"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).

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).

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.

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.

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 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 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.

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 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.

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 $10\ \%$.

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.

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.