17.5 Exercises

EXERCISE 17.1   Prove that the inverse of $\data{A} = (a-b)\data{I}_p+b\undertilde{1}_{p}\undertilde{1}_{p}^{\top}$ is given by

\begin{displaymath}\data{A}^{-1}=\frac{\data{I}_p}{\displaystyle{(a-b) }}
-\fra...
...de{1}_{p}^{\top} }
{{\displaystyle (a-b)\{a+(p-1)b\} } }\cdotp\end{displaymath}

EXERCISE 17.2   The empirical covariance between the 120 returns of IBM and PanAm is $0.0016$ (see Example 17.2). Test if the true covariance is zero. Hint: Use Fisher's $Z$-transform.

EXERCISE 17.3   Explain why in both Figures 17.2 and 17.3 the portfolios have negative returns just before the end of the series, regardless of whether they are optimally weighted or not! (What happened in December 1987?)

EXERCISE 17.4   Apply the method used in Example 17.2 on the same data (Table B.5) including also the Digital Equipment company. Obviously one of the weights is negative. Is this an efficient weighting?

EXERCISE 17.5   In the CAPM the $\beta$ value tells us about the performance of the portfolio relative to the riskless asset. Calculate the $\beta$ value for each single stock price series relative to the ``riskless'' asset IBM.