How do you calculate rate of return ?

Rate of return is calculated by dividing the total gain or loss on an investment by the initial amount of the investment. The formula for calculating rate of return is:

Rate of return = (Ending value of investment - Beginning value of investment) / Beginning value of investment

This calculation will give you the percentage return on your investment.

Simple Rate of Return (ROR)

```
ROR = (Ending Value - Initial Value) / Initial Value
```

Time-Weighted Rate of Return (TROR)

```
TROR = [(1 + R1) - (1 + R2) - ... - (1 + Rn)] - 1
```
where:
- R1, R2, ..., Rn are the returns in each period (annualized)

Dollar-Weighted Rate of Return (DROR)

```
DROR = (Ending Account Value - Initial Account Value + Contributions - Withdrawals) / Initial Account Value
```

Other Considerations:

- Compounding: Returns are reinvested, leading to exponential growth.
- Annualization: Returns are expressed on an annual basis, even if calculated over different time periods.
- Risk-adjusted: Returns can be adjusted for risk, such as using the Sharpe ratio or Sortino ratio.
- Nominal vs. Real: Returns can be stated in nominal terms (with inflation) or real terms (adjusted for inflation).
- Tax implications: Returns may be subject to taxes, which can affect the effective rate of return.

Example:

- Initial investment: $10,000
- Ending value after 5 years: $15,000

Simple ROR:
```
ROR = (15000 - 10000) / 10000 = 0.50 or 50%
```

TROR:
```
TROR = [(1 + 0.10) - (1 + 0.12) - (1 + 0.08) - (1 + 0.15) - (1 + 0.10)] - 1 = 0.57 or 57%
```

DROR: (assuming no contributions or withdrawals)
```
DROR = (15000 - 10000) / 10000 = 0.50 or 50%
```