1. Financial Applications -Annuities (Present Value)
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2. Annuities
An annuity is a sequence of equal payments made at equally spaced intervals of time.
The period of an annuity is the time interval between two consecutive payments.
The term of an annuity is the total time involved in completing the annuity.
Ordinary annuities have payments made at the end of the payment period.

3. Recall: Compound Interest formula
The formula used in compound interest is
Amount (A) .
Principal (P)
Interest rate per period (i)
Number of compounding periods involved (n)

4. Annuities formula Amount of annuity (A) . Regular activity (R)
The formula to calculate the accumulated amount with annuities is:
Amount of annuity (A) .
Regular activity (R)
Interest rate per period (i)
Number of compounding periods involved (n)

5. Example 5 – Annuities (Present Value)
Donna plans to deposit a sum of money in an account which pays 9% compounded annually so that she can make five equal annual withdrawals of $500. How much should she deposit if the first withdrawal is one year later?
Using the Annuities Formula
Now 1 5 4 3 2
$500
$500
$500
$500
$500
Geometric Series with:
Therefore, Donna should deposit $ into the account.

6. Annuities formula (Present Value)
The formula to calculate the Present Value amount with annuities is:
Present Value of annuity (PV) .
Regular activity (R)
Interest rate per period (i)
Number of compounding periods involved (n)

7. Example 6 – Annuities (Present Value)
Mr. Harrison’s life savings total $ He wishes to use this money to purchase an annuity earning interest at 12% compounded semi-annually which will provide him with equal semi-annual payments for 20 years. How much is each semi-annual payment if the first is 6 months from the date of purchase?
Think about the interest earned in 20 years!! R
Now 1 20 19
... 2
...
40Rs in total
Interest earned in 20 years:
Every R includes Interest earned and when all 40 Rs bring back to the beginning, it should be $  P
Therefore, Mr. Harrison is able to get $ per each semi-annual payment.

8. Example 7 – Annuities
A student wishes to buy a car. He can afford to pay $200 per month but has no money for a down payment. If he can make those payments for four years and the interest rate is 12% per annum, what purchase price can he afford?
The $200 is all the student can afford and this includes interest.
So bring the $200 back to the present time will find the affordable price.  P
Therefore, the student can afford a car upto $

9. Example 8 – Annuities A or PV??
Brian wants to buy a motorcycle for $ He has $1000 saved for the down payment and plans to borrow $4000 from the bank and repay the loan on a monthly basis over the next two years. If the bank charges 9% per annum compounded monthly, what is the value of the monthly payment? What is the cost of borrow?
A or PV??
If Brian has enough money, he doesn’t need the loan and $4000 is all he needs.
But however, he needs the loan and therefore all monthly payments that he needs to pay in the future will include interest in it.  P
And if all the monthly payments bring back to present time, it should be $  P
Therefore, Brian’s monthly payment is $182.74

10. Homework:
WS: Single Payments and Annuities
P.453 #1-5, 8-12