Topic 2. Financial mathematics continued (Part 2) 2 per pg(1)

Overview

This notes document focuses on the key concepts and examples of Time Value of Money (TVM) in financial mathematics. The document outlines various approaches to solving different types of TVM problems, provides specific problem examples, showcases the relevant formulas, and illustrates the calculation processes involved.

TVM Concepts and Approaches

Key Objectives

  • Review basic concepts of Time Value of Money (TVM).

  • Solve multiple examples related to TVM problems.

Steps to Address TVM Problems

  1. Identify Problem Type:

    • Determine if it's the Future Value (FV) or Present Value (PV) of a lump sum or annuity.

    • Check if it involves a perpetuity.

  2. Draw a Timeline:

    • Understand cash flows and compounding periods.

    • Assess whether additional timelines are required.

  3. Select the Appropriate Formula:

    • Identify interest rate, time periods, and payment amounts relevant to the calculations.

    • Document the formulas clearly and demonstrate calculations step by step, indicating interest factors and rounding requirements.

Example Problems

Q1: Accumulated Value Calculation

  • Problem: Determine the accumulated value of $11,200 after 3 years at an interest rate of 10% compounded quarterly.

  • Calculations:

    • Effective interest rate: 2.5% per period

    • Compounding periods: 12

    • Resulting FV: $15,062.75

Q2: Effective Annual Rate

  • Problem: Calculate the EAR for a nominal rate of 6% compounded semi-annually.

  • Result: EAR = 6.09%

Q3: Present Value of Future Payments

  • Problem: Calculate the PV of university payments of $6,000 at the start of each semester for three years at 12% annual interest compounded semi-annually.

  • Resulting PV: $22,047.06

Q4: Loan Amortization

  • Problem: Calculate the remaining principal of a $70,000 loan with 12.5% interest compounded monthly after the 1st and 150th payments.

  • Results: Remaining after 1st payment: $69,965.48; after 150th payment: $57,763.34.

Q5: Retirement Account Future Value

  • Problem: Calculate the future value of $100 monthly deposits over 4 years at 12% interest compounded monthly.

  • Result: Total Future Value: $6,183.48

Q6: Present Value of Cash Inflows

  • Problem: Determine how much can be invested now to generate cash inflows of $5,000 every six months for 4 years at a 10% return.

  • Resulting PV: $32,316.06

Q7: Perpetuity Calculation

  • Problem: Find how much to invest now for a scholarship fund yielding $15,000 annually at 6% interest.

  • Result: Required investment: $250,000

Q8: Nominal to Effective Rate Conversion

  • Problem: Convert the APR of 15% into an effective annual rate with monthly compounding.

  • Result: EAR = 16.08%

Q9: Historical Interest Rate Calculation

  • Problem: Calculate the nominal interest rate for an account that grew from $13,000 to $26,401 over 7 years with quarterly compounding.

  • Result: Nominal interest rate: 10.25%

Q10: Projected Cash Inflows

  • Problem: Calculate the future value of variable cash inflows over five years with differing interest rates.

  • Total Cash at Year 5: $5,467.31

Summary

The notes provide a comprehensive overview of TVM problem-solving methods within financial mathematics. By applying the relevant formulas and understanding the calculation steps for different types of financial scenarios, students can enhance their proficiency and prepare effectively for exams.