Redemptions

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Redemption is the systematic repayment of an outstanding principal loan amount ($D_0$ or $P$). Each periodic payment (Annuity, $A_k$) consists of two components:

• Repayment amount ($T_k$): The actual portion reducing the remaining debt.
• Interest payment ($I_k$): The cost of the loan, calculated as the interest rate multiplied by the remaining loan amount from the previous period: $I_k=r\cdot D_{k-1}$.

$A_k=T_k+I_k$
Therefore the remaining debt in period $k$ is $D_k=D_{k-1}-T_k$, and can be expressed as $D_0-{\sum }_{i=1}^N{T}_i$ or $D_0={\sum }_{i=1}^N{A}_k\cdot (1+r{)}^{-k}$. in the beginning.

Repaying a Redemption

There are two common methods for structuring the repayment of a loan. The formulas are available in some way in the formulary.

Instalment Repayment

The repayment portion remains constant across all periods. Because the remaining debt decreases linearly, interest payments drop over time, causing the total annuity to decrease arithmetically. $T_k$ is constant, but $I_k$ and therefore $A_k$ decrease over time.

$$T_k=\frac{D_0}{N}|I_k=r\cdot D_{k-1}|A_k=T_k+I_k$$

The remaining debt after $k$ years is:

$$D_k=D_0\cdot \left(1-\frac{k}{N}\right)$$

Annuity Repayment

The repayment amount remains constant across all periods ($A_k=A$). Over time, the interest share decreases while the principal repayment share increases geometrically. The constant annuity is determined using the present value formula for an annuity-immediate:

$A=D_0\cdot \frac{q^N\cdot (q-1)}{q^N-1}\text{where }q=1+r$

To split interest and repayment components, simply:

$$I_k=r\cdot D_{k-1}|T_k=A-I_k$$

The remaining debt after $k$ years is:

D_k = D_0 \cdot \frac{q^N - q^k}{q^N - 1}$$ ## Periods Without Repayment During times without repayment ($T=0$), there are two cases: - **No payments at all**: The remaining debt grows exponentially due to compounding interest: $D_k = D_0 \cdot (1 + r)^k$. - **Interest-only payments**: The annuity consists solely of interest payments, which are calculated as $A_k = I_k = r \cdot D_{k-1}$. The remaining debt remains constant at $D_k = D_0$ since no principal is repaid. A common example for periods without repayments are student loans, where interest may accrue during the study period, but repayments only start after graduation. Therefore, the future value of the loan has to be determined at the time repayments start (see [[FM 03 Redemptions.pdf#page=28|slide 28]]).