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Solving for a Loan’s Effective Annual Rate from Monthly Payments

Article Quant Q&A · Author: RajSharma
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Summary

The document explains how to infer the interest rate on a loan from its principal, fixed monthly payments, and term. It models the repayments as an ordinary annuity: the loan amount equals the present value of all scheduled payments discounted at the monthly rate. Because the resulting equation has no explicit solution for the rate, the rate must be found numerically, for example by trial and error or a spreadsheet goal-seek function.

Once the monthly rate is found, it can be converted to an effective annual rate by compounding it over twelve months. The answers describe both this monthly-rate approach and a nominal annual-rate formulation, and report an effective annual result for the stated example. The document is a compact numerical-finance illustration rather than a general treatment of lending. Its calculation assumes regular end-of-month payments and does not discuss fees, day-count conventions, irregular schedules, or alternative loan terms.

Key ideas

  • The present value of the monthly repayments must equal the amount initially borrowed.
  • The monthly interest rate is obtained by numerically solving the annuity equation.
  • The monthly rate converts to an effective annual rate through monthly compounding.
  • Spreadsheet goal-seeking and trial and error are presented as practical numerical methods.
  • The setup assumes fixed payments made at the end of each month.

Tags

Fixed incomeStatistics
Full text
# How to calculate interest rate in this problem?


# How to calculate interest rate in this problem?












Problem: A loan of £12,000 is issued and is repaid in instalments of £300 at the end of each month for 4 years. Calculate the effective annual rate of interest for this loan.

What I tried-

But how to solve this equation for i?

## Answer by Neeraj (score 2, accepted)

https://quant.stackexchange.com/a/24968

The annuity expression $a_{4}^{(12)}$is written as: $$a_{4}^{(12)}= \frac{1-(1+i)^{-4}}{i^{(12)}} = \frac{i}{i^{(12)}} a_4$$

where, $i$ is the effective annual rate of interest and $i^{(12)}$ is nominal rate of interest convertible monthly, which is equal to $$i^{(12)}=12((1+i)^{1/12}-1)$$ There is no closed formula to get the interest rate, you have to used hit and trial method to get the effective rate of interest.

To get the initial guess, you may use your annuity table at which $\frac{12000}{3600}$ is nearer to $a_4$.

Alternative method: Assuming $i'$ as effective monthly interest rate, then solve this equation: $$12000 = 300 \left[ \frac{1-(1+i')^{-48}}{i'}\right]$$ After obtaining monthly rate, convert it to effective annual interest rate using: $$i = (1+i')^{12} - 1$$

## Answer by Alex C (score 2)

https://quant.stackexchange.com/a/24965

There are 48 monthly payments. You can use the formula for the Present Value of an annuity:

$12000 = 300 \frac{1}{i/12}[ 1-\frac{1}{(1+i/12)^{48}}]$

to find the interest rate

However there is no explicit solution for i, it is solved by trial and error. The value I get is 9.2418%

## Answer by Liisi (score 2)

https://quant.stackexchange.com/a/58804

you can also solve this by using the PMT function and goalseek in excel. The answer is as follows:

What I did is, I first set the interest rate to 0% and calculated the monthly payment using the PMT function. Then I goalseeked the monthly interest rate such that the monthly payment would be 300.

## Answer by GeaR (score 0)

https://quant.stackexchange.com/a/59220

Soultion can be found using IRR ja EFFECT excel function

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