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 $500 at the end of every six months.
c) An annuity that pays $1,000 at the end of each year.
d) An annuity that pays $500 at the beginning of every six months.

An annuity that pays $1,000 at the beginning of each year has the greatest present value of $6,759

Explanation:

A fix Payment for a specified period of time is called annuity. The discounting of these payment on a specified rate is known as present value of annuity.

Formula for Present value of annuity is as follow

PV of annuity = P x [ ( 1- ( 1+ r )^-n ) / r ]

r = rate of return = 10%

n = number of years = 10 years

a)

It is advance annuity and can be calculated as follows

PV of annuity = P + P x [ ( 1- ( 1+ r )^-(n-1) ) / r ]

P = Annual payment = $1,000

PV of annuity = $1,000 + $1,000 x [ ( 1 - ( 1+ 0.1 )^-(10-1) ) / 0.1 ]

PV of annuity = $6,759

b)

P = Annual payment = $5,00

PV of annuity = $500 x [ ( 1- ( 1+ 0.1/2 )^-(10 x2 ) / 0.1/2 ]

PV of annuity = $6,231

c)

P = Annual payment = $5,00

PV of annuity = $1,000 x [ ( 1 - ( 1+ 0.1 )^-10  / 0.1 ]

PV of annuity = $6,145

d)

PV of annuity = P + P x [ ( 1- ( 1+ r )^-(n-1) ) / r ]

P = Annual payment = $1,000

PV of annuity = $500 + $500 x [ ( 1 - ( 1 + 0.1/2 )^-(20-1) ) / (0.1/2) ]

PV of annuity = $6,543

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