# Return on investment

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.

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## Calculations

There are two methods of calculating the basic ROI. Each has its own mathematical merits.

[itex]V_i[itex] is the initial investment
[itex]V_f[itex] is the final value


### Arithmetic return

In mathematical terms, the arithmetic return is defined as the following.

[itex]ROI_{Arith}=\frac{V_f - V_i}{V_i} = \frac{V_f}{V_i} - 1[itex]

This return has the following characteristics:

• [itex]ROI_{Arith}=+100%[itex] when the final value is twice the initial value
• [itex]ROI_{Arith}>0[itex] when the investment is profitable
• [itex]ROI_{Arith}<0[itex] when the investment is at a loss
• [itex]ROI_{Arith}=-100%[itex] 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.

ROI is also commonly being used by semi-pro poker players. Poker players are keeping track of their ROI to see if their time is being aptly profitable. For instance, I've cashed in 7 of my last 16 $6 3-table NL SNG's on Party. They pay top 5 out of 30. I have two 2nd place cashes, two 3rd place cashes, two 4th place cashes, and one 5th place cash. I have cashed$195 over these 16 tournaments. Since I've spent, or invested, $96, and my profit is$99, that means that my ROI is 103%. MY ITM (In the money) is 43% which is phenomenal when you consider that only 1-in-6 finishes in the money. Once my bankroll hits $300, I'll probably jump up to$11 tournaments, where the same ROI (hopefully) will yield me twice what it does at this level. In a couple years I'll probably be dominating the $1K tournaments with an ROI of 100% and therefore pocketing$1K for about 2 hours of work, while watching TV at the same time.

### 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.

[itex]ROI_{Log} = \ln\left(\frac{V_f}{V_i}\right)[itex].

This return has similar characteristics:

• [itex]ROI_{Log}>0[itex] is profit
• [itex]ROI_{Log}<0[itex] is a loss
• Doubling occurs when [itex]ROI_{Log}=\ln(2)=69.3%[itex]
• Total loss occurs when [itex]ROI_{Log}\to-\infty[itex].

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