MCR3U Unit 8: Financial Applications Lesson 1 Date: Learning goal: I understand simple interest and can calculate any value in the simple interest formula. Simple Interest is the money earned (or owed) only on the borrowed. Simple Interest The original amount invested or borrowed is referred to as the. invested or, or simply the Example 1: Suppose you invest $1000 in a bank that offers you 5% simple interest. What is the ending balance after 5 years? Let s begin by considering ONE YEAR 5% = and 1000( ) = Therefore, the interest, I = for one year. Complete the chart below for this simple interest example. Year Starting that Interest is Calculated On Interest Ending 1 $1000 $1000 I = (1000)(.05) = $50 $1050 2 $1050 $1000 I = 3 4 5 Simple Interest The general formula for simple interest is! =!"# where:! refers to the interest earned (in dollars)! refers to the principal (or initial investment, in dollars)! refers to the interest rate per year (as a decimal)! refers to the length of time the money is invested (in years) The general formula for ending balance is! =! +!"# or! =!(! +!") where! refers to the amount that a simple-interest investment or loan is worth (in dollars) Example 1 (continued): a) Now use the simple interest formula to calculate the interest earned after 1 year. Did you get the same amount as your interest column? b) Now use the ending balance formula. Did you get the same amount as the ending balance in year 5 in the table?
Example 2: $500 is invested at 6.5% simple interest. Find the amount if the investment matures in 7 years. Example 3: A $700 investment doubles in 9 years. Find the simple interest rate. Example 4: $1400 is borrowed at 12.4% simple interest for 4 months. Find the financing charge (interest). Example 5: Tanya invests $4850 at 7.6%/a simple interest. If she wants the money to increase to $8000, how long will she need to invest her money? HW: pg. 49 # 10, 12 (Ans Corr 12e: 15)
MCR3U Unit 8: Financial Applications Lesson 2 Date: Learning goal: I understand the difference between simple and compound interest. I can calculate the future value and interest of a compounded investment. Compound Interest Compound Interest is the money earned (or owed) on the invested or borrowed as well as any that was previously earned or owed during the investment. Example 1: Suppose you invest $1000 in a bank that offers you 5% interest compounded annually. What is the ending balance after 5 years? Complete the chart below for this compound interest example. that Starting Year Interest is Calculated On Interest! =!"# Ending 1 $1000 $1000 I = (1000)(.05)(1) = $50 $1050 2 $1050 $1050 I = 3 4 5 What type of investment would earn you more money, simple or compound? Explain.
SIMPLE vs. COMPOUND INTEREST: WHAT S THE DIFFERENCE? Simple Interest: Complete the difference column. Year Starting that Interest is Calculated on Interest Ending 1 $1000 $1000 $50 $1050 First Differences 2 $1050 $1000 $50 $1100 3 $1100 $1000 $50 $1150 4 $1150 $1000 $50 $1200 5 $1200 $1000 $50 $1250 What type of relationship exists between year and ending balance? Why? Compound Interest: Complete the difference columns. Year Starting that Interest is Calculated on Interest Ending First Differences Second Differences Ratios 1 $1000 $1000 $50 $1050 2 $1050 $1050 $52.50 $1102.50 3 $1102.50 $1102.50 $55.13 $1157.63 4 $1157.63 $1157.63 $57.88 $1215.51 5 $1215.51 $1215.51 $60.78 $1276.29 What type of relationship exists between year and ending balance? Why? SUMMARY For Simple Interest, the relationship between time and the ending balance is. For Compound Interest, the relationship between time and the ending balance is.
RECALL: A general formula for exponential growth is! =!(!)!, where! is the growth factor, and! 1 is the growth rate. Future Value of an Investment (Compound) The general formula for finding the final amount of an investment (compound) is!/!" =!(! +!)! where:! is the final amount (or future value)! is the principal (initial amount invested)! is the interest rate as a decimal! =!!! is the frequency of compounding periods per year! is the number of times interest is paid Compounding Periods if the period is annual ð! =! (interest paid once a year) if the period is semi-annualð! =! (interest paid twice a year) if the period is quarterly if the period is monthly if the period is bi-weekly if the period is weekly if the period is daily ð! =! (interest paid 4 times a year) ð! =!2 (interest paid 12 times a year) ð! =!" (interest paid 26 times a year) ð! =!" (interest paid 52 times a year) ð! =!"# (interest paid 365 times a year) Example 2: Dayton invests $1,400 in an account that gains interest at a rate of 8% p.a., compounded semiannually. His investment matures after 12 years. a) Find the amount of his investment at maturity. b) Calculate the interest he gained.
Example 3: Adam spends $520 on his credit card. The credit company charges 18.5% p.a., compounded monthly. He pays off his loan in full after 2 years. a) Find the amount he must pay back. b) How much interest has he paid? Example 4: Margaret can finance the purchase of a $949.99 refrigerator one of two ways: Plan A: 10%/a simple interest for 2 years Plan B: 5%/a compounded quarterly for 2 years Which plan should she choose? Justify your answer. HW: pg. 70 # 1, 3, 4, 14, 15, 17, 21, pg. 79 # 7
MCR3U Unit 8: Financial Applications Lesson 3 Date: Learning goal: I can calculate the present value, interest rate, and number of compounding periods for a compound investment. FUTURE VALUE vs. PRESENT VALUE Present Value refers to the amount of money needed to invest (the present) so that you will obtain a particular amount in the. In other words, if you know the amount of money you want to have in the future, how much principal should you invest today? RECALL: The compound interest formula! =!(! +!)!, where! represents the starting (principal) amount. If we rearrange this formula to isolate! we obtain the Present Value formula Present Value Formula! =!(! +!)!!! is the final amount (or future value)! is the principal (initial amount invested)! is the interest rate as a decimal! =!!! is the number of compounding periods per year! is the number of times interest is paid! =!"! is the number of years of the investment
Example 1: Melissa would like $10,000 in 3 years to pay for tuition. She has found an investment that yields 4.7%/a compounded biweekly. How much must she deposit now? Example 2: What annual interest rate will cause an investment to triple in nine years if interest is compounded weekly? Show an algebraic solution and give your answer as a percent to the nearest hundredth. HW: pg. 70 #5, 6, 10-13, 23, 25, pg. 80 #8
MCR3U Unit 8: Financial Applications Lesson 4 Date: Learning goal: I can relate the future value annuity formula to a geometric series. I can calculate the future value of an annuity. Annuities Future Value Scenario 1: George invests $300 into an account that pays 2%/a interest compounded monthly. Scenario 2: George deposits $300 each month into an account that pays 2%/a interest compounded monthly. How are the scenarios the same? How are the scenarios different? An is a series of earning compound interest and made at over a fixed period of time. As with most investment types, annuities are often calculated to find future values. In this course (unless otherwise stated) annuities are ordinary, where payments are made at the end of intervals, and the compounding periods coincide with payment periods. Example 1: Joe plans to invest $1000 at the end of each 6-month period in an annuity that earns 6%/a compounded semi-annually for the next 5 years. Draw a timeline to represent his investment. What will be the future value of his annuity? The future value of an annuity is the financial equivalent of the annuity at maturity. TIMELINE Year 0 1 2 3 4 5 Payment $1000 $1000 $1000 $1000 $1000 $1000
Using the formula for a geometric series, here s the formula for calculating the future value of ordinary annuities: Ordinary Annuities Future Value!/!" =![(! +!)!!]!! is the final amount (or future value)! is the payment, the regular amount invested at each interval! is the interest rate as a decimal! =!!! is the number of compounding periods per year! is the number of regular payments! =!!"#$%&!"!"#$%&!!"!"#$%&!"#!"#$! is the number of years of the investment Example 2: Jon plans to invest $1000 at the end of each 6-month period in an annuity that earns 4.8%/a compounded semi-annually for the next 20 years. What will be the future value of his annuity? Example 3: Ms. Marsh wants to retire in 25 years with $1,000,000. She has found an investment that yields 9.6%/a compounded monthly (WOW!). How much should she deposit each month? HW: pg. 152 # 6, 8, 10-13, 18-20 (Ans Corr 8d: $82,826.66)
MCR3U Unit 8: Financial Applications Lesson 5 Date: Learning goal: I can relate the present value annuity formula to a geometric series. I can calculate the present value of an annuity. Annuities Present Value RECALL: An ordinary annuity is a series of equal payments earning compound interest and made at regular intervals over a fixed period of time. The present value of an annuity represents the initial amount that must be deposited so that constant payments may be taken out over an interval of time. A loan is an example of Present value. Example 1: Jen makes $1000 payments every year to pay back a loan at 5%/a compounded annually. It takes her 5 years to pay back the loan. How much was the loan for? TIMELINE Year 0 1 2 3 4 5 Payment $1000 $1000 $1000 $1000 $1000 $1000
Using the formula for a geometric series, here s the formula for calculating the present value of ordinary annuities: Ordinary Annuities Present Value!" =![! (! +!)!! ]!!" is the final present value needed to invest today! is the payment, the regular amount invested at each interval! is the interest rate as a decimal! =!!! is the number of compounding periods per year! is the number of regular payments! =!!"#$%&!"!"#$%&!!"#$%&'(!"#!"#$! is the number of years of the investment Example 2: Shirley has taken a loan to pay for her first car. To repay the loan, her bank is charging her $327.94 per month for 1 year with interest at 9% per year, compounded monthly. What is the actual cost of the car when Shirley purchased it? Example 3: Len borrowed $200 000 from the bank to purchase a yacht. If the bank charges 6.6%/a compounded monthly, he will take 20 years to pay off the loan. a) How much will each monthly payment be? b) How much interest will he have paid over the term of the loan? HW: pg. 163 # 4, 6-11, 13, 18
MCR3U Unit 8: Financial Applications Lesson 6 Learning goal: I can use technology to make financial calculations. Date: Applications of Technology Often these calculations we have been computing throughout the unit are done through a computer. We will be using a TVM solver on ugcloud to do some calculations. The TVM solver does a few things differently than the conventions we have established in our formulae. N is the total number of payment periods, or the number of conversion periods. This is the! we used in our formulae. I% is the annual interest rate as a percent, not a decimal. This is! from our formulae, but as a percent. PV is the present value. Recall it can represent the principal in a compound interest situation. PMT is the regular payment amount. It is the! from our formulae. FV is the future value. Recall it can represent the amount in a compound interest situation. P/Y is the number of payment periods per year. This is! from our formulae C/Y is the number of interest conversion periods per year. For simple annuities and most compound interest questions (all that we have seen), C/Y is the same as P/Y. The different tabs at the bottom allow you to solve for you desired value. DO NOT change the value in the highlighted cell. Example 1: $8000 is invested for 10 years at 8%/a, compounded monthly. Find the amount. Example 2: Ms. Marsh wants to buys a $550,000 house (after down payment) by making biweekly payments. He will pay interest at 5.7%/a. By law, Canadian mortgages can t be compounded more often than semiannually. If Ms. Marsh would like a 25 year mortgage, find his biweekly payment. HW: Picking the Correct Formula Worksheet