# Hypothesis regarding a recursive annuity loan

I have a question regarding an annuity loan calculation and I would like to prove whether the hypothesis I am stating is correct:

Consider an annuity loan $$L_{1}$$, with a principal of $$T_{1} = 100$$ and an interest rate of $$i_{1} = (5.5\% /12)$$ per month and a duration of $$D_{1}=1$$ year. The annuity would be $$J_{1} = \frac{i}{1-(1+i)^{-n}} \cdot T_{1} \approx 85.84$$.

If the borrower is unable to pay $$J_{1,1}$$, but the borrower can pay $$Lim per month, then the lender is willing to provide an additional loan $$L_{2}$$ to cover the difference, with a principal of $$J_{1}-Lim$$ against the same interest rate $$i_{1}$$ and for $$D_{1}$$. For the second month, a new loan is created $$L_{3}$$ to cover the difference of the second month and so on. In addition in the newly created loans, when the borrower in that month cannot pay the loan back, again a new loan is created for that loan used to previously cover the difference in $$Lim$$ and $$J$$. This goes on until the loan is repaid.

Hypothesis: I hypothesize that the "recursive loan coverage" process (first case) is actually equal to the $$L_{1}$$ with only a longer duration and $$J=Lim$$ (second case) and therefore give the lender the same returns (i.e. the total amount of interest paid in the first case is the same as the second case).

Since this is a toy example in my head, I haven't been able to prove this hypothesis yet. But I wonder whether someone can give me insights on how to deal with the recursive loan coverage process. I can imagine that this might not be the same, but then I am wondering whether there is a solution/formula to show this difference.

Using words only:

My hypothesis is consider a loan and a repayment amount and a certain duration. If we cannot pay the repayment amount of this loan, we get a new loan of the remaining amount that we were unable to pay. The conditions of this new loan is the same as the first loan that we could not pay off (i.e. the length / duration and the interest rate). In the next month (second repayment), we repeat the process if we cannot pay back. For the third month the same and so on.

Now what I believe is that, this construction what I just described, is equal to the original loan (that we could not pay) but with a longer duration, i.e. you get more time to pay it back, hence lowering your annuity, hence lowering it to the point that you can pay it back.

The definition of the same is that the total interest payments in both strategies are equal.

• Hi, I have tried to follow your thoughts but I am a little bit lost in your notation. Are monthly payments necessary for your argument? Also, what is $T_1$, what is $T$ and what's their difference? Furthermore, what does the term duration mean: contract duration? Maybe you could add a bit to your example? – Kermittfrog Oct 21 '20 at 7:55
• Hi, $T_1$ is the principal amount of loan $1$, which is actually the base case. The duration, is the duration of the loan. So for an annuity loan of one year, you have $D_1 = 1$ if the loan is also the base case. – Snowflake Oct 21 '20 at 8:29
• Hi, I have added additional information, without notations. I hope this makes my idea more clear. – Snowflake Oct 21 '20 at 8:33
• Basically you are getting new loans for the part that you cannot repay. I also think the monthly payments are necessary for my argument, because that allows you to calculate the part you can or cannot repay. – Snowflake Oct 21 '20 at 8:34

## 1 Answer

All else being equal, the original debt is simply being repaid over a longer time period. Thus a larger interest income will accrue to the bank as the debtor now pays a higher fraction of his regular cashflows towards interest payments (instead of amortization).

At an extreme, an annuity level of $$A=N/r$$ would mean that the debtor will never amortize the loan, thereby effectively providing a steady income stream of $$rN$$ to the bank.

HTH?

• Yeah I understand, but I was wondering whether we can mathematically prove this – Snowflake Nov 1 '20 at 11:36