Study Notes on Compound Interest and Geometric Sequences

Objectives

• Understanding Compound Interest
    - Find the future value and interest earned with compound interest.
    - Calculate the annual percentage yield (APY) for compound interest ventures.
    - Determine the duration required for investments to reach desired amounts.
    - Calculate specific sums and terms related to geometric sequences.

Section 6.2: Compound Interest

• Definition: Compound interest is an investment method where the interest accrued in each period is added to the principal amount, enabling the calculation of interest on the new total in subsequent periods.

• Illustration:
    - Compound interest grows exponentially, likened to a snowball effect.
    - Commonly used in investments, savings accounts, and certain U.S. government bonds.

Compound Interest Example Calculation

• Example Showing Compound Growth Over 5 Years:
    -
    | Year | Beginning Principal | Interest () | Ending Principal |
    |------|--------------------|------------------|------------------|
    | 1 | $20,000 | $2,000 | $22,000 |
    | 2 | $22,000 | $2,200 | $24,200 |
    | 3 | $24,200 | $2,420 | $26,620 |
    | 4 | $26,620 | $2,662 | $29,282 |
    | 5 | $29,282 | $2,928.20 | $32,210.20 |

• Interest Calculation:
     - For Year 1:
     $I = P imes r ext{ where } P=20000, r=0.1$
     $I = 20000 imes 0.1 = 2000$

• Total after Year 5:
  $S = P(1+r)^n$
  Where:

• Future Value Formula:
  $S = P(1+r)^n$

Future Value Calculation Example

• Let’s invest $3,000 for 4 years at an annual interest of 9% compounded annually:
    -
     - Future Value Calculation:
     $S = 3000(1 + 0.09)^4$
     - Calculation Steps:
     $S = 3000(1.36049) = 4081.47$
     - Interest Earned:
     - Total Interest = Future Value - Principal
     $4081.47 - 3000 = 1081.47$

Nominal Interest Rate

• Nominal Annual Rate Definition:
    - The stated interest rate divided by the number of compounding periods in a year denoted as 'm'.
    - Total Compounding Periods = $n = mt$, where
    $i = rac{r}{m}$.
  

Example Interest Rate Calculation

• Example 1:
    - Find interest rate per period and the number of compounding periods for:
    - 12% compounded monthly for 7 years:
      - Let r=0.12, m=12, t=7
      - Calculate:
      $i = rac{0.12}{12} = 0.01$
      - Find $n = 12 imes 7 = 84$ compounding periods.

• Example 2:
    - 7.2% compounded quarterly for 11 quarters:
    - Given: r=0.072
    - Number of periods:
    $n = m = 4, t = 11/4$

Continuous Compounding

• Continuous Compounding allows interest to be calculated instantaneously.

• Formula:
  $S = e^{rt}$

• Example:
    - How much more will you earn if you invest $10,000 for 15 years at 8% compounded continuously compared to quarterly?
    - Calculation steps are done respectively for both methods.
    - Future Value with continuous:
    $S = 10000 imes e^{(0.08 imes 15)}$
    - Different scenarios lead to various APY outputs.

Annual Percentage Yield (APY)

• Definition of APY:
    - It represents the effective yearly rate of return taking into account the effect of compounding interest.

• APY Formula:
     - From periodic compounding:
     $APY = (1 + rac{r}{m})^m - 1$
     - From continuous compounding:
     $APY = e^r - 1$

• Example 6: APY Comparisons
    - For various investment plans:
    - Plan A: 10% compounded annually → APY = 10.00%
    - Plan B: 9.8% compounded quarterly → APY ≈ 10.17%
    - Plan C: 9.65% compounded continuously → APY ≈ 10.13%
    - Conclusion: Best option is Plan B with 9.8% compounded quarterly.

Geometric Sequences

• A sequence formed by multiplying a constant number known as the common ratio (r) with previous term.

• General Form:
     - For geometric sequences: $a_n = a_1 r^{n-1}$

• Sum Formula:
    - The sum of the first n terms:
    $S_n = rac{a_1 (1-r^n)}{1-r} ext{ if } r <br>eq 1$
    - If $r = 1$, then $S_n = a_1 imes n$.

• Example 7:
    - Next three terms for sequences:
      - a) 1, 3, 9…: The common ratio r = 3, leading to ensuing terms like 27, 81
      - b) 3, -6, 12,…: Common ratio r = -2, leading to terms like -24, 48.

In the context of compound interest calculation, 'imes' appears to be a typographical error for 'times', which denotes multiplication. For example, in the formula for interest calculation, it should read 'I = P × r', indicating that the principal (P) is multiplied by the rate (r).