ANNUITIES - PERPETUITIES
An annuity is a financial instrument that pays a constant amount every certain period (eg every year) for an x number of periods. In order to calculate the present value of an annuity the same principle is used as before:
d/(1+r) + d/(1+r)² + d/(1+r)³ + ………..d/(1+r)ⁿ
This equation can also be expressed as d (1 - 1 )
r (1+r)ⁿ
Where d = Payment at the end of each year starting at the end of the year
r = Cost of capital
As can be seen from the above formula if n is increased to infinity the present value is given by the formula 1
r
EXAMPLE
A company with a cost of capital of 14% is considering an investment in a project costing $500.000 that would yield cash inflows of $100.000 a year in perpetuity. Should the project be undertaken?
ANSWER
YEAR Cash flow Disc. Factor Present value
0 -500000 1 -500000
1 - ∞ 100000 1/0.14 = 7.14 714000
NPV 214000
The project should be undertaken since it gives a positive NPV.
EXAMPLE
A project requires an initial capital outflow of $1000.000 and is expected to produce net cash inflows of $90000/year starting at the end of the second year. Calculate the NPV of the project assuming 8% cost of capital.
ANSWER
NPV = -1000000 + 1 x 90000
(1+0.08) 0.08
= -1000000 + 1041667
= 41667
EXAMPLE
A project requires an initial capital outflow of $500000 and is expected to produce net cash inflows of $50000/year starting at the end of the fourth year. Calculate the NPV of the project and advise management as to whether they should go ahead with the project assuming a 9% cost of capital.
ANSWER
NPV = -500000 + 1 x 50000
(1+.09)³ 0.09
= -500000 + 428991
= -71009