Question

Present/future value computations

1. How much must be deposited on January 1, 2013 to accumulate a balance of $50,000 on December 31, 2017? At interest rate of 3.5%
At interest rate of 6%
2. 50,000 is deposited at interest compounded annually what amount will be on hand in seven years at 4%? at 8%
2A. What if 50,000 is deposited at interest compounded semi-annually what amount will be on hand in seven years At 4%? At 8%
3. How much must be deposited at 4.5% interest on January 2, 2019 to pay an annuity of 1,000 per year on December 31 of each year for six years?
3A. What would the amount be if the annuity was paid on January 1 instead of December 21?
4. How much would be on hand if 1,000 were deposited annually at 4.5% for six years?

Answer 1

Present value and future value are related as:

PV = FV / (1+i) ⁿ

Or FV = PV x (1+i) ⁿ

Where,

PV = Present value

FV = Future value

i = Rate of interest

n = No. of periods

1.

FV = $ 50,000

n = 31 Dec 2017 – 01 Jan 2013 = 5 years

if i = 3.5 % or 0.035 compounded annually,

PV = $ 50,000/ (1 + 0.035)⁵

      = $ 50,000/ (1.035)⁵

       = $ 50,000/1.187686306 = $ 42,098.66

if i = 6 % or 0.06 compounded annually,

PV = $ 50,000/ (1 + 0.06)⁵

      = $ 50,000/ (1.06)⁵

       = $ 50,000/ 1.338225578 = $ 37362.91

2.

PV = $ 50,000

n = 7 years

If i = 4 % or 0.04 p.a.

FV = PV x (1+i) ⁿ

       = $ 50,000 x (1 + 0.04)⁷

       = $ 50,000 x (1.04)⁷

       = $ 50,000 x 1.315931779 = $ 65,796.59

If i = 8 % or 0.08 p.a.

FV = $ 50,000 x (1 + 0.08)⁷

       = $ 50,000 x (1.08)⁷

       = $ 50,000 x 1.713824269 = $ 85,691.21

2A.

PV = $ 50,000

n = 7 x 2 = 14 periods (for compounding semiannually)

i = 4 % p.a. or 0.04/2 = 0.02 semiannually

FV = PV x (1+i) ⁿ

       = $ 50,000 x (1 + 0.02)¹⁴

       = $ 50,000 x (1.02)¹⁴

       = $ 50,000 x 1.319478763 = $ 65,973.94

If i = 8 % p.a. or 0.08/2 = 0.04 semiannually

FV = $ 50,000 x (1 + 0.04)¹⁴

       = $ 50,000 x (1.04)¹⁴

       = $ 50,000 x 1.731676448 = $ 86,583.82

3.

Formula for PV of annuity,

PV = P x [1-(1+r)^-n/r]      

P = Periodic Payment = $ 1,000

r = Rate per period = 4.5 % = 0.045

n = Numbers of periods = 6

PV = $ 1,000 x [1-(1+0.045)^-6/0.045]

      = $ 1,000 x [1-(1.045)^-6/0.045]

     = $ 1,000 x [(1- 0.767895738)/0.045]

    = $ 1,000 x (0.232104262/0.045)

    = $ 1,000 x 5.157872483

   = $ 5,157.87

$ 5,157.87 needs to deposit today for the desire cash out flow of $ 1,000.

3A.

PV for annuity due can be calculated as:

PV = PV of ordinary annuity x (1+i)

PV = $ 5,157.87 x (1+0.045)

     = $ 5,157.87 x (1.045) = $ 5,389.98

4.

Formula for future value of annuity is:

FV = P x [(1+r)ⁿ- 1 /r]

FV = Future value of annuity

P = Periodic Payment = $ 1,000

r = Rate per period = 4.5 % or 0.045 p.a.

n = Numbers of periods = 6

FV = $ 1,000 x [(1 + 0.045) ⁶ – 1/0.045]

      = $ 1,000 x [(1.045) ⁶ – 1/0.045]

      = $ 1,000 x [(1.302260125) – 1/0.045]

      = $ 1,000 x (0.302260125/0.045)

       = $ 1,000 x 6.716891663

       = $ 6,716.89

$ 6,716.89 will be on hand if 1,000 were deposited annually at 4.5% for six years.