Time Value of Money: Annuities, Perpetuities, and Amortized Loans

Time Value of Money: Annuities, Perpetuities, and Amortized Loans

  • Context and purpose
    • The lecture covers cash flow components with fixed payments over fixed periods (annuities), perpetuities (payments forever), and amortized loans (fixed or non-fixed payment structures).
    • Key ideas include when to discount cash flows, how to handle payments starting after a delay, and how to compare different saving/borrowing options using rate concepts (APR vs EAR).

Annuity: fixed payments over a fixed period

  • Definition

    • Annuity = fixed dollar amount paid at regular intervals for a fixed number of periods.
    • In the example: first payment occurs two years from today; payments of $100 each year for four years (years 2, 3, 4, 5).

  • Present value of an annuity (as of a given time)

    • If the discount rate is i, and there are n payments of amount a starting at t = 2 (i.e., an annuity-immediate started after one year from now, evaluated at t = 1):
    • The value one period before the first payment (e.g., at t = 1) is
      PVextt=1=a1−(1+i)−ni.PV_{ ext{t=1}} = a \frac{1 - (1+i)^{-n}}{i}.
    • The present value today (t = 0) is
      PV<em>extt=0=PV</em>extt=11+i=a1−(1+i)−ni(1+i).PV<em>{ ext{t=0}} = \frac{PV</em>{ ext{t=1}}}{1+i} = a \frac{1 - (1+i)^{-n}}{i(1+i)}.
    • In the example: a = 100, n = 4, i = 0.09.
    • $$PV_{ ext{t=1}} = 100 imes rac{1 - (1.09)^{-4}}{0.09} \