3: The Time Value of Money3-6c FINDING THE PRESENT VALUE OF AN ANNUITY DUE
We can find the present value of an annuity due in much the same way we found the present value of an
ordinary annuity. Remember that each cash flow for an annuity due occurs one period earlier than for an
ordinary annuity. Thus, an annuity due would have a higher present value than an ordinary annuity with
the same cash flows, discount rate and life.
To find the present value of the annuity due, we use the same method used to find the present
value of an ordinary annuity, with one difference: each of the cash flows of the annuity due occurs one
year earlier – at the beginning rather than the end of the year. The expression for the present value of
an annuity due, shown in Equation 3.8, is similar to the equation for the present value of an ordinary
annuity (PV) given in Equation 3.7.
Eq. 3.8 PVannuitydue
PMT
rr()==×− n r
(^1) ×
1
(1 +)
(1+)
Comparing Equations 3.7 and 3.8, you can see that the present value of an annuity due is merely
the present value of a similar ordinary annuity multiplied by (1 + r).
You have picked out a new car that costs $50,000.
You need to borrow the full amount, and a bank has
offered you a six-year loan with annual end-of-year
payments of $12,000. What is the interest rate that
the bank is charging on this loan?
In this case, we have a five-year annuity with
annual payments of $12,000. Because these
payments fully repay the loan, we know that the
present value of the annuity is equal to the amount
borrowed, $50,000. In principle, we could try to
solve Equation 3.7 for the missing value r. Solving
that equation algebraically is extremely difficult, so
you can use a financial calculator, or you can use
the ‘rate’ function in Excel. To use that function,
enter = rate(6, -12,000,50,000,0,0,0), and Excel
will reveal that the interest rate on the loan is 12%.
In the syntax of the rate function, you enter the
payment as a negative number and the present
value (the amount that must be repaid) as a
positive value.
example
To demonstrate, assume that the Koalaburra
Company wishes to determine the present value
of the five-year, $7,000 service contract at an 8%
discount rate, and assume also that each of the
maintenance expenditures occurs at the beginning
of the year. This means that the first payment for
maintenance expenses would occur today.
The present value of this annuity due is simply
(1 + r) times the value of the ordinary annuity:
PV(annuity due) = $27,948.97 × (1.08) = $30,184.89.
If Koalaburra pays its maintenance costs at the start
of each year, the most it is willing to pay EM for the
service contract increases by more than $2,000 to
$30,184.89.
example