Question

A government project has $500 of initial cost and 62.5 uniform annual benefit. If i = 12% and n = infinity, then the net present worth of this project is most nearly:
a) 20.84

b) 41.67

c) 30.25

d) 50.00
e) 22.40

f) No correct answer above

Answer 1

The correct answer is (a) 20.84

Present Value of Value A received after n years is given by:

PV = A/(1 + r)ⁿ

where n = time and r= interest rate 12% = 0.12

Hence Present Value of all uniform payment received is

PV = 62.5/(1 + 0.12)¹ + 62.5/(1 + 0.12)² + 62.5/(1 + 0.12)³ + 62.5/(1 + 0.12)⁴ + 62.5/(1 + 0.12)⁵ + -------------------- infinity

This is a geometric series of the form a +  ax¹ + ax¹ + ax¹ + ---------------------infinity

whenever absolute value of x < 1 then the above sum equals a/(1 - x)

Here a = 62.5/(1 + 0.12) and r = 1/(1 + 0.12)

Hence,

Hence Present Value of all uniform payment received is

PV = 62.5/(1 + 0.12)¹ + 62.5/(1 + 0.12)² + 62.5/(1 + 0.12)³ + 62.5/(1 + 0.12)⁴ + 62.5/(1 + 0.12)⁵ + -------------------- infinity

=  62.5/(1 + 0.12)/(1 - 1/(1 + 0.12))

= 520.84

PV of the initial cost = Initial Cost = - 500 This negative sign implies that there is an outflow.

Thus Net Present Worth ( or Value) of this project = 520.84 - 500 = 20.84

Hence, the correct answer is (a) 20.84