Question

7 . Present value of annuities and annuity payments 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. An annuity that pays $500 at the end of every six months An annuity that pays $1,000 at the beginning of each year An annuity that pays $1,000 at the end of each year An annuity that pays $500 at the beginning of every six months An ordinary annuity selling at $2,514.15 today promises to make equal payments at the end of each year for the next eight years (N). If the annuity’s appropriate interest rate (I) remains at 8.00% during this time, the annual annuity payment (PMT) will be . You just won the lottery. Congratulations! The jackpot is $10,000,000, paid in eight equal annual payments. The first payment on the lottery jackpot will be made today. In present value terms, you really won —assuming annual interest rate of 8.00%.

Answer 1

Answer:

An annuity that pays $1,000 at the beginning of each year

PTM of the annuity selling for 2,541.15 $   437.50

Present value of the Jackpot:    $62,063,701

Explanation:

Because is at the beginning, the 1,000 will be generating interest right away.

So even the 500 at the beginning will have a slightly higher rate, it cwon't compensate the 1,000 upfront.

Calculate the annual payment:

$PV \div \frac{1-(1+r)^{-time} }{rate} = PTM$

PV  $   2,514.15

time      8 years

rate              8% = 0.08

$2514.15 \times \frac{1-(1+0.08)^{-8} }{0.08} = PTM$

PTM  $   437.50

jackpot present value of an annuity-due (payment at beginning)

$PTM \times \frac{1-(1+r)^{-time} }{rate} (1+r)= PV$

PTM $10,000,000

time          8 years

discount rate 0.08

$10000000 \times \frac{1-(1+0.08)^{-8} }{0.08} (1+0.08)= PV$

PV $62,063,700.5922