Question

Problem 2.9 (Portfolio theory) A portfolio is a row vector in which y is the number of units of asset i held by an investor. After a year, say, the value of the assets will increase (or decrease) by a certain percentage. The change in each asset depends on states the economy will assume, predicted as a returns matrix, R (ri), where riy is the factor by which investment i changes in one year if state j occurs. Suppose an investor has assets in yi land, y2 bonds and y3 stocks, and that the returns matrix is 1.05 0.95 1.0 R1.05 1.05 1.05 1.20 1.26 1.23 Then the total values of the portfolio in one year's time are given by Y R, where (YR), is the total value of the portfolio if state j occurs. (a) Find the total values of the portfolio W-(5000 2000 0) in one year for each of the possible states. (b) Show that U (600 8000 1000) is a riskless portfolio; that is, it has the same value in all states j.

Answer 1

Solution: Gve that Re thonmatix R:(i) Sis he acto by which investment i changes in one estate i ocoige. 120 b . 1-23 t Re tunn matsi ts 3x3 matriv (, ven *the total value 아 Postfolio one yeon for each 아 Possible Stole uoR: (5000 2000 13ILO 126 23

2 : The postoio total volue is : 2130o riven that t.05 095 0 UR- (600 8000 l00o) 50 .os 1x3 1. 20 26 1.23 3X3 +lo(22) - (G30 + guOD +200 510 840o 12 60 60o 84+12 30 1so ) 0230 0230 0230

3 ะ> The 1x3 OR-motrin on toin Some values iuen porthouo