Imagine you have R$ 200,000 to open a coffee shop.After speaking with a consultant, he estimates that the business will generate the following annual net profit:YearProfit (R$)150,000270,000390,0004100,000The first question we usually ask is > How long will it take to recover the invested money?. This question is answered by Payback.What is Payback?Payback represents the time needed to recover the initial investment through the cash flows generated by the project.In other words, Payback is the time required for the money received to equal the money invested. The shorter the Payback, the lower the risk; the faster the capital returns; the greater the liquidity of the investment.Intuitive exampleYou lend R$ 1,000 to a friend. He promises to repay:MonthAmount130023003400After three months you received:\[ 300+300+400=1000 \]Your Payback was 3 months.How to calculate Simple PaybackThe calculation is very simple.Step 1Identify the initial investment.Step 2Add up the cash flows over time.Step 3Find the moment when the investment is recovered.Example 1A company invests R$ 100,000.Expected flows:YearFlow0-100,000130,000240,000350,000Accumulating the receipts:YearCumulative130,000270,0003120,000The investment is recovered during the third year.Finding the fraction of the last periodAfter the second year, there was still missing:\[ 100,000-70,000=30,000 \]In the third year, the following comes in:\[ 50,000 \]Therefore,\[ \frac{30,000}{50,000}=0.6 \]Thus,\[ \boxed{\text{Payback}=2.6\text{ years}} \]Payback formulaWhen recovery occurs in the middle of a period:\[ \boxed{Payback=t+\frac{\text{Amount not yet recovered}}{\text{Flow of the period}}} \]where \(t\) is the last fully recovered period.InterpretationIf the Payback is 2.6 years, it means the investment returns after approximately two years and seven months.Problem with Simple PaybackConsider two projects.Project AYearFlow110021003800Project BYearFlow135023503300Both require an initial investment of R$ 700. Both have a Payback of approximately 2 years. But there is a problem. The money received in the third year is worth less than the money received today. Simple Payback completely ignores this fact.Discounted Payback emergesReceiving R$ 10,000 today is not the same as receiving R$ 10,000 five years from now. Why? Because today we can invest that money and earn interest. Thus, money has time value. This is one of the most important ideas in Financial Mathematics.To correct this limitation, Discounted Payback was created. Now all flows are brought to present value. The rate normally used is the Minimum Attractive Rate of Return (MARR).ExampleSame investment: R$ 100,000. Flows:YearFlow0-100,000130,000240,000350,000MARR: 10% per year.Calculating present valuesYear 1\[ PV=\frac{30,000}{1.10}=27,273 \]Year 2\[ PV=\frac{40,000}{1.10^2}=33,058 \]Year 3\[ PV=\frac{50,000}{1.10^3}=37,566 \]AccumulatingYearCumulative PV0-100,000127,273260,331397,897Notice that even after three years, we still have not recovered the R$ 100,000. Therefore, the Discounted Payback is greater than three years.General ComparisonMethodResultSimple Payback2.6 yearsDiscounted Paybackmore than 3 yearsThis happens because part of the value of the flows was "consumed" by the discounting.Simple PaybackDiscounted PaybackDoes not consider interest.Considers interest.Easier.More realistic.Overestimates the speed of return.Reflects the time value of money.Widely used in quick analyses.Widely used in companies.Limitations of PaybackBoth methods have an important problem. They ignore everything that happens after the investment is recovered.Example: Project A recovers in 2 years; total profit of R$ 300 thousand. Project B recovers in 2 years; total profit of R$ 3 million. Both have the same Payback. Clearly, Project B is much better.When to use it?Payback is recommended when the goal is to answer: How long will it take to recover the investment? Which project has lower risk? Which investment returns more quickly?To evaluate the profitability of a project, however, other indicators are more appropriate, such as: Net Present Value (NPV); Internal Rate of Return (IRR); Modified Internal Rate of Return (MIRR).Exercises1) Mariana wants to open a food truck specializing in artisanal hamburgers. For this, she will need to invest R$ 180,000. Calculate the Simple Payback of the investment, given that she estimates the following annual net flows:YearCash Flow (R$)140,000250,000360,000470,0002) A gym invested R$ 350,000 in new equipment. Calculate the Simple Payback considering the following flows:YearFlow190,0002100,0003120,0004130,0003) A businesswoman intends to open a gourmet coffee shop and will invest R$ 250,000. Consider a MARR of 10% per year. Calculate the present value of each flow and the Discounted Payback. The expected net flows are:YearCash Flow (R$)170,000280,000390,0004100,0004) A park will invest R$ 900,000 in a new attraction. MARR = 8%. Calculate the Discounted Payback. Flows:YearFlow1220,0002250,0003280,0004320,0005350,0005) An industry will purchase a machine for R$ 600,000. Calculate the Discounted Payback. MARR = 15%. Flows:YearFlow1180,0002190,0003200,0004220,000Answer Key1) 3.43 years2) 3.31 years3) 3.77 years4) 4.1 years5) The investment is not recovered.
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