The data are monthly observations on the prices of the largest 136 stocks in Australia from December 1999 to June 2014. Consider a portfolio constructed by holding one share in every one of the \(N\) stocks in the dataset that records a price, \(P_{j t}\) for at every time \(t\) in the sample period.

(a) Compute the simple and log returns to the portfolio over the sample period using the formulae

\[
R(P)=\frac{P_{T}}{P_{1}}-1, \quad r(P)=\log \left(\frac{P_{T}}{P_{1}}\right)
\]

in which

\[
P_{t}=\sum_{j=1}^{N} P_{j t}
\]

Comment on the results.

(b) Compute the portfolio weights of each stock in the portfolio for every time \(t\) using the formula

\[
w_{i t}=\frac{P_{i t}}{\sum_{i=1}^{N} P_{i t}}
\]

in which \(N\) is the number of stocks in the portfolio.

(c) Compute the simple return and log returns to the portfolio in each time period, respectively,

\[
R_{P t}=\sum_{i=1}^{N} w_{i t-1} R_{i t}, \quad r_{P t}=\log \left(\sum_{i=1}^{N} w_{i t-1} e^{r_{i t}}\right)
\]

remembering to use the weight at the beginning of the holding period.

(d) Compare the results obtained in (a) and (c).