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.

(NOTE:- THE SOLUTION IS NOT TO BE DONE ON MS EXCEL)

Answer 1

Solution:

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

Date	Closing Price	Monthly Realized Rates of Returns
30-Aug-13	1.11
30-Sep-13	1.04	(1.04-1.11)/1.11 = -0.06306 or -6.306%
31-Oct-13	1.03	(1.03-1.04)/1.04 = -0.00962 or -0.962%
29-Nov-13	0.92	(0.92-1.03)/1.03 = -0.1068 or -10.68%
31-Dec-13	0.93	(0.93-0.92)/0.92 = 0.01087 or 1.087%
31-Jan-14	0.91	(0.91-0.93)/0.93 = -0.02151 or -2.151%
28-Feb-14	0.82	(0.82-0.91)/0.91 = -0.0989 or -9.89%
31-Mar-14	0.81	(0.81-0.82)/0.82 = -0.0122 or -1.22%
30-Apr-14	0.77	(0.77-0.81)/0.81 = -0.04938 or -4.938%
30-May-14	0.75	(0.75-0.77)/0.77 = -0.02597 or -2.597%
30-Jun-14	0.75	(0.75-0.75)/0.75 = 0 or 0.00%
31-Jul-14	0.89	(0.89-075)/0.75 = 0.186667 or 18.667%
29-Aug-14	0.82	(0.82-0.89)/0.89 = -0.07865 or -7.865%

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

Airthmetic average monthly return = (-0.06306-0.00962-0.1068+0.01087-0.02151-0.0989-0.0122-0.04938-0.02597+0+0.0186667-0.07865)/12 = -0.02238 or -2.238%

Geometric average monthly return =[ {(1-0.06306)x(1-0.00962)x(1-0.1068)x(1+0.01087)x(1-0.02151)x(1-0.0989)x(1-0.0122)x(1-0.04938)x(1-0.02597)x(1+0)x(1+0.0186667)x(1-0.07865)}^ (1/12) ] - 1 = -0.02492 or -2.492%

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

End-of-year stock price = Beginning-of-year stock price X (1+ Geometric average monthly rate of return)12

End-of-year stock price = 1.11 X (1-0.02492)12 = $0.82

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).

Annual rate of return for REX = (Ending share price for the period - Beginning share price for the period) / Beginning share price for the period

Annual rate of return for REX = (0.82-1.11)/1.11 = -0.26126 or -26.126%

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

Compound annual rate of return= (1+ Geometric average monthly rate of return)12 -1

Compound annual rate of return= (1-0.02492)12 -1 = 0.34361 or 34.361%

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

We would use a geometric average of the historical rates of return to estimate the average annual rate of return an investor would have earned over the sample period by holding the shares. The reason for this is because the geometric average takes into account the volatility and compounding that occurs from period to period. So, geometric average is a more accurate measure of returns than the arithmetic mean as the geometric averages accurately reflect the compounded investment returns.