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I'm having trouble transposing the formula for Present Value of an Ordinary Annuity in order to find the interest rate. The formula is:

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Where

P=Present Value of an Ordinary Annuity

PMT=Payment

i=Interest Rate

n=Number of Terms

Not sure if it helps, but I managed to simplify the formula to this:

enter image description here

Thanks in advance.

BlueRhapsody
  • 67

1 Answers1

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It exists no closed-form expression for $i$ if $n>1$. Let $1+i=q$, your second equation becomes

$P=PMT\cdot \frac{q^n-1}{(q-1)\cdot q^n}$

Multiplying both sides by $(q-1)\cdot q^n$

$P\cdot (q-1)\cdot q^n=PMT\cdot (q^n-1)$

Multiplying out the brackets

$P\cdot q^{n+1}-P\cdot q^n=PMT\cdot q^n-PMT$

$P\cdot q^{n+1}-(P+PMT)\cdot q^n+PMT=0$

This is a polynomial with a degree of $n+1$. For $n=1$ it is a quadratic equation. It can be solved by using the quadratic formula. For $n=2$ the cardano formula can be maybe used. But for $n>2$ there exist no closed-form formula. In general you have to apply an approximation method, for instance the Newton-Raphson method.

callculus42
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