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Rate of return

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In finance, rate of return, or ROR, return on investment, ROI, or sometimes just return, is a comparison of the money earned (or lost) on an investment to the amount of money invested.

1 Analysis
2 Average Annual Rate of Return
3 Average Annualized Rate of Return
4 Overall Rate of Return
5 See also
- 5.1 Examples of high yielding investments
6 Further reading
7 External links
8 Sources

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Analysis

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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.

In marketing the notion of "return on investment" is an increasingly important topic for Chief Marketing Officers (CMOs) who need to find ways to justify the high investments made across a broad range of marketing AND sales activities. When put together, these investments can represent between 15 and 40% of a given company's sales. Given the disparity of spend categories, and the inherent difficulty to assess the impact of several of them (e.g. the difficulty to be deterministic when assessing the impact of television on sales), companies have either been developing some intermediary/ surrogate metrics, or proceeded with more or less complex econometric modeling. The latter can however often suffer from (1) appearing as a "black box" to marketeers and (2) being limited to "known" media (and hence not being suited for new media (new per se or not regularly used by a given company or brand). New methods therefore present interesting potential for impact, namely leveraging "heuristics" and (controlled) experiments. This topic can be further expanded/ detailed.

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Average Annual Rate of Return

Also referred as the Arithmetic Average

Add all the percents and divide by the number of RORs

0.0% + 15.3% + (-12.2%) + 9.3% = 12.4%

divide by 4: 12.4%/4 = 3.1% Average Annual ROR

The mathematical logic is as follows: Arithmetic mean

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

 $V i$  is the initial investment
 $V f$  is the final value s

$ROI_{Arith}=\frac{V_f - V_i}{V_i} = \frac{V_f}{V_i} - 1$

This return has the following characteristics:

$ROI_{Arith}=+100\%$ when the final value is twice the initial value
$R O I A r i t h > 0$ when the investment is profitable
$R O I A r i t h < 0$ when the investment is at a loss
$ROI_{Arith}=-100\%$ 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..

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Average Annualized Rate of Return

Annualized returns express the rate of return of a portfolio over a given time period on an annual basis, or a return per year. Reference Source
Also referred as the Geometric Average and Time-Weighted Average Return.

Add 1 to the decimal equivalent of each percent. Multiply all 4 numbers times each other.Take the root of the :number, where the root is equal to the number of observations. In this case you want to take the 4th root, :which is the same thing as raising to the (1/4) power or .25. Finally, subtract 1 and convert to a percent by :moving decimal 2 places to right.

(1 + 0) * (1 + .153) * (1 + (-.122)) * (1 + .093)

(1.0) * (1.153) * (0.878) * (1.093) = 1.10648

Next take the 4th root of this number

(1.10648)^1/4 = 1.025618 [calculator strokes: 1.10648 (y^x) .25 (=)]

1.025618 minus 1.0 = .025618 = 2.5618% Average Annualized (compound) ROR

The mathematical logic is as follows: Geometric mean and Weighted geometric mean.

The above definition solves the problem created by the arithmetic average where a +10% return and a -10% return do add up to 0%. However, starting with $100, a +10% return would result in $110. A subsequent -10% return would result in $99. This is precisely the result obtained by applying the geometric average to returns. Therefore, a +10% and a -10% do not have as result a zero gain but rather a $1 loss.

Academics use is their research natural log return called logarithmic return or continuously compounded return. The continuously compounded return is asymetric thus clearly indicating that the up and down returns are not equal. In the above example the +10% return results in 9.53% continuously compounded return while the -10% results in -10.53%. This clearly indicates that the investment will result in a dollar amount loss corresponding to the difference between the two numbers: 1%.

 $V i$  is the initial investment
 $V f$  is the final value s

$ROI_{Log} = \ln\left(\frac{V_f}{V_i}\right)$ .

This return has similar characteristics:

$R O I L o g > 0$ is profit
$R O I L o g < 0$ is a loss
Doubling occurs when $ROI_{Log}=\ln(2)=69.3\%$
Total loss occurs when $ROI_{Log}\to-\infty$ .

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Overall Rate of Return

Here we’re looking for what would be the ROR you earned over the full 4 year period, not an average annual ROR.
Simple, but incorrect method: The folks on the evening news will simply add the 4 percents together to get 12.4%--which is wrong because it does not take into account the compounding that goes on.
More complicated, but correct method: To do this correctly and take account of the compounding we do some of what was done above in part (B), i.e., convert all the percents to their decimal equivalent, add 1 to each number, and multiply all 4 numbers times each other. Above in (B) when we did that, we got 1.10648. The next step is to subtract 1 and then move the decimal 2 places to right to get 10.648%. So, had you started year 1 with $100,000 in this account, at the end of year 4 your account would 10.648% larger, that is it would now be $110,648.

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