Question

You work for a small investment management firm. You have been provided with the following historical information for three stocks and the market index. The information is shown in the table below.

	AAA Inc.		BBB Inc.		CCC Inc.
	Stock Price	Dividend	Stock Price	Dividend	Stock Price	Dividend	Market Index
2012	$47		$125		$145		20,840
2013	$52	$1.40	$135	$1.50	$160	$3.50	23,320
2014	$57	$1.40	$103	$1.55	$170	$3.60	22,220
2015	$62	$1.70	$130	$1.65	$185	$3.65	25,325
2016	$54	$1.85	$155	$1.75	$180	$3.65	24,500
2017	$64	$1.85	$165	$1.90	$200	$3.75	26,100
2018	$82	$1.90	$135	$2.00	$215	$3.90	27,500
2019	$92	$2.00	$155	$2.20	$225	$4.25	30,000

Using the data provided you are to calculate the following for each of the stocks and the market index.

1. What is the average annual return for the past seven years?
2. What is the geometric average annual return (effective annual rate of return) for the past seven years?
3. What is the population standard deviation and the sample standard deviation?
4. What is the correlation coefficient between Stocks AAA and BBB? Between Stocks AAA and CCC? Between Stocks BBB and CCC?
5. What is the sample covariance between Stocks AAA and BBB, between Stocks AAA and CCC and between stocks BBB and CCC?
6. What is the coefficient of variation for each of the stocks and the market index? You should use the sample standard deviation.
7. What is the beta of each of the stocks? You can use Excel’s “slope” function to estimate this.
8. Assume that the beta of stock AAA is 0.80, the beta of stock BBB is 1.2 and the beta of stock CCC is 1.5. What is the beta of a portfolio where the weight in stock AAA is 20% the weight in stock BBB is 55% and the weight in stock CCC is 25%?
9. Assuming the risk free rate is 2.5% and the expected return on the market is 6% what is the expected return on the portfolio in part (h) above?

Answer 1

1. Average Annual Return for Past 7 Years -

Annual Return on Investment = Dividend per share / Stock Nominal Price x 100

Dividend Per Share = Total Dividends / Number of Shares

Average Annual Return for AAA Inc.

Dividend Per Share = 1.40 + 1.40 + 1,70 + 1.85 + 1.85 + 1.90 + 2.00 / 7 = 12.1/ 7 = 1.73

Stock Nominal Price = 52+ 57+ 62+ 54+ 64+ 82+ 92 / 7 = 463 / 7 = 66

Annual Return = 1.73 / 66 x 100 = 2.62%

Annual Return for AAA Inc. = 2.62%

Average Annual Return for BBB Inc.,

Dividend Per Share = 1.50 + 1.55 + 1.65 + 1.75 + 1.90 + 2.00 + 2.20 / 7 = 12.55 / 7 = 1.79

Stock Nominal Price = 135+ 103+ 130+ 155+ 165+ 135+ 155 / 7 = 978 / 7 = 140

Annual Return = 1.79 / 140 x 100 = 1.28%

Average Annual Return for BBB Inc. = 1.28%

Average Annual Return for CCC Inc.,

Dividend Per Share = 3.50+ 3.60+ 3.65+ 3.65+ 3.75+ 3.90+ 4.25 / 7 = 26.3 / 7 = 3.75

Stock Nominal Price = 160+ 170+ 185+ 180+ 200+ 215+ 225 / 7 = 1335 / 7 = 191

Annual Return = 3.75 / 191 x 100 = 1.96%

Average Annual Return for CCC Inc. = 1.96%

2. Geometric Average Annual Return

The formula is ((1+r 1) x (1+r 2) x ....(1+ r n)) ^1/n - 1

where r = rate of return or dividend rate, n = number of periods

Geometric Annual Return for AAA Inc.,

Annual return = ((1+ 1.40) x (1+ 1.40) x (1+ 1.70) x (1+ 1.85) x (1+ 1.85) x (1+ 1.90) x (1+ 2.00)) ^1/7 - 1

= (2.4 x 2.4 x 2.7 x 2.85 x 2.85 x 2.9 x 3) ^0.14 - 1

= (1098.99) ^0.14 - 1 = 2.66 - 1 = 1.66%

Geometric Annual Return for AAA Inc. = 1.66%

Geometric Annual Return for BBB Inc.,

Annual Return = ((1+ 1.50) x (1+ 1.55) x (1+ 1.65) x (1+ 1.75) x (1+ 1.90) x (1+ 2.00) x (1+ 2.20)) ^1/7 - 1

= (2.50 x 2.55 x 2.65 x 2.75 x 2.90 x 3 x 3.20) ^0.14 - 1

= (1293.38) ^0.14 - 1 = 2.72 - 1 = 1.72%

Geometric Annual Return for BBB Inc. = 1.72%

Geometric Annual Return for CCC Inc.,

Annual Return = ((1+ 3.50) x (1+ 3.60) x (1+ 3.65) x (1+ 3.65) x (1+ 3.75) x (1+ 3.90) x (1+ 4.25)) ^1/7 - 1

= (4.50 x 4.60 x 4.65 x 4.65 x 4.75 x 4.90 x 5.25) ^1/7 - 1

= (54,692.18) ^0.14 - 1 = 4.61 - 1 = 3.61%

Geometric Annual Return for CCC Inc. = 3.61%

3. Population Standard Deviation and Sample Standard Deviation

The formula for Population Standard Deviation is

The formula for Sample Standard Deviation is