Chapter F: Finance Section F.1-F.4 F.1 Simple Interest Suppose a sum of money P, called the principal or present value, is invested for t years at an annual simple interest rate of r, where r is given as a decimal. Then the interest I at the end of t years is given by The future value F at the end of t years is Example 1 You borrow $300 on a credit card that charges simple interest at an annual rate of 12%. What is your interest for 9 months? Example 2 An account with an initial amount of $5000 earns simple interest at a rate of 6% annually. How much is in the account after 4 years? Example 3 Belinda takes out a loan earning simple interest for $1000. a) What simple interest rate is being obtained if the amount F at the end of 8 months is $1050? 1
b) After how many months from the beginning of the loan will you owe $1215? Example 4 How much should be placed into an account paying simple interest of 4% so that after 6 months, the future value of the account will be $2500? Round your answer to the nearest cent. F.2 Compound Interest Usually, interest is not calculated (only) from the original sum of money. A more common situation is one in which interest is calculated, or compounded, periodically and each time interest is calculated, the accumulated amount becomes the new principal. In other words, the interest earns interest. Suppose a principal P earns interest at an annual interest rate of r and interest is compounded m times a year. Then, after t years, the accumulated amount or future value F is Example 5 You put $500 into a savings account that earns 2% interest compounded monthly. How much is in the account after 1 month? 2 months? To calculate compound interest, we will use TVM Solver (we can t use this for simple interest). 2
Compound Interest with TVM Solver on the Calculator: 1. Press APPS. 2. Select Finance... (Option 1) 3. Select TVM Solver (Option 1) total number of compoundings during the time period interest rate (as a percent) initial amount in the account regularly occuring payments future value of the account the number of payments/compoundings per year END/BEGIN (Note: END should be highlighted) 4. Enter the known information. 5. Scroll to the line representing the unknown data. 6. Press SOLVE (ALPHA ENTER) TI-Nspire: 1. Press MENU. 2. Select Finance (Option 8) 3. Select Finance Solver... (Option 1) 4. Enter the known information. 5. Scroll to the line representing the unknown data. 6. Press ENTER Important Note: In the TVM Solver, the values for PV, PMT, and FV will sometimes be negative. This is done to represent the transfer or flow of money. We will usually look at these problems from the standpoint of the investor or borrower. A negative number represents an of money away from the investor or borrower, i.e., when money is leaving your pocket. Use a negative number when: Making payments Depositing money in a bank 3
A positive number represents an of money to the investor or borrower, i.e., when you put money in your pocket. Use a positive number when: You receive a loan from a bank or lender. You receive money from a bank account. Example 6 An account with an initial amount of $5000 earns interest of 6% compounded monthly. How much is in the account after 4 years? How much interest is earned? Example 7 How much should be placed into an account paying interest of 4% compounded quarterly so that after 6 years, the future value of the account will be $2500? How much interest is earned? 4
Example 8 Find the time for an investment of $1000 compounding monthly at an interest rate of 3% per year to double. How much interest is earned? Example 9 Suppose that 4 years ago, I invested $2500 in an account that compounds interest weekly. Right now I have $3625 in the account. What is the interest rate for this account (rounded to 4 decimal places)? Example 10 Suppose $1000 is left for 3 years in an account that earns interest at a rate of 5% per year compounded monthly. How much interest is earned during the third month of the second year? 5
Example 11 Suppose $500 is invested in an account that earns %9 interest compounded monthly. How much interest is earned in 1 year? What percent of the original amount is this? Suppose a sum of money is invested at an annual rate of r expressed as a decimal and is compounded m times a year. The effective yield r eff is Effective Yield with Your Calculator: 1. Press APPS. 2. Select Finance... (Option 1) 3. Select Eff (Option C) 4. Give the arguments as follows: Eff(r, m). Example 12 What is the effective annual yield on an account paying 9% interest per year, compounded monthly? Example 13 Compare the following situations. Which would be the better choice for an investment? Which would be the better choice for a credit card? A: 8% compounded semi-annually B: 7.9% compounded daily 6
Continuous Compound of Interest Example 14 An account with an initial amount of $100 earns 5% interest. How much is in the account after 1 year is interest is compounded a) annually? b) quarterly? c) monthly? d) daily? e) every hour? f) 1 million times per year? Suppose a principal P earns interest at an annual rate of r per year (as a decimal) and interest is compounded continuously. Then, after t years, the accumulated amount (or future value), F, is Example 15 Consider the previous problem. How much interest is in the account after 1 year if interest is compounded continuously? Example 16 If you invest $5000 at 8% per year with interest compounded continuously, how much would you have in your account after 7 months? How much interest is earned during this time? 7
F.3 Annuities and Sinking Funds An time periods. is a sequence of equal payments made at equal An are made at the end of the time periods compounding. is one in which the payments The of an annuity is the time from the beginning of the first period to the end of the last period. The is the total amount in the account, including interest, at the end of the term of an annuity. Example 17 Some examples of annuities are: regular deposits into a savings account, monthly home mortgage payments, monthly insurance payments The future value F V of an ordinary annuity of n payments of P MT dollars paid at the end of each period into an account that earns interest at the rate of i per period is We will use TVM Solver. Example 18 Suppose a person opens up a retirement account in which he/she places $750 each month into an account that earns interest at a rate of 8%/year compounded monthly. a) How much will be in the account when this person retires in 25 years? 8
b) How much interest is earned in total? A to meet a future need. is an account established for accumulating funds Example 19 You find a retirement account that earns interest at a rate of 4% per year compounded monthly. How much should you deposit into your account every month if you want to have $500,000 in the account when you retire in 40 years? Example 20 Dane s parents anticipate that his first year of college will cost $15,000. Knowing Dane s first year of college is 15 years away, determine the amount of money they should deposit into an account each year making 6.25% per year compounded annually if they intend on having the money ready to pay for his first year when he starts college. 9
How much would they have to deposit monthly over the same time period to reach the desired goal if they found an account paying interest at a rate of 6.25%/yr compounded monthly? Example 21 A family wants to save up some money to make a $25,000 down payment on a house in 6 years. a) How much should they deposit each month into an account if the account earns interest at the rate of 2.75%/year compunded monthly? b) How much is in the account after 3 years? How much of this is due to interest? 10
c) If they can afford $500 a month instead, when can they afford the down payment for the house? F.4 Present Value of Annuities and Amortization The present value of an ordinary annuity of n payments of P M T dollars with interest compounded at the rate of i per period is We will use TVM Solver. Example 22 Suppose you win a lottery worth $4,250,000 which is paid out with $170,000 payments for the next 25 years. In order to make these payments to you, how much money must the lottery commission have in an account now if the account earns interest at a rate of 3.25%/year compounded annually? 11
If you are given the option of taking a lump sum of $3,000,000 now or of accepting the payments, which should you take? Assume you are also able to invest the money at an interest rate of 3.25%. Option A: Take the lump sum Option B: Accept the payments is the process of paying off a debt. The periodic payment P MT to be made at the end of each period on a loan of P V dollars that is to be amortized over n periods with interest at the rate of i per period is Example 23 What monthly payment is required to amortize a loan of $75,000 over 10 years if interest at the rate of 5.25% per year, compounded monthly, is charged on the unpaid balance at the end of each month? 12
Example 24 Thaddeus made a down payment of $5000 towards the purchase of a new car. To pay the balance, he secured a loan at the rate of 1.9% per year compounded monthly. Under the terms of his finance agreement, he is required to make payments of $144/month for 36 months. a) What is the cash price of the car? b) How much total interest did Thaddeus pay on the loan? Example 25 A family secured a 15-year bank loan of $162,600 to purchase a house. The bank charges interest at a rate of 2.6% per year, compounded monthly. a) What is their monthly payment? b) How much total interest will they end up paying? 13
on a loan at a given point in time. is the amount you still owe To find the outstanding principal, find the remaining payments. of the Example 26 Consider the scenario from the previous example. a) What is the outstanding principal after 10 years? In other words, how much do they still owe after 10 years? b) How much thave they actually paid in these 10 years? Why couldn t we subtract this from $162,600 to find the outstanding principal? in a loan scenario is how much of the item you actually OWN. It is how much principal you have paid on the original loan plus any down payment (what belongs to you). The interest you pay does NOT count towards your equity. At any moment in time the following is true: 14
Example 27 We again consider the problem from the last two examples. What is their equity after 10 years? Example 28 Five years ago, Janella got a bank loan for the purchase of a home. The home was worth $245,546 and she made a 20% down payment. The interest on the loan was 5.35%/year compounded monthly and the term of the loan was 30 years. a) What is Janella s current monthly mortgage payment? b) After these first 5 years, Janella decides to refinance her home. What is her outstanding principal at this point? Equity? 15
c) Janella refinanced her home by securing a new 20-year loan for the outstanding principal at a new rate of 4%/year compounded monthly. What will her new monthly mortgage payment now? d) How much money will Janella save by refinancing the loan? Example 29 Suppose I take out a 2-year loan in the amount of $5000 at an interest rate of 6.5%/year compounded quarterly in order to buy a used car. a) What is my current quarterly payment? b) How much of the first payment goes toward interest? c) How much goes toward the principal? 16
d) What is the outstanding principal now? Equity? e) How much of the second payment goes toward interest and how much goes toward principal? What is the outstanding principal now? Equity? f) Finish the amortization table for the next two periods: Period Payment Towards Interest Towards Principal Outstanding Principal Equity Earned 0 - - - 5, 000 0 1 2 3 4 17
Example 30 A boat costs $65,000. You make a down payment of $10,000 and finance the remaining balance with a 10-year loan at an interest rate of 5.1%/year compounded monthly. You find that your monthly payment is $586.06 (Check this on your own). a) Fill in the amortization table for the first two monthly payments. Period Payment Towards Interest Towards Principal Outstanding Principal Equity Earned 0 1 2 b) What are the outstanding principal and equity after 7 years? (Hint: Don t analyze all 84 payments that have been made.) 18
Example 31 You have a $3000 credit card bill on a card that charges interest at a rate of 12% per year, compounded monthly, on the unpaid balance. a) If you do not make any additional purchases on the card and make a $71 payment each month, how long will it take you to pay off your bill? How much total interest do you end up paying? b) If you instead plan to pay off this credit card at the end of 3 years, how much will you have to pay each month? How much of your first payment goes toward interest? How much of your first payment goes toward principal (paying off your debt)? 19