Rate of Return Analysis

1. Rate of Return Analysis Internal Rate of Return Calculating Rate of Return Rate of Return Analysis Incremental Cash Flow Analysis

2. Internal Rate of Return Lender’s Viewpoint The interest rate on the balance of a loan such that the unpaid loan balance equals zero when the final payment is made. Year 0 1 2 3 4 5 Cash flow -5000 +1252 +1252 +1252 +1252 +1252 • We know that the PW of five payments of $1252 are equivalent to $5000 when interest rate is 8%. 2. At the end of 5 years, the payments exactly repaid the $5000 debt with interest rate 8%. We say the lender received 8% rate of return.

3. Internal Rate of Return Investor’s Viewpoint The interest rate earned on the unrecovered investment such that the unrecovered investment equals zero at the end of the life of the investment. Year 0 1 2 3 4 5 Cash flow -5000 +1252 +1252 +1252 +1252 +1252 • Investment $5000 in a machine with lift time 5 years and an EUAB of $1252, what is the interest rate we receive on this investment? 2. The cash flow stream is with 8% rate of return.

4. Internal Rate of Return (IRR) By definition: • Given a cash flow stream, IRR is the interest rate i at which the benefits are equivalent to the costs or the NPW=0 • NPW=0 • PW of benefits - PW of costs =0 • PW of benefits = PW of costs • PW of benefits / PW of costs=1 • EUAB-EUAC=0

5. $325 $250 $100 $175 2 3 4 1 0 time $700 Internal Rate of Return (IRR) Suppose you have the following cash flow stream. You invest $700, and then receive $100, $175, $250, and $325 at the end of years 1, 2, 3 and 4 respectively. What is the IRR for your investment? 700 = 100/(1+i) + 175/(1+i)2 + 250/(1+i)3 + 325/(1+i)4.  It turns out that i = 6.09 %. How to calculate IRR?

6. $325 $250 $175 $100 2 3 4 1 0 time $700 Calculating Rate of Return Ways to find the IRR • Compound Interest Tables •  Interest Tables and Interpolation •  Graphically (see the graph of f(c)) •  Numerically (Excel’s IRR, or other root finding methods). If you have a CFS with an investment (-P) followed by benefits (non negative) from the investment. The graph of NPW versus i will have the same general form. It will decrease at a decreasing rate and have a value 0 at some unique value i*. Where the graph has a value 0 defines the IRR. NPW = -700 + 100/(1+i) + 175/(1+i)2 + 250/(1+i)3 + 325/(1+i)4. 

7. $1252 $1252 $1252 $1252 $1252 Interest rate (P/A,i,5) 7% 4.100 2 3 4 5 1 0 time 8% 3.993 9% 3.890 $5000 Example 1 From Compound Interest Tables PWB/PWC = 1 1252(P/A,i,5)/5000 = 1 (P/A,i,5) = 5000/1252 = 3.993 i=8%

8. Example2 :Graphic solution PW of costs =PW of benefits 100=20/(1+i)+30/(1+i)2+20/(1+i)3+40/(1+i)4+40/(1+i)5 NPW=-100+20/(1+i)+30/(1+i)2+20/(1+i)3+40/(1+i)4+40/(1+i)5 i=13.5%

9. Calculating Rate of Return NPW If we have a CFS with borrowed money involved, e.g., (P,-A,-A,-A), the NPW plot would be “flipped around” and would look something like the following one: i* i

10. Some concept about Interest • Interest Convention. If a lender says she is receiving 11% interest, it might seem reasonable to you to say that the borrower is faced with -11% interest. This is not the way interest is discussed.  • Interest is referred to in absolute terms without associating a positive or negative sign with it. A banker might say she pays 5% interest on savings accounts, and charges 11% on personal loans – no sign is associated with the rates. • We implicitlyrecognize interest as a charge for the use of someone else’s money and as a receipt for letting others use our money. In determining the interest rate in a particular situation, we solve for a single unsigned value of it. We then view this value in the customary way, either as a • charge for borrowing money, or •  as a receipt for lending money.

11. Rate of Return Analysis Example statements about a project:  • The net present worth of the project is $32,000.  • The equivalent uniform annual benefit is $2,800. • The project will produce a 23% rate of return The third statement is perhaps most widely understood. Rate of return analysis is probably the most frequently used analysis technique in industry. Its major advantage is that it provides a figure of merit that is readily understood.

12. Rate of Return Analysis Rate of return analysis has another advantage. With NPW or EUAB one must choose an interest rate for using in the calculations. This choice may possibly be difficult or controversial. With ROR analysis no (exterior) interest rate is introduced into the calculations. Instead, we compute a ROR from the CFS. Warning. Relying only on ROR is not always a good idea.

13. Rate of Return Analysis Motivating Example. Banks 1 and 2 offer you the following Deals 1 and 2 respectively: Deal 1. Invest $2,000 today. At the end of years 1, 2, and 3 get $100, $100, and $500 in interest; at the end of year 4, get $2,200 in principal and interest. Deal 2: Invest $2,000 today. At the end of years 1, 2, and 3 get $100, $100, and $100 in interest; at the end of year 4, get $2,000 in principal only. Question. Which deal is the best?

14. Rate of Return Analysis Deal 1: Find out the implicit interest rate you would be receiving; that is, solve for 2000 = 100/(1+i)1 + 100/(1+i)2 + 500/(1+i)3 + 2200/(1+i)4 IRR: i = 10.7844 %. This is the interest rate for the PV of your payments to be $2,000. Deal 2: We find i for which 2000 = 100/(1+i)1 + 100/(1+i)2 + 100/(1+i)3 + 2000/(1+i)4 IRR: i = 3.8194%. Which deal would you prefer?

15. ROR Analysis In summary, we compute the CFS for the difference between the projects by subtracting the cash flow for the lower investment-cost project (A) from that of the higher investment-cost project (B). Then, the decision rule is as follows: IF ΔIRRB-A > MARR, select B IF Δ IRRB-A = MARR, select either A or B IF Δ IRRB-A < MARR, select A. Here, B-A is an investment increment.

16. Why we use ΔIRR in IRR analysis Years 0 1 A -10 15 B -20 28 B-A -10 13 MARR=6% Select A IRR 50% 40% ΔIRRB-A 30% MARR < ΔIRRB-A Select B Select B NPV 3.92 6.05 Although the rate of return of A is higher than B, B got $8 return from the $20 investment and A only got $5 return from $10 investment. Project B: you put $20 in project B to get a return $8. Project A: you put $10 in project A (and $10 in your pocket) to get a return $5.