Math 147 Section 6.2. Application Example

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1 Math 147 Section 6.2 Annual Percentage Yield Doubling Time Geometric Sequences 1 Application Example Mary Stahley invested $2500 in a 36-month certificate of deposit (CD) that earned 9.5% annual simple interest. When the CD matured, she invested the full amount in a mutual fund that had an annual growth equivalent to 20% compounded annually. After 7 years in the mutual fund, how much was her investment worth? 2 1

2 In interest, the interest for each period is added to the principal before interest is calculated for the next period. 3 10,000 invested at 10% per year compounded annually: Year Beginning Principal Interest Ending Principal Note that the amount of interest earned each year is greater than the year before. Notice that the principal at the end of each year is 1.10 times the principal at the beginning of the year. 4 2

3 10,000 invested at 10% per year compounded annually: Year Beginning Principal Interest Ending Principal , Notice that the principal at the end of each year is 1.10 times the principal at the beginning of the year. 10,000 * 1.10 = 11,000 11,000 * 1.10 = 12,100 12,100 * 1.10 = 13, ,000 * 1.10 = 11,000 = 10,000 * 1.10 = 10,000 * ,000 * 1.10 = 12,100 = 10,000 * 1.10 * 1.10 = 10,000 * ,100 * 1.10 = 13,310 = 10,000 * 1.10 * 1.10 * 1.10 = 10,000 * In general if P is invested at an annual rate r compounded annually, the future value, S, at the end of the n th year is 6 3

4 Find the future value of $5000 invested for 10 years at 5% per year compounded annually: 7 Interest compounded annually grows more than simple interest that is paid once at the end of the term. Frequent compounding is good for the investor. The the compounding, the grows. 8 4

5 In general interest rates, r, are stated as annual rates, but interest is often compounded quarterly, monthly, or even daily. The annual rate, r, is the. The, is the nominal rate divided by the number of compounding periods per year. The number of, also called conversion periods, is denoted by n. 9 If P is invested for t years at a nominal interest rate r, compounded m times a year: The total number of compounding periods is The interest rate per compounding period is (expressed as decimal) And future value is 10 5

6 Find the future value of $5000 invested for 10 years at 5% per year compounded annually: Find the future value of $5000 invested for 10 years at 5% per year compounded quarterly: 11 Continuous Compounding The more often we compound, the more we earn; it would seem that if we compounded continuously, we would break the bank. If we invest $1 at 100% annual rate for one year, and the interest is compounded m times a year the future value is: Compounded Annually Monthly Daily Hourly Each Minute Number of periods Future value 12 6

7 Continuous Compounding If we invest $1 at 100% annual rate for one year, and the interest is compounded m times a year the future value is: Compounded Number Future value of periods Annually 1 (1+1/1) 1 = 2 Monthly 12 (1+1/12) 12 = Daily 360 (1+1/360) 360 = Hourly 8640 (1+1/8640) 8640 = Each Minute 518,400 (1+1/518400) = As we continue this process the value increases, but not very fast. In fact the value gets very close to but doesn t quite reach This is one place that the number e occurs naturally. 13 Continuous Compounding In general if $P is invested for t years at nominal rate r compounded continuously the future value is: 14 7

8 Continuous Compounding Find the future value of $5000 invested for 10 years at 5% per year compounded annually: Find the future value of $5000 invested for 10 years at 5% per year compounded quarterly: Find the future value of $5000 invested for 10 years at 5% per year compounded continuously: 15 Annual Percentage Yield We earn more than the nominal annual rate when interest is compounded more frequently that once a year. If we invest $1 at 5% compounded quarterly, in one year our value is: So in one year our $1 earned $ or 5.1%. This is our. For compounding periods of 1 year, the nominal rate is the APY. 16 8

9 Annual Percentage Yield For periodic compounding APY is calculated: For continuous compounding APY is calculated: We can t directly compare two nominal rates with different compounding periods, but we can compare their APYs. 17 APY Suppose a young couple found three different investment companies that offered college savings plans: (a) one at 10% compounded annually (b) another at 9.8% compounded quarterly (c) a third at 9.65% compounded continuously. Find the annual percentage yield (APY) for each of these three plans to discover which plan is best. 18 9

10 APY Suppose a young couple found three different investment companies that offered college savings plans: (a) one at 10% compounded annually 19 APY Suppose a young couple found three different investment companies that offered college savings plans: (b) another at 9.8% compounded quarterly 20 10

11 APY Suppose a young couple found three different investment companies that offered college savings plans: (c) a third at 9.65% compounded continuously. Find the annual percentage yield (APY) for each of these three plans to discover which plan is best. 21 Doubling Time How long does it take an investment of $5000 at 6% compounded quarterly to double? 22 11

12 Geometric Sequences If we invest P at a rate of i per period, compounded at the end of each period, the future value at the end of each succeeding period is: P(1+i), P(1+i) 2, P(1+i) 3,, P(1+i) n, The future value for each period forms a sequence in which each term, after the first, is found by the previous term by the same number. This type of sequence is a. 23 Geometric Sequences A sequence is a geometric sequence, or, if there is a number r, called a, such that: for n >

13 Geometric Sequences What are the next three terms of 5, 25, Geometric Sequences The n th term of a geometric sequence is where a 1 is the first term of the sequence. The sum of the first n terms of a geometric sequence with the first term a 1 and common ratio r is: 26 13

14 Geometric Sequences Find the sum of the first 6 terms of the geometric sequence with first term 1 and common ratio of Application Example Mary Stahley invested $2500 in a 36-month certificate of deposit (CD) that earned 9.5% annual simple interest. When the CD matured, she invested the full amount in a mutual fund that had an annual growth equivalent to 20% compounded annually. After 7 years in the mutual fund, how much was her investment worth? 28 14

15 Application Example Mary Stahley invested $2500 in a 36-month certificate of deposit (CD) that earned 9.5% annual simple interest. When the CD matured, she invested the full amount in a mutual fund that had an annual growth equivalent to 20% compounded annually. After 7 years in the mutual fund, how much was her investment worth? 29 15

Chapter 21: Savings Models

Chapter 21: Savings Models October 14, 2013 This time Arithmetic Growth Simple Interest Geometric Growth Compound Interest A limit to Compounding Simple Interest Simple Interest Simple Interest is interest that is paid on the original