Rate of Return: MARR & IRR

Rate of Return Analysis: MARR & IRR

• Lecture 6 focuses on Rate of Return Analysis, specifically Minimum Acceptable Rate of Return (MARR) and Internal Rate of Return (IRR).
• There is a review of IRR and MARR concepts.
• There is exploration of project cash flow based on IRR.

Internal Rate of Return (IRR)

• IRR is a commonly used comparison method also known as the internal rate of return.
• The IRR for an investment is the interest rate at which the present value of costs equals the present value of the benefits.
• The internal rate of return (IRR) is a metric used in financial analysis to estimate the profitability of potential investments.
• IRR is a discount rate that makes the net present value (NPV) of all cash flows equal to zero in a discounted cash flow analysis.
• The IRR is the interest rate at which the project "breaks even".
• PW = 0, or PW(receipts) = PW(disbursements)
• FW = 0, or FW(receipts) = FW(disbursements)
• AW = 0, or AW(receipts) = AW(disbursements)
• Net cash inflow during period t is Ct
• Total initial investment costs is Co.
• The internal rate of return is IRR
• The number of time periods is t.
• The "internal" in IRR refers to the fact that the return is due to the cash flows "internal" to the project being evaluated.

IRR Calculation Example

• Consider new double-pane windows at Allied Company with an initial cost of $8000 and zero salvage value that save $400 per year in energy over their 30-year life.
• The IRR is determined by finding the interest rate (i*) that makes the present worth (PW) of the investment equal to zero.
• A spreadsheet can be used to calculate and plot the present worth.
• The IRR is between 2% and 3%.
• Through interpolation, the IRR is determined to be approximately 2.85%.

IRR with Interest Formulas

• The document presents some interest formulas
• Capital Recovery Factor = i(1+i)^n/((1+i)^n−1)
• Present Worth Factor =(1+i)^n −1/(i(1+i)^n)
• Arithmetic Gradient Conversion Factor (to uniform series) =(1+i)^n−(1+ni)/(i[(1+i)^n−1])
• Arithmetic Gradient Conversion Factor (to present value) = 1−(1+ni)(1+i)^−n/i^2

IRR - Borrower's Perspective

• From the borrower's perspective, IRR is the interest rate on the balance of a loan where the unpaid loan balance equals zero when the final payment is made.

IRR - Investor's Perspective

• From the investor's perspective, IRR is the interest rate earned on the unrecovered investment, making it equal to zero at the end of the investment's life.
• The calculation of IRR involves finding the discount rate that forces the NPV to zero.

Minimum Acceptable Rate of Return (MARR)

• Investing in a project implies foregoing to invest elsewhere.
• MARR is an opportunity cost.
• Any proposed investment must earn at least as much as elsewhere: MARR is sometimes called the "hurdle rate”.
• Another view: any investment must earn at least enough to pay the "cost of capital.”
• MARR is the interest rate required to accept a project

Differences between MARR & IRR

• The internal rate of return (IRR) is the annual rate of growth that an investment is expected to generate.
• MARR is synonym of cutoff rate, benchmark and cost of capital.
• MARR is the target rate for evaluation of the project investment.
• Project evaluation is accomplished by creating a cash flow diagram for the project, and moving all of the transactions on that diagram to the same point, using the MARR as the interest rate.

MARR Complete Evaluation Formula

• Project value + rate of interest for loans + expected rate of inflation + rate of inflation change + loan default risk + project risk.
• MARR is the minimum required rate of return or target rate that investors are expecting to receive on an investment

Rate of Return Analysis in Practice

• The rate of return is the most frequently used measure of merit in industry.
• Compare the rate of return (usually called the Internal Rate of Return or IRR) to the minimum acceptable rate of return (MARR).
• If IRR > MARR, accept the investment.
• If IRR < MARR, do not accept the investment.
• If IRR = MARR, the investment is marginally acceptable.
• IRR can be calculated graphically or using software like Excel.

Internal Rate of Return Comparisons

• The point is to look at how to use the Internal Rate of Return to decide whether a project should be accepted
• IRR for Independent Projects (A)
• IRR for mutually exclusive projects (B)
• Potential for multiple IRRs (C)
• External Rate of Return ERR (sometimes also called MIRR or Modified Internal Rate of Return).

IRR for Independent Projects

• Select any independent project which has an IRR that is at least the MARR.
• Evaluating independent projects is similar to evaluating projects: where any project with a present worth PW or annual worth AW ≥ 0 is acceptable.
• Returning to the example: If MARR is > than 2.85% the INVESTMENT should NOT be made.

IRR for Mutually Exclusive Projects

• It is more complex than when the analysis is for independent projects.
• You must consider two mutually exclusive investments.
• The first (Project A) costs $10 today and returns $20 in one year.
• The second (Project B) costs $1000 today and returns $1200 in one year.
• MARR is 12%
• The rate of return = 100% Pw = -10 + 20 (P/F,12%,1) = $7.86 Rate of Return = 20% PV = FV (1 + i)-n = 1071.42 $
• The best project will be the one with the higher present value.

Multiple IRR

• Sometimes there is a strange non asymptotic decrease or growth of the IRR as seen in the following figure
• There are 2 zero values, therefore, there are effectively multiple IRRs

Descartes Rule (C)

• This theorem states "The number of positive, real IRRs is less than or equal to the number of sign changes in the cash flow series".
• Using Descartes rule one is able to determine that in some cases with multiple IRRs, they can have 0, 1, or 2 positive real IRRs.
• Positive project balances are usually invested elsewhere - probably at the MARR (Minimum Acceptable Rate of Return), but not at 100% or 200%.

IRR and PW/AW Methods Compared

• IRR: commonly used, facilitates comparisons of projects of different sizes, difficult to calculate, multiple IRR may exist.
• PW and AW: difficult to compare projects of different sizes. PW gives explicit measure of profit. AW (Annual W. / Uniform Series may be more meaningful)
• All are equivalent and thus lead to the same decision.

External Rate of Return (ERR)

• ERR (External RATE of Return or sometimes Economic RATE of RETURN) is the rate of return of the project when reinvest the cash produced at a rate RR (Reinvestment Rate).
• Usually this RR is comparable or equal to the MARR. Essentially no cash is kept unproductive.
• Computing an exact ERR is difficult: The ERR affects project balances, which affect the ERR.
• To get an approximate ERR: Take all net receipts forward at the MARR to the time of the last cash flow. Take all net disbursements forward at the unknown rate iea* (approximate ERR) to the time of the last cash flow. Set FW(receipts) = FW(disbursements) and solve for iea*.

ERR example:

• Consider a project has cash flows of $2500 now, -$12 500 in one year and $15 000 in two years.
• If the MARR is 25%, find the approximate ERR.
• Taking net receipts forward at the MARR and equating to net disbursements taken forward at the unknown rate () gives:
• 2500 (F/P,25%,2) + 15000 = 12500(F/P,i*,1)
• (F/P,i*,1) = 1.5125
• (1+ i*) = 1.5125
• The approximate ERR is 51.25%.

Comparison Methods

• There are Simple but mutually exclusive Projects

Homework for Next Class

• Read and study CHAPT. 7 – 8 - 9 (Newnan / TEXTbook) & CHAPT. 5 (Fraser)