Foundations 30

Unit 7 – Financial Mathematics - Investments

FP 30.1: Demonstrate understanding of financial decision making including analysis of: simple interest, compound interest, investment portfolios

* Adapted from Nelson Foundations of Mathematics

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Unit 7 – Key Terms Date: _____________________________

Key Terms

Term – The contracted duration of an investment or loan.

Interest – The amount of money earned on an investment or paid on a loan.

Fixed Interest Rate – An interest rate that is guaranteed not to change during the term of an investment or loan.

Principal – The original amount of money invested or loaned.

Simple Interest – The amount of interest earned on an investment or paid on a loan based on the original amount (the

principal) and the simple interest rate.

Maturity – The contracted end date of an investment or loan, at the end of the term.

Per Annum – An interest rate per year. The given percent is assumed to be annual unless otherwise stated.

Future Value – The amount, A, that an investment will be worth after a specified period of time.

Rate of Return – The ratio of money earned (or lost) on an investment relative to the amount of money invested, usually

expressed as a decimal or a percent.

Compound Interest – The interest that is earned or paid on both the principal and the accumulated interest.

Compounded Annually – When compound interest is determined or paid yearly.

Compounding Period – The time over which interest is determined; interest can be compounded annually, semi-annually

(every 6 months), quarterly (every 3 months), monthly, weekly, or daily.

Rule of 72 – A formula for estimating the doubling time of an investment.

Present Value – The amount that must be invested now to result in a specific future value in a certain time at a given

interest rate.

Annually – Per year

Semi-Annual – Twice a year

Monthly – Twelve times a year

Quarterly – Four times a year

Daily – 365 times a year.

Annuity – A continual payment of a fixed amount.

i = Prt, where i is the interest, P is the principal, r is the rate, and t is the term. A = P + Prt or A = P(1 + rt) where A is the future value

= 1 + , where A is future value, r is interest rate, n is compounding periods and t is the term.

=

, where A is the future value, P is the payment amount, r is the rate, n is the compounding periods, t is

the term length.

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 1 – Simple Interest Date: _____________________________

Lesson 1 – Simple Interest

Term – The contracted duration of an investment or loan. Interest – The amount of money earned on an investment or paid on a loan. Fixed Interest Rate – An interest rate that is guaranteed not to change during the term of an investment or loan. Principal – The original amount of money invested or loaned. Simple Interest – The amount of interest earned on an investment or paid on a loan based on the original amount (the principal) and the simple interest rate. CSB – Canada Savings Bond GIC – Guaranteed Investment Certificate

i = Prt, where i is the interest, P is the principal, r is the rate, and t is the term. A = P + Prt or A = P(1 + rt) where A is the future value

Calculating Simple Interest 1) Determine your principal, rate as a decimal, and term of the investment 2) Substitute into the simple interest formula 3) Add the original principal to determine the entire amount Example 1: You invest in a $2500 GIC at 2.5% simple interest, paid annually, with a term of 10 years. What is the future value of the investment, and the amount of interest earned. Example 2: You invest $15, 000 in a savings account. You earn a simple interest at a rate of 8%, paid semi-annually, on your investment. You hold the investment for 4.5 years. Fill out the following table for every half year of the investment, then graph and determine an equation for the line.

Year Interest Value of Investment 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

1 2 3 4 5 6 x

10000

12000

14000

16000

18000

20000

y

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 1 – Simple Interest Date: _____________________________ Determining Principal, Rate, and Term for Simple Interest 1) Substitute all the known values into the equation 2) Isolate and solve for the missing value Example 3: a) You invest $5 000 at 8% simple interest, paid annually. How long will it take for the investment to be at $8 000? b) You invest $20 000 into a CSB and it grows to $29 375 in 5 years. What is the interest rate? c) You have a GIC that has grows to $7480 over 8 years at a rate of 4.5%, how much did you initially invest? Rate of Return – The ratio of money earned (or lost) on an investment relative to the amount of money invested, usually

expressed as a decimal or a percent.

Example 4: You have an investment of $4 000 at 8% simple interest. Determine the rate of return after 8 years.

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 1 – Simple Interest Date: _____________________________ Example 5: You invest $25 000 into a simple interest Canada Savings Bond that pays interest annually.

a) If the future value is $29 375 at the end of the five years, what was the interest rate?

b) You want to cash the savings bond in after 4.5 years, what will the value at that time be? Example 6: You invest $30 000 into an account with simple interest at 4.2%, paid daily. You close the account after 90 days and withdraw the money, how much do you withdraw?

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 2 – Compound Interest – Part 1 Date: _____________________________

Lesson 2 – Compound Interest – Part 1

Compound Interest – The interest that is earned or paid on both the principal and the accumulated interest.

Compounded Annually – When compound interest is determined or paid yearly.

Compounding Period – The time over which interest is determined

Number of Compounding Periods in a Year Annually Semi-annually Quarterly Monthly Weekly Daily

= 1 + , where A is future value, r is interest rate, n is compounding periods and t is the term. Calculating Compound Interest 1) Determine the principal, interest rate, number of compounding periods and term length 2) Substitute into the equation and solve. It is easier to calculate inside the bracket first. Example 1: The principal is $4300, r is 3.8%, compounded annually, for 3 years. Show how to get the compound interest formula. Example 2: You invest $23 000 in an account that earns 13.6% compounded semi-annually. The interest arte is fixed for 10 years. You want to use the money for a down payment on a house in 5 to 10 years. What is the future value after 5 years and 10 years?

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 2 – Compound Interest – Part 1 Date: _____________________________ Example 3: A mother, aged 50, and a daughter, aged 18, plan to invest $1500 in an account with an annual interest rate of 9%, compounded monthly. Determine the future value of their investments if they each hold it until they turn 65. Example 4: You invest $3000 for 5 years with the following options. Compare the interest rates for each option: 4.8% compounded annually, semi-annually, monthly, weekly and daily. Rule of 72 – A formula for estimating the doubling time of an investment.

Calculating Doubling Time

1) Divide 72 by the interest rate as a percent.

Example 5: You invest $5000 at 8% interest compounded annually. Estimate the doubling time on the

investment.

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 3 – Compound Interest – Part 2 Date: _____________________________

Lesson 3 – Compound Interest – Part 2

Solving for Principal value, Rate and Term

1) Substitute all values known in

2) Isolate for the remaining value you are looking for

3) If you are solving for the term rate you will need to take the natural logarithm of both sides

Example 1: Using the compound interest formula isolate the P, r and t terms.

Example 2: The principal is $10 000, the rate is 3%, it is compounded quarterly for 5 years to have a future

value of $11 611.84. Manipulate the compound interest formula to show the following values

a) Future Value b) Principal Value

c) Rate d) Term

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 3 – Compound Interest – Part 2 Date: _____________________________ Example 3: Ginny is 18 years old. She wants to invest money so that she can buy a home when she is 30 and married to Harry. She estimates that she will need about $170 000 to buy a home. How much would she have to invest now, at 6.5% compounded annually

Example 4: You want to save for $40 000 for 3 years, invested at a rate of 9.6% compounded quarterly. a)How much should you invest now?

b) How much interest will you earn?

Example 5: A parent has invested $15 500 in a Registered Education Savings Plan (RESP) and wants it to grow to $50 000 by the time their child goes to university in 18 years. What interest rate, compounded anually, would be necessary for it to grow to $50 000?

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 4 – Future Value of Annuity Date: _____________________________

Lesson 4 – Future Value of Annuity

Annuity – A continual payment of a fixed amount.

=

, where A is the future value, P is the payment amount,

r is the rate, n is the compounding periods, t is the term length

Determine a Future Value of an Investment With Annuity - Formula 1) Determine the payment amount, rate, compounding period, and term 2) Substitute into the formula and solve

Example 1: You are saving for a trip in 5 years. You deposit e $500 at the end of each 6-month period, earn 3.8% compounded semi-annually.

a) How much will be in the account at the end of 5 years?

b) How much of this will be interest?

Example 2: Adam made a $200 payment at the end of year into an investment that earned 5% compounded annually. Blake made a single investment at 5% compounded annually. At the end of 5 years their future values were equal. What amount did Blake have to initially invest? Who made more interest?

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 5 – Investments Involving Regular Payments Date: _____________________________

Lesson 5 – Investments Involving Regular Payments

Determining Values Using a Financial Calculator 1) Determine the following information N = total number of payments made (nt) I/Y = Annual interest rate as a percent P/V = Present value (usually 0) PMT = Payment amount as a negative F/V = Future Value P/Y = Number of Payments per year C/Y = Number of Compounds per year 2) Enter each number by pressing the number first then the key for which value it is. For P/y press 2nd I/y, enter the amount then enter, for C/y press up, enter the amount, then enter, then hit quit. 3) Press CPT then the key you are looking to find (FV, I/y, PMT, N, PV)

Example 1: You are saving for a trip in 5 years. You deposit e $500 at the end of each 6-month period, earn 3.8% compounded semi-annually. The future value of the investment is $5449.90. a) Determine N, I/Y, P/V, PMT, F/V, P/Y and C/Y

b)Using the calculator determine the Future Value

c) Using the calculator determine the Annual Rate of Interest

d) Using the calculator determine the Payment amount

d) Using the calculator determine the length of the term

F30 – Unit 7 – Financial Mathematics – Investments Name: ____________________________ Lesson 5 – Investments Involving Regular Payments Date: _____________________________ Example 2: You deposit $750 into an investment at the end of every 3 months. Interest is compounded quarterly, the term is 3 years, and the future value is $10 059.07. What is the annual rate of interest?

Example 3: You want to have $300 000 in 20 years so you can retire. You fund an account that earns a fixed rate of 10.8% calculated annually. How much will you need to invest at the end of year to meet this goal? How much interest do you earn?

Example 4: On your 20th birthday you start making regular $1000 payments into an investment account at the end of every 6 months. You invest at 3.5% compounded semi-annually. At what age will you have enough for a down payment of $18 000?

F30 – Unit 7 – Financial Mathematics – Investments Name: __________________________ Lesson 6 – Analyzing Portfolios Date: ___________________________

Lesson 6 – Analyzing Portfolios

Example 1: You start to build an investment portfolio for your retirement - You purchased a $500 CSB at the end of each year for 10 years. The first five CSBs earned a fixed rate of 4.2% compounded annually. The next five CSBs earned a fixed rate of 4.6% compounded annually. - Three years ago you purchased a $4000 GIC that earned 6%, compounded monthly

a) What is the value of your portfolio 10 years after you start to invest?

b) You find a savings account that earns 4.9% compounded semi-annually. You redeem your portfolio and invest all the money in your savings account. How long will it take to double your money?

Example 2: You are starting a portfolio and you have $2,500 saved so far. You can commit to saving $50 a month form your part time job. You can buy a 6 year GIC that earns 5.1% compounded semi-annually and deposit your monthly savings into a high-interest savings account that earns 4.3% compounded monthly. What will your portfolio be worth in 6 years? How much interest have you earned?

F30 – Unit 7 – Financial Mathematics – Investments Name: __________________________ Lesson 6 – Analyzing Portfolios Date: ___________________________

Example 3: Compare which of the following portfolios has a higher rate of return after 10 years.

- 10 year $2000 GIC that earns 4.2% compounded semi-annually - A savings account that earns 1.8%, compounded weekly, where you save $55 every week - A 5-year $4000 bond that earns 3.9%, compounded quarterly, which he will reinvest in another bond at an interest rate of 4.1% compounded quarterly

- A tax-free savings account that earns 2.2%, compounded monthly, and has a current balance of $5600 - The purchase, at the end of each year, of a 10-year $500 CSB that earns 3.6% compounded annually - A savings account that earns 1.6% compounded monthly, where you save $200 every month.