- Discrete Time Market Models 75
4.1.1 Investment Strategies
The positions held by an investor in the risky and risk-free assets can be altered
at any time step by selling some assets and investing the proceeds in other
assets. In real life cash can be taken out of the portfolio for consumption or
injected from other sources. Nevertheless, we shall assume that no consumption
or injection of funds takes place in our models to keep things as simple as
possible.
Decisions made by any investor of when to alter his or her portfolio and
how many assets to buy or sell are based on the information currently available.
We are going to exclude the unlikely possibility that investors could foresee the
future, as well as the somewhat more likely (but illegal) one that they will
act on insider information. However, all the historical information about the
market up to and including the time instant when a particular trading decision
is executed will be freely available.
Example 4.1
Letm= 2 and suppose that
S 1 (0) = 60,S 1 (1) = 65,S 1 (2) = 75,
S 2 (0) = 20,S 2 (1) = 15,S 2 (2) = 25,
A(0) = 100,A(1) = 110,A(2) = 121,in a certain market scenario. At time 0 initial wealthV(0) = 3,000 dollars is
invested in a portfolio consisting ofx 1 (1) = 20 shares of stock number one,
x 2 (1) = 65 shares of stock number two, andy(1) = 5 bonds. Our notational
convention is to use 1 rather than 0 as the argument inx 1 (1),x 2 (1) andy(1) to
reflect the fact that this portfolio will be held over the first time step. At time 1
this portfolio will be worthV(1) = 20×65 + 65×15 + 5×110 = 2,825 dollars.
At that time the number of assets can be altered by buying or selling some of
them, as long as the total value remains $2, 825 .For example, we could form a
new portfolio consisting ofx 1 (2) = 15 shares of stock one,x 2 (2) = 94 shares of
stock two, andy(2) = 4 bonds, which will be held during the second time step.
The value of this portfolio will beV(2) = 15×75 + 94×25 + 4×121 = 3, 959
dollars at time 2, when the positions in stocks and bonds can be adjusted once
again, as long as the total value remains $3,959, and so on. However, if no
adjustments are made to the original portfolio, then it will be worth $2,825 at
time 1 and $3,730 at time 2.