Formula for calculating the total interest payable over the life of a loan with linear redemption scheme

Question

I am trying to calculate the lifetime interest paid on a loan with a linear redemption scheme. I know that I can enter it into a spreadsheet and take the sum of the interest, but is there a formula to calculate such a value when you know the balance, duration, monthly redemption and the interest rate? I've done a lot of searching, but end op mostly at annuity calculations or loans with a fixed repayment.

P: Principal (amount) of loan

R: The monthly redemption on the loan

T: The term of the loan (i.e. the number of repayments)

r: The annual interest rate

If for example the Principal is $120.000$, the monthly redemption $1.000$, the term 120 months and the nominal yearly interest is $3%$ paid monthly at the end of each month.

Then the interest is $120.000*(0,03/12)$ the first month. The second month it's $(120.000-1000)*(0,03/12)$. The third it's $(120.000-2*1000)*(0,03/12)$ and so on.

What would be the formula to sum this whole series up? Thank you in advance.

Answer 1

Clearly the total principal payments are $120000$.

The total interest payments are: \begin{align*} 120000 &\times \frac{0.03}{12} + (120000-1000) \times \frac{0.03}{12}+(120000-2000)\times \frac{0.03}{12}+\cdots + 1000 \times \frac{0.03}{12}\\ &= \frac{0.03}{12}\times\left(\sum_{k=1}^{120} 1000\times k\right)\\ &= \frac{0.03}{12}\times 1000 \times \sum_{k=1}^{120}k \\ &= \frac{0.03}{12} \times 1000\times \frac{120\times 121}{2} \\ &= 18150. \end{align*}

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Edit 1: In general, the total interest payments will be: $$\frac{r}{12}\sum_{k=1}^T Rk = \frac{r}{12}R\frac{T(T+1)}{2}= \frac{rP(T+1)}{24}$$ (since $P=RT$ - this assumes the loan is repaid in full after $T$ terms).

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Edit 2: If the loan is not paid in full after $T$ terms (i.e., $RT<P$), then the interest in the first $T$ terms is:

\begin{align*} \frac{r}{12}&\left(P + (P-R) + (P-2R) + \cdots + (P-(T-1)R)\right)\\ &=\frac{r}{12} \sum_{k=0}^{T-1}(P-Rk)\\ &= \frac{r}{12}\left(\sum_{k=0}^{T-1} P - R \sum_{k=0}^{T-1}k \right)\\ &= \frac{r}{12}\left(TP - R \frac{(T-1)T}{2}\right)\\ &= \frac{rT}{12} \left(P- \frac{R(T-1)}{2}\right). \end{align*}

You can check that when $RT=P$, this expression coincides with the original answer.