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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-

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But how to solve this equation for i?

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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: $$i = (1+i')^{12} - 1$$

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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%

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