Time value of money calculations are essential for understanding the true value of investments and loans over time. These concepts help you compare financial options and make informed decisions about your money.

Present and future values, perpetuities, and annuities are key components of time value calculations. By mastering these tools and formulas, you'll be able to analyze complex financial scenarios and determine the best course of action for your financial goals.

Time Value of Money Calculations

Present and future values of perpetuities

• Perpetuity involves an infinite series of equal periodic payments (annuity payments)
  • Present Value of Perpetuity (PVP) calculated using formula $PVP = \frac{C}{r}$
    • $C$ represents the periodic payment amount
    • $r$ denotes the interest rate per period (discount rate)
  • Financial calculator inputs for PVP:
    • Enter PMT as the periodic payment amount
    • Input I/Y as the interest rate per period
    • Press CPT PV to compute the present value of the perpetuity
  • Excel formula for PVP: =PMT/RATE
    • PMT is the periodic payment amount
    • RATE represents the interest rate per period
• Growing Perpetuity consists of periodic payments that increase at a constant rate
  • Present Value of Growing Perpetuity (PVGP) determined by formula $PVGP = \frac{C}{r - g}$
    • $C$ signifies the initial periodic payment amount
    • $r$ stands for the interest rate per period
    • $g$ indicates the growth rate of payments per period
  • Excel formula for PVGP: =PMT/(RATE-GROWTH)
    • PMT is the initial periodic payment amount
    • RATE denotes the interest rate per period
    • GROWTH represents the growth rate of payments per period

Solving annuity problems with tools

• Annuity consists of a series of equal periodic payments over a fixed number of periods
  • Present Value of Annuity (PVA) calculated using formula $PVA = PMT \times \frac{1 - (1 + r)^{-n}}{r}$
    • PMT represents the periodic payment amount
    • $r$ signifies the interest rate per period
    • $n$ denotes the number of periods
  • Financial calculator inputs for annuity problems:
    • Enter N as the number of periods
    • Input I/Y as the interest rate per period
    • Enter PMT as the periodic payment amount
    • Press CPT PV to compute the present value of the annuity
    • Press CPT PMT to compute the periodic payment amount
    • Press CPT N to compute the number of periods
  • Excel formulas for annuity problems:
    • PV calculated using =PV(RATE, NPER, PMT)
    • PMT determined by =PMT(RATE, NPER, PV)
    • NPER computed using =NPER(RATE, PMT, PV)
      • RATE represents the interest rate per period
      • NPER signifies the number of periods
      • PMT denotes the periodic payment amount
      • PV stands for the present value of the annuity
  • These calculations are essential for analyzing cash flows in financial decision-making

Effective vs stated interest rates

• Effective Annual Rate (EAR) represents the actual annual interest rate earned when compounding occurs more than once per year
  • EAR formula: $EAR = (1 + \frac{r}{m})^m - 1$
    • $r$ signifies the stated annual interest rate
    • $m$ denotes the number of compounding periods per year (compounding frequency)
  • Excel formula for EAR: =(1+RATE/NPER)^NPER-1
    • RATE represents the stated annual interest rate
    • NPER signifies the number of compounding periods per year
  • Understanding EAR is crucial when dealing with compound interest scenarios

Loan amortization schedules in spreadsheets

• Loan Amortization Schedule provides a breakdown of each loan payment into interest and principal components
  • Components of a loan amortization schedule include:
    1. Payment number
    2. Beginning balance
    3. Payment amount
    4. Interest payment
    5. Principal payment
    6. Ending balance
  • Creating a loan amortization schedule in Excel:
    • Required inputs: Loan amount, interest rate per period, number of periods, and payment amount
    • Use PMT function to calculate the periodic payment amount
    • For each period, follow these steps:
      1. Calculate interest payment by multiplying beginning balance by interest rate per period
      2. Determine principal payment by subtracting interest payment from payment amount
      3. Compute ending balance by subtracting principal payment from beginning balance
  • Interpreting a loan amortization schedule allows you to:
    • Track the remaining loan balance over the life of the loan
    • Identify the portion of each payment allocated to interest and principal
    • Determine the total interest paid over the duration of the loan (sum of interest payments)

Advanced financial analysis techniques

• Net Present Value (NPV) is used to evaluate the profitability of an investment or project
• Internal Rate of Return (IRR) calculates the rate of return that makes the NPV of all cash flows equal to zero
• Financial modeling involves creating a mathematical representation of a financial situation to make projections and analyze various scenarios