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Question

The following table shows that the S&P 500 stock index and Merck using ten years data.

Year S&P 500 Merck

2011 10.7 -29.0

2012 28.2 -11.2

2013 -21.6 -1.2

2014 -12.1 -36.0

2015 -9.8 41.8

2016 20.4 -7.5

2017 28.7 41.3

2018 33.5 35.6

2019 21.8 24.0

2020 38.0 76.5

Calculate the arithmetic average return for S&P and Merck.

(6 marks)

Compute the variance for S&P and Merck.

(6 marks)

Answer 1

Assuming that all of numbers in the S&P 500 and Merck columns represent return values, and in assuming that the phrase "variance" refers to sample variance, it follows that the

Arithmetic Average Return of Stock X =

(∑Return Values of Stock X)/(Number of Return Values for Stock X);

Variance of Returns of Stock X =

[∑(Return Values of Stock X - Arithmetic Average Return of Stock X)²]/(Number of Return Values for Stock X - 1).

Thus, the

a)

Arithmetic Average Return of the S&P 500 Stock Index =

[10.7 + 28.2 + (-21.6) + (-12.1) + (-9.8) + 20.4 + 28.7 + 33.5 + 21.8 + 38.0]/(10) = 13.78 (percent)

Arithmetic Average Return of the Merck Stock =

[(-29.0) + (-11.2) + (-1.2) + (-36.0) + 41.8 + (-7.5) + 41.3 + 35.6 + 24.0 + 76.5]/(10) = 13.43 (percent)

b)

Variance of Returns of the S&P 500 Stock Index =

[(10.7 - 13.78)² + (28.2 - 13.78)² + (-21.6 - 13.78)² + (-12.1 - 13.78)² + (-9.8 - 13.78)² + (20.4 - 13.78)² + (28.7 - 13.78)² + (33.5 - 13.78)² + (21.8 - 13.78)² + (38.0 - 13.78)²]/(10 - 1) = 444.57733

Variance of Returns of the Merck Stock =

[(-29.0 - 13.43)² + (-11.2 - 13.43)² + (-1.2 - 13.43)² + (-36.0 - 13.43)² + (41.8 - 13.43)² + (-7.5 - 13.43)² + (41.3 - 13.43)² + (35.6 - 13.43)² + (24.0 - 13.43)² + (38.0 - 76.5)²]/(10 - 1) = 1296.1134

Note that if instead the phrase "variance" refers to population variance, you divide the sum of squared differences by (Number of Return Values for Stock X), instead of (Number of Return Values for Stock X - 1).



Hope that helps!