💰Intermediate Financial Accounting I Unit 5 – Time Value of Money

Key Concepts

• Time value of money (TVM) recognizes that money available now is worth more than the same amount in the future due to its potential earning capacity
• Present value (PV) represents the current worth of a future sum of money or stream of cash flows given a specified rate of return
• Future value (FV) calculates the value of a current asset at a future date based on an assumed rate of growth
• Discounting determines the present value of future cash flows using a discount rate that reflects the risk and opportunity cost
• Compounding involves reinvesting interest earned on an investment, allowing the initial principal to grow exponentially over time
  • Compound interest can be calculated annually, semi-annually, quarterly, monthly, or daily
  • More frequent compounding leads to higher future values
• Annuities are series of equal payments or receipts occurring at fixed intervals (monthly, quarterly, annually) for a specified period
• Perpetuities are endless streams of equal periodic payments that continue indefinitely

Time Value Formulas

• Present Value (PV): $PV = \frac{FV}{(1+r)^n}$
  • $FV$ = Future value
  • $r$ = Interest rate per period
  • $n$ = Number of periods
• Future Value (FV): $FV = PV(1+r)^n$
• Present Value of an Annuity (PVA): $PVA = PMT \left[\frac{1-\frac{1}{(1+r)^n}}{r}\right]$
  • $PMT$ = Payment amount per period
• Future Value of an Annuity (FVA): $FVA = PMT \left[\frac{(1+r)^n-1}{r}\right]$
• Present Value of a Perpetuity: $PV_{perpetuity} = \frac{PMT}{r}$
• Effective Annual Rate (EAR): $EAR = (1 + \frac{r}{m})^m - 1$
  • $m$ = Number of compounding periods per year
• Rule of 72: $\text{Years to double} \approx \frac{72}{\text{Annual interest rate}}$

Present Value Calculations

• Present value calculations discount future cash flows to their equivalent value today
• Discounting accounts for the time value of money and the opportunity cost of capital
• The discount rate used should reflect the risk associated with the future cash flows
  • Higher risk investments require higher discount rates
  • Risk-free rates (government bonds) use lower discount rates
• Net present value (NPV) sums the present values of incoming and outgoing cash flows over a period
  • Positive NPV indicates a profitable investment
  • Negative NPV suggests an investment should be avoided
• Excel functions like

  PV

  ,

  NPV

  , and

  XNPV

  can simplify present value calculations
• Sensitivity analysis tests how changes in discount rates or cash flows affect the present value

Future Value Applications

• Future value calculations determine the worth of an investment or asset at a later date
• Compound annual growth rate (CAGR) represents the mean annual growth rate of an investment over a specified period
  • $CAGR = \left(\frac{EV}{BV}\right)^{\frac{1}{n}} - 1$
  • $EV$ = Ending value
  • $BV$ = Beginning value
• Rule of 72 estimates the time required to double an investment given a fixed annual rate
• Retirement planning uses future value to project savings growth and determine required contributions
• Loan amortization schedules show the future value of payments, separating principal and interest
• Excel functions like

  FV

  and

  FVSCHEDULE

  automate future value calculations
• Inflation erodes purchasing power over time, so future values must account for expected inflation rates

Annuities and Perpetuities

• Annuities and perpetuities are series of fixed payments over time
• Ordinary annuities have payments occurring at the end of each period
  • Examples include car payments or mortgage payments
• Annuities due have payments occurring at the beginning of each period
  • Examples include rental income or lease payments
• Perpetuities are infinite series of equal payments
  • Examples include preferred stock dividends or consols (bonds with no maturity)
• Present and future value formulas for annuities and perpetuities simplify the calculation process
• Annuity tables provide factors for calculating present and future values based on interest rates and periods
• Annuities and perpetuities help value investments like bonds, leases, and rental properties

Compound Interest vs. Simple Interest

• Simple interest is calculated only on the original principal amount
  • $I = P \times r \times t$
  • $I$ = Interest
  • $P$ = Principal
  • $r$ = Annual interest rate
  • $t$ = Time in years
• Compound interest is calculated on the principal and accumulated interest from previous periods
  • Compounding can occur annually, semi-annually, quarterly, monthly, or daily
  • More frequent compounding results in higher ending balances
• Annual percentage yield (APY) represents the effective annual return with compounding
  • $APY = (1 + \frac{r}{n})^n - 1$
  • $n$ = Number of compounding periods per year
• Continuous compounding assumes interest is compounded infinitely
  • $FV = PV \times e^{rt}$
  • $e$ ≈ 2.71828 (mathematical constant)
• Rule of 72 estimates the time to double an investment with compound interest

Practical Applications in Finance

• Capital budgeting decisions use NPV to evaluate investment projects
  • Projects with positive NPV are accepted
  • Projects with negative NPV are rejected
• Loan and lease agreements rely on time value of money to determine payments and amortization schedules
• Retirement planning uses future value to estimate required savings and investment returns
• Bond pricing employs present value to determine the fair value of fixed-income securities
  • $Bond\ price = \sum_{t=1}^n \frac{Coupon}{(1+r)^t} + \frac{Face\ value}{(1+r)^n}$
• Stock valuation models (dividend discount model) use present value to estimate intrinsic stock prices
• Insurance companies use present value to calculate premiums and reserves
• Real estate valuation discounts future rental income and sale proceeds to determine property values

Common Mistakes and Tips

• Ensure the consistency of compounding periods and interest rates in calculations
  • Convert annual rates to their equivalent periodic rates
• Be cautious when interpreting NPV and IRR for mutually exclusive projects
  • NPV is generally preferred for ranking projects
• Remember to account for the impact of taxes on cash flows and returns
• Use the appropriate discount rate that reflects the risk of the cash flows
  • Higher risk requires higher discount rates
• Consider the limitations of the Rule of 72 as an approximation tool
• Double-check the setup of annuity and perpetuity formulas
  • Confirm whether payments occur at the beginning (annuity due) or end (ordinary annuity) of each period
• Perform sensitivity analysis to assess the impact of changes in assumptions (growth rates, discount rates)
• Understand the distinction between nominal and real returns
  • Nominal returns include inflation
  • Real returns are adjusted for inflation