Question

1. You just had your twentieth birthday. A salesmen offers you the following deal: "Pay our company $75 per month for 20 years starting in one month and when you retire, we'll guarantee you an income of $4,000 per month for 25 years starting one month after your 65th birthday." What is the net present value of this deal if the interest rate is 9%/year compounded monthly?

Answer 1

Net present value of the deal is $ 94.89

Explanation:

Present value of monthly deposit can be computed using formula for PV of annuity as:

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

PV = Present value of monthly deposits

P = Periodic deposit = $ 75

r = Rate of return = 0.09/12 = 0.0075 monthly

n = Number of periods = 20 x 12 = 240 periods

PV = $ 75 x [1- (1+0.0075) -240]/0.0075

     = $ 75 x [1- (1.0075) -240]/0.0075

     = $ 75 x [(1- 0.16641284479638)/0.0075]

     = $ 75 x (0.83358715520362/0.0075)

     = $ 75 x 111.144954027149

     = $ 8,335.87155203619 or $ 8,335.87

Total value of monthly cash withdrawals in 65th birthday can be computed using same formula for annuity.

P = $ 4,000; r = 0.0075/m; n = 25 x 12 = 300

PV = $ 4,000 x [1- (1+0.0075) -300]/0.0075

     = $ 4,000 x [1- (1.0075) -300]/0.0075

     = $ 4,000 x [(1- 0.10628783381341)/0.0075]

     = $ 4,000 x (0.89371216618659/0.0075)

     = $ 4,000 x 119.161622158212

     = $ 476,646.488632846 or $ 476,646.49

PV of $ 476,646.49 can be computed as:

PV = FV/(1+r) n

n = (65 – 20) x 12 = 45 x 12 = 540 periods

PV = $ 476,646.49/ (1+ 0.0075)540

      = $ 476,646.49/ (1.0075)540

     = $ 476,646.49/56.5365885151221

     = $ 8,430.76143288181 or $ 8,430.76

NPV of the deal = PV of cash withdrawals - PV cash deposits

                           = $ 8,430.76 - $ 8,335.87 = $ 94.89