Question

You do not need a lot of money to invest in a mutual fund. However, if you decide to put some money into an investment, you are usually advised to leave it in for (at least) several years. Why? Because good years tend to cancel out bad years, giving you a better overall return with less risk. To see what we mean, let's use a 3-year moving average on the Calvert Social Balanced Fund (a socially responsible fund).

Year	1	2	3	4	5	6	7	8	9	10	11
% Return	1.78	17.79	7.46	5.95	−4.74	25.85	9.03	18.92	17.49	6.80	−2.38

(a) Use a calculator with mean and standard deviation keys to find the mean and standard deviation of the annual return for all 11 years.

x	=	%
s	=	%

(b) To compute a 3-year moving average for 1992, we take the data values for year 3 and the prior 2 years and average them. To compute a 3-year moving average for year 4, we take the data values for year 4 and the prior 2 years and average them. Verify that the following 3-year moving averages are correct.

Year	3	4	5	6	7	8	9	10	11
3-year moving average	9.01	10.40	2.89	9.02	10.05	17.93	15.15	14.40	7.30

(c) Use a calculator with mean and standard deviation keys to find the mean and standard deviation of the 3-year moving average.

x	=	%
s	=	%

(d) Compare the results of parts (a) and (c). Suppose we take the point of view that risk is measured by standard deviation. Is the risk (standard deviation) of the 3-year moving average considerably smaller?

The means are fairly similar, but the standard deviation of moving averages is much lower. This implies the risk of the 3-year moving average is considerably greater.The means are fairly similar, but the standard deviation of moving averages is much higher. This implies the risk of the 3-year moving average is considerably greater.     The means are fairly similar, but the standard deviation of moving averages is much lower. This implies the risk of the 3-year moving average is considerably smaller.The means are fairly similar, but the standard deviation of moving averages is much higher. This implies the risk of the 3-year moving average is considerably smaller.

Answer 1

a)

X	Mean, x̅	x-x̅	(x-x̅)²
1.78	9.45	-7.67	58.8289
17.79	9.45	8.34	69.5556
7.46	9.45	-1.99	3.9601
5.95	9.45	-3.5	12.25
-4.74	9.45	-14.19	201.3561
25.85	9.45	16.4	268.96
9.03	9.45	-0.42	0.1764
18.92	9.45	9.47	89.6809
17.49	9.45	8.04	64.6416
6.8	9.45	-2.65	7.0225
-2.38	9.45	-11.83	139.9489

∑x = 103.95

n = 11

∑(x-x̅)² = 916.381

Mean, x̅ = Ʃx/n = 103.95/11 = 9.45%

Standard deviation, s = √(Ʃ(x-x̅)²/(n-1)) = √(916.381/(11-1)) = 9.57%

b)

Month	Sales	Three month moving Average
1	1.78	-	-
2	17.79	-	-
3	7.46	(1.78+17.79+7.46)/3 =	9.01
4	5.95	(17.79+7.46+5.95)/3 =	10.40
5	-4.74	(7.46+5.95+-4.74)/3 =	2.89
6	25.85	(5.95+-4.74+25.85)/3 =	9.02
7	9.03	(-4.74+25.85+9.03)/3 =	10.05
8	18.92	(25.85+9.03+18.92)/3 =	17.93
9	17.49	(9.03+18.92+17.49)/3 =	15.15
10	6.8	(18.92+17.49+6.8)/3 =	14.40
11	-2.38	(17.49+6.8+-2.38)/3 =	7.30

c)

X	Mean, x̅	x-x̅	(x-x̅)²
9.01	10.6833333	-1.6733333	2.80004444
10.4	10.6833333	-0.2833333	0.08027778
2.89	10.6833333	-7.7933333	60.7360444
9.02	10.6833333	-1.6633333	2.76667778
10.05	10.6833333	-0.6333333	0.40111111
17.93	10.6833333	7.24666667	52.5141778
15.15	10.6833333	4.46666667	19.9511111
14.4	10.6833333	3.71666667	13.8136111
7.30	10.6833333	-3.3833333	11.4469444

∑x = 96.15

n = 9

∑(x-x̅)² = 164.51

Mean, x̅ = Ʃx/n = 96.15/9 = 10.68%

Standard deviation, s = √(Ʃ(x-x̅)²/(n-1)) = √(164.51/(9-1)) = 4.53%

d)

The means are fairly similar, but the standard deviation of moving averages is much lower. This implies the risk of the 3-year moving average is considerably smaller.