Question

1.Deposits of $1,560 are made annually into an account earning i(1)=6.3%. What is the accumulated value right after the 8th deposit is made? Assume the first deposit is made in 1 year.

2.What is the present value of an annuity that pays $1,220 each quarter for 6 years if interest rates are i(4)=4%? Assume the first payment is made in 3 months.

3.You deposit $910 each month into an account earning i(12)=11.9%. Suppose you make the first deposit today, and you make a total of 48 deposits. How much money is in the account 3 years after the last deposit?

Answer 1

1.

FV of ordinary annuity = Periodic payment x FVIFA (i, n)

                                     = $ 1,560 x FVIFA (6.3 %, 8)

                                     = $ 1,560 x [(1+0.63)8 -1/0.063]

                                     = $ 1,560 x [(1.063)8 -1/0.063]

                                    = $ 1,560 x [(1.63029469813447-1)/0.063]

                                    = $ 1,560 x (0.63029469813447/0.063)

                                     = $ 1,560 x 10.0046777481662

                                    = $ 15,607.2972871393 or $ 15,607.30

Accumulated value right after 8th deposit will be $ 15,607.30

2.

Periodic rate = 0.04/4 = 0.01

Number of periods = 6 x 4 = 24

PV of ordinary annuity = Periodic payment x PVIFA (i, n)

                                       = $ 1,220 x PVIFA (1 %, 24)

                                       = $ 1,220 x [1- (1+0.01) -24]/0.01

                                       = $ 1,220 x [1- (1.01) -24]/0.01

                                       = $ 1,220 x [(1- 0.787566127423721)/0.01]

                                       = $ 1,220 x (0.212433872576279/0.01)

                                        = $ 1,220 x 21.2433872576279

                                        = $ 25,916.932454306 or $ 25,916.93

Present value of annuity is $ 25,916.93

3.

Periodic rate = 0.119/12 = 0.009916667

Number of periods = 48

FV of annuity due = Periodic payment x FVIFAD (i, n)

                               = $ 910 x FVIFAD (0.009916667, 48)

                               = $ 910 x (1+0.009916667) x [(1+0.009916667)48 -1/0.009916667]

                               = $ 910 x 1.009916667 x [(1.009916667)48 -1/0.009916667]

                               = $ 910 x 1.009916667 x [(1.60585341396378-1)/0.009916667]

                               = $ 910 x 1.009916667 x (0.60585341396378/0.009916667)

                               = $ 910 x 1.009916667 x 61.0944598587182

                              = $ 56,147.2850781406

The account will have $ 56,147.29 after one month of deposit.

We need to compute account value after three years of last deposit i.e. after 35 periods of last deposit.

FV = PV x (1+i) n

     = $ 56,147.2850781406 x (1+0.009916667)35

       = $ 56,147.2850781406 x (1.009916667)35

      = $ 56,147.2850781406 x 1.4125176555918

      = $ 79,309.0314864195 or $ 79,309.03

The account will have $ 79,309.03 after 3 years of last deposit.