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Chapter 14

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14.1 Time Value of Money

14.2 Installment Buying

14.3 Truth in Lending

14.4 The Costs of Home Ownership

14.1 The time Value of Money

If we borrow an amount of money today, we will repay a larger amount later. This increase in value is known as interest. The money gains value over time.

The amount of a loan or a deposit is called the principal or Present Value.

The interest is usually computed as a percent of the principal. This percent is called the interest rate.

Interest calculated only on principal is called simple interest.

Interest calculated on principal plus any previously earned interest is called compound interest.

Let’s Do It!

Find amount of interest paid to borrow $7500 for 7months at 6%.

What is the total amount repaid (called maturity value of the loan. generally refer to it as the future value)?

Let’s Do It! Applying the formula for Future Value for Simple Interest

George Atkins was late on his property tax payment to the county. He owed $7500 and paid the tax 4 months late. The county charges a penalty of 8% simple interest. Find the total amount required?

|Principle=2500 |Interest | | |

| |Rate=1.23% | | |

|61.5 | | | | |61.8782 | | | |62.0697 | | | | | | | | | | | | | | | | | |

During the first period, the account contains

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At the end of the second period, the account contains

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Consider one more period, namely the third. The account ends the third period containing

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Let’s Do It!

If a$4100 is deposited into an account that earns 5% interest compounded monthly, then how much will be in the account after3 years?

P=_____________, r =__________, m=___________, n=__________

Hence the future value of the $4100 after 3 years is A=

Example Finding Present Value for compound Interest

The Cebelinskis’ daughter will need $17,000 in 5 years to help pay for her college education. What lump sum, deposited today at 7% compounded quarterly, will produce the necessary amount?

Solution

This question requires that we find the present value P based on the following

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17000= P(1+.07/4)20 [pic]

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Let’s Do It! Finding the Present Value of a Future Real Estate Purchase

A California couple is selling their small dairy farm to a developer, but they wish to defer receipt of the money until 2 years from now, when they will be in a lower tax bracket. Find the lump sum that the developer can deposit today, at 5% compounded quarterly, so that enough will be available to pay the couple $1,450,000 in 2 years.

Mathematics Goes To Hollywood

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The second season of The Andy Griffith Show, “Mayberry Goes Bankrupt,” Sheriff Andy Taylor was reluctantly forced to evict kindly old gentlemen Frank Myers from his home due to nonpayment of taxes. But Frank then produced a bond given to him by his grandfather. It was purchased for $100 in 1861 and paid 8.5% interest compounded annually. The bond had been stored away for 100 years. When the town banker told the Mayor he could not pay Frank, Andy explained why this was so.

Well, Mayor … according to the computation machines down at the bank … and they’re good machines … we, that is the town of Mayberry, owe Frank Myers $349,119.27. (To which Frank responds: I’ll take it in cash.)

Was the computation machines down at the bank correct???

Calculator Tutorials:

1) Compounding Interest Using TI-83/ TI-84

Pressing [APPS] and selecting [1:Finance] then [1:TVM Solver...] on the TI-83 Plus/TI-84. You will see a window that looks like the

following

N=Number of compoundings

I%= annual interest rate

PV= present value

PMT= payment

FV= future value

P/Y= payments per year

C/Y= compoundings per year

PMT: END BEGIN

We will always set P/Y and C/Y to the same thing, the number of compounding per year.

Also all payments are made at the end of the compounding period, so END should always be highlighted.

Example 1: If $100 is deposited into an account that earns 5% interest compounded monthly, then how much will be in the account after 3 years?

Solution: Put the following into the calculator. Please note that for the percentage we put in 5 and not .05.

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PV was cash out lay. Cash outlays always go into the calculator as a negative number. As always, make sure that END is highlighted and move the cursor to FV=

(You will not be allowed to leave FV blank until all of the other values are filled in) and press [SOLVE]

([alpha] [ENTER]).

The value of 116.1472231 gets filled in for FV, so the answer is $116.15.

End of section 14.1. Start your online homework on MyMathLab.

14.2 Installment Buying

Borrowing to finance purchases and repaying with periodic payments is called installment buying. In this section we discuss the two general types of consumer installment credit.

1) Closed –End Credit: involves borrowing a set amount up front and paying a series of equal installments (payments) until the loan is paid off. Furniture, appliances, and cars commonly are financed through closed-end credit

2) Open-End Credit: there is no fixed number of installments—the consumer continues paying until no balance is owed. With open-end credit, additional credit often is extended before the initial amount is paid off. Examples of open-end credit include most department store charge accounts and bank charge cards such as MasterCard and VISA.

Closed –End Credit (Add-On loans)

Installment loans set up under closed-end credit often are based on add-on interest.

This means that if an amount P is borrowed, the annual interest rate is to be r, and payments will extend over t years, then the required interest comes from the simple interest formula. You simply “add on” this amount of interest to the principal borrowed

Let’s Do it!

Chris and Heather are a newlywed couple. They buy $3500 worth of furniture and appliances to furnish their first apartment. They pay $700 down and agree to pay the balance at a 7% add-on rate for 2 years. Find

(a) The total amount to be repaid,

(b) The monthly payment,

(c) The total cost of the purchases, including finance charges.

Open-End Credit

With a typical department store account, or bank card, a credit limit is established initially and the consumer can make many purchases during a month (up to the credit limit). The required monthly payment can vary from a set minimum (which may depend on the account balance) up to the full balance.

At the end of each billing period (normally once a month), the customer receives an itemized billing, a statement listing purchases and cash advances, the total balance owed, the minimum payment required, and perhaps other account information.

Finance charges: Any charges beyond cash advanced and cash prices of items purchased. This may include interest, an annual fee, credit insurance coverage, a time payment differential, or carrying charges.

Methods of Computing finance charges:

1) Unpaid balance method: Typically, you would begin with the unpaid account balance at the end of the previous month (or billing period) and apply to it the current monthly interest rate.

Example

Find the finance charge on an open-end account having an unpaid balance of $655.33 with monthly interest rate of 1.21%. Assume interest is calculated on the unpaid balance of the account.

Solution

Finance charge = unpaid balance x interest rate

= $655.33 x .0121

= $7.93

Let’s Do It !

The table below shows Brigit’s Visa account activity for a 4-month period. If the bank charges a monthly interest of 1.1% on the unpaid balance, and there are no other fees or extra charges,

a) Find the missing quantities in the table.

b) Find the total finance charges for the 4 months.

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2) Average daily balance method: It considers balances on all days of the billing period and thus comes closer to charging card holders for the credit they actually utilize.

Example

The activity in Paige Dunbar’s MasterCard account for one billing period is shown below. If the previous balance (on March 3) was $209.46, and the bank charges 1.3% per month on the average daily balance, find

(a) The average daily balance for the next billing (April 3),

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(b) The finance charge to appear on the April 3 billing

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(c) The account balance on April 3.

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Let’s Do It!

For each credit card account, assume one month between billing dates (with the appropriate number of days) and interest of 1.2% per month on the average daily balance. Find

(a) The average daily balance,

(b) The monthly finance charge, and

(c) The account balance for the next billing.

End of section 14.2. Start your online homework on MyMathLab.

14.3 Truth in Lending

The Consumer Credit Protection Act, which was passed in 1968, has commonly been known as the Truth in Lending Act. Truth in Lending standardized the so-called true annual interest rate, or annual percentage rate, commonly denoted APR. All sellers (car dealers, stores, banks, insurance agents, credit card companies, and the like) must disclose the APR when you ask, and the contract must state the APR whether or not you ask. This enables a borrower to more easily compare the true costs of different loans.

How can I tell the true annual interest rate a lender is charging?

For example, how does 1.5% per month at Sears compare to 9% per year add-on interest at a furniture store?

To find APR, use a table provided by the Federal Reserve Bank. It will identify APR values to the nearest half percent from 8.0% to 14.0%, which should do in most cases. (You should be able to obtain a more complete table from your local bank.)

Table shown below is designed to apply to loans requiring monthly payments and extending over the most common lengths for consumer loans from 6 to 60 months. These conditions characterize most closed-end consumer loans.

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Table 6 relates the following three quantities.

APR=true annual interest rate (shown across the top)

n = total number of scheduled monthly payments (shown down the left side)

h = finance charge per $100 of amount financed (shown in the body of the table)

Example

After a down payment on your new car, you still owe $7454. You agree to repay the balance in 48 monthly payments of $185 each. What is the APR on your loan?

Solution

First find the finance charges:

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Now, find the finance charge per $100 financed,

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Now, find the “48 payments” row of Table 6, read across to find the number closest to 19.13, which is 19.45. From there read up to find the APR, which is 9.0%.

Let’s Do It!

Pat bought a horse trailer for $5090. She paid $1240 down and paid the remainder at $152.70 per month for 2.5 years. Find

a) The finance charge.

b) The APR

Let’s Do It!

Mike Karelius still owed $2000 on his new garden tractor after the down payment. He agreed to pay monthly payments for 18 months at 6% add-on interest. Find

a) The finance charge.

b) The APR

End of section 14.3. Start your online homework on MyMathLab.

14.4 The Costs and Advantages of Home Ownership

•Fixed-Rate Mortgages

• Adjustable-Rate Mortgages

Fixed-Rate Mortgages

With a fixed-rate mortgage, the interest rate will remain constant throughout the term, and the initial principal balance, together with interest due on the loan, is repaid to the lender through regular (constant) periodic (we assume monthly) payments.

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Example: Find a Monthly Mortgage Payment

The monthly payment necessary to amortize a $75,000 mortgage at 5.5% annual interest for 15 years:

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Let’s Do It Find the monthly payment needed to amortize principal and interest for each fixed-rate mortgage.

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Preparing an Amortization Schedule

Once the regular monthly payment has been determined, an amortization schedule (or repayment schedule) can be generated. It will show the allotment of payments for interest and principal, and the principal balance, for one or more months during the life of the loan.

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The following steps demonstrate how the computations work.

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This sequence of steps is done for the end of each month. The new balance obtained in Step 3 becomes the Step 1 old balance for the next month.

Example Preparing an Amortization Schedule

The Petersons paid for many years on a $60,000 mortgage with a term of 30 years and a 4.5% interest rate. Prepare an amortization schedule for the first 2 months of their mortgage.

Solution First get the monthly payment. R=$304.01 ( how did we obtain this monthly payment??)

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Starting with an old balance of $59,920.99, repeat the steps for the second month:

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Let’s Do it !Amortization of a Fixed-Rate Mortgage

Complete the first one or two months (as required) of each amortization schedule for a fixed-rate mortgage.

a.

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b.

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Adjustable-Rate Mortgages

An adjustable-rate mortgage (ARM), also known as a variable-rate mortgage (VRM), generally starts out with a lower rate than similar fixed-rate loans, but the rate changes periodically, reflecting changes in prevailing rates.

Your ARM interest rate may change every 1, 3, or 5 years (occasionally more frequently). The frequency of change in rate is called the adjustment period. For example, an ARM with an adjustment period of 1 year is called a 1-year ARM. When the rate changes, your payment normally changes also, these adjustments are caused by fluctuations in an index upon which your rate is based. A variety of indexes are used, including the 1-, 3-, and 5-year U.S. Treasury security rates.To determine your interest rate, the lender will add to the applicable index a few percentage points called the margin. The index and the margin are both important in determining the cost of the loan.

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Example Comparing ARM Payments Before and After a Rate Adjustment

Suppose you pay $20,000 down on a $180,000 house and take out a 1-year ARM for a 30-year term. The lender uses the 1-year Treasury index (presently at 4%) and a 2% margin.

(a) Find your monthly payment for the first year.

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First-year monthly payment = $959.28.

(b) Suppose that after a year the 1-year Treasury index has increased to 5.1%. Find your monthly payment for the second year.

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Let’s Do it! The Effect of Adjustable Rates on the Monthly Payment

The “adjusted balance” is the principal balance at the time of the first rate adjustment.

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(a) Find the initial monthly payment,

(b) Find the monthly payment for the second adjustment period.

(c) Find the change in monthly payment at the first adjustment.

End of section 14.4. Start your online homework on MyMathLab.

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APR= 9.0%

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