The Vanguard European stock fund, Pacific stock fund, and Exxon Mobil reported the following historical dividend-adjusted prices:

Year 1991 1992 1993 1994 1995 1996 VEURX 6.53 7.15 6.91 9.34 9.03 11.17 VPACX 7.18 7.41 6.30 9.52 9.08 9.97 XOM 9.57 10.07 10.88 10.97 15.29 19.18 Year 1997 1998 1999 2000 2001 VEURX 13.50 17.45 21.42 23.38 23.13 VPACX 8.39 7.17 7.01 10.41 8.10 XOM 24.63 30.14 33.94 37.42 34.57 Year 2002 2003 2004 2005 2006 VEURX 17.50 14.42 21.22 24.87 29.53 VPACX 5.64 5.42 7.94 9.08 11.93 XOM 31.50 38.01 48.67 54.41 75.67

(a) Compute the means and covariances of the rates of return on these three assets.

(b) Draw the efficient frontier if you can only invest in VEURX and VPACX.

(c) Now add Exxon Mobil. Use Excel to draw 1,000 random numbers in two columns, called wE and wP. (Create one formula, and copy it into all of the cells.) Each of these 2,000 cells should use the formula ’rand()*3-1’. Create a new column that is 1.0 minus wE and wP, and call it wX.
Now consider these random numbers as investment weights in VEURX, VPACX, and XOM. Compute the risk and reward for each of these portfolios (one portfolio is three numbers: one wE, one wP, and one wX), using the standard deviation and expected rate of return formulas. Finally, create an x-y plot that shows, for each of your wE, wP, and wX portfolios, the riskreward combinations. What does the plot look like?

(d) If the risk-free rate stood at 5% per annum, what would be the tangency portfolio?