240234375
2023-11-13,150.32000732421875,4411.5498046875
2023-11-14,153.13999938964844,4495.7001953125
2023-11-15,153.83999633789062,4502.8798828125
2023-11-16,153.41000366210938,4508.240234375
2023-11-17,152.3000030517578,4514.02001953125
2023-11-20,153.6699981689453,4547.3798828125
2023-11-21,152.9199981689453,4538.18994140625
2023-11-22,153.08999633789062,4556.6201171875
2023-11-24,153.8000030517578,4559.33984375
2023-11-27,153.99000549316406,4550.43017578125
2023-11-28,155.13999938964844,4554.89013671875
2023-11-29,156.00999450683594,4550.580078125
2023-11-30,158.69000244140625,4567.7998046875
2023-12-01,160.7899932861328,4594.6298828125
2023-12-04,162.67999267578125,4569.77978515625
2023-12-05,161.3300018310547,4567.18017578125
2023-12-06,165.39999389648438,4549.33984375
2023-12-07,165.14999389648438,4585.58984375
2023-12-08,164.69000244140625,4604.3701171875
2023-12-11,164.3800048828125,4622.43994140625
2023-12-12,166.1199951171875,4643.7001953125
2023-12-13,168.1699981689453,4707.08984375
2023-12-14,166.89999389648438,4719.5498046875
2023-12-15,164.36000061035156,4719.18994140625
2023-12-18,166.27999877929688,4740.56005859375
2023-12-19,165.25,4768.3701171875
2023-12-20,162.82000732421875,4698.35009765625
2023-12-21,163.27000427246094,4746.75
2023-12-22,165.0800018310547,4754.6298828125
2023-12-26,165.55999755859375,4774.75
2023-12-27,163.24000549316406,4781.580078125
2023-12-28,163.8800048828125,4783.35009765625
2023-12-29,163.41000366210938,4769.830078125
2024-01-02,164.66000366210938,4742.830078125
2024-01-03,163.42999267578125,4704.81005859375
2024-01-04,166.6699981689453,4688.68017578125
2024-01-05,166.72000122070312,4697.240234375
2024-01-08,169.6999969482422,4763.5400390625
2024-01-09,167.19000244140625,4756.5
2024-01-10,167.61000061035156,4783.4501953125
2024-01-11,167.27000427246094,4780.240234375
2024-01-12,167.58999633789062,4783.830078125
2024-01-16,165.17999267578125,4765.97998046875
2024-01-17,165.75999450683594,4739.2099609375
2024-01-18,165.5399932861328,4780.93994140625
2024-01-19,166.38999938964844,4839.81005859375
2024-01-22,168.80999755859375,4850.43017578125
2024-01-23,168.0500030517578,4864.60009765625
2024-01-24,166.6300048828125,4868.5498046875
2024-01-25,168.2899932861328,4894.16015625
2024-01-26,169.57000732421875,4890.97021484375
2024-01-29,169.5399932861328,4927.93017578125
2024-01-30,169.4499969482422,4924.97021484375

Indicator data:
Answer: -0.0836