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:

$\displaystyle x=\frac{1-v^{10}}{i}$

$\displaystyle 1.5x=\frac{1-v^{20}}{i}$

where $\displaystyle v=\frac{1}{1+i}$

So:

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

From here multiply both sides by i and get:

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

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

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

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

Doing so gives x=0.5,1

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

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

Re: Annuity problem solving for i (interest rate)

your approach with the quadratic formula is valid (i haven't checked your working though)

Re: Annuity problem solving for i (interest rate)

Quote:

Originally Posted by

**downthesun01** From here multiply both sides by i and get:

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

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?