Chapter 3 Mathematics of Finance

Chapter 3 Mathematics of Finance Section 2 Compound and Continuous Interest

Learning Objectives for Section 3.2 Compound and Continuous Compound Interest The student will be able to compute compound and continuous compound interest. The student will be able to compute the growth rate of a compound interest investment. The student will be able to compute the annual percentage yield of a compound interest investment. 2

Table of Content Compound Interest Continuous Compound Interest Growth and Time Annual Percentage Yield 3

Terms compound interest rate per compounding period continuous compound interest APY (Annual Percentages Yield) 4

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. 5

Example Suppose a principal of $1,000 was invested in an account paying 8% annual interest compounded quarterly. How much would be in the account after one year? Compounding quarterly means that the interest is paid at the end of each 3-month period and the interest as well as the principal earn interest for the next quarter. 6

Solution Solution: Using the simple interest formula A = P (1 + rt) we obtain: At the end of the first quarter will have: A = P (1 + rt) A = 1000[1 + 0.08(1/4)] A = 1000(1.02) = $1.020 Now the principal is $1,020. At the end of the second quarter will have : A = 1020[1 + 0.08(1/4)] A = 1020(1.02) = $1.040.40 7

Solution (cont.) At the end of the third quarter will have : A = 1.040.40[1 + 0.08(1/4)] A = 1.040.40(1.02) = $1.061.21 At the end of the fourth quarter will have : A = 1.061.21[1 + 0.08(1/4)] A = 1.061.21(1.02) = $1.082.43 Compare with simple interest: A = P(1 + rt) = $1,000[1 + 0.08(1)] A = $1,000[10.08] = $1,080 $2,43 more 8

Solution (cont.) Looking over the calculations A = 1.000(1.02) A = [1.000(1.02)](1.02) = 1,000(1.02) 2 A = [1.000(1.02) 2 ](1.02) = 1,000(1.02) 3 A = [1.000(1.02) 3 ](1.02) = 1,000(1.02) 4 At the end of n quarters: A = 1.000(1.02) n or A = 1.000[1 + 0.08(1/4)] n A = 1.000[1 + 0.08/4] n End 1 st quarter End 2 nd quarter End 3 rd quarter End 4 th quarter End n th quarter Rate per compounding period = annual nominal rate / number of compounding periods per year 9

Solution (cont.) In general A = P(1 + i) A = [P(1 + i)](1 + i) = P(1 + i) 2 A = [P(1 + i) 2 ](1 + i) = P(1 + i) 3. A = [P(1 + i) (n-1) ](1 + i) = P(1 + i) n End 1 st period End 2 nd period End 3 rd period End n th period Then it can be summarized 10

General Formula This formula can be generalized to A P 1 i n r i m n mt P 1 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, A r m mt 11

Compound Interest General Formula A P 1 i where i = r/m A = amount (future amount) at the end of n periods P = principal (present value) r = annual nominal rate m = number of compounding periods per year i = rate per compounding period t = time (years) n = total number of compounding periods = mt n 12

Graphical Illustration of Compound 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.) 2.5 3 Growth of 1.00 compounded monthly at 6% annual interest over a 15 year period 2 1.5 1 0.5 0 13

Example Find the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years. 14

Example Find the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years. Solution: Use A P 1 i Helpful hint: Be sure to do the arithmetic using the rules for order of operations. A 1500 1 A 2929.50 n 0.0675 4 10(4) 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 by A = 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? 17

Changing the number of compounding periods per year To what amount will $1500 grow if compounded daily at 6.75% interest for 10 years? 0.0675 A 1500 1 365 Solution: = 2945.87 10(365) 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. 18

Changing the number of compounding periods per year Example 1 If $1,000 is invested at 8% compound: A. Annually B. Semiannual C. Quarterly D. Monthly What is the amount after 5 years? 19

Changing the number of compounding periods per year Example 1 - Solution If $1,000 is invested at 8% compound, what is the amount after 5 years? A. Annually n = 5 ; I = r = 0.08 A = P(1 + i) n = 1,000(1 + 0.08) 5 = 1,000(1.469328) = $1,469.33 B. Semiannual n = (2)5 = 10 I = r/n = 0.08/2 = 0.04 A = P(1 + i) n = 1,000(1 + 0.04) 10 = 1,000(1.480244) = $1,480.24 20

Changing the number of compounding periods per year Example 1 - Solution If $1,000 is invested at 8% compound, what is the amount after 5 years? C. Quarterly D. Monthly n = (4)5 = 20 I = r/n = 0.08/4 = 0.02 A = P(1 + i) n = 1,000(1 + 0.02) 20 = 1,000(1.485947) = $1,485.95 n = (12)5 = 60 I = r/n = 0.08/12 = 0.006666 A = P(1 + i) n = 1,000(1 + 0.00666) 60 = 1,000(1.489846) = $1,489.85 21

Effect of Increasing the Number of Compounding Periods If the number of compounding periods per year is increased while the principal, annual rate of interest and total number of years remain the same, the future amount of money will increase slightly. 22

Continuous Compound Interest Previously we indicated that increasing the number of compounding periods while keeping the interest rate, principal, and time constant resulted in a somewhat higher compounded amount. What would happen to the amount if interest were compounded daily, or every minute, or every second? 23

Answer to Continuous Compound Interest Question As the number m of compounding periods per year increases without bound, the compounded amount approaches a limiting value. This value is given by the following formula: lim m P(1 r m )mt Pe rt A Pe rt Here A is the compounded amount. 24

Example of Continuously Compounded Interest What amount will an account have after 10 years if $1,500 is invested at an annual rate of 6.75% compounded continuously? 25

Example of Continuously Compounded Interest What amount will an account have after 10 years if $1,500 is invested at an annual rate of 6.75% compounded continuously? Solution: Use the continuous compound interest formula A = Pe rt with P = 1500, r = 0.0675, and t = 10: A = 1500e (0.0675)(10) = $2,946.05 That is only 18 cents more than the amount you receive by daily compounding. 26

Changing the number of compounding periods per year To what amount will $1500 grow if compounded daily at 6.75% interest for 10 years? 0.0675 A 1500 1 365 Solution: = 2945.87 10(365) That is only 18 cents more than the amount you receive by daily compounding. 27

Continuously Compounded Interest Example 2 What amount will an account have after 2 years if $5,000 is invested at an annual rate of 8% compoundly A. daily? A = P(1 + r/m) mt P = 5,000; r = 0.08; m = 365 t = 2 A = 5,000(1+ 0.08/365) (365)(2) = $5,867.55 A. continuously? A = Pe rt P = 5,000; r = 0.08; t = 2 A = 5,000e (0.08)(2) = $5,867.55 28

Growth & Time How much at a future date? What annual rate has been earned? How long take to double the investment? The formulas for the compound interest and the continuous interest can be used to answer such questions. 29

Growth & Time How long will it take for $5,000 to grow to $15,000 if the money is invested at 8.5% compounded quarterly? 30

Growth & Time Solution How long will it take for $5,000 to grow to $15,000 if the money is invested at 8.5% compounded quarterly? 1. Substitute values in the compound interest formula. 2. divide both sides by 5,000 3. Take the natural logarithm of both sides. 4. Use the exponent property of logarithms 5. Solve for t. Solution: 0.085 15000 5000(1 ) 4 3 = 1.02125 4t 4t ln(3) = ln1.02125 4t ln(3) = 4t*ln1.02125 ln(3) t 13.0617 4ln(1.02125) 31

Finding Present Value Growth & Time Example 3 How much will you invest now at 10% to have $8,000 toward the purchase of a car in 5 years if interest is A. compounded quarterly? P =? A = 8,000 i = 0.10/4 n = 4(5) = 20 A = P(1 + i) n 8,000 = P(1 + 0.025) 20 P = 8,000 / (1 + 0.025) 20 P = 8,000 / (1.638616) = $4,882.17 B. compounded continuously? P =? A = 8,000 r = 0.10 t = 5 A = Pe rt 8,000 = Pe (0.10)(5) P = 8,000 /e (0.10)(5) P = $4,852.25 32

Computing Growth Rate Growth & Time Example 4 An investment of $10,000 in a growth-oriented fund over a 10- years period would have grown in $126,000. What annual nominal rate would produce the same growth if interest was: C. compounded quarterly? A = P(1 + i) n 126,000 = 10,000 (1+ r) 10 12.6 = (1+ r) 10 Raiz10(12.6) = 1+ r r = Raiz10(12.6) 1 r = 0.28836 ó 28.836% D. compounded continuously? A = Pe rt 126,000 = 10,000e 10r 12.6 = e 10r ln 12.6 = 10r r = ln 12.6/10 r = 0.25337 ó 25.337% 33

Growth & Time Example 5 Computing Growth Time How long will it take $10,000 to grow to $12,000 if it is invested at 9% compounded monthly? Using logarithms & calculator: A = P(1 + i) n 12,000 = 10,000 (1+ 0.09/12) n 1.2 = 1.0075 n Solve for n by taking logarithms of both sides: ln 1.2 = ln 1.0075 n ln 1.2 = nln 1.0075 n = ln 1.2 / ln 1.0075 n = 24.40 ó 25 months or 2yr 1mth 34

Annual Percentage Yield The simple interest rate that will produce the same amount as a given compound interest rate in 1 year is called the annual percentage yield (APY). To find the APY, proceed as follows: Amount at simple interest APY after one year = Amount at compound interest after one year m r P(1 APY ) P 1 m r 1 APY 1 m APY m m r 1 1 m This is also called the effective rate, as well as APR. 35

Annual Percentage Yield Example What is the annual percentage yield for money that is invested at 6% compounded monthly? 36

Annual Percentage Yield Example What is the annual percentage yield for money that is invested at 6% compounded monthly? General formula: Substitute values: Effective rate is 0.06168 = 6.168 APY m r 1 1 m APY 0.06 1 1 0.06168 12 12 37

Computing the Annual Nominal Rate Given the APY What is the annual nominal rate compounded monthly for a CD that has an annual percentage yield of 5.9%? 38

Computing the Annual Nominal Rate Given the APY What is the annual nominal rate compounded monthly for a CD that has an annual percentage yield of 5.9%? 1. Use the general formula for APY. 2. Substitute value of APY and 12 for m (number of compounding periods per year). 3. Add one to both sides 4. Take the twelfth root of both sides of equation. 5. Isolate r (subtract 1 and then multiply both sides of equation by 12. m r APY 1 1 m 12 r 0.059 1 1 12 r 1.059 1 12 12 12 12 12 1.059 1 12 r 1.059 1 12 r 1.059 1 12 0.057 r r 39

Annual Percentage Yield EXAMPLE 6 - Using APY to Compare Investments. Find the APY for each of the bank in the table and compare these CD s. Certificates of Deposits (CDs) Bank Rate Compounded Advanta 4.93% monthly DeepGreen 4.95% daily Charter One 4.97% quarterly Liberty 4.94% continuously 40

Annual Percentage Yield EXAMPLE 6 - Using APY to Compare Investments (contd.) Find the APY for each of the bank in the table and compare these CD s. Bank Calculations [APY = (1 + r/m) m 1] APY Advanta = (1 + 0.0493/12)^12 1 = 0.05043 5.043% DeepGreen = (1 + 0.0495/365)^365 1 = 0.05074 5.074% Charter One = (1 + 0.0497/4)^4 1 = 0.05063 5.063% Liberty = e^(0.0494) 1 = 0.05064 5.064% 41

Example of Annual Percentage Yield EXAMPLE 7 - Computing the Annual Nominal Rate given the APY A savings & loans wants to offer a CD with a monthly compounding rate that has an APY of 7.5%. What annual nominal rate compounded monthly should they use?. 42

Example of Annual Percentage Yield EXAMPLE 7 - Computing the Annual Nominal Rate given the APY (continued) APY = (1 + r/m) m - 1 0.075 = (1+ r/12) 12-1 1.075 = (1+ r/12) 12 Raiz12(1.075) = 1+ r/12 Raiz12(1.075) 1 = r/12 r = 12(Raiz12(1.075) 1) Using the calculator: r = 0.072539 r = 7.254% 43

Chapter 3 Mathematics of Finance Section 2 Compound and Continuous Interest END Last Update: Marzo 19/2013