Unit 4: Investments, Loans, Annuities and Networks
Core concepts
Unit 4 applies recurrence relations to finance and introduces graph theory. For compound interest, use A = P(1 + r)ⁿ where r is the interest rate per period and n the number of periods; match the compounding period to the rate (e.g. monthly rate = annual/12). Understand effective annual rate as a way to compare investments with different compounding frequencies.
For loans and annuities, model balances with recurrence relations. A reducing-balance loan satisfies Bₙ₊₁ = Bₙ(1 + r) − R, where R is the regular repayment; an annuity (investment with regular deposits) satisfies Bₙ₊₁ = Bₙ(1 + r) + D. Know that the future value of an annuity is the accumulated total and the present value is what a stream of payments is worth today. Be able to interpret an amortisation table and identify the interest component of a payment.
For networks, know the vocabulary: vertices, edges, degree, path, walk, cycle, connected graph, tree (connected, no cycles, n − 1 edges). Understand Eulerian trails (uses every edge once; exists when exactly 0 or 2 vertices have odd degree) and Hamiltonian paths (visits every vertex once). Be able to find a minimum spanning tree using Kruskal's or Prim's algorithm and a shortest path using Dijkstra's algorithm.
Traps
- Using an annual rate in a monthly recurrence without dividing by 12.
- Forgetting to subtract the repayment after applying interest in a loan recurrence.
- Confusing Eulerian (edges) with Hamiltonian (vertices) conditions.
- Applying Kruskal's algorithm and accidentally creating a cycle.
Worked rule of thumb
A loan of $10 000 at 0.5% per month with repayments of $300: after one month, interest = 10 000 × 0.005 = $50, so B₁ = 10 000 + 50 − 300 = $9 750. Always apply interest first, then subtract the repayment.