## Summary

Since investment portfolios have multiple cash flows, their performance is typically measured by the internal rate of return (“IRR”)refs 1-3.  IRRs are widely used to plan and analyze financial projects.  The planning process called capital budgeting won’t be discussed in this article.  The purpose of this article is to describe the analytical use of IRRs in evaluating profitability.  Generally speaking, the positive IRR reflects a profit and the negative IRR reflects a loss.  Higher IRRs infer more profitable investments, but the analyst is cautioned to examine the investment’s return as well as its IRR refs 1-2!!  There are three steps to computing an IRR.

1. calculate the present value of every cash flow
2. find the net present value
3. find the IRR (the IRR is a specific discount rate that sets the net present value to 0).

## Present value

Analysts evaluate the history of multiple cash flows by finding the time value for each cash flow.  Time value is measured by converting the future value of each cash flow to its present value.  In hindsight, the present value is the initial cash payment and all future values are subsequent cash flows.  Equation 1 shows how one present value is estimated from one future value over the time span labeled N.  The present value depends on its discount rate, R.

present value = future value/(1+R)N                    Eq. 1

## Discount rate

The process of discounting an item means to reduce its price or market value.  In equation 1, the discount rate (R) is the rate at which the known future value reverts to its theoretical present value.  The practical significance of the discount rate depends on its intended use.  In financial planning it reflects the risk of an investment as influenced by interest rates, inflation, and the uncertainty of time ref 2.  In the hindsight analysis of a portfolio, the discount rate represents the rate of return for a given cash flow.

## Net present value (NPV)

The sum of all present values in a portfolio is the theoretical cash balance called net present value (NPV)refs 2-3.  The positive NPV reveals a profit and the negative NPV reveals a loss.

## Internal rate of return (IRR)

The IRR is a specific discount rate that sets the net present value to 0.  As such, it represents the time value of all cash flows in a portfolio.  It also reflects the rate of return of the portfolio.  The IRR is calculated by a trial-and-error process of computing net present values for different discount rates.  In the appropriate set of trial discount rates, net present values will vary from negative to zero to positive or positive to zero to negative depending on the cash flows.

## Example

Suppose \$1,000 was invested every 6 months and the stockbroker charged a \$7 trading fee each time.  After 21 months, the total market value grew to \$4,436.46.  Was the IRR 10%?

 Time span in years (N) Item Cash flow 0 Investment + trading fee -1,007 0.5 Investment + trading fee -1,007 1 Investment + trading fee -1,007 1.5 Investment + trading fee -1,007 1.75 Market value 4,436.46

NPV = sum of present values

= (PV at N=0) + (PV at N=0.5) + (PV at N=1) + (PV at N=1.5) + (PV at N=1.75)

= (-1,007/(1+.10)0) – (1,007/(1+.10)0.5) – (1,007/(1+.10)1) –( 1,007/(1+.10)1.5) + (4,436.46/(1+.10)1.75)

= -1,007 -960.79 -915.75 -873.51 +3,757.05

= 0

Yes, the IRR was 10%.  The NPV was \$35.13 at 9% IRR, \$0 at 9.985% IRR, and -\$35.18 at 11% IRR.

## Applications

Periodic reports.  The IRR increases when cash inflow increases, cash outflow decreases, and time is compressed ref 1.   The passage of time will decrease the IRR when all cash flows are static.  Consequently, any increase in 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.

Comparisons.   Be cautious about using the IRR to compare different investments ref 1.   For one reason, higher IRRs don’t always identify higher returns.  Two projects with different cash flows may have the same IRR, yet one project yields a higher return at the time of comparison.  For another reason, the compression of time tends to raise the IRR and promote a false sense of security.  Project A’s exceptionally high IRR for a brief time period may not be sustainable in the long run.  Project B’s lower IRR over a longer time period may be sustainable.  Be sure to examine the payouts as well as the IRRs when comparing investments ref 1!!

## U.S. Tax Code

The calculation of IRR is indifferent to tax rules for reporting an investment’s cost basis.  The LIFO and FIFO rules have no effect on calculations of IRR.

## Miscellaneous

IRR vs CAGR.  Both the IRR and CAGR measure an investment’s rate of return.  The CAGR measures an initial and final cash flow over one time period.  The IRR is a more flexible measure due to its capability of analyzing multiple cash flows over time ref 4.

ERR.  The IRR is sometimes called the economic rate of return (ERR)ref 3.

IRR computation.  The trial-and-error determination of IRR is applicable in all situations, but it can be simplified to a single step when all cash flows are constant ref 2.

Two IRRs.  For mathematical reasons, an investment project with delayed cash outflows may have two IRR’s of widely different values ref 1.   The practical significance of the higher IRR is uncertain.

## References

1. Baker, Samuel L. Perils of the internal rate of return.  Economics interactive tutorials, University of South Carolina.  12/5/2009.  ©2000.
2. A.A. Groppelli and Ehsan Nikbakht. Barron’s FinanceFifth Edition.  2006, Barron’s Educational Series, New York.
3. Grayson, Linda.  Internal Rate Of Return: An Inside Look.  © 2014, Investopedia, LLC.
4. Fuhrmann, Ryan C.   What are the main differences between  compound annual growth rate (CAGR) and internal rate of return (IRR)?  © 2014, Investopedia, LLC.