## Friday, April 19, 2013

### The Many Faces of ROI (Return on Investment)

Learning how to calculate the return on investment (ROI), or rate of return (ROR) happens early on in finance training, and it can be a deceptively simple introduction.
Simple because the ROI calculation involves subtleties, and these are often glossed over during the introductory learning, not clearly spelled out later, or not practiced enough such that the original knowledge atrophies.
For these reasons, it is beneficial to revisit this topic from time to time in order to mitigate the risk that we may misinterpret information or neglect to ask appropriate questions.
Understanding ROI is important for an organization to analyze its:
§  potential strategic and tactical investments,
§  current and historical operating performance
§  current and historical stock performance
In addition, comparisons of the above with those of its competitors can also provide valuable information with respect to the status of the organizational strategy.

From the Beginning…
The ROI calculation involves three main elements:
§  How much money we made or project to make (the “return”),
§  How much money we invested or anticipate investing (the “investment”),
§  How long of a time span is involved
Figure A
Basic ROI Equation
0 = Beginning Time Period
T = Ending Time Period
t = T - 0
Figure A shows two forms of the basic ROI equation. In Equation 1, the return is in the numerator, the investment is in the denominator and the time span involved is denoted by the subscripted items (0,t and T). In Equation 2, the value at the end of the time period (T) is divided by the value at the beginning of the time period (0), with 1 then being subtracted.
Note that the subscript terms, 0 and T, are points in time, while t is a period of time. For example, if it is April 1, and we simulate something until May 1, then 0=April 1, T=May 1, and t= 1 month or 30 days, depending on how we are measuring units.
So if we have an investment of ²100 (for new readers the ² symbol stands for Treasury Café Monetary Units, or TCMU’s, freely exchangeable with any currency at any rate of your choosing) and we made ²2 during the month, then using Equation 1 our t-period return is 2%.
The calculation of returns in this manner is called the “simple return”.

…Through Time…
Because time periods may vary, rates of returns are often expressed in annualized amounts. The advantage of annualizing is that it creates an “apples to apples” comparison between:
§  Investments in different assets, projects or firms over unequal time periods, or
§  Investments in the same asset over different time periods
Figure B
Annualized Rate of Return
Figure B shows the formulas for annualizing ROI. The giant Greek symbol tells us that we are to multiply each of the n terms together. Each time period is to be measured in terms of its proportion to an annual unit. This means that if period 1 (i.e. i=1) is 3 months, then ti for that period would be 0.25 since 3 months is a quarter of a year. If period 2 (i=2) is 6 months, then ti for that period would be 0. 5, since 6 months is a half of a year.
Figure C
Annualized 1 Month Return Example
For returns where we only have 1 time period available, Equation 1 is simplified because the Greek symbol can be ignored as we do not have anything to multiply together - our only input is the simple return calculated for a single time period. Figure C shows the Excel output example where we annualize the 2% monthly simple return example introduced in the last section.
Figure D
Annualized Return From 12 Monthly Caculations Example
The two shaded boxes represent the cumulative totals from Figure B's Equation 1 and Equation 3 which are used as the final values in the formula
Figure D shows the situation where multiplication is required. The two columns on the right represent the cumulative totals of Figure B’s Equation 1 and 3 (the sum of the products and the sum of time periods). In this example we see that each monthly time period has a 2% ROI. Not surprisingly, we end up with the same result as we did in Figure C, which should be the case, as an annualized 1 month return of 2% (which means we earn 2% every month of the year) should equal the one year return if we calculate it out month by month.
Figure E
Annualized 5 Year Return Example
The two shaded boxes represent the cumulative totals from Figure B's Equation 1 and Equation 3 which are used as the final values in the formula
Figure E shows the calculation where we have three time periods, in this case each longer than 1 year. We made 16% (using the simple return calculation) over a 2-year time period, then made 18% over the next 2 years, and earned 10% during the final year, making our total periods 5 (their proportion to an annual unit). The annualized rate of return for this 5 year investment is 8.529%.
When annualized, the return is sometimes referred to as the “Compound Annual Growth Rate”, or CAGR (to be discussed further a little later).

…And Back Again
Figure F
ROI Conversion from Annual to Other Periods
By using algebra we can re-arrange the annualized ROI into any other time period we choose. Figure F shows the final equations involved. Using Equation 1, Figure G shows the Excel output for a one month return (ti = 1/12 = .08333) if the annual ROI is to be 26.82%. Our calculation indicates that we need to make a 1-month return of 2% in order to achieve this annualized amount.
Figure G
Consistent with the aim of creating ways to check our work, we calculated Figure G in order to validate its results vs. those of Figures C and D, since we know what the result should be in that case if our formula is working correctly.
By using algebra we can re-arrange the annualized ROI into any other time period we choose.

Compounding
Why is a 2% monthly return not equivalent to a 24% (12 * 2%) annual return? The answer is due to what we call “compounding”. This term is used to identify the fact that we earn “interest on our interest” as we go through time.
The cumulative column in Figure D shows this compounding occurring. In period 1 we begin with 1 and end with 1.02. In period 2’s total, there are three items: a) we retain the .02 from last period, b) our original 1 earns another .02, and c) the .02 from last period earns 2% as well.
This process repeats every period, where the return from the last period earns return in the next in addition to our prior amounts.
Compounding can occur at different frequencies. Many corporate bonds earn interest semi-annually, so the annualized rate of return needs to reflect 2 periods throughout the year (t=0.5). Dividends on stocks are paid quarterly, to t=0.25 in this circumstance. So sometimes we will see that rates are given as “annual” rates but because they are compounded the annualized rate is different.
Figure H
Compounding Conversion Equation
Figure H shows the formula for annualizing these rates that compound over some frequency during the year. Say for example that we own a bond with a face value of ²100 and that it pays a semi-annual interest payment of ²5. Rather than say “I earn 5% semi-annually” some will quote their return on a simple annual basis, in this case 10% (5% x 2 times per year). In order to reflect earning interest on our interest for 6 months, using Equation 1 in Figure H, we would calculate that our annualized return is 10.25%.
Note that this is the same result we get using Figure B’s Equation 1, using the original 5% as ri (with t = .5).
Since we know how to convert a periodic to annual, and an annual to periodic, we are always in a position where we can convert from one periodic state to any another via the annualizing process. For example, if we need to convert a quarterly rate to a semi-annual equivalent, we can annualize the quarterly using the equations in Figure B, and then calculate the semi-annual rate using the equations in Figure F. The annualized rate is a “stepping stone” to any other periodic rate we wish to calculate.

A Brief Detour Down e Street
Figure I
ROI Formulas under Continous Compounding
0 = Beginning Time Period
T = Ending Time Period
t = T - 0
Up to this point we have dealt in what financial mathematicians call “discrete time” – a distinct and identifiable number of days, months, or years.
There is also a concept known as “continuous time”, which means the units are “infinitely small”. In the time it takes you to read this sentence infinite amounts (i.e. more than millions and billions) of very small time units have passed by.
When using continuous time we make use of a particular mathematical number called e, which is called the exponential function, and its companion the natural logarithm, which is denoted by ln (Excel has functions for both of these: EXP() and LN()).
Figure I shows the Return on Investment formulas using these functions, and Figure J shows an example calculation. In order to get a sense of the inter-relationship, the first two values in Figure D were used to calculate the example. The final ROIDiscrete value matches the calculation in this exhibit (as it should!).
Why is this detour significant? After all, in our lives we generally think about returns in discrete time increments, attempting to answer questions such as “how much did we make last year?”
Figure J
Example Continuous Compounding Calculation
Excel functions are LN() and EXP()
The reason is that as we get into higher order analytics we find that using continuous time units makes some of our calculations easier or provide better models for how the world works, such as an investment through time or in situations where we wish to perform a Monte Carlo analysis.

Whenever we encounter a ROI calculation, there are several questions we need to ask in order to understand exactly what is being communicated.
What time period does this represent?
Simply describing an investment as having made 10% is insufficient. We need to know the time period involved, as that can significantly impact our judgment of the rate’s attractiveness. Making 10% in the last month is obviously better than making it over the last 20 years!
Does the rate reflect compounding?
As we saw in the Compounding discussion, if someone tells us the return is 10%, but it reflects compounding, if we are to think about it in annual terms we know we need to make an adjustment so that it is 10.25%.
Is the calculation consistent amongst comparisons?
Since we often use ROI in order to make comparisons, we need to ensure that we do not mix our rates. For example, if we are told about rates for the dividend yield of our stock and those of our bonds, we need to recognize that we potentially need to reflect different compounding frequencies - quarterly vs. semi-annual - in order to be “apples to apples”.

Key Takeaways
The Return on Investment is a widely used figure in discussions about an organization’s finances, operations, and investments. Correct understanding of this information requires us to be aware of the different methods that may have been used to calculate it, so that we can reconcile this with our interpretations.
Questions
How often do you use one of the formulas discussed in this post?
Can you describe any incidents where misunderstandings have occurred?