Business Mathematics Chapter 12

The interest on $6,000 at 6% compounded semiannually for eight years is (use table in the handbook):

$9,628.20 - 6,000.00 = $3,628.20 (Pages 345 - 346 Table 11-1)

Jim Moore opens a new savings account. He deposits $12,000 at 12% compounded semiannually. At the start of the fourth year, Jim deposits an additional $50,000 that is also compounded semiannually at 12%. At the end of six years, the balance in Jim Moore's account is (use the tables in the handbook):

$12,000 × 1.4185 = $17,022 + $50,000 = $67,022 $67,022 × 1.4185 = $95,070.71.

Compounding interest daily is seldom used in comparison to compounding once a year.

False

True or False: The APY (annual percentage yield) is different from the effective rate.

False Banks use the APY and the effective rate interchangeably

Compound value = $ amount divided by table factor.

False Compound value = $ amount multiplied by table factor.

Interest = principal × rate divided by the time.

False I = P x R x T

The nominal rate is really the true rate

False The effective rate is really the true rate.

Future Value

Final amount of the loan or investment at the end of the last period. Also called compounding

When interest is compounded quarterly, interest is calculated how many times per year?

Four times per year

$100,000 for 20 years compounded at 4% annually results in a rate per period of:

4% 4%/1 = 4%.

Using the rule of 72, how many years will it take to double your investment at 12% per year?

72 / 12 = 6 years

Compounding Interest

The interest that is calculated periodically and then added to the principal. The next period (old principal plus interest)

In the table of present value of one dollar, all table factors are less than $1.

True

Compounding looks at what $1 today will be in the ____

future

Compounding

involves the calculation of interest periodically over the life of the loan (or investment). After each calculation, the interest is added to the principal. Future calculations are on the adjusted principal (old principal plus interest)

$1 is compounded semiannually for 5 years at 2% interest. How many periods will this result in?

Semiannual compounding results in twice a year. Therefore, 2 x 5 = 10 periods.

Jose invested $50,000 at 12% for 4 years compounded annually. What is the maturity value at the end of year 3?

$50,000 x (1 + 12)^3 $50,000 x 1.404928 $70,246.40 maturity value at the end of year 3 = $70,246.40

Calculating compound amount and interest manually

1. Calculate the simple interest and add it to the principal. Use this total to figure next year's interest. 2. Repeat for the total number of periods. 3. Compound amount- Principal Compound interest

Calculating compound amount by table lookup

1. Find the periods. Years multiplied by number of times interest is compounded in 1 year. 2. Find the Rate. Annual rate divided by number of times interest is compounded in 1 year 3. Go down the period column of the table to the number of periods desired; look across the row to find the rate. At the intersection of the two columns is the table factor for the compound amount of 1$ 4.Multiply the table factor by the amount of the loan. This gives the compound amount.

$1 is compounded annually for 3 years at 24% interest. What is the interest rate per period?

24% x 1 = 24% Rate x Period

Rate for each period

Annual interest rate divided by the number of times the interest is compounded per year. Compounding changes the interest rate for annual, semiannual, and quarterly periods as follows: Annually: 8% ÷ 1 = 8% Semiannually: 8% ÷ 2 = 4% Quarterly: 8% ÷ 4 = 2%

Consider which option results in a higher effective rate, Bank A offers 4% compounded annually Bank B offers 4% compounded quarterly

Bank B's effective rate is 4.06%. 4% quarterly pays 1% four times per year = 4.06%

Compounding:

Compounding calculates interest periodically.

Earl Ezekiel wants to retire in San Diego when he is 65 years old. Earl is now 50. He believes he will need $300,000 to retire comfortably. To date, Earl has set aside no retirement money. Assume Earl gets 6% interest compounded semiannually. How much must Earl invest today to meet his $300,000 goal?

Future Value: $300,000 Time: 15 years Interest: 6% compounded semiannually 15 years x 2 = 30 periods 6% / 2 = .03 = 3% Table Factor: .4120 $300,000 x .4120 = $123,600

Tony Ring wants to attend Northeast College. He will need $60,000 4 years from today. Assume Tony's bank pays 12% interest compounded semiannually. What must Tony deposit today so he will have $60,000 in 4 years? (Use the Table provided.)

Future Value: $60,000 Time: 4 Years Interest: 12% compounded semiannually 4 years x 2 = 8 periods 12% / 2 = .06 = 6% Table Factor: .6274 $60,000 x .6274 = $37,644 Present Value: $37,644

Jim Ryan, an owner of a Burger King restaurant, assumes that his restaurant will need a new roof in 7 years. He estimates the roof will cost him $9,000 at that time. What amount should Jim invest today at 6% compounded quarterly to be able to pay for the roof? (Use the Table provided.)

Future Value: $9,000 Time: 7 Years Interest: 6% compounded Quarterly 7 years x 4 = 28 6% / 4 = 0.015 = 1.5% Table Factor: .6591 $9,000 x .6591 = $5,931.90 Present Value: $5,931.90

The International Monetary Fund is trying to raise $500 billion in 5 years for new funds to lend to developing countries. At 6% interest compounded quarterly, how much must it invest today to reach $500 billion in 5 years? (Use the Table provided.)

Future Value: 500 Billion Time: 5 Years Interest: 6% compounded quarterly 5 years x 4 = 20 Periods 6% / 4 = 0.015 = 1.5% quarterly Table Factor: .7425 $500 billion x .7425 = $371.25 billion Present Value: $371.25 Billion

Given the equation, A=P(1 + I)^m match the abbreviation to the respective compounding term.

I = Interest rate per period A = Compound Amount N = Number of periods P = Principal

Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill's simple interest?

I = P x R x T I = $80 x .08 x 48/12 I = 25.60 $80 + 25.60 = $105.60 Bill's Simple Interest: $105.60

Feliz borrowed $2,000 for 5 years at 6%. Using the simple interest formula, how much will he need to pay at the end of the loan.

Maturity Value = Interest / (P x R x T) 2,000 -------------------------- = $600 $2,000 x .06 x 5 years $2,000 + 600 = 2,600

Wright invested $500 at 7% compounded daily for 6 years. What is the future value of his investment.

Number of periods: 6 Interest: 7% Principal: $500 Table Interest on 1$ deposit compounded 6 years at 7% = 1.5219 $500 x 1.5219 = $760.95 Future Value: $760.95

Number of periods

Number of years multiplied by the number of times the interest is compounded per year. For example, if you compound $1 for 4 years at 8% annually, semiannually, or quarterly, the following periods will result: Annually: 4 years × 1 = 4 periods Semiannually: 4 years × 2 = 8 periods Quarterly: 4 years × 4 = 16 periods

Gabrielle deposited $1,000 in a savings account at 8% compounded quarterly for 1 year. What is the future value of his savings account?

Number of years x compounding per year 1 x 4 = 4 I = Annual Interest / Compounding I = 8% / 4 = 2 $1,000 x 1.0824 table factor = $1,082.40

Present value does not:

Present value does not know the present dollar amount.

Rule of 72

The number of years it takes for a certain amount to double in value is equal to 72 divided by its annual rate interest.

Compound interest results in ____ interest over time than simple interest

True

Bill Smith deposited $80 in a savings account for 4 years at an annual compounded rate of 8%. What are Bill's compound amount and interest?

Year 1: Year 2: Year 3: Year 4 80.00 86.40 93.31 100.77 x .08 .08 x .08 .08 ---------------------------------------------- $6.40 $6.91 $7.46 8.06 + $80.00 +$86.40 + $93.31 100.77 ---------------------------------------------- $86.40 $93.31 $100.77 108.83 Calculate Compound Interest: $100.00 - 80.00 = $28.83 or: 6.40 + 6.91 + 7.46 + 8.06 = $28.83

Continuous compounding results in _______ interest than daily compounding.

more continuous results in a fraction of a percentage more than daily

Compounding is when interest is earned on

the principal and prior periods' interest