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In finance, the return on investment (ROI) or just return is a calculation used to determine whether a proposed investment is wise, and how well it will repay the investor. It is calculated as the ratio of the amount gained (taken as positive), or lost (taken as negative), relative to the basis.
The analysis of the return on investment is either done by static or dynamic formal methods, which may be distinguished by the role of time in the model chosen. Dynamic models take account of the fact that a later date of payment may be valued inferior in a model with interest rates. In other words, static approaches can be regarded as sufficient, if the distribution of payments in each period may be assumed as equal to others. All basic ROI-Models are deterministic, for instance the well-known Total Cost of Ownership Model by the Gartner Group. Deterministic models assume the security of prediction. Abandoning this leads into the wide sphere of risk-aware-models, that are inspired by the mathematics of insurances.
Calculations
There are two methods of calculating the basic ROI. Each has its own mathematical merits.
<math>V_i<math> is the initial investment
<math>V_f<math> is the final value
Arithmetic Return
In mathematical terms, the arithmetic return is defined as the following.
<math>ROI_{Arith}=\frac{V_f - V_i}{V_i} = \frac{V_f}{V_i} - 1<math>
This return has the following characteristics:
- <math>ROI_{Arith}=+100%<math> when the final value is twice the initial value
- <math>ROI_{Arith}>0<math> when the investment is profitable
- <math>ROI_{Arith}<0<math> when the investment is at a loss
- <math>ROI_{Arith}=-100%<math> when investment can no longer be recovered
Interestingly, to compensate for a negative ROI, one needs a positive ROI that is higher in magnitude. For example, to recoup a 50% loss one needs to realize a 100% gain.
Logarithmic Return
The above definition is problematic in that a +10% return and a -10% return do not add up to 0%. For example, starting with $100, a +10% return would result in $110. A subsequent -10% return would result in $99.
To correct this, academics use a natural log return called logarithmic return or geometric return.
<math>ROI_{Log} = \ln\left(\frac{V_f}{V_i}\right)<math>.
This return has similar characteristics:
- <math>ROI_{Log}>0<math> is profit
- <math>ROI_{Log}<0<math> is a loss
- Doubling occurs when <math>ROI_{Log}=\ln(2)=69.3%<math>
- Total loss occurs when <math>ROI_{Log}\to-\infty<math>.
See also
External links
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