Discrete Math B: NOTES CHAPTER 5 – Mathematics of Finance 1
Chapter 5: The Mathematics of Finance
You want to buy some new furniture for your first apartment. You have a steady job, but not enough to pay in full up front, so you decide you want to apply for a personal loan. One bank will loan you $1000 at 8% compounded annually that must be paid in 10 monthly payments. Another local bank will loan you $1000 compounded monthly at 7.9% that must be paid in 15 monthly payments. Your parents will loan you $1000 at a simple interest rate of 8%, but you have to pay it back within 6 months. Which offer should you take?
In the very near future…
some of you may be making major purchases that require financing
some of you may begin to save money for retirement or start investing for your future
some of you may get very compelling offers from credit card companies that seem too good to be true (they are)
You need to know how money works. In this chapter we are going to talk about a few basic ideas.
5.1 Simple and Compound Interest
Interest is the money you are charged as a fee for borrowing money from a lender.
Interest is also the money that you earn when you invest money.
Simple Interest
Usually used for loans of ______year or ______.
You are charged (or paid) interest only on the amount you borrowed (or invested) and not on past interest.
Principle - the amount ______or ______.
Rate - ______of interest per year, always expressed as a ______.
Time - the amount of ______money is earning interest.
You also need to know the units of time in a year.
In a year
52 weeks
12 months
360 days
Yes. 360 days.
Historically, to simplify calculations, we assume twelve 30-day months = 360 days.
Interest found using a 360 day year is called ______interest.
Interest found using a 365 day year is called ______interest.
Example 1: Find the Simple Interest. Round your answer to the nearest cent.
a) $3000 at 10% for 7 monthsb) $12,500 at 4.3% for 32 weeksc) $530.35 at 1.25% for 65 days
More terms:
“A” = Future or Maturity Value of your principle – the amount of your principle after r% interest for t years.
Example 2: Find the maturity value and the amount of simple interest earned.
a) A loan of $2500 to be repaid in 8 months with interest of 9.2%
b) A loan of $11,280 for 85 days at 11% interest.
Example 3: Your friend is willing to loan you $450 today if you are willing to pay him back $500 in 6 months. What was the simple interest rate?
Compound Interest
Usually used for loans of ______than one year.
You are charged (or paid) interest not only on the money you borrowed (or invested) but only on interest as well.
For example:
IDEA: SIMPLE INTEREST
Consider $1000 invested at 5% interest for 2 years.
Value at the end of year 2: ______
IDEA: COMPOUND INTEREST
Consider $1000 invested at 5% interest for 1 year. Reinvest at same rate for another year:
Value at end of year 1: ______Value at end of year 2:______
“Compounding Periods”Term Times Per Year
annually1
bi-annually, semi annually 2
quarterly4
monthly12
weekly52
Example 4: Find the compound interest of each deposit and the amount of interest earned.
a) $450 at 5.6% interest compounded quarterlyb)$5600 at 12% interest compounded annually
for 10 yearsfor 5 years
Example 5:
What is the present value (how much should be invested now) at 5% compounded semi-annually, to accumulate 3000 in 10 years?
Example 6:
What is $1.00 worth after being invested at 5% compounded monthly?
nominal/stated rate – interest that you expect to earn (what you are told you will earn)r
effective rate – interest that you ACTUALLY earn. (Usually called APR or APY)re
Example 7:
Which is a better deal?Borrowing money at 8% interest compounded semi-annually or borrowing at 7.9% interest compounded monthly?
“THE RULE OF 72”
If you invest $1000 at 8% annual interest, how long will it take your money to double?
For interest rates between 4% and 20%:
The doubling time for an amount being compounded can be estimated quickly using the rule of 72.
Divide 72 by interest rate (expressed as a percent)
≈ approximate time to double.
For interest rates less than 4% or for daily compounding, use 70 instead of 72.
Example: About how long will it take $5000 to double if it is invested at 12% compounded quarterly?
5.2 Future Value of an Annuity
- annuity - a sequence of equal payments made at equal periods of time
Examples:
- ordinary annuity - an annuity where the frequency of the payments are the same as the frequency of the compounding
- payment period - the time between payments
- term – time between the first payment period to the last payment period.
- future value of an annuity - the final sum on deposit; the sum of all the compounded amounts of all the payments.
Example 1:
Suppose $1500 is deposited at the end of each year for the next 6 years in an account paying 8% per year compounded annually.
What this looks like:
Notice that the first $1500
earns interest 5 times!
Notice the last payment earns no interest.
Total Future Value of the Annuity:
1500(1.08)5+1500(1.08)4+1500(1.08)3+1500(1.08)2+1500(1.08)+1500
Suppose $1500 is deposited at the end of each year for the next 6 years in an account paying 8% per year compounded annually.
Suppose $1500 is deposited at the end of each year for the next 6 years in an account paying 8% per year compounded annually. What is the future value of the annuity?
Example 2:
Karen Scott is an athlete who believes that her playing career will last for 7 years. To prepare for her future, she deposits $22,000 at the end of each year for 7 years in an account paying 6% compounded annually. How much will she have on deposit after 7 years? How much is from payments? How much is from interest?
Example 3:
Mrs. Leahy deposits $100 at the end of each month in a college savings account for her son. If the account earns 5% interest compounded monthly, how much will the account be worth in 16 years? How much is from payments? How much is from interest?
Sinking Funds
- sinking fund – a fund set up to receive period payments designed to produce a given some at some point in the future.
I need $2000 in 5 years… how much should I save each month if I can get 6% interest compounded monthly?
Example 4:Sarah believes she needs $130,000 in 20 years for her retirement. What payment should she make each month if she can earn 7.2% interest compounded monthly? How much of the $130,000 is from payments? How much is from interest?
5.3 Present Value of an Annuity; Amortization
Recall annuity – series of payments
- Present value of an annuity – the amount of money that would have to be deposited in one lump sum today (at the same compound interest rate) to produce exactly the same value at the end of 10 years.
Example 1:
Mr. Rex and Miss Cera both pledge to invest equal amounts in the Children’s Museum Endowment in 5 years. The endowment fund earns 6.5% interest compounded annually. Mr. Rex prefers to make a contribution of $500 at the end of year. Miss Cera prefers to give one lump sum payment today. What about will she have to contribute to be worth the same as Mr. Rex’s donation?
One Possible Strategy
1. Use “Future Value of Annuity” formula to determine the future value of Mr. Rex’s investment.
2. Use “Future Value Compound Interest” formula. Let A = answer #1. Solve for “P” for Miss Cera’s lump sum.
Another Possible Strategy
Amortization
- “Amortized”- when a loan’s principal and interest are paid in a sequence of equal periodic payments.
Basically, you need to solve the Present Value equation for “R” to determine the payment.
Example 2:
The Perez family buys a house for $275,000 with a down payment of $55,000. They take out a 30-year mortgage for $220,000 at an annual interest rate of 6%. Find the amount of the monthly payment needed to pay off this loan. What is the total amount of interest paid?
Amortization Schedules: Explanation of how much of each monthly payment is actually going to principal and how much is going to interest.
Example 3: Susan Stewart borrows $1000 for 1 year at 12% annual interest compounded monthly. After making three payments, she decides to pay off the remaining balance all at once. How much must she pay?
Susan’s Amortization Table
Example 4: Mrs. Leahy decides to finance $200,000 on a 30-year mortgage on a new home. The amount of interest charged at the end of each month for any unpaid balance is 6.5%. Create an amortization table for the first four payments of the loan.