## Return on Investment

##### Published: May 2006

ROI is a financial metric used to measure the relative efficiency of an investment. Comparing the ROI of different investments reveals the relative success of various investment schemes.

ROI is calculated by comparing the net return of an investment to its initial cost. This can be defined as

$$ROI=\frac{V_f\:-\:V_t}{V_t}\:$$which can be simplified by multiplying by vt throughout to $$ROI={\frac{V_f}{V_t}\:-\:1}$$

where Vf is the final sum returned by an investment and Vt is the sum originally invested.

This calculation will return a value in the range -1 to < ∞, with-1 being an investment which can no longer be recovered (and close to 8 representing a very good investment!). To express this calculation as a percentage, simply multiply the result by 100.

However, establishing the relationship between ROIs calculated using the above equation is not particularly straightforward. For instance, an investment which yields an ROI of 0.2 one year and depreciates by the same amount the following year (ie -0.2) would yield -0.04 overall, rather than the 0 which may be apparent at first glance. This is because the relationship between Vf and
Vi described by the above equation is not linear – the variable Vf is not independent of the variable Vi.

This confusion can be remedied by applying logarithms to the ROI calculation, thus ROI can also be expressed as:

$$logROI={log\frac{V_f}{Vt}=logV_f\:-\:log\:V_t}$$

This calculation will return a value in the range – ∞ → ∞. By allowing an equal range of expression on either side of 0, the logarithmic ROI calculation is balanced and symmetrical. Thus ROIs can be directly compared.

The concept of ROI is used for many other applications outside the purely financial expression outlined above. For example, IT and Human Resources departments often talk about the ROI of a new technology or training methodology they plan to implement.

The method used to calculate ROI will vary according to the situation to which it is applied. Whilst the basic principle of comparing the net benefits of an investment to the original investment will always apply, other factors will also be incorporated into the equation. For instance, an IT department looking to calculate the ROI of purchasing and implementing a new system would need to incorporate the depreciation in value of any new technology purchased (new technology is generally deemed obsolete after three years of use) in such an ROI assessment as well as estimates of the benefits of time saved.