Question

After graduating from university last year with a degree in accounting and finance, Jim Hale took a job as a trainee analyst for an investment company in Melbourne. Jim’s 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 Jim’s first day, his boss called Jim in and told him he wanted to do some rudimentary analysis of the investment returns of the regional airline Regional Express Holdings Ltd (REX). Specifically, Jim was given the following month-end closing prices for the company spanning the months from September 2019 to August 2020:

Date	Closing Price	Date	Closing Price
30 Aug 13	1.11	31 Mar 14	0.81
30 Sep 13	1.04	30 Apr 14	0.77
31 Oct 13	1.03	30 May 14	0.75
29 Nov 13	0.92	30 Jun 14	0.75
31 Dec 13	0.93	31 Jul 14	0.89
31 Jan 14	0.91	29 Aug 14	0.82
28 Feb 14	0.82

Jim was then instructed by his boss to complete the following tasks using the REX price data (note that REX paid no dividend during 2008).

1. Compute the monthly realized rates of return earned by REX for the entire year.

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

3. Calculate the year-end price for REX, computing the compound value of the beginning-of-year price of $ 1.11 per share for 12 months at the geometric average monthly rate of return calculated earlier: End-of-year stock price = Beginning-of-year stock price X (1+ Geometric average monthly rate of return)¹²

4. Compute the annual rate of return for REX using the beginning share price for the period and the ending price (i.e. $1.11 and $0.82).

5. Use the geometric average monthly rate of return and the following relationship to calculate the annual rate of return: Compound annual rate of return= (1+ Geometric average monthly rate of return)¹² -1

6. If you were given annual rate of return data for REX or any other company’s shares and you were asked to estimate the average annual rate of return an investor would have earned over the sample period by holding the shares, would you use an arithmetic or 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

1. Monthly realized rates of return

Monthly realized rates of return is computed with the following formula = (Current Month Price/Last Month price)-1

2. Average monthly rate of return

Average monthly rate of return using arithmetic average = -2.24%

This is just the average of the 12 months realised rates of return computed in (1) above.

Average monthly rate of return using geometrical average = -2.49%

This is computed using the formula:

((1+Monthly Realised Return of Month 1)*(1+Monthly Realised Return of Month 2)*......*((1+Monthly Realised Return of Month 11)*(1+Monthly Realised Return of Month 12))^(1/12)-1

3. Year End Price = $0.82

This is computed by the formula (as given in question) = Beginning-of-year stock price ($1.11) X (1+ Geometric average monthly rate of return)^12

4. Annual rate of return = -26.13%

This is computed by the fomula = (Ending Price ($0.82) - Beginning Price ($1.11))/Beginning Price ($1.11)

5. Annual rate of return using geometric average monthly rate of return = -26.13%

This is computed by the fomula =  (1+ Geometric average monthly rate of return)^12 -1

6. If I were given annual rate of return data for REX or any other company’s shares and were asked to estimate the average annual rate of return an investor would have earned over the sample period by holding the shares, I would you use an geometric average of the historical rates of return rather than using arithmetic average.

While arithmetic return is simply the average of the returns, geometric average considers the compounding between the periods. Geometric returns are more appropriate measure in case of a company share where there is a correlation with multiple variables including period, market risk premiums, yield, etc. Also, the returns are volatile (as it increases and decreases between the period) and hence arithmetic return will not provide the right number when there is volatility.

Workings: