Question

Mini-Case

After graduating from college last spring with a major in accounting and finance, Jim Hale took a job as an analyst trainee for an investment company in Chicago. His first few weeks were filled with a series of rotations throughout the firm’s various operating units, but this week he was assigned to one of the firm’s traders as an analyst. On his first day, Jim’s boss called him in and told him that he wanted to do some rudimentary analysis of the investment returns of a semiconductor manufacturer called Advanced Micro Devices, Inc. (AMD). Specifically, Jim was given the following month-end closing prices for the company spanning the period of November 1, 2011, through November 1, 2012:

Date	Closing Price	Date	Closing Price
1-Nov-11	$5.69	1-Jun-12	$5.73
1-Dec-11	5.4	2-Jul-12	4.06
3-Jan-12	6.71	1-Aug-12	3.72
1-Feb-12	7.35	4-Sep-12	3.37
1-Mar-12	8.02	1-Oct-12	2.05
2-Apr-12	7.36	1-Nov-12	1.88
1-May-12	6.08

He was then instructed by his boss to complete the following tasks using the AMD price data (note that AMD paid no dividend during the period being analyzed).

Questions

1. Compute AMD’s monthly realized rates of return for the entire year.

2. Calculate the average monthly rate of return for AMD using both the arithmetic and the geometric averages.

3. Calculate the year-end price for AMD, computing the compound value of the beginning-of-year price of $5.69 per share for 12 months at the monthly geometric average rate of return calculated earlier:

   End-of-YearStockPrice=Beginning-of-YearStockPrice(1+GeometricAverageMonthlyRateofReturn)12End-of-Year  Stock Price=Beginning-of-Year         Stock  Price(1+Geometric  AverageMonthly  Rate of Return)12

4. Compute the annual rate of return for AMD using the beginning and ending stock prices for the period (i.e., $5.69 and $1.88).

5. Now calculate the compound annual rate of return using the geometric average monthly rate of return:

   CompoundAnnualRateofReturn=(1+GeometricAverageMonthlyRateofReturn)12−1Compound Annual   Rate  of Return=(1+Geometric AverageMonthly Rate of Return)12−1

6. If you were given annual rate of return data for AMD’s or any other company’s stock and you were asked to estimate the average annual rate of return an investor would have earned over the sample period by holding the stock, would you use an arithmetic or a geometric average of the historical rates of return? Explain your response as if you were talking to a client who has had no formal training in finance or investments.

Answer 1

monthly realized rates of return each month = (current month price - previous month price) / previous month price

Arithmetic average is calculated using AVERAGE function in Excel

Geometric average is calculated by adding 1 to each month's realized return, calculating the geometric mean of that series using GEOMEAN function in Excel, and then subtracting 1 from the GEOMEAN calculated.

End-of-YearStockPrice = Beginning-of-YearStockPrice * (1 + GeometricAverageMonthlyRateofReturn)¹²

End-of-YearStockPrice = $5.69 * (1 + (-8.82%))¹² = $1.88

End-of-YearStockPrice = Beginning-of-YearStockPrice * (1 + GeometricAverageMonthlyRateofReturn)¹²

$1.88 = $5.69 * (1 + GeometricAverageMonthlyRateofReturn)¹²

GeometricAverageMonthlyRateofReturn = ($1.88 / $5.69)^1/12 - 1

GeometricAverageMonthlyRateofReturn = -8.82%

CompoundAnnualRateofReturn = (1 + GeometricAverageMonthlyRateofReturn)¹² − 1

CompoundAnnualRateofReturn = (1 + (-8.82%))¹² − 1

CompoundAnnualRateofReturn = -66.96%

I would use geometric average of the historical rates of return. This is because the geometric average more accurately captures the real rate of return earned. The geometric average calculates the rate of return based on each month's change in price relative to the original price, whereas the arithmetic average takes a simple average of the entire period. Therefore, the real rate of return is more accurate with a geometric average, and less accurate with an arithmetic average.