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.