Exercises: Investment appraisal: payback, NPV, and IRR

You are offered a choice between receiving $1.000\ \text{€}$ today or receiving $1.000\ \text{€}$ in one year's time. a) Explain which you would choose and why, using the concept of the time value of money (you can invest it today and have more in a year's time). b) Relate this concept to compounding (taking today's money into the future) and discounting (bringing future money to its present value).

An investment requires an initial outlay of $12.000\ \text{€}$ and generates different cash flows each year: $5.000\ \text{€}$ in the first, $4.000\ \text{€}$ in the second, and $4.000\ \text{€}$ in the third. a) Calculate its payback period by adding up the cash flows year by year until the outlay is recovered, and express it in years and months. b) Explain in your own words what the payback period measures and why companies with low liquidity favour it so much.

The figure represents an investment on a timeline: an initial outlay today and three annual cash inflows. We know that the investment recovers the outlay right at the end of Year 3. a) Calculate the Year 3 cash flow, marked with "?". b) State the payback period for this investment.

A company has $5.000\ \text{€}$ sitting idle and can place them for a year at an interest rate of $4\ \%$. a) Calculate how much money it will have after one year (capitalisation). b) Explain, with that result, why receiving $5.000\ \text{€}$ in one year is not equivalent to receiving $5.000\ \text{€}$ today, and what the opposite operation to capitalising is called (discounting).

A car wash invests $15.000\ \text{€}$ in a wash tunnel that will generate a cash flow of $5.000\ \text{€}$ each year. a) Calculate its payback period. b) If the company only accepts investments that are recovered in less than 4 years, decide whether it will carry out this one and explain what criterion it is applying.

A copy shop invests $9.000\ \text{€}$ in a large-format printer that will generate flows of $3.000\ \text{€}$ in the first year, $4.000\ \text{€}$ in the second and $5.000\ \text{€}$ in the third. a) Calculate the payback period reasoning how much remains to be recovered at the end of each year and express it in years and months (assume the flows are generated evenly throughout the year). b) Explain why the period is refined by calculating the months instead of settling for whole years only.

Explain what a project's cash flow for a year is (receipts minus payments for the year, which is not the same as the accounting profit) and why investment appraisal methods work with cash flows and not with profits. Apply it to this case: a machine that is paid for today in cash is depreciated in the accounts over five years; state in which year the cash outlay occurs and over how many years the accounting expense is spread.

An industrial patisserie is considering three packaging machines, all with the same outlay of $8.000\ \text{€}$: machine 1 returns $4.000\ \text{€}$ a year for two years; machine 2, $2.000\ \text{€}$ a year for five years; machine 3, $8.000\ \text{€}$ in the first year and nothing more. a) Calculate the payback of each. b) If the company requires recovering the investment in less than three years, state which machines pass the filter. c) Rank the machines from best to worst according to the payback criterion.

A project requires an initial outlay of $18.000\ \text{€}$ and will generate cash flows of $11.000\ \text{€}$ in one year and $12.100\ \text{€}$ in two years. The discount rate is $10\ \%$. a) Calculate the NPV: $VAN = -A + \frac{F_1}{(1 + k)} + \frac{F_2}{(1 + k)^2}$. b) Interpret the sign of the result and decide whether the project should be accepted, giving reasons for your answer.

An investment of $24.000\ \text{€}$ promises increasing cash flows: $11.000\ \text{€}$ in the first year, $12.100\ \text{€}$ in the second year, and $13.310\ \text{€}$ in the third year, with a discount rate of $10\ \%$. a) Calculate the NPV by discounting the three cash flows. b) Interpret the result: how much value does the project create after recovering the investment and providing a return on capital of $10\ \%$ per annum?

A project costs $10.000\ \text{€}$ today and will return a single payment of $11.500\ \text{€}$ in one year's time. a) Calculate its IRR, that is, the rate $r$ that makes the NPV equal to zero: set up $-10.000 + \frac{11.500}{1 + r} = 0$ and solve for $r$. b) If the company requires a minimum required rate of return of $12\ \%$, decide whether to accept the project by comparing the IRR with that rate. c) What if the required rate of return were $18\ \%$? State the IRR decision rule in your own words.

An electric bicycle workshop is studying a project with an outlay of $15.000\ \text{€}$ and flows of $8.800\ \text{€}$ in the first year and $9.680\ \text{€}$ in the second, with a discount rate of $10\ \%$. a) Calculate the NPV: $VAN = -A + \frac{F_1}{(1 + k)} + \frac{F_2}{(1 + k)^2}$. b) Interpret the result and decide whether the workshop should accept the project, giving reasons.

A fruit shop is considering buying a refrigerated van for $18.000\ \text{€}$ that would bring it a cash flow of $10.000\ \text{€}$ in each of the next two years, with a discount rate of $10\ \%$. a) Calculate the NPV, rounding to cents. b) Decide whether the purchase is worthwhile and interpret the sign of the result: what the project lacks to create value. c) Explain, without doing the calculation, what would happen to the NPV if the discount rate fell.

An investment costs $5.000\ \text{€}$ today and will return a single payment of $5.400\ \text{€}$ in one year. a) Set up the IRR, that is, the rate $r$ that makes the NPV zero: $-5.000 + \frac{5.400}{1 + r} = 0$, and solve for its value. b) Decide whether the project should be accepted when the company finances itself at a cost of $6\ \%$ a year, and state the decision rule you have applied. c) Reason whether a company financing itself at $9\ \%$ would also accept it.

A publishing house can choose only one of two projects, with a discount rate of $10\ \%$: the Dictionary project requires $20.000\ \text{€}$ and returns $13.200\ \text{€}$ in the first year and $14.520\ \text{€}$ in the second; the Atlas project requires $30.000\ \text{€}$ and returns $19.800\ \text{€}$ in the first year and $21.780\ \text{€}$ in the second. a) Calculate the NPV of each project. b) Decide which one creates more value and reason whether the bigger project is automatically better for having a higher NPV. c) Calculate the payback of both and check whether the two criteria rank the projects the same way.

A gym is studying two machines with the same outlay of $12.000\ \text{€}$: machine X will generate flows of $8.000\ \text{€}$, $5.000\ \text{€}$ and $3.000\ \text{€}$ over the next three years; machine Y will generate $3.000\ \text{€}$, $5.000\ \text{€}$ and $8.000\ \text{€}$ (its flows grow because it takes time to become popular). a) Calculate the payback of each. b) Compare the total flows of both, state which machine the payback would choose and which you would choose with reasoning, bearing in mind that early money is worth more. c) Relate your choice to the idea of the time value of money.

An investment requires $20.000\ \text{€}$ today and yields $11.000\ \text{€}$ after one year and $12.100\ \text{€}$ after two years. a) Verify numerically that with a discount rate of $10\ \%$ the NPV is exactly zero, making this rate the project's IRR. b) Explain in your own words what the IRR means: what annual yield the invested money returns to the project. c) If the company can obtain financing at $8\ \%$, reason whether it should accept the project; what if financing costs $12\ \%$?

A company is evaluating two projects with the same initial outlay of $11.000\ \text{€}$ and a discount rate of $10\ \%$: the SHORT project returns $11.000\ \text{€}$ in Year 1 and $2.420\ \text{€}$ in Year 2; the LONG project returns nothing until Year 3, when it yields $16.500\ \text{€}$ (it is a plantation that takes three years to produce its first harvest). a) Calculate the payback period for each project, in years and months. b) Calculate the NPV of each project, rounding to the nearest euro. c) Verify that the two criteria rank the projects in opposite order, explain why this happens, and justify which criterion is more reliable for decision-making and why.

A Chief Financial Officer boasts: "I only use the payback period; if the money comes back quickly, the project is good, and the rest is redundant." Prove him wrong using this example: Project A costs $10.000\ \text{€}$ and yields $6.000\ \text{€}$, $6.000\ \text{€}$, and $0\ \text{€}$ in the first three years; Project B costs the same and yields $4.000\ \text{€}$, $4.000\ \text{€}$, and $8.000\ \text{€}$. a) Calculate the payback period for both and check which one that CFO would choose. b) Compare the total cash flows generated by each project and highlight the two things that the payback period ignores: what happens after the initial outlay is recovered and the time value of money. c) Conclude why NPV is a more comprehensive criterion.

An investment requires $40.000\ \text{€}$ today and will return $21.600\ \text{€}$ after one year and $23.328\ \text{€}$ after two. a) Set up the IRR equation: $-40.000 + \frac{21.600}{1 + r} + \frac{23.328}{(1 + r)^2} = 0$, and solve it (hint: let $x = 1 + r$ and apply the quadratic formula). b) Interpret the result: what annual return does the project offer? c) Decide whether it is worthwhile with a financing cost of $5\ \%$ and reason what would happen to your decision if financing cost $10\ \%$.

An agri-food company compares two projects with the same outlay of $25.000\ \text{€}$ and a discount rate of $10\ \%$: the VEGETABLE GARDEN project returns $16.500\ \text{€}$ in the first year and $12.650\ \text{€}$ in the second; the OLIVE GROVE project returns nothing in the first year and generates $31.460\ \text{€}$ in the second (olive trees take time to produce). a) Calculate the payback of each project, in years and months. b) Calculate the NPV of both, rounding to the nearest euro. c) Check which project each criterion chooses, explain what the disagreement is due to and reason which criterion is more reliable for the decision.

A business group is undecided between two incompatible projects. The SMALL project costs $1.000\ \text{€}$ and returns $1.500\ \text{€}$ in one year; the LARGE project costs $10.000\ \text{€}$ and returns $12.000\ \text{€}$ in one year. a) Calculate the IRR of each. b) Calculate the NPV of each with a discount rate of $10\ \%$. c) Check that the IRR chooses the small one and the NPV the large one, explain why the IRR can be misleading when comparing projects of different sizes and reason which criterion is suitable if the goal is to create more value.

A hotel chain can only finance two of these three projects, with a discount rate of $10\ \%$: SPA, outlay of $10.000\ \text{€}$ and flows of $3.300\ \text{€}$ in the first year and $10.890\ \text{€}$ in the second (the spa takes time to fill up); RESTAURANT, outlay of $20.000\ \text{€}$ and flows of $13.200\ \text{€}$ and $12.100\ \text{€}$; CAR PARK, outlay of $15.000\ \text{€}$ and flows of $12.100\ \text{€}$ and $6.050\ \text{€}$. a) Calculate the NPV of each project. b) Calculate the payback of each. c) Decide which two projects to carry out according to the NPV and check whether the payback would have chosen the same portfolio; comment on the result.

A project requires $10.000\ \text{€}$ today and will return $6.000\ \text{€}$ in one year and $6.000\ \text{€}$ in two. a) Calculate its NPV with a discount rate of $5\ \%$. b) Calculate its NPV with a rate of $15\ \%$. c) Explain, with the two results, how the discount rate affects the NPV and what the rate for which the NPV would be exactly zero represents. d) Reason what a company that decides on investments must consider when interest rates rise.