💳Principles of Finance Unit 8 – Time Value of Money: Multiple Payments

Key Concepts

• Time value of money (TVM) principle states that money available now is worth more than an identical sum 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) is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today
• Annuity is a series of equal payments or receipts that occur at evenly spaced intervals over a fixed period of time
  • Ordinary annuity has payments occurring at the end of each period
  • Annuity due has payments occurring at the beginning of each period
• Perpetuity is a constant stream of identical cash flows with no end
• Discount rate is the rate of return used to discount future cash flows back to their present value
• Compounding is the process in which an asset's earnings are reinvested to generate additional earnings over time

Time Value Basics

• Money has a time value because of the opportunity to earn interest or a return on investment over time
• A dollar today is worth more than a dollar in the future because of its potential to earn interest
• The time value of money is a fundamental concept in finance that underlies investment decisions, capital budgeting, and valuation
• Factors influencing time value include interest rates, inflation, and risk
  • Higher interest rates lead to a higher present value of future cash flows
  • Inflation reduces the purchasing power of money over time
  • Riskier investments require a higher rate of return to compensate for the additional risk
• The time value of money is typically considered on a nominal basis, which includes the effect of inflation
• Real interest rates remove the effect of inflation to measure the true cost of borrowing or return on investment

Types of Cash Flows

• Single cash flows involve one-time payments or receipts at a specific point in time (lump sum deposit or withdrawal)
• Annuities are a series of equal cash flows occurring at fixed intervals for a specified period (monthly rent payments)
  • Ordinary annuities have cash flows occurring at the end of each period
  • Annuities due have cash flows occurring at the beginning of each period
• Perpetuities are a series of equal cash flows that continue indefinitely (preferred stock dividends)
• Uneven cash flows are a stream of cash flows that vary in amount or timing (dividends that grow at a constant rate)
• Deferred annuities are annuities where the first cash flow occurs at a later date than the valuation date (retirement annuity)
• Loans involve an initial cash inflow followed by a series of cash outflows to repay the principal and interest (mortgage)

Calculating Present Value

• Present value calculation discounts future cash flows to their equivalent value today using a specified rate of return
• The general formula for the present value of a future cash flow is: $PV = FV / (1 + r)^n$
  • PV = Present value
  • FV = Future value
  • r = Discount rate per period
  • n = Number of periods
• For an ordinary annuity, the present value formula is: $PV = PMT × [(1 - (1 + r)^(-n)) / r]$
  • PMT = Periodic payment amount
• For a perpetuity, the present value formula simplifies to: $PV = PMT / r$
• The net present value (NPV) is the sum of the present values of all cash inflows and outflows of an investment
  • A positive NPV indicates that an investment is expected to be profitable
• Excel functions for calculating present value include

  PV

  ,

  NPV

  , and

  XNPV

Calculating Future Value

• Future value calculation determines the value of a cash flow or series of cash flows at a future point in time
• The general formula for the future value of a present cash flow is: $FV = PV × (1 + r)^n$
• For an ordinary annuity, the future value formula is: $FV = PMT × [(((1 + r)^n) - 1) / r]$
• The future value of a series of uneven cash flows can be calculated by finding the future value of each individual cash flow and summing them
• Continuously compounded interest uses the formula: $FV = PV × e^(r × n)$
  • e ≈ 2.71828 (mathematical constant)
• Excel functions for calculating future value include

  FV

  and

  FVSCHEDULE

Annuities and Perpetuities

• An annuity is a series of equal cash flows occurring at fixed intervals for a specified period
  • Examples include loan payments, lease payments, and fixed-term investments
• Perpetuities are a series of equal cash flows that continue indefinitely
  • Examples include preferred stock dividends and consols (government bonds with no maturity date)
• The present value of an annuity is the sum of the present values of each individual cash flow in the series
• The future value of an annuity is the sum of the future values of each individual cash flow in the series
• Annuities due have cash flows occurring at the beginning of each period, while ordinary annuities have cash flows at the end of each period
  • The formulas for annuities due are adjusted to account for the earlier timing of cash flows
• Deferred annuities have a delay between the valuation date and the first cash flow
  • The present value is calculated by discounting the present value of the annuity back to the valuation date

Real-World Applications

• Retirement planning involves estimating the present value of future expenses and saving enough to fund those expenses
• Loan amortization schedules show the breakdown of each loan payment into principal and interest over the life of the loan
• Capital budgeting decisions use the net present value of a project's cash flows to determine its profitability
• Bond pricing uses the present value of the bond's future cash flows (coupon payments and face value) to determine its market price
  • The yield to maturity is the discount rate that equates the bond's price with the present value of its cash flows
• Stock valuation models, such as the dividend discount model, use the present value of future dividends to estimate a stock's intrinsic value
• Lease vs. buy decisions compare the present value of the cash flows associated with leasing an asset to the cost of purchasing it outright

Common Pitfalls and Tips

• Make sure to use the correct discount rate for the time period of the cash flows (annual rate for yearly cash flows, monthly rate for monthly cash flows, etc.)
• Be consistent with the compounding frequency of the discount rate and the timing of the cash flows
• Remember to account for any initial investment or cash outlay when calculating net present value
• Consider the impact of taxes on cash flows and use after-tax discount rates when appropriate
• Be aware of the limitations of the time value of money concepts, such as the assumption of constant discount rates and the sensitivity of results to changes in assumptions
• Use sensitivity analysis to test the robustness of your results by varying key inputs and assumptions
• Double-check your calculations and use Excel functions or financial calculators to minimize errors
• When comparing investment alternatives, make sure to use the same time frame and discount rate for all options