Question

                                                            Continuous Probability Distributions

You must draw diagram for each question and show all workings. Answers without clearly labelled diagrams and workings will not score full marks.

Before we can use the Z tables, we have to standardise the normal variable (X) first, i.e. converting X into Z score. THA 4 will assess you on your ability of applying the standardisation formula and using the Z and inverse Z tables.

Question 1 Stock Returns Stock returns are often assumed to be normally distributed. This means that the full probability distribution of returns can be explained by the mean and variance of returns, i.e. we can calculate probabilities by knowing the mean and variance. Assume that a stock has a mean annual return of 7.5% and an annual standard deviation of returns of 15%*,

a) What is the probability that you will return between 0% and 15% in any given year?

b) What is the probability that you will return exactly 7.5% in any given year?

c) What is the probability of losing money in any given year, i.e. getting a negative return?

d) To have a really good year, you would want returns to be in the top 5% of returns. What return must you earn to conclude that you have had a really good year?

e) Imagine that there was another stock with mean annual return of 10% but its standard deviation of returns was only 30%. In relation to your answer in (c), outline why you might prefer your old stock over this new stock if you were concerned about losing money (negative returns).

Answer 1

Mean = 7.5%

Standard deviation of returns = 15%

a) Probability that you will return between 0% and 15% in any given year = P(0% < X < 15%)   

Since the returns are normally distributed

z-score = (X - Mean) / Standard deviation of returns

z-score of 0% = (0% - 7.5%) / 15%

z-score of 0% = -0.5

z-score of 15% = (15% - 7.5%) / 15%

z-score of 15% = 0.5

P(0% < X < 15%) =  P(-0.5 < z < 0.5)

P(-0.5 < z < 0.5) = P( 0 > z > - 0.5) + P(0 < z < 0.5)

P(0 < z < 0.5) = P(z > 0) + P(z < 0.5)

P(0 < z < 0.5) = 1 - P(z = 0) + P(z = 0.5) - 1

From full normal distribution table P(z = 0) = 0.5 & P(z = 0.5) = 0.6915

P(0 < z < 0.5) = 1 - 0.5 + 0.6915 - 1

P(0 < z < 0.5) = 0.1915

Since P( 0 > z > - 0.5) = P(0 < z < 0.5)

P(-0.5 < z < 0.5) = 0.1915 + 0.1915

P(-0.5 < z < 0.5) = 0.383 or 38.3%

P(0% < X < 15%) = 38.3%

b) Probability that you will return exactly 7.5% in any given year = P(X = 7.5%)

Since Normal distribution is a continuous distribution   P(X = 7.5%) = 0%

c) Probability of losing money in any given year, i.e. getting a negative return = P(X < 0%)

z-score = (X - Mean) / Standard deviation of returns

z-score of 0% = (0% - 7.5%) / 15%

z-score of 0% = -0.5

P(X < 0%) = P(z < -0.5)

Since P(z < -0.5) = P(z > 0.5)

P(z > 0.5) = 1 - P(Z = 0.5)

From full normal distribution table P(z = 0.5) = 0.6915

P(z > 0.5) = 1 - 0.6915  

P(z > 0.5) = 0.3085

P(X < 0%) = 0.3085 or 30.85%

d) Probability of being in the the top 5% of the returns = P(95%)

z-score for being top 5% returns i.e. P(95%) = 1.645

z-score = (X - Mean) / Standard deviation of returns

1.645 = (X - 7.5%) / 15%

0.24675 = (X - 7.5%)

X = 0.32175 or 32.175%

I must have a return of greater than 32.175% to conclude that I have had a good year which implies that I have generated enough returns to be the top 5% of the returns.

e) The Probability of losing money in the new stock i.e. getting a negative return = P(X < 0%)

Mean = 10%

Standard deviation of returns = 30%

z-score = (X - Mean) / Standard deviation of returns

z-score of 0% = (0% - 10%) / 30%

z-score of 0% = -0.33

P(X < 0%) = P(z < -0.33)

Since P(z < -0.33) = P(z > 0.33)

P(z > 0.33) = 1 - P(z = 0.33)

From full normal distribution table P(z = 0.33) = 0.6293

P(z > 0.33) = 1 - 0.6293  

P(z > 0.33) = 0.3707

P(X < 0%) = 0.3707 or 37.07%

The probability of losing money in the new stock is greater than the old stock (37.07% > 30.85%) since the volatility of the new stock is higher