The present value of an annuity is the sum of the discounted value of all future cash flows.
you have the opportunity to invest in several annuities. which of the following 10-year annuities has the greatest present value (pv)? assume that all annuities earn the same positive interest rate.
(a) an annuity that pays $1, 000 at the beginning of each year
(b) an annuity that pays $1, 000 at the end of each year
(c) an annuity that pays $500 at the end of every six months
(d) an annuity that pays $500 at the beginning of every six months

Ans. The best choice is:

A) An annuity that pays $1, 000 at the beginning of each year (PV=$6,759.02 with a discount rate of 10% annual)

Explanation:

Hi, well, I think that the best way of explaing is by discussing  each option. In order to be alot more clear in the emplanation, let´s consider a 10% annual discount rate for all cases:

(A) $1, 000 at the beginning of each year, for 10 years

This is pretty straight forward, but please consider this, one of the payments is made in the present, therefore, when calculating the present value of the annuity, do not use 10 periods, use 9 and add 1000 to this calculation, everything should look like this.

(B) $1, 000 at the end of each year, for 10 years

We do the same as we did in A), but this time, we count 10 annuities and we don´t add 1000 at the beginning. Everything should look like this.

(C)  $500 at the end of every six months, for 20 semesters

This option requires that we transform the rate (10% Effective annual) into semi-annual terms. That is as follows.

Therefore

Now, this is our discount rate, the one we need to use if the payments are made every six months. Let´s see how the math to this should look like.

(D) $500 at the beginning of every six months, for 20 semesters (10 years)

We need to use the discount rate of C) (4.88% semi-annual), but the process is just like in A)

Best of Luck.