Unit 9: Borrowing Money 1
Financial Vocab Amortization Table A that lists regular payments of a loan and shows how much of each payment goes towards the interest charged and the principal borrowed, as the balance of the loan is reduced to 0. Collateral An asset that is held as security against the repayment of the loan The asset is often the item that you used the loan to purchase In the even that you cannot repay your loan, the item used as collateral, will be taken Compound interest A is the future value, P is the principle amount, i represents the interest rate per compounding period and n represents the number of compounding periods. Increases exponentially NOTE: 3% interest monthly does not mean 3% per month. It means 3% 12 = 0.25% per month Simple Interest A=P(1+rt) or A=P+Prt A is the future value, P is the principal amount, t represents time in years, and r represents the interest rate per annum. Increases linearly (same amount of interest each time) 2
Most people will need to take out a loan sometime in their lives (or borrow money in some form.) Few people can afford expensive purchases such as a car or a house without borrowing money from a financial institution. In unit 6 we discussed investments/loans where interest is calculated one of two ways: Simple interest or compound interest. Simple vs. Compound Interest Simple Interest: the amount of interest that you receive (if investing) or pay on a loan is calculated ONLY on the amount of money that you borrow. Usually loans from personal friends, family For simple interest, only the initial principal earns interest. The formula for simple interest is: A= P(1+rt) where A represents the amount present P represents the principal amount r = interest rate divided by 100 t represents the number of years We can calculate the interest only by using the formula: I = Prt 3
Example 1 Becca borrowed some money from her sister at 3% simple interest, with the interest calculated annually. If the loan was for 6.5 years and she had to pay back $2360, what was the principal? 4
Example 2: Ralph invested his summer earnings of $6000 at 4% simple interest, paid annually. Determine the relation that models this situation and determine when the investment will be worth $8640. 5
Compound Interest: Recall that compound interest is beneficial to the person receiving the interest, as interest gained on interest. But it costs the person paying the interest on a loan/credit card a lot of money, for the same reason. Therefore, the longer it takes to pay back the loan, the more you pay. Most financial institutions charge compound interest. For example; Loan Credit Cards Mortgage Borrowing Investing Savings Accounts Chequing Accounts GIC** Line of Credit Canada Savings Bond ** Student Loan ** GICs and Canada Savings Bonds can have simple or compound interest. Most loans are repaid by making regular payments over the term of the loan. However, some loans can be repaid in a single payment at the end of a term. For compound interest, the initial principal and accumulated interest also gain interest. The formula is where P is the principal amount i is the interest rate per compounding period n is the number of compounding periods Notice that i is the interest rate per compounding period. If interest is compounded x times each year, then the given percentage must be divided by x to come up with i. 6
7
Ex: A person takes out a $2000 loan, compounded at 10% semi annually. It takes the person 4 years to pay back the loan. How much does he he owe if he pays it back in one lump sum? 8
Ex: Which investment would yield a greater return for 10 years Option 1: $1000 at 3.5% annual simple interest Option 2: $1000 at 3% annual compound interest 3% 9
Ex: James intends to go to university. His Grandmother would like to invest $2000 in a GIC [Guaranteed Investment Certificate] A) How much will the GIC be worth if it is invested at 3% simple interest for 5 years? B) How much would it be worth if the interest was compounded annually? C) Which option is better for the bank? For James? 10
Ex: Annette wants to borrow money to renovate her kitchen. Her bank will charge her 3.6% compounded quarterly. Annette wants to pay back the money with one lump payment after 3 years, and wants this payment to be at most $20000. Determine the maximum amount of money she can borrow. 11
When a financial institution lends money, it will always negotiate the terms of the loan, including the interest rate and how it wants the money paid back. We will consider two cases: Paying Back Loans (Part 1) 1. A loan is paid off using a single payment at the end of the term. 2. A loan is paid off by making regular loan payments (only cases in which payment frequency matches the compounding period). We will start off by looking at loans that are paid off using a single payment at the end of the term. Examples: a farmer making a single lump sum payment on his loan after his crop has been harvested a payday loan offered by certain financial service providers. We have already dealt with these types of problems. Note: Interest Paid = A - P OR I = A - P 12
Ex: Trina s employer loaned her $10000 at a fixed interest rate of 6%, compounded annually, to pay for college tuition and textbooks. The loan is to be repaid in a single payment on the maturity date, which is at the end of 5 years. Determine how much interest Trina will pay on the loan. 13
You Try! Ex: Mary borrows $1000 at 10% interest, compounded semi annually. Sean borrows $1000 at 10% interest compounded annually. How much interest will each pay at the end of two years? 14
The more frequent the compounding, the more interest will be charged. When making financial decisions, it is important to understand the rate of interest charged, as well as the compounding, as these can create large differences over long periods of time. Ex: Which represents the lowest interest that would be paid? (A) 10% compounded daily (B) 10% compounded monthly (C) 10% compounded annually 15
Ex: Which represents the lowest interest that would be paid? (A) 8% compounded daily (B) 12% compounded monthly 16
Paying Back Loans Regular/Multiple Loan Payments However, in the majority of loans, the lender wants scheduled payments, not just one lump sum. This is common for mortgages and vehicle loans. There are three common types of regular payment schedules:. Monthly Biweekly Semi Monthly Accelerated Biweekly 17
Ex. For a loan that has a $600 per month payment, determine how much will be paid out at the end of 3 years, using each of the 3 payment options. Monthly: Bi weekly: (every 2 weeks) (26 times a year) 1. Find bi weekly payment first. 2. Find the amount paid out in 3 years Accelerated Bi weekly: (monthly payment )2) 26 times a year 1. Find accelerated bi weekly payment first. 2. Find amount paid out in 3 years. 18
Ex: 130 biweekly payments are required to pay off a loan. How many years does this represent? Ex: 288 semi monthly payments are required to pay off a loan. How many years does this represent? When people purchase a vehicle, they often link their loan payment schedule to their payroll schedule. Why is this the case? 19
The formulas that we learned previously (i.e A = P (1 + rt) and A = P(1 + i) n ) ONLY apply to single loan payments at the end of a term, and thus CANNOT be used in situations in which there is a regular loan payment. What we will do for these types of questions is refer to a table (can be manually created or using software) which shows the payment, interest principal and balance. 20
Ex: Mark is buying an ATV for the summer. The bank offers him a loan of $7499.99 to pay for his ATV with an interest rate of 4.5% compounding monthly. If Mark makes 36 monthly payments of $223.10, calculate the total interest paid at the end of the loan. (A) Complete the first three rows of the Amortization table... 21
1. Interest Paid = (interest rate per year )# compounding periods) H Outstanding Balance 2. Principal Paid = Monthly Payment Interest 3. Current Balance = Previous Balance Principal Paid during Current Payment Period The entire table (as created using software) is shown below: 22
Determining the Cost of a Loan Using Technology To calculate a loan payment requires a more complicated formula. Usually, we use a TVM Solver (TVM stands for Time Value of Money). The TI 83 has a great TVM Solver in the Apps/Finance menu. Hit "Apps" then "Finance", choose TMV Solver. Hit Enter N is the number of payments I% is the annual interest rate PV is the present value of the loan PMT is the payment. FV is the future value of the loan P/Y is the payments per year. CY is the compounding periods per year. NOTES: if a variable is not being used in a question, or it is being solved for, put in zero for its value. For our purposes we will always leave PMT at END To solve for a variable: Make sure you put in 0 for its' value. Scroll down to the variable you want to solve for and hit "Enter". "FV" is negative when borrowing money 23
Examples: 1. Consider Lars, who borrowed $12000 from a bank at 5% compounded monthly to purchase a new snowmobile. The bank requires him to pay $350 per month until the loan is paid off. A) How long it will take to pay off at least half of the loan? To solve for N, we use the arrow keys to navigate to N and select SOLVE (press ALPHA then ENTER). B) How long it will take to pay off the loan? C) How much interest will Lars have paid by the time the loan is paid off? (Both estimated value and exact value to find exact amount of interest we have to use our Finance App.) Number of Months to Pay off Loan Total Amount Paid = Approximate Interest Paid = Exact amount of interest paid = 24
2. When you were born your grandparents deposited $5,000 in a special account for your 21st birthday. The interest was compounded monthly at 5%. A) How much will it be worth on your 21st birthday? B) Suppose they also added $ 10 every month. How much will it be worth on your 21 st birthday? 25
3. Brittany takes out a loan for $100 000 at 6% annual interest. She takes 20 years to repay the loan. A) Use a financial application to determine the amount of each monthly payment. B) How much interest will she have paid at the end of the 20 years? 26
4. A bank offers a mortgage rate of 3.75% compounded semi annually for a 25 year period. Connie purchases a $300 000 house with a 10% down payment. A) Use TMV solver to determine her monthly payment. B) How much interest does she pay? 27
5. Which is the better option? Explain. Option 1: Cost of a mortgage is $300 000, interest rate 2.5%, monthly payments, 25 year amortization. Option 2: Cost of mortgage is $300 000, interest rate is 3.5%, monthly payments, 20 year amortization. 28
A note about accelerated bi weekly payments Another option that banks often offer to the borrower is the option to pay back a loan with accelerated bi weekly payments. Consider a loan of $100000 in which a person has a $600 per month payment. The interest is compounded annually at 5%. Using our TVM Solver, we see that it takes about 280 months (or 23.33 years) to pay off the loan, with total interest paid being $67695.50. In the span of one year, this person would pay 12 x $600 = $7200. If this person decided to go on a bi weekly payment plan, we would determine his payment per period by dividing $7200 by the 26 periods: Even though this person is paying the exact same money per year, he will save interest charges and lower his amortization date the time taken to pay off a loan because he is saving on interest charges by paying every two weeks instead of waiting until the end of the month. Using our TVM Solver, we see it takes 605 periods, which is equivalent to 1210 weeks (or 23.27 years). The total interest is $67356.53. The person has not saved much time or much money considering the length of time of the loan. An accelerated bi weekly payment combines the two types of payments. The person pays half of the original monthly loan payment (in this case $300) every two weeks. At this rate, the loan will be paid off in 525 periods. This is 1050 weeks or 20.19 years. This saves over 3 years of paying off a loan. The total interest paid is $57274.12, which is a savings of over $10000! The lesson pay off your loans as quick as possible! 29
30