You purchase a hot stock for \(1000. The stock either gains 30% or loses 25% each day, each with probability 0.5. Its returns on consecutive days are independent of each other. You plan to sell the stock after two days.

a. What are the possible values of the stock after two days, and what is the probability for each value? What is the probability that the stock is worth more after two days than the \)1000 you paid for it?

b. What is the mean value of the stock after two days? Comment: You see that these two criteria give different answers to the question “Should I invest ?"

The stock is valued at $1,000.

Every day, Stock either increases by 30% or decreases by 30%.

Alternatively, the stock may lose 25% of its value.

In each situation, the probability is 0.5.

After the first day, how much is the stock worth?

Gain: Since the stock has increased by 30%,

$\mathrm{\$}1000+\mathrm{\$}1000\times 0.30=\mathrm{\$}1300$

Loss: Since the stock has lost 25% of its value,

$\mathrm{\$}1000-\mathrm{\$}1000\times 0.25=\mathrm{\$}750$

After the second day, the stock's value has increased,

2 Gains:

$\mathrm{\$}1300+\mathrm{\$}1300\times 0.30=\mathrm{\$}1690$

Gain + Loss :

Since it is unknown if the stock will gain or lose money on the first day, as well as whether it will lose or gain money on the second day.

We'll say the stock gains 30%on the first day and loses 25% on the second.

$\mathrm{\$}1300-\mathrm{\$}1300\times 0.25=\mathrm{\$}975$

Two losses:

Since the stock has lost 25% of its value, the stock is now worth 750 dollars with a 25% loss on the first day and continues to lose 25% on the second day.

$750-\mathrm{\$}750\times 0.25=\mathrm{\$}562.50$

Now, let's look at the chances of each outcome.

For two gains

$\mathrm{P}\left(\mathrm{\$}1690\right)=\frac{1}{4}=0.25$

For Loss + Gain (or Loss + Gain):

Keep in mind that the gain on the first day and the loss on the second day will be the same as the loss on the first day and the gain on the second day.

$\mathrm{P}\left(\mathrm{\$}975\right)=\frac{2}{4}=0.50$

Two losses:

$\mathrm{P}(\mathrm{\$}562.50)=\frac{1}{4}=0.25$

We learned from the aforementioned odds that there is only one potential conclusion for a stock valued more than $ 1000 after two days:

$\mathrm{P}(\mathrm{X}>\mathrm{\$}1000)=\mathrm{P}\left(\mathrm{\$}1690\right)=0.25$

The stock is valued at $1,000.

Every day, Stock either increases by 30% or decreases by 30%.

Alternatively, the stock may lose 25% of its value.

In each situation, the probability is 0.5.

We have The likelihood of each result, for two gains, from Part (a).

$\mathrm{P}\left(\mathrm{\$}1690\right)=\frac{1}{4}=0.25$

For Loss + Gain (or Loss + Gain):

Keep in mind that the gain on the first day and the loss on the second day will be the same as the loss on the first day and the gain on the second day.

$\mathrm{P}\left(\mathrm{\$}975\right)=\frac{2}{4}=0.50$

Two losses:

$\mathrm{P}(\mathrm{\$}562.50)=\frac{1}{4}=0.25$

The expected value is now the sum of each alternative multiplied by its likelihood.

As a result, the stock's mean value after two days is

$\mathrm{\mu }=\sum \mathrm{xP}\left(\mathrm{x}\right)=\mathrm{\$}1690\times 0.25+\mathrm{\$}975\times 0.5+\mathrm{\$}562.50\times 0.25=\mathrm{\$}1050.625$