Compound Interest and Present Value

nominal, yield, rates, reveal, 100, 365,

Banks often advertise their annual (____) interest rates and not their true or effective rate (annual percentage ____, or APY). --> This has made it difficult for investors and depositors to determine the actual (A)____ of interest they were receiving. --> The Truth in Savings law forced savings institutions to ____ their actual (A)____ of interest. --> The APY is defined in the Truth in Savings law as the percentage rate expressing the total amount of interest that would be received on a $____ deposit based on the annual rate and frequency of compounding for a ____-day period.

P = $20,000, R = 12%, T = 24(4 x 6yrs) 12/4 = 3% 20,000 x 0.4919 = $9,838

Bill Blum needs $20,000 6 years from today to attend V.P.R. Tech. How much must Bill put in the bank today (12% quarterly) to reach his goal?

P = $80, R = 8%, T = 4yrs I = 80 x 1.08 = 86.4 = 1yr I = 86.4 x 1.08 = 93.31 = 2yr I = 93.31 x 1.08 = 100.77 = 3yr I = 100.77 x 1.08 = 108.83 = 4yr

Bill Smith deposited $80 in a savings account for 4 years at an annual compounded rate of 8%. Calculate Bill's compound amount and interest in the simplified process for each year.

B = 80 x 1.3605 = 108.84

Bill Smith deposited $80 in a savings account for 4 years at an annual compounded rate of 8%. Use the table to calculate Bill's compound amount in 4 yrs.

P = $80, R = 8%, T = 4yrs NB = (80 x 0.08) + 80 = 86.4 = 1yr NB = (86.4 x 0.08) + 86.4 = 93.31 = 2yr NB = (93.31 x 0.08) + 93.31 = 100.77 = 3yr NB = (100.77 x 0.08) + 100.77 = 108.83 = 4yr

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 for each year?

P = $80, R = 8%, T = 4yrs I = P x R x T I = 80 x 0.08 x 4 = 25.6 80.00 + 25.60 = $105.60

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

P = $108.84, R = 8%, T = 4(1 x 4yrs) 4 periods, 8% = 0.735 108.84 x 0.735 = $80.00

Bill Smith knew that in 4 years he wanted to buy a bike that cost $108.84 (future). Bill's bank pays 8% interest compounded annually. How much money must Bill put in the bank today (present) to have $108.84 in 4 years (use the table)?

P = $24,000, R = 8%, T = 16(4 x 4yrs) 8/4 = 2% 0.7284 x 24,000 = $17,481.60

Bob Fry wants to buy his grandson a Ford Taurus in 4 years. The cost of a car will be $24,000. Assuming a bank rate of 8% compounded quarterly, how much must Bob put in the bank today?

9000 x 4.4811 = $4,032.99

Calculate interest compounded daily on $900 at 6% per year for 25 years.

P = $1,500, R = 7%, T = 5yrs 1.4190 x 1,500 = $2,128.5

Calculate using table: What is $1,500 compounded daily for 5 years will grow to at 7%?

P = $15,000, R = 10%, T = 20(1 x 20yrs) A. 20 B. 10/1 = 10% C. 0.1486 D. 15,000 x 0.1486 = 2,229

Future Amount Desired $15,000 T = 20yrs R = 10% Compounded: Annually A. Period(s)? B. Rate used? C. PV factor D. PV Amount

P = $7,000, R = 6%, T = 12(2 x 6yrs) 6/2 = 3% A. 12 B. 6/2 = 3% C. 0.7014 D. 0.7014 x 7,000 = 4,909.80

Future Amount Desired $7,000 T = 6yrs R = 6% Compounded: Semiannually A. Period(s)? B. Rate used? C. PV factor D. PV Amount

daily, continuous

Although many banks add interest to each account quarterly, some banks pay interest that is compounded D____, and other banks use C____ compounding.

R = 12% T = 4(4 x 1yr) 12/4 = 3% 4 x 1 = 4 Periods 1.1255 - 1 = 12.55%

Find the effective rate (APY) for the year: principal, $7,000; interest rate, 12%; and compounded quarterly.

year, present, 1.08, 1.17, 1.36 (Round the final answer) 1 x .08 = 0.08 + 1 = 1.08 = 1 yr 1.08 x 0.08 = 0.0864 + 1.08 = 1.17 = 2 yr 1.17 x 0.08 = 0.0864 + 1.17 = 1.26 = 3 yr 1.26 x 0.08 = 0.0864 + 1.26 = 1.36 = 4 yr

$1 will grow if it is calculated for 4 years at 8% annually. This means that the interest is calculated on the balance once a ____. --> We start with $1, which is the (A)____ value (PV). --> After year 1, the dollar with interest is worth $_.___. --> At the end of year 2, the dollar is worth $_.___. By the end of year 4, the dollar is worth $_.___. --> Note how we start with the (A)____ and look to see what the dollar will be worth in the future. --> Compounding goes from (A)____ value to future value.

Present

(A)____ value (PV) starts with the future and tries to calculate its worth in the (A)____ ($80). For example, we assume Bill Smith knew that in 4 years he wanted to buy a bike that cost $108.84 (future)

once, 6, 3, month, day, multiplied, 4, 8, 16, divided, 8%, 4%, 2%

Compounded annually: Interest calculated on the balance ____ a year. Compounded semiannually: Interest calculated on the balance every __ months or every 1/2 (or 0.5) years. Compounded quarterly: Interest calculated on the balance every __ months or every 1/4 (or 0.25) years. Compounded monthly: Interest calculated on the balance each ____. Compounded daily: Interest calculated on the balance each ____. Number of periods: Number of years ____ 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 = __ periods Semiannually: 4 years × 2 = __ periods Quarterly: 4 years × 4 = __ periods Rate for each period: Annual interest rate ____ 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 = __% Semiannually: 8% ÷ 2 = __% Quarterly: 8% ÷ 4 = __%

life, added, plus, Future, final, last,

Compounding involves the calculation of interest periodically over the ____ of the loan (or investment). --> After each calculation, the interest is ____ to the principal. --> Future calculations are on the adjusted principal (old principal (A)____ interest). --> Compound interest, then, is the interest on the principal (A)____ the interest of prior periods. --> ____ value (FV), or the compound amount, is the ____ amount of the loan or investment at the end of the ____ period.

nominal, P=8000, R=8%, T= 4(4 x 1yr) 1 - Blue Bank APY = 8/4 = 2%, 4 Periods, 2% (Table) = (1.0824) - 1 = (0.0824) = 8.24% 2 - Sun Bank P=8000, R=8%, T= 2(2 x 1yr) 8/2 = 4%, 2 Periods, 4% = (1.0816) - 1 = (0.0816) = 8.16% The long way to calculate: 1 - Blue Bank P=8000, R=8%, T= 4(4 x 1yr) R = 8/4 = 2% 4 Periods, 2% (Table) = 1.0824 I = 8000(1.0824 - 1) = 695.20 APY = 695.20/8000 = 8.24% 2 - Sun Bank P=8000, R=8%, T= 2(2 x 1yr) R = 8/4 = 2% 2%, 4 Periods, 2% (Table) = 1.0824 I = 8000(1.0816 - 1) = 652.80 APY = 652.80/8000 = 8.16%

Let's study the rates of two banks to see which bank has the better return for the investor. 1 - Blue Bank pays 8% interest compounded quarterly on $8,000. 2 - Sun Bank offers 8% interest compounded semiannually on $8,000. --> The 8% rate is the ____ rate, or stated rate, on which the bank calculates the interest. Calculate the effective rate (annual percentage yield, or APY)

P = 6000, R = 3%, T = 16(2 x 8yr) 3/2 = 1(1/2)% 2 x 8 = 16 Periods 1.2690 x 6000 = $7,614

Lionel Rodgers deposits $6,000 in Victory Bank, which pays 3% interest compounded semiannually. How much will Lionel have in his account at the end of 8 years?

present

Note that the table factor for compounding is over 1 (1.3605) and the table factor for (A)____ value is less than 1 (.7350). --> The compound value table starts with the (A)____ and goes to the future. --> The (A)____ value table starts with the future and goes to the (A)____.

A. 4 periods 8/4 = 2% 1yr = 200 × 1.02 = 204 2 yr = 204 × 1.02 = 208.08 3 yr = 208.08 × 1.02 = 212.24 4 yr = 212.24 × 1.02 = 216.48 B. B = 200 x 1.02 = 216.48 C. I = 216.48 - 200 = $16.48

P = $200 T = 1 yr R = 8% Compounded: Quarterly A. Number of periods? B. Total amount? C. Total interest?

1.0824 x 200 = 216.48

P = $200 T = 1 yr R = 8% Compounded: Quarterly Use the table to find the FV

P = $8,000 R = 6% T = 20(4 x 5yrs) T = 4 x 5 = 20 = Periods R = 6%/4 = 1.5% = 1(1/2) = R B = 8,000 x 1.3469 = $10775.20 = 5yr

Pam Donahue deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will be the balance of her account at the end of 5 years?

year, 360,

Remember that continuous compounding sounds great, but in fact, it yields only a fraction of a percent more interest over a ____ than daily compounding. --> Today, computers perform these calculations. Table 12.2 is a partial table showing what $1 will grow in the future by daily compounded interest, ____-day basis.

P = $20,000, R = 8%, T = 16(4 x 4yrs) 8/4 = 2% 16, 2% = 0.7284 20,000 x 0.7284 = 14,568 = PV

Rene Weaver needs $20,000 for college in 4 years. She can earn 8% compounded quarterly at her bank. How much must Rene deposit at the beginning of the year to have $20,000 in 4 years?

P = $6,000 R = 10% T = 5yrs Periods = 2 x 5 = 10 R = 10/2 = 5% 10, 5% = 1.6289 using the table I = 6,000 x (1.6289 - 1) = ? 6,000 x 0.6289 = ($3,773.40) = I

Using the table find the interest on $6,000 at 10% compounded semiannually for 5 years.