Question

An investor plans the following investments for the next 20 years: 13.00 years of $10,725.00 per year, and 7.00 years of $13,125.00 per year.

She thinks his investments will earn 8.00% a year for the first 13.00 years, and then earn 10.00% per year for the last 7.00 years. How much would the investor have to set aside today if she wants to fund the entire account?

Answer 1

Today investor needs to set aside an amount which is same as the sum of present value of two deposit stream.

Computation of PV of deposit stream 1:

PV of cash outflow can be computed using formula for PV of annuity as:

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

P = Periodic cash = $ 10,725

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

n = Numbers of periods = 13

PV = $ 10,725 x [1-(1+0.08)^-13/0.08]

     = $ 10,725 x [1-(1.08)^-13/0.08]

     = $ 10,725 x [(1-0.36769792467)/0.08]

     = $ 10,725 x (0.63230207533/0.08)

     = $ 10,725 x 7.9037759416

    = $ 84,768.00

Computation of PV of deposit stream 2:

PV of cash outflow can be computed using formula for PV of annuity as:

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

P = Periodic cash = $ 13,125

r = Rate per period = 10 % or 0.1 p.a.

n = Numbers of periods = 7

PV = $ 13,125 x [1-(1+0.1)^-7/0.1]

     = $ 13,125 x [1-(1.1)^-7/0.1]

     = $ 13,125 x [(1-0.51315811823)/0.1]

     = $ 13,125 x (0.48684188177/0.1)

     = $ 13,125 x 4.8684188177

     = $ 63,898.00

$ 63,898 is the PV of second cash deposit stream in year 7.

PV of this single sum now can be computed as:

PV = FV/ (1+r) ⁿ

= $ 63,898/ (1+0.08)¹³

= $ 63,898/ (1.08)¹³

= $ 63,898/ 2.719623726165

= $ 23,495.16088 or $ 23,495.16

Fund size needed today = $ 84,768.00 + $ 23,495.16 = $ 108,263.16

The investor needs $ 108,263.16 to keep aside for facilitating the two cash deposit streams.