AP/ADMS 3531, Winter 2010
Solutions to homework assignment #1
Guidelines for rounding: If you are using percents, for example 1.23%, then you should use two
decimal places for your calculations. If the same number is converted to decimal form, as
0.0123, then use four decimal places throughout your calculations. Dollar amounts, for example
a share price of $12.34, should be rounded to the nearest cent. However, if there are no pennies,
such as 100 shares worth $12 per share, then use integer dollar amounts.
1. (15 points) Stocks X and Y have normally distributed returns, with the following returns
during four previous years:
Year Stock X
Stock Y
1
10%
5%
2
4
28
3
-3
53
4
15
34
Which stock is more likely to give a negative rate of return next year? Show calculations to
support your answer, and do not make any additional assumptions beyond the information
presented above. Your answer should be based solely on our textbook; you do not need to use
any other references.
Begin by calculating the average returns. The average return for X is (10 + 4 – 3 + 15) / 4 =
6.5%.
The average return for Y is (5 + 28 + 53 + 34) = 30%.
Then use equation 1.4 to calculate the variance. For X, the variance is {(10-6.5)
2
+ (4-6.5)
2
+
(-3-6.5)
2
+ (15-6.5)
2
} / 3 = 181 / 3 = 60.33. For Y, the variance is {(5-30)
2
+ (28-30)
2
+
(53-30)
2
+ (34-30)
2
} / 3 = 1174 / 3 = 391.33. Notice that we divide by 3, not 4, because you use
N-1 when calculating the variance of historical returns.
Finally, take square roots to get the standard deviations. The standard deviation of X is 7.77 %,
and for Y it is 19.78%.
Now we need to determine the distance between 0% and the average of each stock, using the
standard deviations of each stock.
To assess the likelihood of a negative return, we need to compare 0% and the average return on
each stock, measured in the number of standard deviations. For X, 0% is 6.5/7.77 = 0.84
standard deviations below the mean. For Y, 0% is 30/19.78 = 1.52 standard deviations below the
mean.
Therefore X is more likely to give a negative rate of return. For any normally distributed
variable, there is a higher probability of ending up 0.84 standard deviations below the mean,
than there is of ending up 1.52 standard deviations below the mean. It is not necessary to
determine the exact probabilities to answer the question.