Question

Derek will deposit $2,185.00 per year for 14.00 years into an account that earns 8.00%. Assuming the first deposit is made 6.00 years from today, how much will be in the account 32.00 years from today?

Derek will deposit $5,800.00 per year for 27.00 years into an account that earns 16.00%, The first deposit is made next year. How much will be in the account 37.00 years from today?

Derek will deposit $1,404.00 per year for 16.00 years into an account that earns 13.00%. The first deposit is made today. How much will be in the account 16.0 years from today?

Derek will deposit $6,877.00 per year for 12.00 years into an account that earns 14.00%, The first deposit is made next year. He has $19,281.00 in his account today. How much will be in the account 46.00 years from today?

Derek will deposit $8,190.00 per year for 23.00 years into an account that earns 4.00%, The first deposit is made next year. How much will be in the account 40.00 years from today?

Answer Format: Currency: Round to: 2 decimal places

Answer 1

i)

Formula for FV of annuity is:

FV = P x [(1+r) ⁿ – 1/r]

P = Periodic cash flow = $ 2,185

r = Rate per period = 8 % or 0.08 p.a.

n = Numbers of periods = 14

FV = $ 2,185 x [(1+0.08)¹⁴ – 1/0.08]

      = $ 2,185 x [(1.08)¹⁴ – 1/0.08]

      = $ 2,185 x [(2.937193624258 – 1)/0.08]

      = $ 2,185 x (1.937193624258/0.08)

      = $ 2,185 x 24.21492030322

      = $ 52,909.600862537 or $ 52,909.60

Fund size in year 14 is $ 52,909.60, Future value of this fund in year 32 can be computed as

FV = PV x (1+r) ⁿ

n = Numbers of periods = 32 – 14 – 6 = 12

FV = $ 52,909.600862537 x (1+0.08)¹²

      = $ 52,909.600862537 x (1.08)¹²

     = $ 52,909.600862537 x 2.518170116819

     = $ 133,235.375784861 or $ 133,235.38

There will be $ 133,235.38 in the account 32 years from today.

ii)

Formula for FV of annuity is:

FV = P x [(1+r) ⁿ – 1/r]

P = Periodic cash flow = $ 5,800

r = Rate per period = 16 % or 0.16 p.a.

n = Numbers of periods = 27

FV = $ 5,800 x [(1+0.16)²⁷ – 1/0.16]

      = $ 5,800 x [(1.16)²⁷ – 1/0.16]

      = $ 5,800 x [(55.000382414179– 1)/0.16]

      = $ 5,800 x (54.000382414179/0.16)

      = $ 5,800 x 337.50239008862

      = $ 1,957,513.8625140 or $ 1,957,513.86

Fund size in year 27 is $ 1,957,513.86, Future value of this fund in year 37 can be computed as

FV = PV x (1+r) ⁿ

n = Numbers of periods = 37 – 27 =10

FV = $ 1,957,513.8625140 x (1+0.16)¹⁰

      = $ 1,957,513.8625140 x (1.16)¹⁰

     = $ 1,957,513.8625140 x 4.411435078650

     = $ 8635445.32003774 or $ 8,635,445.32

There will be $ 8,635,445.32 in the account 37 years from today.

iii)

Formula for FV of annuity due is:

FV = (1+r) x P x [(1+r) ⁿ – 1/r]

P = Periodic cash flow = $ 1,404

r = Rate per period = 13 % or 0.13 p.a.

n = Numbers of periods = 16

FV = (1+0.13) x $ 1,404 x [(1+0.13) ¹⁶ – 1/0.13]

      = (1.13) x $ 1,404 x [(1.13) ¹⁶ – 1/0.13]

      = $ 1,586.52 x [(7.067325526798 – 1)/0.13]

      = $ 1,586.52 x (6.067325526798/0.13)

      = $ 1,586.52 x 46.67173482152

       = $ 74,045.6407290 or $ 74,045.64

There will be $ 74,045.64 in the account 16 years from today.

iv)

FV of annuity in year 12:

FV = P x [(1+r) ⁿ – 1/r]

P = Periodic cash flow = $ 6,877

r = Rate per period = 14 % or 0.14 p.a.

n = Numbers of periods = 12

FV = $ 6,877 x [(1+0.14)¹² – 1/0.14]

      = $ 6,877 x [(1.14)¹² – 1/0.14]

      = $ 6,877 x [(4.817904819829 – 1)/0.14]

      = $ 6,877 x (3.817904819829/0.14)

      = $ 6,877 x 27.27074871306

      = $ 187,540.938899718 or $ 187,540.94

FV of $ 187,540.94 in year 46:

FV = $ 187,540.938899718 x (1+0.14) ^{(46 – 12)}

     = $ 187,540.938899718 x (1.14) ³⁴

     = $ 187,540.938899718 x 86.05278799290

     = $ 16,138,420.65512670

FV of single sum $ 19,281 in year 46:

FV = $ 19,281 x (1+0.14)⁴⁶

      = $ 19,281 x (1.14)⁴⁶

      = $ 19,281 x 414.594142030667

      = $ 7,993,789.65249329

Total fund size in year 46 = $ 16,138,420.65512670 + $ 7,993,789.65249329

                                            = $ 24,132,210.30761990 or $ 24,132,210.31

There will be $ 24,132,210.31 in the account 46 years from today.

v)

FV of annuity in year 23:

FV = P x [(1+r) ⁿ – 1/r]

P = Periodic cash flow = $ 8,190

r = Rate per period = 4 % or 0.04 p.a.

n = Numbers of periods = 23

FV = $ 8,190 x [(1+0.04)²³ – 1/0.04]

      = $ 8,190 x [(1.04)²³ – 1/0.04]

      = $ 8,190 x [(2.464715543165 – 1)/0.04]

      = $ 8,190 x (1.464715543165/0.04)

      = $ 8,190 x 36.61788857913

      = $ 299,900.50746306 or $ 299,900.51

FV of $ 299,900.51 in year 40:

FV = $ 299,900.50746306 x (1+0.04) ^{(40 – 23)}

     = $ 299,900.50746306 x (1.04) ¹⁷

     = $ 299,900.50746306 x 1.94790049555628

     = $ 584,176.347104882 or $ 584,176.35

There will be $ 584,176.35 in the account 40 years from today.