FX Quant > Rate of Return Formulas and Calculations
In finance, rate of return (ROR), or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested. The money invested may be referred to as the asset, capital, principal, or the cost basis of the investment.
For purposes of measuring ROR, the initial value Vi and final value Vf must be clearly stated.
In mathematical terms, the arithmetic return is defined as the following:
where Vi is the initial investment value and Vf is the final investment value.
Along with the arithmetic return, there is also a natural log return called logarithmic return or continuously compounded return, which we will not use.
Annual and Annualized Rate of Return
An Annual Rate of Return is the return on an investment over a one-year period, such as January 1st 2006 through December 31st 2006. An Annualized Rate of Return is the return on an investment over a period other than one year (such as a month, or two years) multiplied or divided to give a comparable one-year return. For instance, a one-month ROR of 1% could be stated as an annualized rate of return of 12%. Or a two-year ROI of 10% could be stated as an annualized rate of return of 5%.
In financial economics, the term compound ROR indicates a rate of return that is based on compounding, reinvestment, and/or the changing market value of an account. Compound ROR indicates that the value of the investment increases or decreases during the investment period. To calculate the compound rate of return, the investor includes the reinvested profits in the total investment. The yield depends on the frequency of compounding.
| Annual Compounded ROR Based on Frequency of Compounding | |||
|---|---|---|---|
| Non-compounded Annual ROR | Semi-Annualy | Quarterly | Monthly |
| 1% | 1.002% | 1.004% | 1.005% |
| 5% | 5.062% | 5.095% | 5.116% |
| 10% | 10.250% | 10.381% | 10.471% |
| 15% | 15.563% | 15.865% | 16.075% |
| 20% | 21.000% | 21.551% | 21.939% |
| 30% | 32.250% | 33.547% | 34.489% |
| 40% | 44.000% | 46.410% | 48.213% |
| 50% | 56.250% | 60.181% | 63.209% |
Calculating Annualized Rates of Return
Value Added Monthly Index (VAMI) - Hypothetical Growth of $1,000
The Value Added Monthly Index (VAMI) is an index that tracks the monthly performance of a hypothetical $1,000 investment. The calculation for the current month's VAMI is:
Profit Reinvestment Strategies
Different VAMI figures are obtained depending on the profit reinvestment strategy. Profit can be reinvested 1.) monthly, 2.) annually, or 3.) it can be not reinvested at all.
If the profit is reinvested, larger position sizes should be traded - the initial trading size should be multiplied by a monthly reinvestment factor (MRF).
The reinvestment factor can be 1.) decreased after a losing month, or 2.) not decreased - increase the MRF or keep it constant, but never decrease it. The latter reinvestment strategy (never decrease the MRF) yields the best results - see the example below.
How to Reinvest Monthly
In our trading signals we post position sizes and position size changes normalized for a $10,000 nominal account value. On the first day of each month you should divide your account Net Asset Value, or NAV (NAV = account balance + unrealized profit/loss), by $10,000. This ratio will be your Monthly Reinvestment Factor (MRF). To obtain your position size changes, multiply the values from the trading signals by your MRF (for that particular month).
If the NAV increases on the first day of the next month, you will calculate a new MRF. If not, you will retain the same MRF (do not decrease it)
The same principle remains valid in the case of annual reinvestment, except the Reinvestment Factor should be calculated on January 1 and kept constant for the rest of the year.
Example
|
1 |
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|
Month # |
Monthly ROR, %, on fixed capital (beginning $1,000) |
Compound monthly ROR, % relative to previous month capital |
VAMI (reinvested annually) |
Annual ROR, (reinveste annually) |
Monthly reinvest-ment factor |
VAMI (reinvested monthly) |
Annual ROR, % (reinvested monthly) |
Growth of $1,000 without reinvestment |
| 1 | 1.96 | 1.96 | 1019.60 | 1.020 | 1019.60 | 1019.60 | ||
| 2 | 5.08 | 4.98 | 1070.40 | 1.071 | 1071.40 | 1070.40 | ||
| 3 | (9.35) | (8.74) | 976.90 | 1.071 | 971.22 | 976.90 | ||
| 4 | (5.62) | (5.75) | 920.70 | 1.071 | 911.01 | 920.70 | ||
| 5 | 21.75 | 23.62 | 1138.20 | 1.144 | 1144.04 | 1138.20 | ||
| 6 | 7.68 | 6.75 | 1215.00 | 1.232 | 1231.90 | 1215.00 | ||
| 7 | (2.27) | (1.87) | 1192.30 | 1.232 | 1203.93 | 1192.30 | ||
| 8 | (15.47) | (12.97) | 1037.60 | 1.232 | 1013.36 | 1037.60 | ||
| 9 | 4.75 | 4.58 | 1085.10 | 1.232 | 1071.87 | 1085.10 | ||
| 10 | (10.21) | (9.41) | 983.00 | 1.232 | 946.10 | 983.00 | ||
| 11 | 22.10 | 22.48 | 1204.00 | 1.232 | 1218.35 | 1204.00 | ||
| 12 | 31.50 | 26.16 | 1519.00 | 51.90 | 1.606 | 1606.40 | 60.64 | 1519.00 |
| 13 | 4.98 | 4.98 | 1594.65 | 1.686 | 1686.39 | 1568.80 | ||
| 14 | 1.27 | 1.21 | 1613.94 | 1.708 | 1707.81 | 1581.50 | ||
| 15 | (3.69) | (3.47) | 1557.89 | 1.708 | 1644.79 | 1544.60 | ||
| 16 | 4.70 | 4.58 | 1629.28 | 1.725 | 1725.06 | 1591.60 | ||
| 17 | (16.94) | (15.79) | 1371.96 | 1.725 | 1432.83 | 1422.20 | ||
| 18 | 8.85 | 9.80 | 1506.39 | 1.725 | 1585.50 | 1510.70 | ||
| 19 | 13.71 | 13.82 | 1714.65 | 1.822 | 1822.01 | 1647.80 | ||
| 20 | 9.24 | 8.19 | 1855.00 | 1.990 | 1990.36 | 1740.20 | ||
| 21 | (9.30) | (7.62) | 1713.74 | 1.990 | 1805.26 | 1647.20 | ||
| 22 | 12.88 | 11.42 | 1909.38 | 2.062 | 2061.62 | 1776.00 | ||
| 23 | 5.88 | 4.68 | 1998.70 | 2.183 | 2182.84 | 1834.80 | ||
| 24 | 10.50 | 7.98 | 2158.20 | 42.08 | 2.412 | 2412.04 | 50.15 | 1939.80 |
Legend:
2 - The monthly rate of return (ROR) on fixed capital is calculated by dividing the monthly $ profit/loss by the account value at beginning of trading..
3 - The compound monthly ROR, is calculated by dividing the monthly $ profit by the previous month capital. This method of calculation is required by CFTC Regulation 4.25(a)(7)(i)(F) for reporting purposes. The compound monthly ROR will be different from the ROR on fixed capital since the account value changes (due to profit/loss from trading or reinvestments).
If the profit is not reinvested monthly, the compound monthly ROR is calculated as:
where RORcompn
is the compound monthly ROR in the n-th month, RORfixedn
is the monthly ROR on fixed capital in the n-th month. Since the trading
size is adjusted on January 1 each year, the compound monthly ROR is reset to
monthly ROR on fixed capital each January.
If trading profit is reinvested monthly and the trading size is adjusted, the compound monthly ROR, relative to previous month capital, will be identical to the ROR on fixed capital:
4 - The Value Added Monthly Index (VAMI) is the value of $1,000 investment as of the end of each month based upon the compound monthly RORs. The VAMI calculation, with annual reinvestments, would be as follows:
Alternatively, VAMI can be calculated by multiplying compound monthly RORs (hence the name compound ROR)
5 - Annual ROR, if reinvested annually:
6 - The Monthly Reinvestment Factor (MRF) is calculated as:
7 - VAMI, with monthly reinvestments, would be as follows:
8 - Annual ROR, with monthly reinvestments, is calculated as above (5 - Annual ROR, with annual reinvestments), except VAMI with monthly reinvestments (6) is used.
9 - Growth (Monthly Index - MI) of $1,000 without reinvestments is obtained by summing monthly RORs on fixed capital:
Please note the main difference between the MI and VAMI: the monthly growth without reinvestments is obtained by summing RORfixed percentages, while the VAMI is obtained by multiplying (1+RORcomp).
See also this Excel worksheet example.
