Page 31 - DMGT207_MANAGEMENT_OF_FINANCES

Management of Finances

Notes 2.4 Perpetuities

An annuity that goes on for ever is called a perpetuity. The present value of a perpetuity of C
amount is given by the simple formula: C/i where i is the rate of interest.
This is because as the length of time for which the annuity is received increases, the annuity
discount factor increases but as length gets very long, this increase in the annuity factor slows
down.

!

Caution As annuity life becomes infinitely long the annuity discount factor approaches an
upper limit. Such a limit is 1/i.

Example: Many business problems are solved by use of compound interest and present
value tables. For example, B Corporation is investigating two possible investments. Project A is
the purchase of a mine for 20,00,000 which will give an expected income from sale of ore of
480,000 per year for 10 years, after which the property will be sold at an estimated price of
600,000. Project B is the purchase of an office building that is leased for 15 years. The lease
provides annual receipts of 4,00,000 at the end of the each of the next 4 years, and annual
receipts of 4,50,000 for the remaining life of the lease. The purchase price is 20,00,000. B
Corporation requires a 20 per cent return on its investments. Which investment is preferable?
Solution:

To evaluate Project A we need to find the present value of the future income stream of 4,80,000
per year for 10 years plus the present value of the future sales price of 6,00,000, both discounted
to the present at the company's required rate of return of 20 per cent.
PV of annuity of 4,80,000 ( n = 10, i = 20%) = 480,000 × 4.19247 20,12,386
PV of 6,00,000 at the end of 10 years = 600,000 × 0.16151 96,906
Total present value of Project A cash inflows 21,09,292
The problem can be broken down into two separate annuities, one with receipts of 4,50,000 per
year for 15 years and the other with payments of 50,000 for 4 years. The present value of the two
annuities can be found by computing the present value of 4,50,000 for 15 years at 20 per cent
minus an annuity of 50,000 for 4 years at 20 per cent.
PV of annuity of 4,50,000 ( n = 15, i = 20 per cent) = 450,000 × 4.67547 21,03,961
PV of annuity of 50,000 ( n = 4, i = 20 per cent) = 50,000 × 2.58873 (1,29,437)
Total present value of project B cash inflows 19,74,524
By discounting each project at the company's required rate of return, we find the Project A.
Cash inflows have a present value of 12,09,292 and Project B cash inflows have a present value
of 19,74,524. Since the asking price of each project is 20,00,000, project B should not be
accepted. The value of project A is greater than the asking price, therefore the company should
acquire Project A.

Task Calculate the present value of cash flows of 700 per year for ever (in perpetuity)
(a) Assuming an interest rate of 7%
(b) Assuming an interest rate of 10%

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