Annuity Loan

Repayment type: annuity

Also known as: Annuitätendarlehen, level-payment loan, constant-installment loan

The installment stays the same for the whole term. Each installment contains interest on the remaining balance plus principal. Because the balance falls, the interest share falls and the principal share rises from one installment to the next. This is the standard form for German real-estate and investment loans.

How it works

1. Formula

$$A = B_0 \cdot \frac{r}{1 - (1 + r)^{-n}}$$

where $A$ is the installment, $B_0$ the principal, $r$ the interest rate per period and $n$ the number of installments.

2. With the terms

$$\text{Installment} = \text{Principal} \cdot \frac{\text{Rate per period}}{1 - (1 + \text{Rate per period})^{-\text{Number of installments}}}$$

3. With the example numbers: 100,000 EUR, 4.00 % p.a. with one installment per year, 5 installments.

$$A = 100{,}000 \cdot \frac{0.04}{1 - 1.04^{-5}} = 100{,}000 \cdot \frac{0.04}{0.178073} = 22{,}462.71$$

Splitting each installment

In each period $t$ the installment splits into interest on the remaining balance and principal:

$$I_t = B_{t-1} \cdot r$$ $$P_t = A - I_t$$ $$B_t = B_{t-1} - P_t$$

where $I_t$ is the interest, $P_t$ the principal share and $B_t$ the remaining balance after period $t$.

With the terms:

$$\text{Interest} = \text{Previous balance} \cdot \text{Rate per period}$$ $$\text{Principal} = \text{Installment} - \text{Interest}$$ $$\text{New balance} = \text{Previous balance} - \text{Principal}$$

With the example numbers, year 1:

$$I_1 = 100{,}000 \cdot 0.04 = 4{,}000.00$$ $$P_1 = 22{,}462.71 - 4{,}000.00 = 18{,}462.71$$ $$B_1 = 100{,}000 - 18{,}462.71 = 81{,}537.29$$

Example

100,000 EUR at 4.00 % p.a., 5 years, one installment per year.

Annuity with an initial repayment rate

German contracts often define the installment not as an amount but as "interest rate plus x % initial repayment" (anfängliche Tilgung), for example 4 % interest + 2 % repayment on the principal per year. The installment is still level, but because it follows from the repayment rate and not from the term, a remaining balance is left at the end of the rate lock.

1. Formula

$$A = B_0 \cdot \frac{i + t_0}{m}$$

where $A$ is the installment, $B_0$ the principal, $i$ the interest rate p.a., $t_0$ the initial repayment rate p.a. and $m$ the number of installments per year.

2. With the terms

$$\text{Installment} = \text{Principal} \cdot \frac{\text{Interest rate p.a.} + \text{Initial repayment rate p.a.}}{\text{Installments per year}}$$

3. With the example numbers: 4 % interest + 2 % initial repayment, one installment per year.

$$A = 100{,}000 \cdot \frac{0.04 + 0.02}{1} = 6{,}000.00$$

The principal share grows by the saved interest each year, which is why an annuity with 2 % initial repayment clears in about 28 years, not 50. The remaining balance at the end of the rate lock is refinanced; see Rate Lock and Prolongation.

In LoanMonitor

• Wizard, Terms: repayment type Annuity.
• Plan Annuity to maturity: the installment follows from rate, term and payment frequency, and the loan is fully repaid at maturity.
• Plan Fixed installment: enter the installment as amount or as initial repayment % p.a. The wizard previews the installment per period and the remaining balance at maturity.
• Extra repayments on an annuity loan can either lower the installment or shorten the term; see Extra Repayments.

Related

• Loan Types overview
• Linear Loan
• Balloon Loan