- Introduction: A Simple Market Model 3
The returns on bonds or stock are particular cases of the return on a portfolio
(withx=0ory = 0, respectively). Note that becauseS(1) is a random
variable, so isV(1) as well as the corresponding returnsKSandKV.The
returnKAon a risk-free investment is deterministic.
Example 1.
LetA(0) = 100 andA(1) = 110 dollars. Then the return on an investment in
bonds will be
KA=0. 10 ,
that is, 10%. Also, letS(0) = 50 dollars and suppose that the random variable
S(1) can take two values,
S(1) =
{
52 with probabilityp,
48 with probability 1−p,for a certain 0<p<1. The return on stock will then be
KS=
{
0 .04 if stock goes up,
− 0 .04 if stock goes down,that is, 4% or−4%.
Example 1.
Given the bond and stock prices in Example 1.1, the value at time 0 of a
portfolio withx= 20 stock shares andy= 10 bonds is
V(0) = 2, 000dollars. The time 1 value of this portfolio will be
V(1) =
{
2 ,140 if stock goes up,
2 ,060 if stock goes down,so the return on the portfolio will be
KV=
{
0 .07 if stock goes up,
0 .03 if stock goes down,that is, 7% or 3%.