Appendix A: Derivation of the Loan Amortisation Formulae | RDP 2021-10: The Rise in Household Liquidity

RDP 2021-10: The Rise in Household Liquidity Appendix A: Derivation of the Loan Amortisation Formulae

Gianni La Cava and Lydia Wang

November 2021

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The loan amortisation formulae for the outstanding debt balance and scheduled payment can be derived in a series of steps. For an amortisation schedule, define the function D ( t ), which represents the principal amount outstanding at period t. A formula for the principal amount outstanding in any given period can be derived given a scheduled payment ( m ) and contracted nominal interest rate (R = (1 + i)).

D (0) = D_{0}

D (1) = D (0) R = D_{0} R - m

This process generalises to the principal amount outstanding in period t:

D (t) = D_{0} R^{t} - m \sum_{k = 0}^{t - 1} R^{k}

Given a geometric progression for the sum of the interest rate terms, this can also be written as:

D (t) = D_{0} R^{t} - m \frac{R^{t} - 1}{R - 1}

Assuming that the loan is fully repaid after n periods, the outstanding balance will be completely paid off in the last payment period, which gives:

D (n) = D_{0} R^{n} - m \frac{R^{n} - 1}{R - 1} = 0

Solving for the scheduled payment ( m ) gives:

m = D_{0} \frac{i (R^{n} - 1)}{R^{n} - 1} = 0

After substituting for the scheduled payment and simplifying, the outstanding balance in period t is:

D (t) = D_{0} (1 - \frac{R^{t} - 1}{R^{n} - 1}) = D_{0} (\frac{R^{n} - R^{t}}{R^{n} - 1})