Evaluating Investment Returns: Arithmetic vs. Geometric Average Return

Introduction

in unit trusts is a common way to grow one’s wealth over time. In this scenario, we’ll examine the performance of an investment in the “BML Fund” managed by Maluti Asset Managers (MAM) over a seven-year period. We’ll calculate and compare the arithmetic average return (AAR) and geometric average return (GAR) of this investment, and discuss which average MAM would prefer to use in their performance report. Additionally, we’ll explore the outcomes of selling the investment after three years versus holding it for the full seven years, analyzing the holding period returns (HPR) in both cases.

Arithmetic vs. Geometric Average Return

The AAR is calculated by summing the annual returns and dividing by the number of years, providing a simple average. On the other hand, the GAR considers the compounding effect of returns and is calculated by taking the nth root of the product of (1 + return percentages), where ‘n’ is the number of years.

To calculate the AAR: (7.55% + 9.05% + 7.11% – 15.2% – 17.1% – 6.7% + 0.66%) / 7 = -0.66%

To calculate the GAR: GAR = [(1 + 7.55%) * (1 + 9.05%) * (1 + 7.11%) * (1 – 15.2%) * (1 – 17.1%) * (1 – 6.7%) * (1 + 0.66%)]^(1/7) – 1 ≈ -1.37%

Comparing AAR and GAR

The AAR is -0.66%, while the GAR is approximately -1.37%. MAM would prefer to use the AAR in their performance report because it reflects the simple average return over the seven years, providing a clearer picture of the investment’s performance to potential investors. The GAR, which accounts for the compounding effect, tends to be lower and is typically used for internal purposes to assess the actual growth of the investment.

Selling the Investment After Three Years

If you decide to sell your investment in the BML Fund after three years, you would have experienced the following returns: 7.55%, 9.05%, and 7.11%.

Total value after three years = R1 million * (1 + 7.55%) * (1 + 9.05%) * (1 + 7.11%) ≈ R1,309,491.05

Holding Period Return (HPR) after three years: HPR = (Ending Value – Beginning Value) / Beginning Value HPR = (R1,309,491.05 – R1,000,000) / R1,000,000 ≈ 30.95%

Selling the Investment After Seven Years: If you decide to sell your investment after seven years, the BML Fund would have experienced all seven annual returns.

Total value after seven years = R1 million * (1 + 7.55%) * (1 + 9.05%) * (1 + 7.11%) * (1 – 15.2%) * (1 – 17.1%) * (1 – 6.7%) * (1 + 0.66%) ≈ R831,172.64

Comparing Three-Year and Seven-Year Outcomes

The BML Fund would be worth less after seven years (R831,172.64) than it was worth after three years (R1,309,491.05). This outcome is due to the negative returns experienced in years four, five, and six. The 7-year HPR would be:

HPR = (Ending Value – Beginning Value) / Beginning Value HPR = (R831,172.64 – R1,000,000) / R1,000,000 ≈ -16.83%

Conclusion

In conclusion, when assessing the performance of investments, it’s essential to consider both the arithmetic and geometric average returns. The AAR provides a simple average, suitable for performance reporting, while the GAR accounts for compounding and is useful for internal analysis. In this case, the BML Fund’s AAR was -0.66%, suggesting a negative average return over seven years. If you sell the investment after three years, you would receive approximately R1,309,491.05 with a HPR of 30.95%. However, if you hold for seven years, the investment’s value would be approximately R831,172.64, resulting in a negative HPR of -16.83% due to the negative returns in years four to six.