Question

6. Shares in Peachum and Associates (PA) stand at $100. A model for the price change is that it will either increase by $1 or decrease by $1 at the end of each week. The probability of an increase is constant at 0.55. (a) What are the greatest and least possible prices for a share in PA after one year (52 weeks)? (b) What is the expected value of the share price after one year? (c) What is the standard deviation in the number of price increases after one year? (d) Using the BINOMDIST function in Excel, or otherwise, find the probability that the share price of PA will be below 90 in one year.

Answer 1

Part a:

The price of share can either increase by $1 or decrease by $1 at the end of each week.

The current price of PA share is $100.

If the prices increase by $1 for all of the 52 weeks, it will be the greatest possible price, which is .

($100 + 152 + $1}} = $152

So the greatest possible price of PA share after 1 year will be $152.

If the prices decrease by $1 for all of the 52 weeks, it will be the least possible price, which is .

($100 - (52 * $1)) = $48

So the least possible price of PA share after 1 year will be $48.

Part b:

Let us denote,

X as the number of times price gets increased in 52 weeks.

If we say that getting price increased is a success, then, we can say probability of success in each trial is 0.55

Then according to the definition of binomial distribution,
X ~ Binomial numberoftrials, n = 52, probabilityof success, p=0.55),

X = {0, 1, ..., 52)

According to binomial distribution, the expected value of X will be

E(X) = n*p= 52 * 0.55 = 28.6

Then out of 52 weeks we are expecting 28.6 times increasing of share price by $1. Since for other weeks in the year we will be expecting decrease of share price by $1, we are expecting 23.4 times decreasing of share price by $1 as well.

So, expected value of share price after 1 year will be,

$100 + (28.6 * $1)-(23.4 * $1) = $105.2

Expected value of share price after 1 year will be $105.2.

Part c:

Let us denote,

X as the number of times price gets increased in 52 weeks.

If we say that getting price increased is a success, then, we can say probability of success in each trial is 0.55

Then according to the definition of binomial distribution,
X ~ Binomial numberoftrials, n = 52, probabilityof success, p=0.55),

X = {0, 1, ..., 52)

According to binomial distribution, the standard deviation of X will be

SD(X) = n*p* (1 - p) = 52*0.55 +0.45 = 3.587478

So the standard deviation in the number of price increases after 1 year will be 3.587478.

Part d:

Let us say price gets increased in a year x times, then price gets decreased in a year by (52-x) times.

Then the final price will be

100 + ((2 * $1)) - ((52 - 1) * $1) = $(48 + 2.1)

We want to find the probability that the share price of PA will be below $90 after a year. That is, .

P(48 + 2.1 < 90), i.e. Plr <21)

Now, if we say that getting price increased is a success, then, we can say probability of success in each trial is 0.55

Then according to the definition of binomial distribution,
X ~ Binomial numberoftrials, n = 52, probabilityof success, p=0.55),

X = {0, 1, ..., 52)

We have to find

Per <21) = P(X < 20)

Using BINOMDIST function in Excel, [ BINOMDIST(20,52,0.55,TRUE) ], we get,

P(X < 20) = 0.01210936

Thus, the probability of the share price of PA will be below $90 after a year is 0.01210936.

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