Question

A random walk on Wall Street? The “random walk” theory of stock prices holds that price movements in disjoint time periods are independent of each other. Suppose that we record only whether the price is up or down each year, and that the probability that our portfolio rises in price in any one year is 0.65. (This probability is approximately correct for a portfolio containing equal dollar amounts of all common stocks listed on the New York Stock Exchange.)

(a) What is the probability that our portfolio goes up for three consecutive years?

(b) What is the probability that the portfolio’s value moves in the same direction (either up or down) for three consecutive years?

Short answer

Part (a) The probability is $0.275$

Part (b) The probability is $0.3175$

Step-by-step solution

Part (a) Step 1. Given

$P(Riseinoneyear)=0.65$

Part (a) Step 2. Concept

$Theprobabilityofanevent=\frac{numberoffavorableoutcomes}{totalnumberofoutcomes}$

Part (a) Step 3. Calculation 

The probability that your portfolio will increase in value over the next three years is determined as follows: $P(Riseinthreeyears)={P(Riseinoneyear)}^3$$0.275$ is the required probability.

Part (b) Step 1. Calculation

The likelihood of a portfolio moving in the same direction in three years can be computed as follows:

The likelihood of a one-year fall is computed as follows:

$$\begin{gathered} P(Fallinoneyear)=1-P(Riseinoneyear) \\ =1-0.65 \\ =0.35 \end{gathered}$$

It is clear from the preceding section that $P(Riseinthreeyears)=0.275$

The chance of a drop in three consecutive years can now be computed as follows:

$$\begin{gathered} P(Fallinthreeconsecutiveyears)={0.35}^3 \\ =0.043 \end{gathered}$$

The probability that is required is: P (Same direction) $=0.275+0.043=0.3175$

As a result, the chance is $0.3175$