Question

P12-7 Calculating Returns and Variability [LO1] Using the following returns, calculate the arithmetic average returns, the variances, and the standard deviations for X and Y. Returns Year 11 % NMto 24 % 2945 18 -10 -24 53 Requirement 1: (a)Calculate the arithmetic average return for X. (Click to select) (b)Calculate the arithmetic average return for Y. (Click to select) Requirement 2: (a)Calculate the variance for X. (Do not round intermediate calculations.) (Click to select) (b)Calculate the variance for Y. (Do not round intermediate calculations.) (Click to select) Requirement 3: (a)Calculate the standard deviation for X (Do not round intermediate calculations.) (Click to select) (b)Calculate the standard deviation for Y. (Do not round intermediate calculations.) (Click to select)

Answer 1

Requirement 1:

Arithmetic average return = Sum of annual returns /Number of years

(a)

Arithmetic average return for X = (11 % + 29 % + 18 % - 19 % + 20 %)/5

                                                 = (0.11 + 0.29 + 0.18 – 0.19 + 0.2)/5

                                                 = 0.59/5 = 0.118 or 11.8 %

(b)

Arithmetic average return for Y = (24 % + 45 % - 10 % - 24 % + 53 %)/5

                                                  = (0.24 + 0.45 - 0.10 – 0.24 + 53)/5

                                                  = 0.88/5 = 0.176 or 17.6 %

Requirement 2:

(a)

Year (n)	Returns (ri)	ri - rm	(ri - rm)2
1	0.11	-0.008	0.000064
2	0.29	0.172	0.029584
3	0.18	0.062	0.003844
4	-0.19	-0.308	0.094864
5	0.20	0.082	0.006724
Sum ∑ ri	0.59	∑(ri - rm)2	0.135080
Average (rm)	0.118	∑(ri - rm)2/(n-1)	0.033770

Variance for X = ∑(r_i - r_m)²/(n-1) = 0.13508/(5 -1) = 0.13508/4 = 0.03377

(b)

Year (n)	Returns (ri)	ri - rm	(ri - rm)2
1	0.24	0.064	0.004096
2	0.45	0.274	0.075076
3	-0.10	-0.276	0.076176
4	-0.24	-0.416	0.173056
5	0.53	0.354	0.125316
Sum ∑ ri	0.88	∑(ri - rm)2	0.453720
Average (rm)	0.176	∑(ri - rm)2/(n-1)	0.113430

Variance for Y = ∑(r_i - r_m)²/(n-1) = 0.45372/ (5 -1) = 0.45372/4 = 0.11343

Requirement 3:

(a)

Standard deviation for X = √Variance = √0.03377 = 0.183766155752358 or 0.183766

(a)

Standard deviation for Y = √Variance = √0.11343 = 0.336793705404362 or 0.336794