# Thread: Annuity problem solving for i (interest rate)

1. ## Annuity problem solving for i (interest rate)

The present value of a 10-year annuity-immediate with level annual payments and interest rate i is x. The present value of a 20-year annuity-immediate with the same payment and interest rate is 1.5x. Find i.

So far I have:

$x=\frac{1-v^{10}}{i}$
$1.5x=\frac{1-v^{20}}{i}$
where $v=\frac{1}{1+i}$

So:

$1.5(\frac{1-v^{10}}{i})=\frac{1-v^{20}}{i}$

From here multiply both sides by i and get:

$1.5-1.5v^{10}=1-v^{20}$

From here I'm not sure what to to do.

I know that $v^{10}=0.5 => v=0.933032992 => i=0.071773462$

I was thinking that I could substitute $x^{2}=v^{20}$ and $x=v^{10}$ to setup and solve quadratic formula.

Doing so gives x=0.5,1

Therefore $v^{10}=0.5, 1$

But I'm not sure that's a proper way of solving the problem. Thanks

3. ## Re: Annuity problem solving for i (interest rate)

Originally Posted by downthesun01
From here multiply both sides by i and get:

$1.5-1.5v^{10}=1-v^{20}$
OK; that simplifies to:
2v^20 - 3v^10 + 1 = 0
Let x = v^10; then:
2x^2 - 3x + 1 = 0; solve to get x = 1 or x = 1/2 ; reject x = 1.

So v^10 = 1/2
(1 / (1 + i))^10 = 1/2
1 / (1 + i) = (1/2)^(1/10)
Solve for i ; OK?