1. Ordinary Annuity
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2. Annuity a sequence of equal payments made at equal time intervals
Examples: daily wages, periodic payments of installment purchases, monthly rent, annual insurance premiums
Payment interval – the time between successive payments
Term – the time between the first and last payment intervals
Periodic payment (R) – the amount of each payment
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3. R
Last payment interval
First payment interval
Term …
The time between two consecutive payments of an annuity is called payment interval or payment period.
Payments maybe made monthly, quarterly, semi-annually, annually or every 2 months or every 4 months.
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4. Types of Annuity
Simple annuity – an annuity for which the payment period is the same as the interest period
Example: An annuity for which the interest rate is compounded monthly and payments are also made monthly
General annuity – interest and payment periods do not coincide with one another
- Example: An annuity for which the interest rate is
compounded quarterly while payments are made
monthly
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5. Classification of Annuity
Annuity certain – an annuity where payments begin and end at fixed times.
Example: installment payments for certain purchase made
Contingent annuity – payments are dependent on an event that can not be foretold
- Example: premium on life insurance policy
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6. 3 kinds of Annuity Certain (Simple Annuity)
Ordinary annuity – an annuity where payments
are made at the end of each payment interval
Ordinary annuity of n payments R n
n-1
n-2 4 3 2 1
Annuity due – an annuity where payments are made at the beginning of each payment interval
Annuity due of n payments R n
n-1
n-2 4 3 2 1
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7. Deferred annuity - an annuity wherein the first payment interval does not coincide with the first interest period. The first payment is put off to some later date.
Deferred annuity of n payments
d+n
d+(n-1)
d+(n-2)
d-1
d-2 2 1 R d
d+1
d+2
1st payment starts on the (d+1)th period
NO payment for d periods
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8. Ordinary Annuity Amount of an Ordinary annuity (F) – value of annuity in one lump sum amount at the end of the term – value of annuity on the last payment date – sum of the accumulated values of all payments at the end of the term or on the last payment date F R n
n-1
n-2 4 3 2 1
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9. SUM=F R n
n-1
n-2 4 3 2 1
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10. F is the amount of an ordinary annuity of n payments
R is the periodic payment of the annuity
i is the interest rate per period
n is the total number of payments or periods
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11. Ex 1 Find the amount of an annuity of P3000 paid at the end of every 6 months for 20 years if money is worth 6.24% converted semi-annually. F
= 232,457.71
3000 40 39 38 4 3 2 1
(semi-annual periods)
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12. Ex 2 In order to create a fund for his forth coming business venture, Matthew decides to deposit P5000 in a fund at the end of each month. If the bank pays 4% compounded monthly on his deposits, how much is in the fund at the end of 2 years? F
= 124,714.44
5000 24 23 22 4 3 2 1
(months)
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13. Ordinary Annuity Present Value of an Ordinary annuity (P) – value of annuity in one lump sum amount at the beginning of the term – value of annuity at one period before 1st payment date – sum of the discounted values of all payments at the beginning of the term or at one period before the 1st payment date P R n
n-1
n-2 4 3 2 1
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14. P= SUM R n
n-1
n-2 4 3 2 1
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15. P F P is the present value of an ordinary annuity of n payments
value of annuity at
one period before 1st payment date P F
value of annuity on
last payment date R n
n-1
n-2 4 3 2 1
P is the present value of an ordinary annuity of n payments
R is the periodic payment of the annuity
i is the interest rate per period
n is the total number of payments or periods
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16. Ex 4 Find the present value of an annuity of P3000 paid at the end of every 6 months for 20 years if money is worth 6.24% converted semi-annually.
68, = P F
= 232,457.71
fr Ex 1
3000 40 39 38 4 3 2 1
(semi-annual periods)
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17. P Cash value (CV) = Down payment (D) + Present Value (P)
Ex 5. An LCD TV is purchased with down payment of P30,000 and P at the end of each month for two years to discharge all principal and interest at 15% converted monthly. Find the cash price of the TV set.
Cash value (CV) = Down payment (D) + Present Value (P) P 24 23 22 4 3 2 1
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18. Ex 6. Linda is paying P10,000 every 3 months for 3 years for a loan she acquired. If she is being charged an interest of 5% converted quarterly, how much was her original loan? P
10000 12 11 10 4 3 2 1
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19. Ex 7 . Jane deposits P14,000 every 3 months in a savings account that pays 6% compounded quarterly. Assuming that she does not withdraw any amount, how much would she have in her account at the end of 4 years? F
14000 16 15 14 4 3 2 1
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20. Ex 8 A multimedia workstation is for sale at Php28,000 every six months for two years at 12% compounded semi-annually or at Php20,000 down and Php7, each month for the next 12 months at 15% compounded monthly. Which terms should you choose?
cheaper,
1st offer better P P 12 11 10 4 3 2 1
28000 4 3 2 1
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21. Ex 9 . In preparation for the college education of his son, Mr Yanga will deposit P2400 at the end of each month for 5 years in a fund earning 4.5% compounded monthly. How much is in the fund (a) just after 15th deposit ? (b) just after the last deposit?
2400 60 16 15 4 3 2 1
2400 59
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22. Ordinary Annuity Finding periodic payment (R) of an Ordinary annuity P
Formulas for the amount F, present value P of an ordinary annuity: P F R n
n-1
n-2 4 3 2 1
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23. Ex 1 A newly-formed business bought a property worth P14. 5M
Ex 1 A newly-formed business bought a property worth P14.5M. They paid a downpayment of P3M with an agreement to pay the balance in 10 years at 12% compounded quarterly. How much is the quarterly payment?
11,500,000 = P R 40 39 38 4 3 2 1
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24. Ex 2 Jim and his partners want to have P4. 2M in 3 years
Ex 2 Jim and his partners want to have P4.2M in 3 years. They make semi-annual deposits in an account which pays interest at 7% compounded semi-annually. Find their semi-annual deposit.
F = 4,200,000 R 6 5 4 3 2 1
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25. Ex 3 On his retirement at age 60, Ric receives P800,000 as a share of a pension fund. His heir invests this sum at 6.36% compounded quarterly. How much could Ric or his heir regularly withdraw at the end of each 3 months for the next 25 years?
800,000 = P R
100 99 98 4 3 2 1
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26. Ex 4 At the end of each year for 10 years, a corporation will deposit equal sums in a depreciation fund to provide for the replacement of machinery worth P500,000. If the fund accumulates at 8% effective, how much must each deposit be?
F = 500,000 R 10 9 8 4 3 2 1
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27. Ex 5 A loan of P50,000 with interest at 15% compounded every 4 months is to be repaid by 24 equal payments made at the end of every 4 months. Find the size of each payment.
50,000 = P R 24 23 22 4 3 2 1
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