## 8.9 Problems: Delta Method and Resampling

Exercise 8.1 (Delta method)
Exercise 8.2 (Gradients for example functions)
Exercise 8.3 (Standard error of Sharpe ratio)
• Look at determinants of se of Sharpe ratio. See discussion from Lo’s paper
Exercise 8.4 (Correlation between estimated Sharpe ratios)
• Look at the formula for the covariance between two Sharpe ratios
Exercise 8.5 (Bootstrap percentile confidence interval)

Illustrate the transformation perserving property of percentile interval using the example of computing the confidence interval for $$\sigma$$ based on the percentile interval for $$\sigma^2$$. Show that asymptotic confidence interval is not transformation preserving.

Exercise 8.6 (Bootstrap variability)

Repeat the bootstrap simulations from Example 8.16 using 5 different random number seeds: 124, 125, 126, 127 and 128. Make a table to compare the bootstrap estimated standard errors for the example functions across the 5 bootstrap simulations. Briefly comment on your results.

Exercise 2.1 (Bootstrap accuracy)

Repeat the Monte Carlo simulation in 8.7 using 1,000 simulated samples and 10,000 bootstrap simulations for the bootstrap calculations. That is, set R=10000 in the call to boot() inside the Monte Carlo simulation loop.

1. How does the increase in simulations impact the bootstrap bias estimates?

2. How does the increase in simulations impact the bootstrap standard error estimates?