Lesson 3 of 8 · All lessons in this module
  1. Present value and discount (zero-coupon) bondsVideo lesson
  2. Coupon bonds and perpetual bonds
  3. Level-payment instruments: mortgages and amortizing loansVideo lesson
  4. Valuing shares with dividend discount models
  5. Implied returns on bonds
  6. Required return, implied growth and the P/E ratio
  7. Cash flow additivity, no arbitrage and forward rates
  8. No-arbitrage pricing of FX forwards and options

Lesson 3 of 8 · 12 min

Level-payment instruments: mortgages and amortizing loans

A fully amortizing loan is repaid with equal payments whose PV equals the amount borrowed; each payment is part interest, part principal.

In short

  • Level-payment instruments pay the same amount A every period until maturity: mortgages, amortizing loans and annuities.
  • Find A by setting the PV of the payments equal to the principal: A=r×PV1−(1+r)−tA = \frac{r \times PV}{1-(1+r)^{-t}}.
  • Interest each period = outstanding balance × periodic rate. Principal repaid = A − interest.
  • Over time the interest share falls and the principal share rises, although A never changes.
  • The outstanding balance at any date = PV of the payments still to come.
  • Monthly loans: annual rate ÷ 12, years × 12.
  • Compounding forward gives the future value of level payments, A[(1+r)t−1]/rA[(1+r)^t - 1]/r; payments at the start of each period (annuity due) are worth (1 + r) times more.

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