5.12 A simple bootstrap example

• We want to invest a fixed sum of money in 2 financial assets that yield returns of X and Y respectively
• We will invest a fraction of our money ${\alpha}$ in X
• And invest the rest $(1-{\alpha})$ in Y
• Since there is variability associated with the returns on the 2 assets, we want to choose ${\alpha}$ to minimize the total risk (variance) of our investment
• Thus, we want to minimize: $Var({\alpha}X + (1-{\alpha})Y)$
• The value of ${\alpha}$ that minimizes the risk is given by:

$${\alpha} = \frac{{\sigma}_{Y}^{2}-{\sigma}_{XY}}{{\sigma}_{X}^{2}+{\sigma}_{Y}^{2}-2{\sigma}_{XY}}$$ where ${\sigma}_{X}^{2} = Var(X)$, ${\sigma}_{Y}^{2} = Var(Y)$, and ${\sigma}_{XY} = Cov(X, Y)$

• The quantities $Var(X)$, $Var(Y)$ and $Cov(X, Y)$ are unknown but we can estimate them from a dataset that contains measurements for X and Y.

• Simulated 100 pairs of data points (X, Y) four times and got four values for $\hat{\alpha}$ ranging from 0.532 to 0.657.

• Now, how accurate is this as an estimate of ${\alpha}$?

  • Get the standard deviation of $\hat{\alpha}$
  • Same simulation process as above but done 1,000 times to get 1,000 values of $\hat{\alpha}$ (i.e., 1,000 estimates for ${\alpha}$)
  • True known value of ${\alpha}$ is 0.6
  • The mean over all 1,000 estimates of ${\alpha}$ = 0.5996
  • Std dev of the estimate is:

$$\sqrt{\frac{1}{1000-1}\sum_{r=1}^{1000}(\hat{\alpha}_{r}-\bar{\alpha})^{2}} = 0.083$$ - Gives fairly good estimate of accuracy of ${\alpha}$ (we expect $\hat{\alpha}$ to differ from ${\alpha}$ by ~ 0.08)