Lecture Notes on Time Value of Money
Stefan Arping Amsterdam Business School University of Amsterdam

1

Roadmap
• compounding and discounting • annuities and perpetuities • growing annuities and perpetuities • net present value • internal rate of return • real world complexities

2

Compounding
• suppose you invest $100 for four years at 10% interest • how does your investment evolve over time? beginning year balance year 1 year 2 year 3 year 4 100.00 110.00 121.00 133.10 interest 10.00 11.00 12.10 13.31 end year balance 110.00 121.00 133.10 146.41 =100 × (1 + 0.10) =100 × (1 + 0.10)2 =100 × (1 + 0.10)3 =100 × (1 + 0.10)4

• the future value (FV) of $100 invested for four years at 10% is 100 × (1 + 0.1)4 • general principles of compounding:
– convert dollars today into dollars in N years: multiply with (1 + r)N – convert dollars in N years into dollars in N + m years: multiply with (1 + r)(N +m)−N = (1 + r)m

3

Discounting
• conversely, the present value (PV) of $146.41 to be received in four years is 146.41 = 100 (1 + 0.10)4 • general principles of discounting:
– convert dollars in N years into dollars today: divide by (1 + r)N – convert dollars in N + m years into dollars in N years: divide by (1 + r)(N +m)−N = (1 + r)m – PV of stream of future cash flows: convert all future cash flows into cash flows today and take sum

• the PV of future cash flows is equivalent in value to the future cash flows in the sense that if you had the PV today you could transform it into the future cash flows by investing the PV at the discount rate

4

Equivalence of PV and Future Cash Flows
• say, you receive 0, 100, and 200 in 1, 2, and 3 years, respectively • PV @ 10% 0 100 200 + + = 232.91 1.101 1.102 1.103

• investing 232.91 @ 10% allows you to replicate the stream of future cash flows: year 1 2 3 beginning year balance 232.91 256.20 181.82 interest