Question

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 $500 at the beginning of every six months

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

An ordinary annuity selling at $10,538.38 today promises to make equal payments at the end of each year for the next twelve years (N). If the annuity’s appropriate interest rate (I) remains at 6.50% during this time, the annual annuity payment (PMT) will be     .

You just won the lottery. Congratulations! The jackpot is $35,000,000, paid in twelve 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 6.50%. (Note: Round intermediate calculations to the nearest whole number.)

Answer 1

Answer 1) Option B) An annuity that pays $1000 at the beginning of each year.

Working

Assuming common interest rate of 5% p.a. for all annuities.

Present Value is the sum of discounted annuities over the period of time at the discount rate.

Plan	A	B	C	D
Payment Structure	End	Beginning	Beginning	End
Anuity	500	1000	500	1000
N (*2 in case of 6 months)	20	10	20	10
FV	0	0	0	0
Rate (/2 in case of 6 months)	2.50%	5%	2.50%	5%
Present Value	-7794.5811	-8107.8217	-7989.4457	-7721.7349

An annuity that pays $1000 at the beginning of each year has the greatest PV of 8107.82

Answer 2) 1,291.67

Using PMT function in excel or financial calculator and inputting the following values:

Present Value	-10,538.38
N	12
Rate	6.50%
Future Value	0
PMT	1,291.67

Answer 3) Payments on lottery will be at beginning of each year. In present value terms, we have won 25,343,040.52 USD.

Working

Using PV function in excel or calculator using beginning of period setting and inputting the following values, we get PV as:

PMT (35 million / 12)	2,916,667
N	12
Rate	6.50%
Future Value	0
Present Value	-25,343,040.52