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I have just signed a lease that requires me to repay $\$100,000$ over $10$ years with payments starting today. What is my annual repayment if the effective interest rate is $7\%$ per annum?

I did (100000)/(1-(1/(1+0.07^10))/0.07 * 1.07 which equals 15234. However, the real answer is 13,306.31. What did I do wrong?

Rodrigo de Azevedo
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Jackie Ao
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1 Answers1

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The equation for a loan, where the repayments are made $\texttt{at the beginning}$ of each period is

$C_0\cdot q^n=r \cdot q \cdot \frac{1-q^n}{1-q}$

$C_0$ is the value of the loan at $t=0$ (present value). r are the constant repayments at the beginning of each period.

$q=1+i$ And $i$ is the interest rate.

This equation can be solved for r.

First the equation has to be didided by q.

$C_0\cdot q^{n-1}=r \cdot \frac{1-q^n}{1-q}$

Multiplying both sides by the reprocical of the fraction. The fraction on the RHS disappears, because $\frac{a}{b}\cdot \frac{b}{a}=1$

$C_0\cdot q^{n-1}\cdot \frac{1-q}{1-q^n}=r $

Inserting the values gives

$100,000 \cdot 1.07^{10-1}\cdot \frac{1-1.07}{1-1.07^{10}}=13,306.31$

callculus42
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