2 1 ordinary-annuity

2.1 Ordinary Annuity2.1 Ordinary Annuity

Ordinary Annuity


• Annuity – a sequence of equal payments made regularly

• Examples: daily wages, monthly installments, 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


R R R

Last payment intervalFirst payment interval

Term


Types of Annuity• Simple annuity – an annuity for which the

interest period is the same as the payment 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


Types of Annuity• Ordinary annuity – an annuity for which

payments are made at the end of each interest period

• Deferred annuity - an annuity wherein the first payment does not coincide with the first interest period


Formulas for the amount S, present value A and periodic payment R of an ordinary annuity:

€

S = R(1+ i)n −1

i

⎡

⎣ ⎢

⎤

⎦ ⎥

€

A = R1− (1+ i)−n

i

⎡

⎣ ⎢

⎤

⎦ ⎥

€

R =Si

(1+ i)n −1

€

R =Ai

1− (1+ i)−n


Formulas for the present value Adef and periodic payment R of a deferred annuity:

€

Adef = R1 − (1+ i)−n

i

⎡

⎣ ⎢

⎤

⎦ ⎥(1+ i)−d

€

R =Adef i

1 − (1+ i)−n[ ](1+ i)−d


1. Find S given R = Php12,500, i = 3.75%, and n = 16.

€

=12,5001.037516 −1

.0375

⎡

⎣ ⎢

⎤

⎦ ⎥

€

=Php267,409.27€

S = R(1+ i)n −1

i

⎡

⎣ ⎢

⎤

⎦ ⎥


5. Find A given R = Php13,210, j = 14%, m = 2 and t = 7 years.

€

=Php115,527.63€

A = R1 − (1+ i)−n

i

⎡

⎣ ⎢

⎤

⎦ ⎥

€

=13,2101 −1.07−14

.07

⎡

⎣ ⎢

⎤

⎦ ⎥


9. Find R given S = Php246,916.3, j = 9%, m = 12 and t = 3 years.

€

=Php6,000€

R =Si

(1+ i)n −1

€

=(246,916.3)(.0075)

1.007536 −1


17. Jane deposits Php14,000 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?

€

=14,0001.01516 −1

.015

⎡

⎣ ⎢

⎤

⎦ ⎥

€

=Php251,053.18

€

S = R(1+ i)n −1

i

⎡

⎣ ⎢

⎤

⎦ ⎥


19. 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 Php7,040.15 each month for the next 12 months at 15% compounded monthly. Which terms should you choose?


19. Which terms should you choose?

€

1 .Offer is better

€

1Offer :

€

=Php97,022.96

€

A = 28,0001−1.06−4

.06

⎡

⎣ ⎢

⎤

⎦ ⎥

€

2Offer :

€

CP = 20,000 + A

€

A = 7,040.151−1.0125−12

.0125

⎡

⎣ ⎢

⎤

⎦ ⎥

€

=Php98,000.02


23. A business process outsourcing (BPO) company bought a new office space for Php720,000 downpayment and monthly installments of Php40,000 at the end of each month for 3 ½ years. What is the cash equivalent of the property if money is worth 12% compounded monthly?

€

=Php2,086,324.33

€

=720,000 + 40,0001−1.01−42

.01

⎡

⎣ ⎢

⎤

⎦ ⎥

€

CP = DP + A


24. CooKing offers a kitchen package at Php328,000 quarterly for two years at 8%, m = 4. Caterking offers almost the same package at Php190,000 down and Php201,857 each month for the next 12 months (j = 13%, m = 12). Which offer should you choose?


24. Which offer should you choose?

€

CooKing' .s offer is better

€

CooKing:

€

=Php2,402,757.91

€

A = 328,0001 −1.02−8

.02

⎡

⎣ ⎢

⎤

⎦ ⎥

€

CaterKing:

€

CP =190,000 + A

€

A = 201,8571− .13

12( )−12

.1312

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

=Php2,449,999.52


27. A newly-formed business bought a property worth Php14.5M. They paid a downpayment of Php3M with an agreement to pay the balance in 10 years at 12% compounded quarterly. How much is the quarterly payment?

€

=Php497,517.35

€

R =Ai

1− (1+ i)−n

€

=(11,500,000)(.03)

1−1.03−40


29. Jim and his partners want to have Php4.2M in 3 years. They make semi-annual deposits in an account which pays interest at 7% compounded semi-annually. Find their semi-annual deposit.

€

=Php641,206.48

€

R =Si

(1+ i)n −1

€

=(4,200,000)(.035)

1.0356 −1


31. To start an IT business, Elmer and Erwin borrowed Php4.5M with interest at 10% compounded quarterly. They will repay the debt by making 12 equal quarterly payments. If the first payment is due at the end of 18 months, how much is the quarterly payment?

€

=Php496,339.81

€

R =Adef i

1 − (1+ i)−n[ ](1+ i)−d

€

=(4,500,000)(.025)

1 −1.025−12[ ](1.025)−5


37. Find the present value of a Php120,000 annuity payable every year for 10 years but deferred 3 years. Money is worth 8% effective.

€

=Php639,201.47€

Adef = R1− (1+ i)−n

i

⎡

⎣ ⎢

⎤

⎦ ⎥(1+ i)−d

€

=120,0001−1.08−10

.08

⎡

⎣ ⎢

⎤

⎦ ⎥(1.08)−3


39. If money is worth 12% compounded quarterly, find the present value of 28 quarterly payments of Php18,000 each, the first of which is due at the beginning of the 4th year.

€

=Php244,000.64€

Adef = R1− (1+ i)−n

i

⎡

⎣ ⎢

⎤

⎦ ⎥(1+ i)−d

€

=18,0001 −1.03−28

.03

⎡

⎣ ⎢

⎤

⎦ ⎥(1.03)−11

2 1 ordinary-annuity

Economy & Finance