Math 348: Spring 2007

Portfolios and

Investment Strategies

Example: Albert builds a portfolio

SupposeS1(0) = 50;S2(0) = 100 and A(0) = 100.

Then x1(1) = 20;x2(1) = 10 and y(1) = 10.

At time t = 0 theportfolio is given by

V(0) = x1(1)S1(0) +x2(1)S2(0) +y(1)A(0)

= 20¢50+10¢100+10¢100:

Albert invests Aunt Agatha's $3000 equally in 2 risky and 1 riskless assets.

Time Passes

At t = 1, we¯nd S1(1) = 60;S2(1) = 90 and A(0) = 110.

Then

V(1) = x1(1)S1(1) +x2(1)S2(1) +y(1)A(1)

= 20¢60+10¢90+10¢110

= 3200:

Albert rebalances the portfolio by selling 3 shares of S1 and buying twoshares of S2.

Then x1(2) = 17;x2(2) = 12 and y(2) = 10.

More Time Passes

At t = 2, we¯nd S1(1) = 54;S2(1) = 95 and A(0) = 121.

Then

V(2) = x1(2)S1(2) +x2(2)S2(2) +y(2)A(2)

= 17¢54+12¢95+10¢121

= 3268:

Inadmissible Strategies

SupposeS1(0) = 50;S2(0) = 100 and A(0) = 100.

Albert has $100.

If x1(1) = ¡ 20, and x2(1) = 10, then

V(0) = x1(1)S1(0) +x2(1)S2(0) +y(1)A(0)

100 = ¡ 20¢50+10¢100+y(1) ¢100:

so y(1) = 1.

Time Passes

At t = 1, we¯nd S1(1) = 60;S2(1) = 90 and A(0) = 110.

Then

V(1) = x1(1)S1(1) +x2(1)S2(1) +y(1)A(1)

= ¡ 20¢60+10¢90+1¢110

= ¡ 190:

Albert has gonebrokeand whoever let himshort S1 without any deposit isout of a job.

Wesay Albert's strategy is inadmissible.