Discounting annuity

Question

Suppose that the present value of an annuity will provide 16 payments of $3100 once a year. If the first payment will come 15 years from now, and the annual rate of interest is 11.6%, what is the present value of the annuity?

Wording in this question seems a little vague.

How I approached this was: Since the first payment is in 15 years and you need to find present value,

S = 3100(((1+0.116/16)^16)-1)/(0.116/16)) = 52390.44

Then discount back 16 years to present time: 52390.44(1.116)^-16 = 9049.51

Although that answer is wrong, any help appreciated!

EDIT: Found answer, interpreted it wrong, the person wants 3100 each year for 16 years starting 15 years from the present

3100(((1.116)^16)-1/.116)*(1.116^-30) = 4756.10

Answer 1

As usual, ambiguities are resolved by writing out the cash flow. if $v = 1/(1+i)$ is the annual present value discount factor, then the present value of the payments is given by $$PV = 3100(v^{15} + v^{16} + \cdots + v^{30}).$$ We can write this in actuarial notation as: $$PV = 3100 v^{14} (v + v^2 + \cdots + v^{16}) = 3100 v^{14} \frac{1-v^{16}}{i} = 3100 {}_{14|} a_{\overline{16}\rceil i}.$$ Substituting $v = 1/(1.116) = 0.896057$, we get $$PV = 3100(0.896057)^{14} \frac{1 - (0.896057)^{16}}{0.116} \approx 4576.10.$$