## Preview

The simple *rate-of-return *( **R **) is a measure of your investment’s profitability for any chosen time interval. By comparison, the **CAGR**** **and** IRR **are rates of return that measure your investment’s profitability as if it were an orderly process with respect to time. **CAGR** is the acronym for “compound annual growth rate”. It is the constant rate at which an investment’s market value grows every year in a cumulative fashion. **IRR** is the acronym for “internal rate of return”, which describes the performance of all cash flows in a financial project such as the individual investor’s program of dollar-cost-averaging or an investment club’s program of portfolio management. **IRR** is an annualized rate-of-return when all time intervals are measured in years.

## Return

Any profit from your investment is called a *return.* There are 2 types of return: *realized *and *unrealized. **Realized* *returns* are cash payments from dividends, interest, and sales. *Unrealized returns* are the market values of reinvested dividends and unsold holdings.

*return ***= market value – cost** [equation 1]

Example 1: Suppose you invested $100 and held the stock for 5 years until its market value grew to $201. From equation 1, you determine that your return is $101. If you sell it, it’s a *realized* return; otherwise, it’s an *unrealized* return.

## Simple rate-of-return

The simple *rate-of-return *( **R **) is a measure of your investment’s profitability for any chosen time interval, but time is not an essential factor in the calculation (equation 2).

** R**** = return/cost** [equation 2]

**R** is reported as a decimal number or a percentage.

Example 2: The cost of an investment was $100 and 5 years later the return was $101. From equation 2, **R** = $101/$100 = 1.01. Multiply the answer by 100 to find the percentage. **R** = 100×1.01 = 101%.

## CAGR

The **CAGR **is a rate-of-return that measures your investment’s profitability as a growth rate. Time is a factor in the calculation of **CAGR** (equation 3).

**rate ****= (final/initial) ^{(1/N)} -1** [equation 3]

*N* is the number of events or time periods between

the initial and final values.

Example 3, simple **R** versus **CAGR**: The cost of an investment was $100 and 5 years later its final value was $201. We know from example 2 that the simple **R** is 101%. Using the growth rate formula from equation 3, we find that the **CAGR** is 15%.

*Significance: CAGR is the acronym for “compound annual growth rate”. It is the constant rate at which an investment’s market value grows every year in a cumulative fashion.*

MATH: **CAGR **is a *growth rate* that describes the ‘future’ (or final) value of a single cash payment. In contrast, the *discount rate* devalues a cash flow. Both rates represent a *common ratio* that generates a *geometric series* of points aligned to a smooth curve** ^{ref 1}**. Chart 1 illustrates the geometric series of an inflated and devalued investment.

Chart 1. Geometric series.

In chart 1, the **black circle** represents a single investment. The **blue curve** is a series of theoretical values related to the investment by a common ratio called the *discount rate* or the *growth rate* depending on the particular application. The *discount rate* devalues the investment to lower values as a function of the time-period N. The *growth rate *inflates the investment to higher values. Both rates are calculated by the formula in equation 3.

## IRR

Equation 3 is also used to calculate the **IRR**, an acronym for the “*internal rate of return*”. The **IRR **is used to measure the profitability of investments with multiple cash flows. It is a discount rate that balances all devalued cash flows in a financial project.

MATH: In the field of Finance, a devalued cash flow is called the *present value*. The present value is found by revising equation 3 to calculate the initial value for time period N at a given discount rate. The *net present value *of the project is the sum of all present values. The **IRR **is the discount rate that sets the net present value to zero** ^{ref 2}**. It

**is the best-fit discount rate found by an iterative process of trial and error. The significance of the**

**IRR**will be discussed after working through example 4.

Example 4, **IRR**: An investor paid $100 each year for 4 years to purchase and accumulate shares of a particular stock. After 5 years the market value of all shares was $735. Since the purchases were multi year cash flows, the **IRR** is a good choice for analyzing this investment. In this example, the trial discount rate is 13.1%. Table 1 (below) illustrates the analysis:

Table 1. IRR analysis of the investment in example 4.

Row 1, N displays the time period in years for factor N of equation 3. Row 2, ACTUAL is the series of investments that began with a $100 payment at time 0. Additional $100 payments were made at the end of years 1 through 4 for a total cash outflow of $500. The total market value of the investment was $735 at the end of the 5^{th} year. To determine the **IRR**, the *present value*** ^{ ref 2}** of every cash flow was calculated with the trial discount rate of 13.1% after rearranging equation 3 to solve for the initial value. Row 3, DISCOUNTED is the series of

*present values*for each cash flow in row 2. Notice that the total

*present value*of all cash outflows equals the discounted cash inflow of $396.65. Therefore, the

*net present value*is $0 and the 13.1% discount rate is the investment’s

**IRR**. Row 4, PROJECTED is the final value for each present value in row 3. The final value is predicted by rearranging the terms of equation 3 and using the

**IRR’s**13.1% as a growth rate for the remaining time. It’s no accident that the sum of final values in row 4 equals the $735 cash inflow in row 2. Chart 2 (below) illustrates the growth curves for projected values.

Chart 2. Projected values for every cash outflow in example 4.

In chart 2, N is the time period in years. Each **black square** depicts an investment of $100. Each **blue curve** shows the predicted growth of the investment. Every point on a curve is a *future value* and the endpoint at year 5 is the final value. The final values are listed in row 4 of table 1. They decreased as the years progressed because there was less time remaining for growth.

Significance: The **IRR** is a rate-of-return that describes the performance of all cash flows in an investment. The **IRR** is an annualized rate-of-return when all time intervals are measured in years.

## Time distortion

A positive **CAGR **or** IRR** always shows a profit. Conversely, a negative **CAGR **or** IRR** always shows a loss. Higher CAGRs and IRRs imply more profitable investments, but beware that those with short holding periods may grossly misrepresent the long term performance of an investment.

Example 5, time distortion: Suppose that four different $100 investments grew to $200 apiece. From equation 1, we know that the return was $100 for every investment. If the holding periods were 10, 5, 1, and 1/5^{th} years, what were the annualized rates of return?

Table 2. Annualized- and Simple Rates of return

for different holding periods

Legend. Equation 3 is used to calculate the annualized rate-of-return when the unit of time is in years. For this equation, the “Holding period” is the value of N and “Final/Initial” is the quotient of $200 divided by $100. The 4^{th} column is the annualized rate-of-return calculated by equation 3. The 5^{th} column is the simple rate of return calculated by equation 2.

High annualized rates are desirable, but don’t feel exuberant about an exceptionally high annualized rate-of-return. As shown in table 2, the annualized rate-of-return might temporarily be inflated by a brief holding period. It’s unlikely that a short term investment could sustain the 3,100%, or even 100%, annualized rate-of-return in the long run.

Significance: The passage of time decreases an annualized rate-of-return when cash flows are static. ANY increase in the **CAGR **or** IRR** over time indicates a profitable increase of cash inflow relative to cash outflow. It’s always wise to verify this impression by checking the payout of the project.

## Conclusions

In the investment world, rates of return are measurements of profitability. Positive rates indicate profits and negative rates indicate losses. All rates of return are sensitive to the volatility of market prices; they rise and fall with the market. The annualized rates of CAGR and IRR are exquisitely sensitive to short time periods; don’t get exuberant about high annualized rates before checking the time period and potential payout. In the long term, annualized rates tend to decline unless supported by dividend payments and capital gains. An IRR that is holding steady during the passage of time is revealing an underlying growth in market value.

**Copyright © 2015 Douglas R. Knight **

## References

- Donna Roberts, Geometric sequences and series. Copyright 1998-2012. http://www.regentsprep.org/regents/math/algtrig/atp2/geoseq.htm
- A. Groppelli and Ehsan Nikbakht. Barron’s Finance. Fifth Edition. 2006, Barron’s Educational Series, New York.