 # Compound Interest and Present Value

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Compound Interest and Present Value

Learning unit objectives
LU 19-1: Compound Interest (Future Value) – The Big Picture Compare simple interest with compound interest. Calculate the compound amount and interest manually, using algebraic formulas and with a financial calculator. Explain and compute the effective rate (APY). LU 19-2: Present Value -- The Big Picture Compare present value (PV) with compound interest (FV). Compute present value using algebraic formulas and with a financial calculator. Check the present value answer by compounding. 12-

Compound Interest (Future Value)
Compounding – Involves the calculation of interest periodically over the life of the loan or investment Compound Interest – The interest on the principal plus the interest of prior periods Future Value (compound amount) – The final amount of the loan or investment at the end of the last period Present Value – The value of a loan or investment today 19-3 12-

Compounding Terms Compounding Periods Interest Calculated
Compounding Annually Compounding Semiannually Compounding Quarterly Compounding Monthly Compounding Daily Once a year Every 6 months Every 3 months Every month Every day 19-4 12-

Future Value of \$1 at 8% for Four Periods (Figure 19.1)
Compounding goes from present value to future value Future Value After 4 periods, \$1 is worth \$1.36 After 1 period, \$1 is worth \$1.08 After 2 periods, \$1 is worth \$1.17 After 3 periods, \$1 is worth \$1.26 Present value \$1.00 \$1.08 \$1.1664 \$1.2597 \$1.3605 Number of periods 19-5 12-

Tools for Calculating Compound Interest
Number of periods (N) Number of years multiplied by the number of times the interest is compounded per year Rate for each period (I) Annual interest rate divided by the number of times the interest is compounded per year If you compounded \$1 for 4 years at 8% annually, semiannually, or quarterly, what is N and I? Periods Rate Annually: Semiannually: Quarterly: 4 x 1 = 4 Annually: Semiannually: Quarterly: 8% ÷ 1 = 8% 4 x 2 = 8 8% ÷ 2 = 4% 4 x 4 = 16 8% ÷ 4 = 2% 19-6 12-

Simple Versus Compound Interest
Compounded 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 maturity value? Bill Smith deposited \$80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s interest and compounded amount? I = P x R x T I = \$80 x .08 x 4 I = \$25.60 MV = \$80 + \$25.60 MV = \$105.60 Interest: \$ \$80.00 = \$28.83 19-7 12-

Calculating Compound Amount using formula
Step 1. Find the periods n: Years multiplied by number of times interest is compounded in 1 year. Step 2. Find the rate i: Annual rate divided by number of times interest is compounded in 1 year. Step 3. Plug the PV amount, n, and i into the following formula: FV = PV(1 + i)n Step 4. Solve. This gives the compound amount. 19-8 12-

Calculating Compound Amount using the compound interest formula
Bill wants to know the value of \$80 in 4 years at 8%. He begins by identifying the PV, n, and i: PV = \$80 n = 4 (4 years x 1 compounding period per year) i = 8% (8% divided by 1 compounding period) Calculator keystrokes for this problem are: ( ) yx 4 x 80 = \$108.84 19-9

Calculating compound amount using YOUR TI BA II PLUS calculator
Remember to clear the TVM each time you work with new data: 2ND CLR TVM To solve the future value of \$80 at 8% compounded annually for 4 years, using your calculator, follow these steps: Step 1: Input 4 and then press N. Step 2: Input 8 and then press I/Y. Step 3: Input 80, press +/- and then press PV. Step 4: Input 0, and then press PMT. Step 5: Press CPT FV = 19-10

Compounding (fv) figure 19.2
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CALCULATING THE COMPOUND INTEREST
FV – PV = Compound interest \$ \$80.00 = \$28.84 19-12

Nominal versus Effective Rates annual Percentage yield (APY)
Truth in Savings Law Annual Percentage Yield Nominal Rate (stated rate) – The rate on which the bank calculates interest Effective rate (APY)4 = Interest for 1 year Principal 19-13 12-

Calculating Effective Rate (APY)
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Nominal and Effective Rates (APY) of Interest Compared (Figure 19.3)
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Present Value of \$1 at 8% for Four Periods (Figure 19.4)
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Relationship of compounding (fv) to Present value (pv) – bill smith example
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Calculating Present Value using formula
Step 1. Find the periods n: Years multiplied by number of times interest is compounded in 1 year. Step 2. Find the rate i: Annual rate divided by number of times interest is compounded in 1 year. Step 3. Plug the FV amount, n, and i into the following formula: PV = FV (1 + i)n Step 4. Solve. This gives the present value. 19-18 12-

Calculating Present Value using formula
Since Bill knows the bike will cost \$ in the future, he completes the following calculation: PV = \$108.84/( )4 = \$80.00 Calculator keystrokes for this problem are: ( ) yx 4 = STO ÷ RCL 1 = \$80.00 19-19

Calculating Present Value using A FINANCIAL CALCULATOR
Remember to clear the TVM each time you work with new data: 2ND CLR TVM Since Bill knows the bike will cost \$ in the future, he completes the following calculation: Step 1: Input 4 and then press N. Step 2: Input 8 and then press I/Y. Step 3: Input and then press FV. Step 4: Input 0 and then press PMT. Step 5: Press CPT PV = 19-20

Comparing Compound Interest (FV) with Present Value (PV)
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Comparing compound interest (fv) with Present value (pv)
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Problem 19-11 Lynn Ally, owner of a local Subway shop, loaned \$40,000 to Pete Hall to help him open a Subway franchise. Pete plans to repay Lynn at the end of 8 years with 6% interest compounded semiannually. How much will Lynn receive at the end of 8 years? LU 19-1(2) Solution: 6% 2 = 3% \$40,000 x (1 = .03)16 = \$64,188.26 8 years x 2 = 16 periods Step 1: Input 16 and then press N. Step 2: Input 6/2 = and then press I/Y. Step 3: Input 40,000 +/- and then press PV. Step 4: Input 0 and then press PMT. Step 5: Press CPT FV = 64, 19-23 12-

Problem 19-13 Melvin Indecision has difficulty deciding whether to put his savings in Mystic Bank or Four Rivers Bank. Mystic offers 10% interest compounded semiannually. Four Rivers offers 8% interest compounded quarterly. Melvin has \$10,000 to invest. He expects to withdraw the money at the end of 4 years. Which bank gives Melvin the better deal? Check your answer. LU 19-1(2) Solution: Mystic 4 years x 2 = 8 periods Four Rivers 4 years x 4 = 16 periods 10% 2 = 5% 8% 4 = 2% FV = \$10,000( )8 = \$14,774.55 \$14, \$10,000 = \$4,774.55 FV = \$10,000(1 +.02)16 = \$13,727.86 \$13, \$10,000 = \$ 19-24 12-

Problem 19-14 Lee Holmes deposited \$15,000 in a new savings account at 9% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional \$40,000 at 9% interest compounded semiannually. At the end of 6 years, what is the balance in Lee’s account? LU 19-1(2) Solution: 3 years x 2 = 6 periods 9% 2 = 4.5% \$15,000( )6 = \$19,533.90 + 40,000.00 \$ 59,533.90 \$59,533.90( )6 = \$77,528.62 19-25 12-

Problem 19-23 Paul Havlik promised his grandson Jamie that he would give him \$6,000 8 years from today for graduating from high school. Assume money is worth 6% interest compounded semiannually. What is the present value of this \$6,000? LU 19-2(2) Step 1: Input 16 and then press N. Step 2: Input 6/2 = and then press I/Y. Step 3: Input 6000 and then press FV. Step 4: Input 0 and then press PMT. Step 5: Press CPT PV = Solution: 8 years x 2 = 16 periods 6% 2 = 3% \$6,000( )16 = \$3,739.00 19-26 12-