Financial Mathematics: Annuity Calculations and Problem Solving

This document provides a detailed explanation of annuities, covering both future and present value calculations. It begins with an introduction to annuities, defining them as a series of equal payments made at equal intervals, and highlights their relevance in financial planning, investment, and loan estimations. The document then delves into the future value of ordinary annuities, providing formulas and worked examples to illustrate how to calculate the future value of periodic payments, including scenarios involving forborne annuities. Following this, the present value of ordinary annuities is explored, with formulas and examples demonstrating how to determine the present value of future payments, including deferred annuity situations. The assignment includes several examples where students are required to calculate annuity payments and interest rates. Additionally, the document showcases how to calculate different variables associated with annuities, such as the number of payments. Overall, the assignment offers a practical guide to understanding and solving a range of annuity problems.

2.4 Annuity
MAHANI FSKM

Learning
objectives
Student should be able to:
• find the future and present value of annuity
• solve for annuity payment, R
no. of payments, n
interest rate, k
• identify the problems of present and future value

2.4.1
Introduction to
Annuity and Types
of Annuities

Introduction
• Annuity is a series of equal payments made at equal intervals of time
• Equal payments are deposited or paid periodically over time (at the end
of payment period)-ordinary annuity certain
• Ex: shop rental, insurance policy premiums, annual dividends received
and instalment payments
• Annuity enable people to plan for future investments, savings and
retirement.
• Use by financial consultants to advise people on investments
• Use by property agent to estimates the repayments of housing and car
loans

2.4.2
Future Value of
Ordinary Annuity
Certain

Future value
• sum of all the future values of the periodic payments.
• 𝑆 = 𝑅 1+𝑖 𝑛−1
𝑖 or 𝑺 = 𝑹 𝟏+𝒌
𝒎
𝒎𝒕
−𝟏
𝒌
𝒎
• where
• R=equal payments/periodic payments
• i=interest rate per interest period
• n =term of investment
• S=future value
• 𝑰 = 𝑺 − 𝒏𝑹

Example 1
Solution:
RM100 is deposited every month for 2 years 7 months at 12% compounded monthly. What is the future
and how much interest is earned?
𝑅 = 100, 𝑡 = 2
7
12 = 31
12, 𝑘 = 0.12, 𝑚 = 12
𝑆 = 𝑅
1 + 𝑘
𝑚
𝑚𝑡
− 1
𝑘
𝑚
=100 1+0.12
12
12(31
12)
−1
0.12
12
=𝑅𝑀3613.27
𝐼 = 𝑆 − 𝑛𝑅
=3613.27 − 31 100
=𝑅𝑀513.27

Example 2
Solution:
RM150 was deposited every month in an account for 15 years at 3.6% compounded monthly. Find the
amount in the account at the end of 15 years and the interest earned.
𝑅 = 150, 𝑡 = 15, 𝑘 = 0.036, 𝑚 = 12
𝑆 = 𝑅
1 + 𝑘
𝑚
𝑚𝑡
− 1
𝑘
𝑚
=150 1+0.036
12
12(15)
−1
0.036
12
=𝑅𝑀35731.01
𝐼 = 𝑆 − 𝑛𝑅
=35731.01 − 12(15) 150
=𝑅𝑀8731.01

Example 3: Forborne annuity-annuity earns interest for some periods after the last payment
Solution:
RM100 was invested every month in an account for 2 years at 12% compounded monthly. After two yea
no more deposit was made. Find the amount of the account at the end of the five years and the interest
𝑅 = 100, 𝑡 = 2, 𝑘 = 0.12, 𝑚 = 12 𝑆 = 𝑅
1 + 𝑘
𝑚
𝑚𝑡
− 1
𝑘
𝑚
=100 1+0.12
12
12(2)
−1
0.12
12
=𝑅𝑀2697.35
𝑆 = 𝑃 1 +𝑘
𝑚
𝑚𝑡
=2697.35 1 +
0.12
12
12(3)
=𝑅𝑀3859.28
𝐼 = 𝑆 − 𝑛𝑅
=3859.28 − 24 100
=𝑅𝑀1459.28
no deposit 50 2annuity

2.4.3
Present Value of
Ordinary Annuity
Certain

Present value (discounted value)
• sum of all present values of the periodic payments.
• 𝐴 = 𝑅1− 1+𝑖 −𝑛
𝑖 or 𝑨 = 𝑹𝟏− 𝟏+ 𝒌
𝒎
−𝒎𝒕
𝒌
𝒎
• where
• R=equal payments/periodic payments
• i=interest rate per interest period
• n =term of investment
• A=present value
• 𝑰 = 𝒏𝑹 − 𝑨

Example 4
Solution:
Raymond has to pay RM300 every month for 24 months to settle a loan at 12% compounded monthly.
What is the original value of the loan and the total interest that he has to pay?
𝑅 = 300, 𝑡 = 2, 𝑘 = 0.12, 𝑚 = 12
𝐴 = 𝑅
1 − 1 + 𝑘
𝑚
−𝑚𝑡
𝑘
𝑚
=300 1− 1+0.12
12
−12(2)
0.12
12
=𝑅𝑀6373.02
𝐼 = 𝑛𝑅 − 𝐴
=24 300 − 6373.02
=𝑅𝑀826.98