Solving for interest rates without a known term in annuity immediate and annuity due

Question

Sylvestre receives an annuity-immediate with monthly payments of 100. Susan receives an annuity-due with annual payments of 1,165 and the same term. The value of Sylvestre’s annuity is 97.09% times the value of Susan’s.

So first I set up the system of equations like so (Annuities are already expanded due to not knowing how to format them):

$$1165(1+i)\times(\frac {1-v^n}{i})=1200(.9709)\times\frac {1-v^n}{i}\times\frac{i}{i^{(12)}} $$

This is done by converting the annuity immediate with monthly payments into a yearly annuity with annual payments, but instead divided by the nominal monthly interest rate. The way I am going about the question is based on the assumption that the two annuities have the same annual effective interest rate. If so, then by simplifying the result is:

$$i^{(12)}=\frac i{1+i}$$

However when I try to convert either the effective or the nominal interest rate into the other one to isolate a variable, trying to solve for one of them always results in me getting the interest rate to be 0%, which is not not possible given the annuity formula. Am I going about this problem incorrectly?

Answer 1

Zero is the right answer. The present value of an income stream discounted at a rate of zero is simply the sum of the payments. If they both receive the same amount of money, the discount rate that makes the present values equal is $0$. (Or rather, it's one such rate. There could be others. If the two streams are identical, for instance, any rate works.)

Answer 2

Write out the cash flows and the equation of value. Let $n$ be a whole number of years representing the term of the annuities.

$$100(v+v^2+\cdots+v^{12n}) = (0.9709)(1165)(1+v^{12}+v^{24}+\cdots+v^{12(n-1)}),$$ where $v = 1/(1+j)$ is the effective monthly present value discount factor. Then $$100\frac{1-v^{12n}}{j} = (1131.1)\frac{1-v^{12n}}{1-v^{12}},$$ hence $$\frac{1-v^{12}}{j} = 11.30985.$$ This admits the approximate numerical solution $$j \approx 0.00921663,$$ or an effective annual rate of $i = (1+j)^{12} - 1 = 0.116382$. Notice that as long as the term is a whole number of years, the length of the term is irrelevant.

I would recommend reviewing your solution and comparing against the above, since there is more than one error in your computation.