# Use of Arithmetic Mean and Geometric Mean in Return Calculation

## Introduction

• Returns are a crucial part of an investment, returns on an investment decision the fund allocation to that investment. It is important to know the actual returns on investment before investing in it. However, many of us do not know how returns on investment actually work, lack of visualization makes us belive in what others say.
• Normally we insist on taking an average of any set rather than considering each element, this does not hold good for a rate of return on investment as the process to calculate average here is slightly different, let us see the same in an illustration.

#### Illustration

Illustration: Ram has invested Rs.100000 in a mutual fund for 5 years, the return of the mutual fund in 5 years is 10%,15%, -10%,12%,8% respectively. Let us see the calculation of the
return.
Year 1st Year 2nd Year 3rd year 4th Year 5th Year
Rate 10% 15% -10% 12% 8%
100000 ₹ 1,10,000.00 ₹ 1,26,500.00 ₹ 1,13,850.00 ₹ 1,27,512.00 ₹ 1,37,712.96
• Ram wants to know what is the average returns for the 5 years, hence he takes the mean of the rates:
• Mean = [10 + 15+ (-10) + 12 + 8]/5 = 7%
• Hence, Ram thinks that he has got a 7% return per year.

Let us calculate the maturity of Rs. 100000 after 5 years at the rate of 7% per annum
Year 1st Year 2nd Year 3rd year 4th Year 5th Year
Rate 7% 7% 7% 7% 7%
100000 ₹ 1,07,000.00 ₹ 1,14,490.00 ₹ 1,22,504.30 ₹ 1,31,079.60 ₹ 1,40,255.17
• As clearly visible both the returns differ after by (140255.13 – 137712.96) = Rs. 2542,17
• What is the reason behind the difference? As we know that in investment, total return after a year is the principal amount for the next year i.e. the rates of return are not dependent only on the current rate, but instead on the previous year’s rate also.
• If we increase or decrease the rate for a year, the principal amount of the next year will also vary and hence the total returns, hence Arithmetic mean cannot to used to determine the average rate of return over few years.
• In the above scenarios i.e. calculating returns over a period, Geometric Mean proves to be more accurate. Let us see how.
• Geometric mean = • In case of returns calculation that may involve negative terms, the above formula gets modified to:
• Geometric Mean = • Substituting with rates in every year to obtain the Mean Rate
• Mean Rate = = 1.066 – 1 = 0.066 or 6.6%
• Now, calculating the returns with the rate of interest as 6.6%
Year 1st Year 2nd Year 3rd year 4th Year 5th Year
Rate 6.6% 6.6% 6.6% 6.6% 6.6%
100000 ₹ 1,06,600.00 ₹ 1,13,635.60 ₹ 1,21,135.55 ₹ 1,29,130.50 ₹ 1,37,653.11
• Now, if we see the difference between the actual return obtained after 5 years and return calculated after 5 years at the rate of 6.6%,
• Difference = (137653.11 – 137712.96) = Rs. 59.85
• We can clearly see that in the case of average returns, Geometric Mean turns to be more accurate and realistic than the Arithmetic Mean.
• To generalize in a better way, cases where the return has to be calculated for certain years, the returns are not only dependent on the rate of next year but rather on the amount of this year Geometric Mean is a better method than Arithmetic Mean, Geometric Mean is more accurate than the Arithmetic Mean. 