Chapter 6.2, Problem 76E

(a)

To determine

Possible values of the stock after two days and probability for each value. Also, the probability for the stock is worth more after two days than the $1000.

Answer to Problem 76E

The probability for the stock is worth more after two days than the $1000 is 0.25.

Explanation of Solution

Given information:

Stock price is $1,000.

Each day,

Stock either gains 30%.

Or

Stock loses 25%.

Probability in each case is 0.5.

Calculations:

Worth of the stock after first day,

Gain:

Since stock gains 30%,

  $\$1000+\$1000\times 0.30=\$1300$

Loss:

Since stock losses 25%,

  $\$1000-\$1000\times 0.25=\$750$

Worth of the stock after second day,

2 Gains:

Since stock worth $1300 on the first day, and on the second day the stock continue to gain 30% on $1300.

  $\$1300+\$1300\times 0.30=\$1690$

Gain + Loss (or Loss +Gain):

Since it is not known that stock either gains or losses on the first day as well as either losses or gains on the second day.

We will assume the stock to gain 30% on the first day and there will be the loss of 25% on the second day.

  $\$1300-\$1300\times 0.25=\$975$

2 Losses:

Since stock losses 25%, the stock worth $750 with 25% loss on first day and stock continuous to loss 25% on second day as well.

  $\$750-\$750\times 0.25=\$562.50$

Now,

The probability of each outcome,

For 2 Gains:

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

For Loss + Gain (or Gain + Loss):

Note that

Gain on first day and loss on second day will be same as loss on first day and gain on second day.

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

For 2 Losses:

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

From the above probabilities,

We came to know that

Only 1 possible outcome is there for the stock worth more than $1000 after two days:

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

(b)

To determine

Mean value of the stock after two days.

Answer to Problem 76E

Mean value of the stock after two days is $1050.625.

Explanation of Solution

Given information:

Stock price is $1,000.

Each day,

Stock either gains 30%.

Or

Stock loses 25%.

Probability in each case is 0.5.

Calculations:

From Part (a),

We have

The probability of each outcome,

For 2 Gains:

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

For Loss + Gain (or Gain + Loss):

Note that

Gain on first day and loss on second day will be same as loss on first day and gain on second day.

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

For 2 Losses:

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

Now,

The expected value is the sum of the product of each possibility with its probability.

Thus,

Mean value of the stock after two days,

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