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Why Frequent Repayments Raise a Loan’s Effective Interest Rate

Article Quant Q&A · Author: newbie
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Summary

The document explains how to calculate the cost of a short-term loan that is repaid through equal installments at regular intervals. It endorses solving for the periodic rate that discounts each payment to the initial amount advanced, as in an annuity present-value calculation. The example compares ten repayments over 100 days with a single repayment at the end of that period.

The key intuition is that early installments return principal before the borrower has had the full term to use it. Although the total repaid is unchanged, the average amount of capital outstanding over time is lower when payments begin sooner, so the implied rate can be much higher than a simple comparison of total repayment with the original advance suggests. The response accepts the calculator’s result but does not independently derive it or specify day-count conventions or whether the quoted rate is effective or nominal; those choices matter when annualizing a periodic rate.

Key ideas

  • Solve for the periodic discount rate that equates the present value of installments with the amount advanced.
  • Early repayments reduce the principal available to the borrower over the loan term.
  • Comparing total repayment with the initial advance understates the cost when installments are frequent.
  • Annualized rates depend on the chosen compounding and day-count conventions.
  • The example provides intuition but no independent derivation of the calculator output.

Tags

Fixed incomeStatistics
Full text
# Given cash flows, what is the interest rate of the following contract?


# Given cash flows, what is the interest rate of the following contract?












I am presented with an investment opportunity where I am given #481,000 on day 1. Thereafter, every 10 days, I am required to give back #50,000 every for 100 days (10 * 50000 = 500000).

How do I calculate the interest rate I am paying?

I am guessing I have to use the present value of annuity problem to find out the interest rate.

So, my present value is #481,000. My "annuity" is 50000 every 10 days. First payment is due on the 10th day. Last payment is due on day 100.

Plugging the above values in wolframalpha I get .7107% for interest rate. I divide that by 10 to get per day interest rate and multiply by 365 to get 25.94%

I am surprised to see the above answer. It is a lot more than 14% which is what the rate would be if I were to pay 500000 at the end of 100 days. Is my reasoning incorrect?

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/19146

The annuity method is the correct method. I am not familiar with wolframalpha but I assume it is correct.

Look at it this way: in the second case (take out 481,000, repay 500,000 after 100 days) you have full use of the borrowed 481,000 for 100 days.

In the first case (takeout 481,000, repay with an annuity of 10 payments over 100 day) you effectively borrow less because they force you to start repaying some of it after just 10 days. In a rough sense you only borrow half as much over time (in term of dollar days) so it is not surprising that it is a much more costly loan. The amount you repay is the same (500,000) but the amount available to you is far less, resulting in (roughly) twice the cost.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.