1. Present Value of an Annuity with Non-Annual Payments
Dr. Craig Ruff
Department of Finance
J. Mack Robinson College of Business
Georgia State University
© 2014 Craig Ruff

2. Present Value of an Annuity – Non-Annual Payments
The present value of an annuity follows very much the same pattern as the future value of an annuity with non-annual compounding, except that now we are summing a series of present values.
Using the same two-year annuity, the question is now: What is the present value of an annuity that pays you $100 every six months for the next two years? The discount rate is 10%, compounded semi-annually.
© 2014 Craig Ruff

3. Present Value of an Annuity – Non-Annual Payments
1/2
100 1
100
1 1/2
100 2
100
© 2014 Craig Ruff

4. Present Value of an Annuity – Non-Annual Payments
First, find the present value of the $100 arriving at t = ½…$95.23
Second, find the present value of the $100 arriving at t = 1..$90.70
Third, find the present value of the $100 arriving at t = 1½..$86.38
Fourth, find the present value of the $100 arriving at t = 2…$82.27)
Sum up the four pieces to find the present value of the whole annuity…$ ( ) .
1/2
100 1
100
1 1/2
100 2
100
95.23
90.70
86.38
105
82.27
© 2014 Craig Ruff

5. Present Value of an Annuity – Non-Annual Payments
Like the other annuities, there are comparable summation and computational formulas:
© 2014 Craig Ruff

6. Calculating the Present Value of an Annuity – Non-Annual Payments
On the calculator, with this example, you would enter…
Buttons
Numbers to Enter PV
???? FV I 5 N 4
PMT
-100
Again, your calculator does not know years, months, days, etc. It only know periods.
Since the payments are occurring every six months, the period is six months. There are 4 payments (N=4) and the rate is 5% period (I=5). Note that the FV is set to zero.
© 2014 Craig Ruff

7. Examples
© 2014 Craig Ruff

8. Example: Present Value of an Annuity – Non-Annual Payments
Now, let’s calculate the payment on a traditional mortgage. You plan to borrow $200,000 today via a standard, fixed-rate, 30-year mortgage. The rate is 4%. Payments are made monthly, with the first payment due in one month.
(Comment: It is standard convention in the U.S.A., that we would state the rate as 4%. This is the nominal
rate. Properly speaking, the rate should be stated as 4%, compounded monthly. And, the effective rate can be quickly calculated as 4.074%).
So, the annual monthly ‘payment’ is $
If you have yet to buy a house, please don’t be
deceived. Your monthly mortgage payment will
actually be somewhat higher, as taxes and home
insurance are typically collected from you on on a
monthly basis by the lender. The lender wants to
make sure that your taxes and insurance are paid up;
thus, the lender collects the money from you and
directly pays the taxes and insurance.
The $ would only cover what is referred to
as “P&I” for principal and interest.
Buttons
Numbers to Enter PV FV I
4/12 = N
30 * 12 = 360
PMT
????  
© 2014 Craig Ruff

9. Example: Present Value of an Annuity – Non-Annual Payments
Working with the same mortgage, it is now one year later. So far, you have made all payments on time. You want to know your outstanding balance the instant after you make your twelfth payment.
Following the same logic as when we found the initial payments, the outstanding balance will be the present value of the remaining payments.
Buttons
Numbers to Enter PV
????
196, FV I
4/12 = N
360 – 12 = 348
PMT
So, the balance at t=12 months, is $196,
This is the original payment we found at t=0 when the mortgage was initiated. See previous slide.
© 2014 Craig Ruff

10. Example: Present Value of an Annuity – Non-Annual Payments
As a check,
here is part of
a spreadsheet
(amortization
table) that
follows the
mortgage
through the
first twelve or
so months…
The balance at t=12 months, is $196,
© 2014 Craig Ruff

11. Perpetuity
© 2014 Craig Ruff

12. Perpetuity A perpetuity is an annuity that lasts forever.
Ideally, the perpetuity is calculated as the sum of present values of the individual payments. As a perpetuity lasts forever, ,at first glance, it might appear that there is no solution. Fortunately, as the payments get further and further away, the present values of these payments get closer and closer to zero.
Indeed, the present value of a perpetuity converges to the very simple formula of:
© 2014 Craig Ruff

13. Perpetuity
As an example, what is the present value of a perpetuity that will pay $100/year forever, with the first payment due in one year? The discount rate is 10%.
One way to see the reasonableness of this answer is to imagine that you deposit $1,000 today in a bank account paying 10%, compounded annually. At the end of the first year, you could withdraw $100. At the end of the second year, you could withdraw $100. At the end of the third year, you could withdraw $100. And on and on forever. The point is that
the $100 withdraws every year look just like the the perpetuity. And what did it take you to get these annual $100 withdraws forever? It took a deposit of $1000 today.
© 2014 Craig Ruff

14. Perpetuity Suppose you forget the perpetuity formula.
You could approximate the perpetuity value by using a huge value for N. Here we use 10,000.
Buttons
Numbers to Enter PV
???? 1, FV I 10 N
10,0000
PMT
-100
Here we are telling the calculator that the number of payments is 10,000.
Notice that the calculator returns an answer of 1,000,000, which matches the
answer we got using the formula on the previous slide.
© 2014 Craig Ruff

15. End