Obtain the SPY, UPRO, and SDS daily closing prices (unadjusted) for the periods from their inception through 2018. SPY tracks the S\&P 500; UPRO daily returns track three times the S\&P 500 daily returns; and SDS daily returns track twice the negative of the S\&P 500 one day returns. (SPY is a SPDR ETF begun in 1993, UPRO is a ProShares ETF begun in 2009, SDS is also a ProShares ETF begun in 2006.) Obtain all daily closes for their longest periods.

Also obtain the -GSPC prices for the same length of time as that of the oldest of the ETFs (SPY)

Note that the data series are of different lengths.

Now compute the daily returns for these four series and produce a correlation matrix of the daily returns.

For each correlation, use all of the relevant data. State the numbers of observations used for each correlation.

Are the correlations as would be expected?