Free response question

IEO past paper: 2023 International Economics Olympiad, open questions

Open Question 5. Internal Rate of Return In case you forgot the formula: $$w^n + w^{n-1} + w^{n-2} + \dots + w = \frac{w^{n+1} - w}{w - 1}$$ Let $P_0, P_1, P_2, \dots, P_k$ be a future payment flow. This means that at the time instant 0 we pay $P_0$, at the time instant 1 we pay $P_1$, and so on. If a payment is negative, usually $P_0$, this means that we receive money instead of paying. The Net Present Value of this flow, denoted by NPV(r), is defined as the sum of all present values of the above payments, for a given interest rate r. The rate r is the rate which is applied at each time instant. To find the Internal Rate of Return (IRR) of this flow, we follow the next steps: 1. We define a set of interest rates $r_0, r_1, r_2, \dots, r_n$. 2. We calculate the quantities $NPV(r_0), NPV(r_1), NPV(r_2), \dots, NPV(r_n)$. 3. We find successive $r_a, r_b$ such that $NPV(r_a) > 0$ and $NPV(r_b) < 0$. 4. We find a straight line joining the points $(r_a, NPV(r_a))$ and $(r_b, NPV(r_b))$. 5. We find the value $r^*$ where this straight line meets the horizontal r-axis. 6. $IRR = r^*$. A company takes, at the "zero" time instant, a loan equal to 71 money units and plans to pay the first month 15 money units, the second 28 and the third 30, that is $P_1 = 15, P_2 = 28, P_3 = 30$.