Homework 4 — bring it to the next session

Bring it to class, done on paper. Nothing to hand in or send: the first 15–20 minutes of next session are spent going through it together at the board — you walk us through your working, we correct it there and then. Mistakes are the useful part, so bring what you tried even where you got stuck.

Two parts. Part 1 on paper, with a calculator. Part 2 typed. Expect 2–3 hours.

Money to the cent, rates to two decimals in a percentage. Show the formula before the number every time: the exam gives marks for the method.

Part 1 — Drill set (paper, show all work)

H1. €12,000 is invested for 5 years at 7%. (a) Final amount with simple interest. (b) With annual compounding. (c) The difference, and one line on where it comes from.

H2. €20,000 at 6% nominal for 3 years, compounded (a) annually (b) quarterly (c) monthly. Give the three final amounts.

H3 (EAR). Compute the effective annual rate of 9% nominal compounded (a) semi-annually (b) quarterly (c) monthly (d) continuously. Then say in one line why a borrower should always ask for the EAR.

H4 (resit form). Select the correct alternative and justify in one line. Two loans quote the same nominal rate. Loan A compounds monthly, loan B annually. For the borrower: (i) A is cheaper (ii) B is cheaper (iii) they cost the same

H5 (present value). (a) What is €25,000 receivable in 6 years worth today at 8%? (b) How much must be invested today at 5% to have €50,000 in 10 years?

H6 (annuity). A lease requires €9,000 at the end of each year for 7 years. Your cost of capital is 6%. (a) The present value of the lease. (b) The seller offers to sell the asset outright for €52,000. Which is cheaper in today’s money, and by how much?

H7 (perpetuity). A building is let for €36,000 a year, for ever, first payment in one year. (a) Its value at a 6% required return. (b) At 8%. (c) One line on why the two differ so much.

H8 (NPV — the decision). A project costs €900,000 now and returns €260,000, €300,000, €340,000 and €360,000 at the end of the next four years. The cost of capital is 10%. (a) Lay out a table: year, flow, discount factor, present value. (b) The NPV. (c) Accept or reject, in one sentence. (d) Recompute at 18%. What has changed, and what is the name of the rate between the two where the sign flips?

H9 (growing perpetuity — what a firm is worth). A small competitor expects a free cash flow of €240,000 next year, growing 2.5% a year for ever. Your required return is 10%. (a) Its value today. (b) Its value if growth is 3.5% instead. By how much, in euros, does one point of growth move the price? (c) One line: why does the formula give nonsense if someone assumes growth of 10% or more?

H10 (challenge — a function study, on money). For the project in H8, write NPV(r) as a function of the discount rate, r ≥ 0, and study it: NPV(0) and what it means; the limit as r → ∞ and the asymptote; why it is decreasing; where it changes sign (you have its value at 10% and at 18%); a sketch with those points. Finish with one sentence on what the sketch says about the project’s margin of safety.

Part 2 — Memo to the Board (150–250 words, typed)

The Board of Arno Bikes has been offered the assembly line from class: €2.4M now, returning €700k, €800k, €900k and €1,000k over four years. A Board member says: “It returns €3.4M on a €2.4M outlay. That is a million of profit — obviously we do it.”

Write the memo that corrects this, politely and with numbers. Say what is wrong with adding the flows up, give the NPV at the firm’s 9% cost of capital, and state the rate at which the answer would change. End with a recommendation and one condition you would attach to it.

No formulas in the text. Put the discounting table in an appendix if you want it seen.

Self-check before submitting

  • Every compound-interest answer states m (how many times a year) explicitly.
  • Every EAR is computed from the nominal rate, never compared to it directly.
  • Every present value shows the discount factor separately from the flow.
  • The NPV table has a row for year 0, with a factor of 1.
  • The memo names the number that makes the Board member’s reasoning wrong, and does it in one clause.

How this feeds the assessments

  • H1–H4 are the resit’s multiple-choice-with-justification block, in the exact form.
  • H5–H7 are the present-value machinery every financial-maths question rests on.
  • H9 goes one step beyond the syllabus: the same perpetuity, with growth — how a whole firm is priced.
  • H10 is the week-3 function study applied to the NPV curve: the final and the resit each ask one of the two, and this is where they meet.
  • H8 and the memo are the NPV question, and the register the midterm asks for: a decision, defended with numbers, for people who will not read an appendix.