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Investing Interests

Investing Interests. Simple and Compound Interests. Simple Interest. Interest is calculated based on the rate and the principle and rewarded annually. I = p*r Ex. $400 invested at 5% rate will give I=400*0.05 = $20 as interest. The money earned as interest is not reinvested.

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Investing Interests

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  1. Investing Interests Simple and Compound Interests

  2. Simple Interest • Interest is calculated based on the rate and the principle and rewarded annually. I = p*r • Ex. $400 invested at 5% rate will give I=400*0.05 = $20 as interest. • The money earned as interest is not reinvested. • If the money is invested for a period of more than one year the amount of money earned as interest will be multiplied by the number of years I=p*r*t • Ex for 5 years the interest earned will be I=400*0.05*5=$100

  3. Total Amount During Simple Interest • Let say we invest p dollars as principles • The interest will be I = p*r*t • The total amount of money in the bank will be A = I + p or A = p + p*r*t or A = p(1 + r*t)

  4. Example • Tony invested $1200 in a simple interest investment with a 10 year term and earns 5% interest annually. How much money will he have at the end of the investment term • Solution A = p(1+rt) A = 1200(1+0.05*10) = 1800 In 10 years Tony will have $1800

  5. Compound Interest • Unlike simple interest, compound interest on an amount accumulates at a faster rate than simple interest. The basic idea is that after the first interest period, the amount of interest is added to the principal amount and then the interest is computed on this higher principal. • The latest computed interest is then added to the increased principal and then interest is calculated again. This process is completed over a certain number of compounding periods. • The result is a much faster growth of money than simple interest would yield.

  6. Calculating compound interest • For simplicity we’ll consider the compounding period as 1 year. • After the first year the amount will be A = p(1+rt) t = 1 so A = p(1+r) • After the second year all the money will be reinvested A = p(1+ rt) but p = p(1+r) so A = p(1+r)(1+r) = p(1+r)2 • After the third year A = p(1+ rt) but p = p(1+r)2 so a = p(1+r)2(1+r) = p(1+r)3 • After n years A = p(1+r)n

  7. Compounding Periods • So far we assumed that the interest calculations happened annually, so compounding period was 1 year. • If the interest are calculated in shorter periods of time, ex semi-annually, quarterly, monthly etc. The formula is the same, but the interest has to reflect the compounding period. • Ex if the interest is 6% annually and the compounding period is quarterly then the interest rate for the compounding period will be 6/4=1.5% and the formula would be A = p(1+r/m)n Where m number of compounding periods per year , n - number of the total compounding periods (not in a year, but for the entire time the money is invested)

  8. Suppose a principal of $1.00 was invested in an account paying 6% annual interest compounded monthly. How much would be in the account after one year? Example

  9. Solution • Solution: Using the simple interest formula A = P (1 + rt) we obtain: • amount after one month • after two months • after three months After 12 months, the amount is (1.005)12= 1.0616778. With simple interest, the amount after one year would be 1.06. The difference becomes more noticeable after several years.

  10. Graphical Illustration ofCompound Interest Growth of 1.00 compounded monthly at 6% annual interest over a 15 year period (Arrow indicates an increase in value of almost 2.5 times the original amount.)

  11. General Formula • In the previous example, the amount to which 1.00 will grow after n months com- pounded monthly at 6% annual interest is • This formula can be generalized to where A is the future amount, P is the principal, r is the interest rate as a decimal, m is the number of compounding periods in one year, and t is the total number of years. To simplify the formula,

  12. Example • Find the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years.

  13. Example • Find the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years. • Solution: Use • Helpful hint: Be sure to do the arithmetic using the rules for order of operations.

  14. Changing the number of compounding periods per year To what amount will $1500 grow if compounded daily at 6.75%interest for 10 years? Solution: = 2945.87 This is about $15.00 more than compounding $1500 quarterly at 6.75% interest. Since there are 365 days in year (leap years excluded), the number of compounding periods is now 365. We divide the annual rate of interest by 365. Notice, too, that the number of compounding periods in 10 years is 10(365)= 3650.

  15. Same Problem Using Simple Interest Using the simple interest formula, the amount to which $1500 will grow at an interest of 6.75% for 10 years is given byA = P (1 + rt) = 1500(1+0.0675(10)) = 2512.50 which is more than $400 less than the amount earned using the compound interest formula.

  16. Changing the number of compounding periods per year To what amount will $1500 grow if compounded daily at 6.75%interest for 10 years? Barnett/Ziegler/Byleen Finite Mathematics 11e

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