Theory of Interest Basic Concepts

Page 9: Interest Rate of a Period

• Definition: Interest is the increase in value over time

Page 10: Interest Rate Calculation

• Let X = initial value, Y = ending value

• Formula for interest: j = (Y - X) / X or Y = X(1 + j)

• Interest earned = Y - X = jX

Page 11: Example of Interest Accumulation

• Interest rates: i1 (Year 1), i2 (Year 2), i3 (Year 3)

• With an initial deposit of $100, determine the accumulated amount A at the end of Year 3

Page 12: Accumulated Amount Calculation

• Formula: A = 100(1 + i1)(1 + i2)(1 + i3)

• Timeline: 100 → 100(1 + i1) → 100(1 + i1)(1 + i2) → A

Page 13: Compound Interest Example

• If i1 = i2 = i3 = i, then A = 100(1 + i)^3

Page 14: Interest Calculation Assignment

• Given i1 = i2 = i3 = 4%, calculate interest for each year

Page 15: Interest Amount Calculation

• Year 1: Initial amount 100, ending amount 104, interest = 4

• Year 2: Initial amount 104, ending amount 108.16, interest = 4.16

• Year 3: Ending amount ≈ 4.33

Page 16: Additional Example on Principal Calculation

• Given accumulated amount A = 100 after 2 years, determine principal P

Page 17: Principal Calculation Formula

• Formula: P = 100 / ((1 + i1)(1 + i2))

Page 18: Present Value Calculation

• If i1 = i2, then P = 100(1 + i)^-2

Page 19: Present Value Definition

• P: Present value, A: future accumulated amount

Page 20: Notation for Accumulated Amount

• α(t): Accumulated amount factor after t periods

• A(t): Accumulated amount at time t

• Rate calculation between periods: A(t2) - A(t1) / A(t1)

Page 21: Effective Annual Rate

• Determine equivalent annual rate if the period isn't a year

• Example: Monthly interest of 0.5% yields an effective annual rate of 6.17%

Page 22: Example on Effective Annual Rate Comparison

• Comparing r1 (0.5% monthly) with r2 (3% every 6 months)

Page 23: Effective Annual Rate Calculation

• Solutions yield effective rates of approximately:

  • r1: 6.17%

  • r2: 6.09%

Page 24: Simple Interest Definition

• Simple interest formula: α(t) = 1 + t * i, A(t) = A(0)(1 + t * i)

Page 25: Simple Interest Example

• Calculate interest on a $100 deposit at 4% annual simple rate over 3 years

Page 26: Simple Interest Calculations

• Yearly interest amounts derived from simple growth formula

• Interest remains constant each year

Page 27: Effective Rate Calculation at Year 5

• Given a 6% simple annuity for 5 years, find effective rate for year 5.

Page 28: Effective Rate Calculation Solution

• Calculate using formula: i5 = (α(5) − α(4)) / α(4)

Page 29: Present Value Basics

• PV as the value at a given time, typically at time 0 (A(0))

Page 30: Present Value Calculation

• Formula: A(0) = A(t)(1 + i)^-t

Page 31: Example of Present Value Calculation

• Determine the deposit required today to fund future payments at 7.5% interest.

Page 32: Solution to Example

• Present value determined through discounting future payments

Page 33: Retirement Accounts Comparative Example

• Account conditions for Sarah's two retirement funds

Page 34: Solution to Retirement Accounts

• Determine the principal amount using growth formulas for both accounts.

Page 35: Fund Accumulation Calculation

• Money accumulates at different effective rates; compute the value of a deposit in a combined scenario.

Page 36: Solution for Fund Calculation

• Accumulated values evaluated through exponential growth formulas.

Page 37: Nominal Rate Basics

• NI: Annual compounding details for interest calculation laid out clearly.

Page 38: Deposits and Accumulation Over Time

• Example calculation for deposited amount at 4% nominal rate over 10 years.

Page 39: Accumulated Amount at Quarterly Compounding

• Demonstrations of compound interest under various compounding frequencies.

Page 40: Monthly and Daily Compounding Examples

• Comparison of amounts accumulated under different compounding frequencies.

Page 41: Savings Account Comparison Example

• Eric and Maia's interest calculations based on different compounding and interest approaches.

Page 42: Solution for Savings Account Comparison

• Solve for interest earned based on accumulated growth rates.

Page 43: Discount Rate Overview

• Introduction of discount rate concepts and calculations for financial assessment.

Page 44: Discrepancy Between i and d

• Describe relationships between interest and discount rates.

Page 45: Example of Nominal and Discount Rate Application

• Apply combined interest and discount rate concepts to real example.

Page 46: Solution to Discount Rate Calculation

• Worked through combined interest calculations, demonstrating how to apply formulas.

Page 47: Continuous Interest Intro

• Discuss how nominal rates behave under continuous compounding conditions.

Page 48: Continuous Interest Annual Rates Example

• Evaluate how consistently compounding rates translate into effective annual rates.

Page 49-51: Continuous Interest Annual Effective Rates

• Summary of results for varying compounding frequencies under continuous conditions.

Page 52: Continuous Interest Calculations

• Formula structures for transitioning from nominal to continuous interest.

Page 53: Continuous Interest Rate Calculation

• Mathematical details for creating interest factors using continuous methodologies.

Page 54-57: Force of Interest

• Definition and mathematical treatments of interest force in continuous contexts; equations derived from integral formulations.

Page 58: Example on Present Value Calculations

• Demonstrates how to calculate total present value against future cash flows using found formulas.

Page 59-62: Further Examples with Present Values

• Expanding on present value calculations to derive equivalence in different contexts.

Page 63-66: Example Contextual Calculations

• Dive into specific simulations of deposits using varied interest and discount approaches to evaluate accumulated values.

Page 67-68: Final Example for Fund Comparison

• Calculation leading to finding rates through accrued values of different funds.

Page 69-70: Bonus Example Challenge

• Opportunity for extra practice in interest simplifications.

Page 71-72: Summary of Concepts

• Recap of key formulas and applications.

Page 73: Suggested Problems

• Recommendations for practice and exam preparation from SoA problems.