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1846Residual After 2 Rebalances 1A residual spread follows X (t+1) = 2/3 X t + epsilon (t+1) with E[epsilon (t+1)] = 0. If today's residual is 9 bp, what is E[X 2 | X 0 = 9 bp]?统计简单derivation未尝试免费1851Cumulative Mean-Reversion Carry 1A desk is short a positive residual and books one unit of carry each day equal to that day's expected residual. If X (t+1) = 3/4 X t + epsilon (t+1) with zero-mean shocks and X 0 = 12 bp, what is the total expected carry over the next 3 days?统计简单essay未尝试免费1856Horizon Risk Budget Ratio 1A stationary mean-reverting spread obeys X (t+1) = 1/2 X t + epsilon (t+1), where Var(epsilon (t+1)) = 4. Starting from the current level, what fraction of the same-horizon random-walk forecast-error variance does the 4-step mean-reverting forecast-error variance represent?统计简单derivation未尝试免费1861Long-Run Residual Variance 1A mean-reverting residual follows X (t+1) = 1/2 X t + epsilon (t+1) with Var(epsilon (t+1)) = 4. What is the stationary variance of X t?统计简单essay未尝试免费1866RW Versus MR Diagnosis 1A residual's variance keeps growing roughly linearly with horizon and never appears to plateau. Which description fits better: random walk or mean reversion?统计简单essay未尝试免费1867RW Versus MR Diagnosis 2After a 10 bp shock, the expected residual is only 2 bp four days later. Does that behavior point more toward random walk or mean reversion?统计简单数值题未尝试免费1868RW Versus MR Diagnosis 3A five-day variance ratio comes in well below 1. What does that suggest about serial dependence in returns?统计中等essay未尝试面试订阅1869RW Versus MR Diagnosis 4If a spread's conditional mean always equals today's level, regardless of horizon, which model is the closer description?统计中等derivation未尝试面试订阅1870RW Versus MR Diagnosis 5A desk notices that expected residual carry from holding a spread for longer horizons quickly saturates instead of growing linearly forever. Is that more consistent with random walk or mean reversion?统计困难derivation未尝试面试订阅6035Half-Life of a Mean-Reverting SpreadA residual spread follows X (t+1) = 0.8 X t + epsilon (t+1) with zero-mean shocks. In trading days, what is the half-life of mean reversion, i.e. the horizon h at which the expected residual has decayed to half its current value?统计简单数值题未尝试免费6036Implied AR(1) Coefficient From a Target Half-LifeA desk wants a mean-reverting signal whose shocks lose half their expected size every 5 trading days. If the signal is modeled as AR(1), X (t+1) = phi X t + epsilon (t+1), what value of phi is implied?统计中等数值题未尝试面试订阅6037Random-Walk Risk Scaling Over a HorizonA price follows a driftless random walk whose daily increments are iid with standard deviation 2 bp. By what multiple does the standard deviation of the cumulative move grow when the horizon increases from 1 day to 9 days?统计简单数值题未尝试免费6038Sign and Size of Lag-1 AutocorrelationA stationary spread obeys X (t+1) = -0.4 X t + epsilon (t+1) with iid zero-mean shocks. What is the lag-1 autocorrelation of X t, and what does its sign say about period-to-period dynamics?统计中等数值题未尝试面试订阅6039Ornstein-Uhlenbeck Speed to Discrete CoefficientA spread is modeled in continuous time as a mean-reverting Ornstein-Uhlenbeck process with reversion speed kappa = 0.5 per day. If you sample it once per day and fit an AR(1), what discrete coefficient phi should you expect?统计中等数值题未尝试面试订阅6040Two-Period Variance Ratio of an AR(1)Returns are generated by a stationary AR(1) with autoregressive coefficient 0.5. The Lo-MacKinlay variance ratio at lag 2 is VR(2) = Var(r t + r (t+1)) / (2 Var(r t)). Compute VR(2) and state whether it signals momentum or mean reversion.统计困难数值题未尝试面试订阅