11.3 Explain the Time Value of Money and Calculate Present and Future Values of Lump Sums and Annuities

Money has a time value, affecting financial decisions in business. Understanding future and present values of lump sums and annuities is crucial for managers. These concepts help evaluate investments, compare financing options, and assess project feasibility.

Calculating future and present values involves formulas and tables for both lump sums and annuities. Managers must consider the timing and nature of cash flows to make informed decisions. Additional factors like inflation, risk-free rates, and compounding frequency also impact financial analysis.

Time Value of Money

Future value of lump sums and annuities

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• Future value (FV) represents the value of an investment at a specific future date
  • Calculated by applying compound interest over a period of time
  • Formula: $FV = PV(1 + i)^n$
    • $PV$ represents the present value or initial investment
    • $i$ represents the interest rate per period (annual, quarterly, monthly)
    • $n$ represents the number of periods (years, quarters, months)
• Future value of a lump sum involves a single payment or investment made at a specific point in time
  • Example: Investing $10,000 today at 6$17,908.48
    • $FV =$10,000(1 + 0.06)^{10} = $17,908.48$
• Future value of an annuity involves a series of equal payments made at regular intervals over a specified period
  • Formula: $FV = PMT[(1 + i)^n - 1] / i$
    • $PMT$ represents the payment amount per period (annual, quarterly, monthly)
  • Example: Investing $2,000 annually at 5$75,837.56
    • $FV =$2,000[(1 + 0.05)^{20} - 1] / 0.05 = $75,837.56$
• Future value tables provide pre-calculated factors for various interest rates and time periods
  • Simplify calculations by multiplying the lump sum or annuity payment by the appropriate factor from the table
  • Example: Using a future value table, the factor for 6% interest over 10 years is 1.7908, so $10,000 × 1.7908 =$17,908

Present value of lump sums and annuities

• Present value (PV) represents the current value of a future sum of money or stream of payments
  • Calculated by discounting future cash flows at a specific rate of return
  • Formula: $PV = FV / (1 + i)^n$
• Present value of a lump sum involves discounting a single future payment to its current value
  • Example: The present value of receiving $25,000 in 5 years, discounted at 4$20,833.65
    • $PV =$25,000 / (1 + 0.04)^5 = $20,833.65$
• Present value of an annuity involves discounting a series of equal future payments to their current value
  • Formula: $PV = PMT[1 - (1 + i)^{-n}] / i$
  • Example: The present value of receiving $5,000 annually for 10 years, discounted at 3$41,199.26
    • $PV =$5,000[1 - (1 + 0.03)^{-10}] / 0.03 = $41,199.26$
• Present value tables provide pre-calculated factors for various interest rates and time periods
  • Simplify calculations by multiplying the lump sum or annuity payment by the appropriate factor from the table
  • Example: Using a present value table, the factor for 4% interest over 5 years is 0.8219, so $25,000 × 0.8219 =$20,547.50

Lump sums vs annuities in business decisions

• Lump sum payments are single, one-time cash flows occurring at a specific point in time
  • Examples: Capital investments (purchasing equipment), loan disbursements (receiving a business loan), or one-time purchases (buying inventory)
  • Calculations involve using the basic future value or present value formulas
• Annuities are series of equal cash flows occurring at regular intervals over a specified period
  • Examples: Loan repayments (monthly mortgage payments), lease payments (annual rent), or recurring investments (quarterly contributions to a retirement account)
  • Calculations involve using the future value of an annuity or present value of an annuity formulas
• Understanding the difference between lump sums and annuities is crucial for:
  1. Evaluating investment opportunities (comparing lump sum investments vs annuities)
  2. Comparing financing options (lump sum loan vs installment payments)
  3. Assessing the feasibility of projects (estimating future cash flows as lump sums or annuities)
• Managers must consider the timing and nature of cash flows to make informed decisions
  • Example: When deciding between two investment options, a manager should compare the present value of each option's future cash flows (lump sums and annuities) to determine which provides the highest return
  • This comparison helps assess the opportunity cost of choosing one investment over another

Additional Considerations in Time Value of Money

• Inflation affects the purchasing power of money over time, reducing the real value of future cash flows
• The risk-free rate represents the theoretical rate of return on an investment with no risk of financial loss
• Nominal interest rates are the stated rates, while real interest rates account for inflation
• The effective annual rate considers the impact of compounding frequency on the overall return
• The Rule of 72 is a quick estimation tool to determine how long it takes for an investment to double, calculated by dividing 72 by the annual interest rate