MAP4C - Grade 12 Mathematics - Lesson 16 Annuities Assignment

This document provides a detailed solution to the MAP4C Lesson 16 Annuities assignment, covering a range of financial concepts. The assignment focuses on applying annuity formulas to solve various problems. The questions cover calculating future value, present value, monthly payments for loans, and the amount needed for investments. Students are asked to determine the future value of an annuity, the monthly payment for a vehicle loan, the monthly deposit needed to save a specific amount, and the present value of income streams. The assignment also explores how to calculate the total amount needed to repay a loan, the interest paid over the life of a loan, and the minimum amount needed in an account for living expenses. Additionally, it includes problems on calculating the amount to invest for retirement income and the amount needed to deposit monthly to reach a financial goal. The solutions include the strategy used, the calculations, and the final answer for each question. The assignment is designed to reinforce understanding of annuities and their application in personal finance.

MAP4C Lesson Assignment 16
In this assignment you are going to focus on: Annuities
Total Marks: 60 marks
Instructions:
After completing the lesson action section of your home page, you are ready to complete your lesson assignment. Follow
these steps.
a) In the textbox given, explain the strategy that you used to determine your answer.
b) Fill in your answer. The mark for each question is given at the introduction to the questions to be completed.
c) When finished answering all your questions, save your file and upload the file into the appropriate dropbox.
Part A (Putting the Concepts to Work): Completion (60 marks): Using your reading, answer the following
questions. Give your answer as well as your strategy for determining your answer (this will include: the information
given in the question, the formula to be used, the calculations and the final answer.
As part of your answer, give the page number in the reference material or the website that you used to find the answer to
the question.
1. (four marks) An amount of $800 is invested at the end of each month for 12 months into an account that pays
3% per year, compounded monthly. Determine the future value of this annuity.
Solution:
PMT= 800, i = 3/12 % = 0.4 %, n = 12
FV = PMT ∗( 1+ i ) n −1
i
FV = $ 9814.04
____ 2. (four marks) An amount of $1500 is deposited semi-annually for two years into a fund that pays 10.5% per
year, compounded semi-annually. Use a TVM Solver to determine the future value of this annuity.
c. $ 6489.25
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a. $6079.21 c. $6489.25
b. $5871.01 d. $3078.75
____ 3. (four marks) Determine the monthly payment for a five-year loan on a $19 000 vehicle at 7% per year,
compounded monthly.
$ 376.22
P = 19000, r = 7% , n =12, Y = 5
PMT = ( P∗r
n )/ [ 1−( 1+ r
n )
−nY
]
PMT = $376.22
4.(four marks) Determine the monthly deposit needed to save $10 000 in three years if interest is earned at 3.5% per year,
compounded monthly.
Solution:
$ 9004.62
A=10000, Y = 3, n =12, r = 3.5%
PMT = (Ar/n) * ( 1+ r
n )
ny
−1
Solving we have,
PMT = $9004.62
5. (four marks) Determine the amount that should be invested now to generate an semi-annual income of $2000
for three years from a fund that earns 9% per year, compounded semi-annually.
PMT = 2000, n = 2*3 = 6 , i = 9/2 % = 4.5%
P=PMT∗[ 1− ( 1+i )−n
i ]
P = 10315.74
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6. (four marks) Determine the present value of monthly car payments of $590 at 6.9% per year, compounded
monthly for two years.
PMT = 590 , n = 24, i = 6.9/12 % = 0.575 %
PV =PMT∗[ 1− ( 1+i )−n
i ]
( PV ) = $13191.06
7. (four marks) Colin recently started a part-time job. He is saving to travel to Europe. He plans to deposit $500
at the end of each month into an account that pays 2.4% per year, compounded monthly. How much will
Colin have saved at the end of eight months?
PMT= 500, i = 2.4/12 % = 0.2 %, n = 18
FV = PMT ∗( 1+ i )n −1
i
FV = $ 9154.64
8. (four marks) Hayden recently graduated from college and owes $24 000 on a student loan that she must
begin to repay. Payments are to be made at the end of each month for the next five years. Interest is 8.5% per
year, compounded monthly. Calculate the total amount needed to repay the loan.
P = 24000, n =12 , Y = 5 , r = 8.5 %
PMT = ( P∗r
n )/ [ 1−( 1+ r
n )
−nY
]
Monthly payment : 492.40
Total amount needed to pay off the loan : 492.40 * 12 *5 = $29544
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9. (four marks) Amir would like to borrow $3300 to purchase furniture. His bank offered him a two-year
personal loan with monthly payments. The interest rate on the loan is 7.5% per year, compounded monthly.
Calculate the total amount Amir will have to pay to the bank.
P = $3300 , n = 12, Y = 2, r = 7.5
PMT = ( P∗r
n )/ [ 1−( 1+ r
n )
−nY
]
PMT = $ 148.50
Total amount that Amir has to pay back : 148.5 * 24 = 3564
10. (four marks) Maya would like to borrow $4500. Her bank offered her a two-year personal loan with monthly
payments. The interest rate on the loan is 8% per year, compounded monthly. How much interest will Maya
pay over the life of the loan?
P= 4500, r = 8, n = 12, Y = 2
PMT = ( P∗r
n )/ [1−(1+ r
n )−nY
]
PMT = 384.48
Interest that Maya has to pay back = 203.52 * 12*2 – 4500 = $384.48
11. (four marks) Katie has a part-time job at a local convenience store and earns $700 per month. She plans to
deposit $350 at the end of each month into an account that pays 3.75% per year, compounded monthly. How
much will be in Katie’s account after three years?
PMT= 350, i = 3.75/12 % = 0.3125 %, n = 36
FV = PMT ∗( 1+ i ) n −1
i
FV = $ 13314.11
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12. (four marks) Roger plans to take an 18-month college program. He estimates that he will need $750 per month
for living expenses. Currently, his savings are in an account paying 3.6% per year, compounded monthly.
What is the minimum amount Roger will need in his account when he starts college so he can pay his living
expenses each month?
PMT = 750 , n = 18, i = 3.6/12 % = 0.3 %
P V =PMT∗[ 1− ( 1+i )−n
i ]
( PV ) = $13122.83
13. (four marks) What amount should be invested now at 6% per year, compounded annually, to generate a
retirement income of $35 000 per year for 15 years?
PMT = 35000 , n = 15, i = 6 %
P=PMT∗[ 1− ( 1+i )−n
i ]
Amount that should be invested( P ) = $339,928.71
14. (four marks) What income should be invested now at 9% per year, compounded semi-annually, to generate a
retirement income of $16 000 every six months for 20 years?
R = 16000 , n = 40, i = 4.5 %
P=PMT∗[ 1− ( 1+i )−n
i ]
Amount that should be invested( P ) = 294,425.35
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15. (four marks) Issac can invest his money in an account that pays 3.9% per year, compounded monthly.
Determine the amount he must deposit at the end of each month to have $12 000 in two years.
r = 3.9, A = 12000, n = 12 , Y = 2
PMT = ( A∗r
n )/ [ 1− ( 1+ r
n )
−nY
]
= $520.57
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