That's it. Now for any number of time periods, N, you can compute the future value of a set of equal cash flows, C, at a periodic compound rate of r.
Notice that with the formula for FVA, if you know the other variables, you can solve for the cash payment, C, or the term, N. Unfortunately, it is difficult to calculate r directly. How is r computed?
Now, consider the general form of the future value of an annuity due.
Notice that the FVA_{Due} equation is just like the FVA equation, except for the extra term (1 + r) on the far right. The term (1 + r) in the future value of an annuity due formula values the cash flow stream of an ordinary annuity one period later. What does this imply?
Present Value of Annuities
What if you wanted to find the present value of an annuity
stream? As illustrated in the solution to Challenge A, this can be
solved by using the PVA formula. The formula for finding the present
value of an annuity will let you find the present value of a stream of
cash flows simultaneously.
Start by considering the present value of an ordinary annuity, stated in general form.
As was done above in computing the future value, remember that the present value of a single cash flow t periods into the future at a discounting interest rate of r percent is
For a series of N cash flows, the present value is
Recall from Time Value of Money that the negative exponent in the PV
equation is equivalent to the reciprocal of the number raised to the
positive of the exponent. In other words,
The purpose of the present value formula is to calculate today's
value of a future cash payment. Certainly you would pay less than a
dollar today for a dollar in the future. To obtain a future value for a
current amount, you multiply the current amount by a factor. Similarly,
you can divide a future value by a factor to compute the present value.
This is represented mathematically as
Using the same procedure used before to obtain the formula for the
future value of an annuity, you derive the following formula for the
present value of an annuity.
 
PVA = present value of an ordinary annuity
C = the fixed annuity payment
r = the periodic interest rate
N = the total number of annuity payments

Solving the PVA equation for C would give you the annual cash payment needed to pay a fully amortizing loan over N periods in the amount of PVA at a periodic interest rate of r.
Example: Ordinary annuity Your company needs a
substantial amount of money today, and your boss has asked you to
ascertain how much the company can borrow. You learn that the company's
operating cash flow will be sufficient to support quarterly debt
service payments of $45,000 for 20 years. The company's bank is willing
to offer a 20year loan requiring quarterly payments and an annual
interest rate of 13.5 percent. What is the maximum amount the company
can borrow?

Notice this is an annuity, because you plan to make equal periodic
debt service payments of $45,000 per quarter for 20 years. For this
problem, C is the $45,000 quarterly payment; r is the 13.5 percent annual rate divided by the compounding frequency, m, which is 4, so r = 3.375 percent or .03375. N is equal to the payment frequency, 4, times the number of years, 20, or 80.


Construct a time line. Remember that each payment, a cash outflow,
occurs at the end of a period. The first payment occurs at the end of
period 1, and the last payment occurs at the end of period 80. The time
line would show 80 endofperiod payments of $45,000, occurring one per
quarter for 20 years. For reference, the end of the first period is at
the mark for time period 1, and the end of the last period is at the
mark for time period 79.


Substitute the given values in the formula for present value of an ordinary annuity, and solve.

Therefore, assuming that the company can make quarterly
payments of $45,000 each over 20 years, it can borrow $1,239,643 at
13.5 percent compounded quarterly. The first payment will occur one
quarter after the company borrows $1,239,643.
As with the FVA problems, you can solve this problem with Excel.
Using either the PVA equation or Excel's builtin PV function, discount all 80 payments to t = 0.
Reconsider the above situation with each $45,000 payment being made at the beginning of each period. This is an example of the present value of an annuity due, denoted as PVA_{Due}. The general form of PVA_{Due} is
Similar to FVA_{Due} and FVA, the PVA_{Due} equation is just like the PVA equation except for the extra term (1 + r) on the far right. The term (1 + r) in the PVA_{Due}
equation serves to value the cash flow stream one period forward, or
into the future. Thus, to find the present value of an annuity due, you
calculate the present value as if it were an ordinary annuity, then
adjust the present value forward one period by the periodic interest
rate.
Example: annuity due Now find the present value of the $45,000 annuity due.

For this calculation, C is the $45,000 quarterly payment; r is the 13.5 percent annual rate divided by the compounding frequency, m, which is 4, or 3.375 percent; and N
is equal to the payment frequency, 4, times the number of years, 20, or
80. All the terms are exactly the same as they were for the ordinary
annuity calculation.


You may wish to make a time line representing the annuity's cash flows and their timing at the beginning of each period.
Note that the first payment is made at the beginning of the first
period, which is the same as the end of time period 0. The last payment
is made at the beginning of period 80, which is at the mark for the end
of time period 79. The time values of the payments, denoted as cash
outflows for the company, all need to be discounted to the beginning of
the first period, time period 0.


Insert the given values into the formula for the present value of an annuity due, and solve.

Assuming that the company can make quarterly payments of $45,000
starting at the beginning of the first period, over 20 years, it can
borrow $1,281,480 at 13.5 percent compounded quarterly over 20 years.
Thus, the PVA_{Due} equation gives you a lumpsum present value
on the date of the first payment. This implies that the first payment
is not discounted at all, and that the last payment is discounted back
79 periods.
You can solve this problem with the help of an
Excel worksheet.
Summary
Before leaving this section, you may want to review the following essential concepts.
 When you compute the FV of an ordinary annuity, the result is the equivalent lumpsum payment on the date of the last annuity payment. 
 When you compute the FV of an annuity due, the result is the equivalent lumpsum payment one period after the last annuity payment. 
 When you compute the PV of an ordinary annuity, the result is the equivalent lumpsum payment one period before the first annuity payment. 
 When you compute the PV of an annuity due, the result is the equivalent lumpsum payment on the date of the first annuity payment. 
Always use an interest rate appropriate for the length of the period
between annuity payments. For example, if the payments are annual, use
a oneyear interest rate; if they are monthly, then use a monthly
interest rate.
How many decimal places should you use?


2. Your company buys structured
settlements or annuities from people who want their money now. If
your company requires an annual rate of 28 percent compounded quarterly, how much
would it pay for an inheritance that will generate 180 quarterly
payments of $15,000 starting today?
Solution 2
3. You are an investment attorney, and a new client who just won $60
million in the lottery seeks your advice. Lottery officials have given
her two payment options. Under the first option, the lottery would pay
your client a onetime payment of $29,888,572.92 today. Under the
second option, the lottery would pay your client $250,000 per month for
the next 20 years, with the first payment to be made immediately.
If your client believes that she can earn a guaranteed 8 percent
return, compounded monthly over the next 20 years, which option should
she select?
Solution 3
4.
Having landed a large signing bonus upon completing your MBA and taking
a job, you have decided to lease a new car for three years. The dealer
offers a 36month lease that requires you to pay $550 per month and a
$2,000 down payment due at lease signing. If you were able to earn 3
percent compounded monthly in a money market account, how much would
you have to put in the account today in order to meet your lease
obligations? (Include the amounts due today.)
Solution 4
5. Assume
a government bond has a face value of $1,000 (it pays $1,000 at
maturity), a coupon of 6 percent (it pays interest at an annual rate of
6 percent), semiannual payments of interest ($30 twice a year), and a
fiveyear maturity. If the market interest rate for such a bond is 5
percent, how much is the bond worth? (Discount the cash flows at an
annual rate of 5 percent, compounded semiannually.) How much is it
worth at a market rate of 8 percent? (Discount the cash flows at an
annual rate of 8 percent compounded semiannually.)
Solution 5

You can use Excel to solve PV.
PV Excel Tutorial
