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Internal Rate of Return Andrew Jain and Ravinder Saidha What We Will Cover • What is Internal Rate of Return? • Formula to calculate IRR for: • Projects / Common Stocks • Zero-Growth Models • Constant Growth Models • Multiple Growth Models • Crossover Rate • Independent & Mutually Exclusive Projects • Advantages and Disadvantages of IRR • Conclusion What is Internal Rate of Return? • Another way of making a capital budgeting decision • Is calculated when the Net Present Value is set equal to Zero • There are four model types we will cover: • Projects / Common Stocks • Zero Growth • Constant Growth • Multiple Growth IRR for Common Stocks • Formula CF1 CF2 CFN NPV CF0 ... 0 (1 IRR ) 1 (1 IRR ) 2 (1 IRR ) N N CFt 0 t 0 (1 IRR ) t Sample Question Time Period: 0 1 2 3 4 Cash Flows: -1,000 500 400 300 100 PV of the inflows -1,000 discounted at IRR NPV = 0 Sample Question Continued • Can only find IRR by trial and error CF1 CF2 CFN NPV CF0 ... 0 (1 IRR ) (1 IRR ) 1 2 (1 IRR ) N 500 400 300 100 0 1000 (1 IRR ) (1 IRR ) 1 2 (1 IRR ) 3 (1 IRR ) 4 • IRR = 14.49% Practice Question Professor Stephen D'Arcy is planning to invest $500,000 in to his own insurance company, but is unsure about the return he will gain on this investment. He produces estimated cash flows for the following years: • Year 1: $200,000 • Year 2: $250,000 • Year 3: $300,000 How do you find his internal rate of return for this investment? 200 ,000 250 ,000 300 ,000 • A 500 ,000 (1 IRR )1 (1 IRR ) 2 (1 IRR )3 200 ,000 250 ,000 300 ,000 • B 500 ,000 (1 IRR )1 (1 IRR ) 2 (1 IRR )3 200 ,000 250 ,000 300 ,000 • C 500 ,000 (1 IRR )3 (1 IRR ) 2 (1 IRR )1 200 ,000 250 ,000 300 ,000 • D 500 ,000 (1 IRR )1 (1 IRR ) 2 (1 IRR )3 • E This is a trick question IRR for Zero Growth Models • A zero growth model is when dividends per share remain the same for every year • Formula: D1 IRR P • Where: • D1 = Dividend paid • P = Current price of stock Sample Question • Andrew is prepared to pay his stockholders $8 for every share held. The current price that his stock is currently held for is $65. What is his internal rate of return? $8 IRR $65 • IRR = 12.3% IRR for Constant Growth Models • A constant growth model is when the dividend per share grows at the same rate every year • Formula is similar to zero growth, except you have to add growth: D1 IRR g P Sample Question • Rav paid $1.80 in dividends last year. He has forecasted that his growth will be 5% per year in the future. The current share price for his company is $40. What is his IRR? What is D1? Do * (1 + Growth Rate) $1.80 * (1+5%) = $1.89 $1.89 IRR 0.05 $40 IRR = 9.72% IRR for Multiple Growth Model • A multiple growth model is when dividends growth rate varies over time • The focus is now on a time in the future after which dividends are expected to grow at a constant rate g • Unfortunately, a convenient expression similar to the previous equations is not available for multiple-growth models. You need to know what the current price of the stock is to find IRR • Formula: N D D P t t 1 t 1 (1 IRR ) t ( IRR g )(1 IRR )T • Where: • Dt = Dividend payments before dividends are made constant • Dt+1 = Dividend payment after dividends are set to a constant rate • t = time dividends are paid at • T = time that dividends are made constant • P = Current price of stock Sample Question • The University of Illinois paid dividends in the first and second year amounting to $2 and $3 respectively. It then announced that dividends would be paid at a constant rate of 10%. The current price of the stock is $55. • We know: • D1 = $2 • D2 = $3 • P = 55 • T = 2 (as after second year, dividends become constant) • We need to find D3: • $3 * (1+10%) = $3.30 $2 $3 $3.30 55 (1 IRR )1 (1 IRR ) 2 ( IRR 0.1)(1 IRR ) 2 • IRR = 14.9% Practice Question • Professor Stephen D'Arcy is the CEO of a large insurance firm, AIG. He is prepared to pay $10 in dividends for the first three years, in which after the third year, the growth rate in dividends will be 10%. If the stock currently sells for $100, how do you find his internal rate of return? $10 $10 $10 $11 • A 100 (1 IRR )1 (1 IRR ) 2 (1 IRR )3 ( IRR 0.1)(1 IRR ) 4 $10 $11 $12 .1 $13 .31 • B 100 (1 IRR )1 (1 IRR ) 2 (1 IRR )3 ( IRR 0.1)(1 IRR ) 4 $10 $10 $10 $11 • C 100 (1 IRR )1 (1 IRR ) 2 (1 IRR )3 ( IRR 0.1)(1 IRR )3 $10 $10 $10 $10 • D 100 (1 IRR )1 (1 IRR ) 2 (1 IRR )3 ( IRR 0.1)(1 IRR )3 • E I have no idea what you want me to do Crossover Rate • The crossover rate is defined as the rate at which the NPV’s of two projects are equal. Source: http://people.sauder.ubc.ca/phd/barnea/documents/lecture%202%20-%202004.pdf Internal Rate of Return • Advantages • Doesn’t require a discount rate to calculate like NPV calculations • Disadvantages • Lending vs. Borrowing • Multiple IRRs • Mutually Exclusive projects. Disadvantages • Lending vs. Borrowing • Example: Suppose you have the choice between projects A and B. Project A requires an investment of $1,000 and pays you $1,500 one year later. Project B pays you $1,000 up front but requires you to pay $1,500 one year later. Project C_0 C_1 IRR NPV at 10% A -1,000 +1,500 +50% +364 B +1,000 -1,500 +50% -364 Disadvantages Continued • Multiple IRR’s • In certain situations, various rates will cause NPV to equal zero, yielding multiple IRR’s. • This occurs because of sign changes in the associated cash flows. • In a case where there are multiple IRR’s, you should choose the IRR that provides the highest NPV at the appropriate discount rate. Disadvantages Continued • Mutually exclusive projects can be misrepresented by the IRR rule. • Example: Project C requires an initial investment of $10,000 and yields a inflow of $20,000 one year later. Project D requires an initial investment of $20,000 and yields an inflow of $35,000 one year later. It would appear that we should choose project C due to its higher IRR. Project D, however, has the higher NPV. Project C_0 C_1 IRR (%) NPV at 10% C -10,000 +20,000 100 +8,182 D -20,000 +35,000 75 +11,818 Conclusion • There are various types of models for calculating IRR including common stock, zero growth, constant growth, and multiple growth. • Despite the disadvantages covered, IRR is still a much better measure than the payback method or even return on book. • When applied correctly, IRR calculations yield the same decisions that NPV calculations would. • In cases where IRR causes conflicts in decision-making, it is more useful to use NPV. Questions?