3: The Time Value of Money3-6b FINDING THE PRESENT VALUE OF AN ORDINARY ANNUITY
The present value of an ordinary annuity is found in a manner similar to that used for a mixed stream.
Discount each payment and then add up each term to find the annuity’s present value.
An equipment supplier has approached Koalaburra
Company, a producer of plastic toys, with an
intriguing offer for a service contract. Extruding
Machines Pty Ltd (EM) offers to take over all of
Koalaburra’s equipment repair and servicing for five
years in exchange for a one-time payment today.
Koalaburra’s managers know their company spends
$7,000 at the end of every year on maintenance, so
EM’s service contract would reduce Koalaburra’s cash
outflows by this $7,000 annually for five years.
Because these are equal annual cash benefits,
Koalaburra can determine what it should be willing
to pay for the service contract by valuing it as a five-
year ordinary annuity with a $7,000 annual cash flow.
If Koalaburra requires a minimum return of 8% on all
its investments, how much is it willing to pay for EM’s
service contract? The time line in Figure 3.11 shows
the present value calculation for this annuity.
We find the present value of this ordinary annuity
by using the same method used in the preceding
section to find the present value of a mixed stream.
That is, we discount each end-of-year $7,000 cash
flow back to time 0, and then sum the present values
of all five cash flows. As Figure 3.11 shows, the
present value of this annuity (EM’s service contract)
is $27,948.97. If Koalaburra were to initially deposit
$27,948.97 into an account paying 8% annual
interest, then it could withdraw $7,000 at the ends of
years 1 through 5. After the final withdrawal (at the
end of year 5), the account balance would exactly
equal zero.
Therefore, if EM offers the service contract to
Koalaburra for a lump-sum price of $27,948.97 or
less, Koalaburra should accept the offer. Otherwise,
Koalaburra should continue to perform its own
maintenance.exampleFIGURE 3.11 PRESENT VALUE OF A FIVE-YEAR ORDINARY ANNUITY DISCOUNTED AT 8%The present value of the five-year, $7,000 ordinary annuity discounted at 8% is $27,948.97, which is merely the sum of the
present values of the individual cash flows shown at the end of the arrows.
0 1 2 3 4 5
End of year