Loan Amortization (Annuities)

An annuity is a series of equal payments made at equal intervals. This is a common approach to loan amortization. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by mathematical functions known as “annuity functions”. Amortization usually means gradually reducing the value of a loan or an intangible asset. For loans, it’s about paying off the debt through regular payments that cover both interest and principal, ensuring the loan is fully repaid by its due date.

⋅(0$1$2$3$n$n$PF⋅pMMMM−MMVV⋅eri1ods) — Figure 2.2: An Annuity Is a Series of Regular Payments

$P_{0}$ — Initial loan size (present value ( $PV$ ) of loan)
$R$ — Interest rate described as an annual number (%, 0-1), e.g., 0.06 for 6%
$r$ — $R ∕ 12$ — Interest rate per payment(month) (%, 0-1), e.g., 0.005 for 6%/12=0.5%
$M$ — Monthly payment
$n$ — Number of payments (months), e.g., 360 for 30 years loan
$FV$ — Future value

The future value of the loan at time $n$ is:

F V_{loan} = P_{0} {(1 + r)}^{n}

The future value of $n$ monthly payments $M$ is:

F V_{payments} = M [\frac{{(1 + r)}^{n} - 1}{r}]

Monthly payment is derived from the requirement that at the end of the term, the $F V_{payments}$ should be equal to $F V_{loan}$ :

M = P_{0} \times \frac{r {(1 + r)}^{n}}{{(1 + r)}^{n} - 1}

If you want to find number of periods $n$ when given payment $M$ will pay the loan, you can solve for $n$ :

n = \frac{- ln (1 - \frac{r P_{0}}{M})}{ln (1 + r)}

Difference between $F V_{loan}$ and $F V_{payments}$ gives remaining balance of the loan at the $t$ -th period:

F V_{remaining} (t) = \underset{F V_{loan}}{\underset{⏟}{P_{0} {(1 + r)}^{t}}} - \underset{F V_{payments}}{\underset{⏟}{M [\frac{{(1 + r)}^{t} - 1}{r}]}}

On Figure 2.3 you can see how all these values are changing with time. $F V_{payments}$ approaches $F V_{loan}$ at the end of the loan term, and at $t = n$ , $F V_{remaining}$ =0.

F Vloan
F Vpayments
F Vremaining
0511223012345Y$05050,0,0,0,0,0e i00000an00000rTthootuasla inndtesrest paid — Figure 2.3: Amortization of a 30-year loan for $1,000,000 at 5% interest

To get an impression of interest rate’s impact on loan amortization take a look at Figure 2.4 and Figure 2.5. While remaining balance over time changes very similar, the monthly payments and total paid interest are very different.

3%
4%
5%
6%
0511223012357Y$05050,4,3,4,0,1e i51621an73826r7th%ousands — Figure 2.4: Total Interest Paid on a $1,000,000 Loan at Various Interest Rates

3% @ $4,216/m
4% @ $4,774/m
5% @ $5,368/m
6% @ $5,996/m
051122301234567891Y$05050000000000,0e i0000000000an0r7th%ou@sa$n6d,s653/m — Figure 2.5: Remaining Balance for $1,000,000 Loan at Various Interest Rates

More details on Wikipedia.

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