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Return on investment is a crucial indicator for evaluating investment performance, but investing involves a continuous process of different transactions such as buying, selling, stock-split, and dividends. Even if the final profit amount is the same, the calculation of returns varies with different time points and investment amounts. We will illustrate the definitions, calculation methods, applicable scenarios, and relevant explanations for these return rates using an example (excluding stock splits, dividends, transaction fees, and related taxes for emphasis on the meaning of returns).

### A sample set of investment return calculation transaction data

Date | Annual Return on A-Stocks | John - Hold to Maturity | Joe - Buy at the Same Price | Ray - Buy at Different Price | May - Sell at Different Price |

2001/1/1 | 0.00% | $16,000 investment, bought 1600 A shares at $10 | $10,000 investment, bought 1000 A shares at $10 | $10,000 investment, bought 1000 A shares at $10 | $16,000 investment, bought 1600 A shares at $10 |

2002/1/1 | -10.00% (2002) | A stock, price $10 | $6,000 investment, bought 600 A shares at $10 | A stock at $10 | A stock at $10 |

2003/1/1 | 66.67% (2003) | A stock at $9 | A stock at $9 | $5,400 investment, bought 600 A shares at $9 | A stock at $9 |

2004/1/1 | 6.67% (2004) | A stock at $15 | A stock at $15 | A stock at $15 | Sold 900 A shares at $15 total of $13,500 |

2005/1/1 | A stock price is $16, 1600 shares, market value $25,600 | A stock price is $16, 1600 shares, market value $25,600 | A stock price is $16, 1600 shares, market value $25,600 | A stock price is $16, 700 shares, market value $11,200 | |

Total Return | Geometric Cumulative Return | 60.00% | 60.00% | 66.23% | 54.38% |

Unrealized Return | 60.00% | 60.00% | 63.75% | 60.00% | |

Arithmetic Mean Return | 15.00% | 15.00% | 16.56% | 13.60% | |

MWRR | Geometric Mean Return 12.47% | 12.47% | 13.78% | 16.30% | 13.46% |

TWRR | Geometric Mean Return 12.47% | 12.47% | 12.47% | 12.47% | 12.47% |

Explanation of Relevant Fields:

- A Stock Annual Return: (Next Year’s Initial Stock Price – Current Year’s Initial Stock Price) / Current Year’s Initial Stock Price, representing the performance of A stock for the year.
- John: Purchased at the beginning and held until the end.
- Joe: Compared to John, bought at the same price each year and held until the end
- Ray: Compared to John, bought at different prices each year and held until the end.
- May: Compared to John, bought at the same price each year and sold at different prices in different years.

### Total Return

The Total Rate of Return refers to the percentage of investment gains or losses relative to the total investment amount during the investment period.

Total Return=(Invest Gain/Total Investment Amount) = (Present Value+Selling Amount−Total Investment Amount)/Total Investment Amount = ((13,500+11,200)-(10,000+6,000))/(10,000+6,000) = 54.38%

Using the values provided:

**Characteristics of Total Return:**

- John’s Total Return is the Same as A Stock’s Total Return: This indicates that the Total Return of A Stock is the return rate from the initial purchase to holding until maturity.
- John’s Total Return is the Same as Joe’s: This signifies that the Total Return does not differ due to different investment time points, neglecting the time value.

### Unrealized Return

The unrealized return rate is the ratio of the investment gain/loss to the market value of the remaining shares that have not been sold up to the current point. Various methods, including weighted average cost, FIFO (First-In-First-Out), moving average, etc., can be used for calculating the cost per share. Joe’s weighted average cost = (1000 * 10 + 600 * 10) / 1600 = 10.

The characteristics of unrealized return

- John and May have the same unrealized return rate: This means that if all inventory is sold now, the total return rate that can be obtained, but it does not consider the already earned portion of the return rate, such as dividends and the gains/losses from sold positions. Most financial institution websites display the return rate in the investment inventory list as the unrealized return rate, without considering the realized gains, which cannot truly reflect your total return rate, and it does not take into account the time value.

### Arithmetic Average Return

Arithmetic average return represents the annual return, calculated using arithmetic mean. It shares similar characteristics with the total return but cannot accurately represent the average return of the underlying asset.

The total return rate, unrealized return rate, and arithmetic average return rate discussed above do not take into account the time value of money. In simple terms, if ten years ago, Joe invested 10,000 and the current investment value is 15,000, the total return rate is 50%. Ray invested 10,000 a year ago, and the current investment value is also 15,000, with a total return rate of 50%. From the perspective of total return rate, without considering the time value, the investment performance of the two individuals is completely identical.

### Money-Weighted Rate of Return(MWRR)

ow to incorporate the concept of time value of money into investment returns, and then introduce the concept of the time value of currency: Please refer to the Wikipedia page on the Time Value of Money.

If the calculated rate of return per period makes the present value of investment and recovery amounts (selling amount, dividends, investment present value) zero, it can be considered as the average rate of return per period during that period. This is defined as the Internal Rate of Return (IRR). Please refer to the Wikipedia page on Internal Rate of Return for more information.

The calculation of the internal rate of return (IRR) involves weighting each period’s cash flow by the present value factor of the currency. Therefore, within the domain of calculating investment returns, this rate of return is also known as the Money-Weighted Rate of Return (MWRR).

The calculated Money-Weighted Rate of Return (MWRR) for individuals, including John, Joe, Ray, and May, reveals the following characteristics:

- Joe’s Money-Weighted Rate of Return is the same as the annual return rate of A stock: This implies that the Money-Weighted Rate of Return, calculated based on the weighted return of the investment from the initial purchase until the end, is equivalent to the performance of the underlying asset.
- Different entry and exit points for each individual lead to distinct Money-Weighted Rates of Return: The Money-Weighted Rate of Return provides a comprehensive view of the real return achieved by considering the amounts, timing, and decisions related to buying and selling the underlying asset.
- The Money-Weighted Rate of Return is influenced by investment amounts at different time points: When used to assess the performance of fund investments, it can be affected by the purchases and redemptions made by investors. Therefore, it may not be suitable as a performance indicator for comparing funds or indices.

The Money-Weighted Rate of Return is primarily applied in loan, investment, and other financial decision-making scenarios. Additionally, it can be used to analyze the feasibility of financial goals. Please refer to the following article for more information.

Money-Weighted Rate of Return – Loan and Investment Programs Selection

### Time-Weighted Rate of Return(TWRR)

How to consider the time value of money and remove the impact of investment amounts? Regardless of whether there were buy or sell transactions in a particular period, calculate the rate of return for each period, then use these period returns to calculate the cumulative return and the geometric average return. This approach removes the influence of investment amounts, showcasing the performance of the investment portfolio. This rate of return is known as the Time-Weighted Rate of Return (TWRR). The calculation for May’s return rate in 2004 is as follows:

The formula for Time-Weighted Rate of Return (TWRR) over a specific period is given as:

In provided example

This represents the Time-Weighted Rate of Return over the specified period.

- The geometric average return of A stock and the Time-Weighted Rate of Return (TWRR) for individuals (John, Joe, Ray, and May) are the same. This indicates that regardless of the different amounts invested at different times, the TWRR is not affected by the timing and amount of investments; it is only related to the performance of the investment positions.
- If you have multiple assets in your portfolio, you can consider them as a single investment portfolio. In this case, TWRR calculates the average operational performance of the entire investment portfolio.
- The standard deviation of the periodic returns can serve as a measure of risk for the investment portfolio.
- TWRR can be used for comparing the operational performance of different investment portfolios. Furthermore, it can be compared to market indices, and various indicators such as Alpha, Beta, and the Sharpe ratio can be calculated using the periodic returns.

Investment Return Calculation Methods Comparison

According to the above discussion, the characteristics of various return calculation methods are summarized in the table below:

Total Return | Unrealized Return | MWRR | TWRR | |

Unrealized Profit on Inventory | O | |||

Actual Earnings | O | O | ||

Considering Time Value | O | O | ||

Reflecting Investment Selection Skills | O | |||

Used to Calculate Investment Risk | O | |||

Can be used as a basis for comparison. | O |

### Application areas of Return Calculation Methods

Summarizing the applicability of various methods for calculating investment return:

- Total Return: If I want a simple calculation to understand how much money I’ve earned from my investment.
- Unrealized Return: If I don’t want to bother with my past gains or losses and just want to know the current profit or loss in my portfolio to set stop-loss and take-profit targets.
- Money-Weighted Rate of Return: If I have financial goals and want to calculate the internal rate of return based on future expected cash flows. It requires assessing whether my investments have achieved the internal rate of return for my financial goals up to the current point.
- Time-Weighted Rate of Return: If I want to know the operational performance of my investments, compare it with others or indices, and evaluate the return and risk. It serves as a basis for deciding whether to invest independently, choose indices, funds, or asset management services.

Software for calculating Time-Weighted Rate of Return