Compound Interest and Present Value

19-1: Compound Interest (Future Value) – The Big Picture

Simple vs. Compound Interest
- Simple interest is calculated only on the principal amount.
- Compound interest is calculated on the principal and accumulated interest from prior periods.
Calculating Compound Amount and Interest
- Manually.
- Using algebraic formulas.
- With a financial calculator.
Effective Rate (APY)
- Explanation of effective rate.
- Computation of effective rate.

19-2: Present Value - The Big Picture

Present Value (PV) vs. Compound Interest (FV)
- Comparison of present value and compound interest.
Computing Present Value
- Using algebraic formulas.
- With a financial calculator.
Checking Present Value
- Verification of the present value answer by compounding.

19-3 Compound Interest (Future Value)

Compounding: Periodic calculation of interest over the loan or investment's duration.
Compound Interest: Interest on the principal plus accumulated interest from prior periods.
Future Value (Compound Amount): Final value of the loan or investment at the end of the last period.
Present Value: Value of a loan or investment today.

19-4 Compounding Terms

Compounding Periods and Frequency
- Annually: 1 time a year.
- Semiannually: 2 times each year.
- Quarterly: 4 times each year.
- Monthly: 12 times each year.
- Weekly: 52 times each year.
- Daily: 365 times each year.

Future Value Illustration

Future Value of $1 at 8% for Four Periods (Figure 19.1)
Compounding goes from present value to future value
Demonstrates how the value of $1 grows over four periods with compounding.

19-6 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.
Example: Compounding $1 for 4 years at 8%
- Annually: N = 4 × 1 = 4, I = 8% ÷ 1 = 8%
- Semiannually: N = 4 × 2 = 8, I = 8% ÷ 2 = 4%
- Quarterly: N = 4 × 4 = 16, I = 8% ÷ 4 = 2%

19-7 Simple Versus Compound Interest (1 of 2)

Simple Interest Example: Bill Smith deposited $80 for 4 years at 8% annual interest.
- Interest Calculation: I = P × R × T = $80 × .08 × 4 = $25.60
- Maturity Value: MV = $80 + $25.60 = $105.60

19-8 Simple Versus Compound Interest (2 of 2)

Compound Interest Example: Bill Smith deposited $80 for 4 years at 8% annual interest.
Year-by-year breakdown of compound interest:
- Year 1: Beginning balance $80.00, Interest $6.40, End of year $86.40
- Year 2: Beginning balance $86.40, Interest $6.91, End of year $93.31
- Year 3: Beginning balance $93.31, Interest $7.46, End of year $100.77
- Year 4: Beginning balance $100.77, Interest $8.06, End of year $108.83
Total Interest: $108.83 − $80.00 = $28.83

19-9 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 formula:
Step 4: Solve. This gives the compound amount.

19-10 Calculating Compound Amount using the Compound Interest Formula

Bill wants to know the value of $80 in 4 years at 8%.
Identify:
- PV = $80
- $n = 4$ (4 years × 1 compounding period per year)
- $i = 8%$ (8% divided by 1 compounding period)
Calculator keystrokes are provided (but not detailed in this slide).

19-11 Calculating Compound Amount using Your TI BA II PLUS Calculator

Clear the TVM: 2ND CLR TVM
Steps to solve the future value of $80 at 8% compounded annually for 4 years:
- 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 = 108.84

19-12 Compounding (FV)

Figure 19.2: Compounding starts with the present value and looks to the future.

19-13 Calculating Effective Rate (APY) (1 of 2)

Blue, 8% compounded quarterly
- PV = $8,000
- $n = 4$ (1 year × 4 compounding periods per year)
- $i = 2%$ (8% divided by 4 compounding periods)

19-14 Calculating Effective Rate (APY) (2 of 2)

Sun, 8% compounded semiannually
- PV = $8,000
- $n = 2$ (1 year × 2 compounding periods per year)
- $i = 4%$ (8% divided by 2 compounding periods)

19-15 Nominal and Effective Rates (APY) of Interest Compared (Figure 19.3)

Comparison of nominal and effective rates based on compounding period:
- Annual: Nominal rate 6%, End balance $1,060.00, APY 6.00%
- Semiannual: Nominal rate 6%, End balance $1,060.90, APY 6.09%
- Quarterly: Nominal rate 6%, End balance $1,061.36, APY 6.14%
- Daily: Nominal rate 6%, End balance $1,061.83, APY 6.18%

19-16 Present Value Illustration

Present Value of $1 at 8% for Four Periods (Figure 19.4)
Present value goes from the future value to the present value
Demonstrates how the present value of $1 decreases over four periods.

19-17 Relationship of Compounding (FV) to Present Value (PV) – Bill Smith Example

Present value starts with the future and looks to the present

19-18 Calculating Present Value using Formula (1 of 2)

Step 1: Find the periods $(n)$ : Years multiplied by the number of times interest is compounded in 1 year.
Step 2: Find the rate $(i)$ : Annual rate divided by the number of times interest is compounded in 1 year.
Step 3: Plug the FV amount, $(n)$ , and $(i)$ into the following formula:
Step 4: Solve. This gives the present value.

19-19 Calculating Present Value using Formula (2 of 2)

Bill knows the bike will cost $108.84 in the future. Calculations are shown, calculator keystrokes are mentioned.

19-20 Calculating Present Value Using a Financial Calculator

Clear the TVM: 2ND CLR TVM
Bill knows the bike will cost $108.84 in the future, calculate the present value:
- Step 1: Input 4 and then press N.
- Step 2: Input 8 and then press I/Y.
- Step 3: Input 108.84 and then press FV.
- Step 4: Input 0 and then press PMT.
- Step 5: Press CPT PV = −80.00

19-21 Comparing Compound Interest (FV) with Present Value (PV) (1 of 3)

Compound Amount: We know the present dollar amount and find what the dollar amount is worth in the future.

19-22 Comparing Compound Interest (FV) with Present Value (PV) (2 of 3)

Present Value: We know the future dollar amount and find what the dollar amount is worth in the present.

19-23 Comparing Compound Interest (FV) with Present Value (PV) (3 of 3)

The present value is what we need now to have $20,000 in the future

19-24 Problem 19-11

Lynn Ally loaned $40,000 to Pete Hall, to be repaid in 8 years with 6% interest compounded semiannually. How much will Lynn receive at the end of 8 years?
Solution:
- 8 years × 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 = 64,188.26

19-25 Problem 19-13

Melvin Indecision is deciding between Mystic Bank (10% interest compounded semiannually) and Four Rivers Bank (8% interest compounded quarterly). He has $10,000 to invest for 4 years. Which bank is better?

19-26 Problem 19-13 Solution: Mystic

N= tm (4 years × 2) = 8 periods
i= j/m ( 10%/2) = .05

19-27 Problem 19-13 Four Rivers

N= tm (4 years × 4) = 16 periods
i= j/m (8%/2) =.02

19-28 Problem 19-14

Lee Holmes deposited $15,000 at 9% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional $40,000 at 9% interest compounded semiannually. What is the balance at the end of 6 years?
Solution: 3 years × 2 = 6 periods

19-29 Problem 19-23

Nana promised Lesly $6,000 in 8 years for graduating. Money is worth 6% interest compounded semiannually. What is the present value of this $6,000?
Solution:
- 8 years × 2 = 16 periods
- 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 = -3739.00