Principles of Finance

💳Principles of Finance Unit 7 – Time Value of Money: Single Payments

The Time Value of Money (TVM) concept is crucial in finance, emphasizing that money's worth changes over time. This unit focuses on single payments, exploring how present and future values are calculated and compared using discount rates and interest rates. Understanding TVM helps in making informed financial decisions, from personal investments to corporate finance. Key applications include retirement planning, loan analysis, and capital budgeting, all of which rely on accurately assessing the value of money across different time periods.

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 value of a future sum of money or stream of cash flows given a specified rate of return
  • Future value (FV) is the value of a current asset at a future date based on an assumed rate of growth
  • Discount rate is the interest rate used to calculate the present value of future cash flows
  • Compounding involves the reinvestment of earnings at a given rate of return to constantly increase the principal amount year over year
  • Discounting determines the present value of a future cash flow by a given discount rate
  • Opportunity cost represents the benefits an individual, investor, or business misses out on when choosing one alternative over another

Time Value Basics

  • Money has a time value because of the following reasons:
    • Potential to earn interest
    • Effects of inflation over time
    • Uncertainty and risk associated with future cash flows
  • The sooner money is received, the more it is worth due to the time value of money principle
  • Interest can be earned on the initial principal balance and the accumulated interest from prior periods (compounding)
  • Inflation erodes the purchasing power of money over time, reducing the real value of future cash flows
  • Risk and uncertainty increase for cash flows further into the future, reducing their present value
  • The rate of return used in TVM calculations should account for these factors (interest, inflation, risk)
  • Individuals prefer current consumption over future consumption, leading to positive time preference and requiring compensation (interest) to defer consumption

Present Value (PV)

  • Present value (PV) is the current worth of a future sum of money or stream of cash flows given a specified rate of return
  • PV is calculated by discounting the future value (FV) by the discount rate over a specific period
  • The PV formula is: PV=FV(1+r)nPV = \frac{FV}{(1+r)^n}, where FV is the future value, r is the discount rate, and n is the number of periods
  • A higher discount rate results in a lower PV, while a lower discount rate results in a higher PV
  • PV is used to compare the value of money received at different points in time
  • Helps in making investment decisions by determining if the present value of future cash flows is greater than the initial investment
  • PV is affected by the timing and magnitude of cash flows, as well as the discount rate applied

Future Value (FV)

  • Future value (FV) is the value of a current asset at a specified date in the future based on an assumed rate of growth (interest rate)
  • FV is calculated by compounding the present value (PV) by the interest rate over a specific period
  • The FV formula is: FV=PV(1+r)nFV = PV(1+r)^n, where PV is the present value, r is the interest rate, and n is the number of periods
  • A higher interest rate results in a higher FV, while a lower interest rate results in a lower FV
  • FV is used to estimate the growth of an investment over time
  • Helps in setting financial goals and determining the required investment to reach a desired future value
  • Commonly used in retirement planning to estimate the future value of current savings and investments

Discount Rates and Interest Rates

  • Discount rate is the interest rate used to calculate the present value of future cash flows
    • Represents the rate of return that could be earned on an investment in the financial markets with similar risk
    • Reflects the opportunity cost of the funds, considering factors such as interest, inflation, and risk
  • Interest rate is the amount charged, expressed as a percentage of the principal, by a lender to a borrower for the use of money
    • Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal and accumulated interest from previous periods
  • Discount rates and interest rates are inversely related to the present value and future value of money
    • A higher discount rate or interest rate results in a lower present value and a higher future value
    • A lower discount rate or interest rate results in a higher present value and a lower future value
  • The choice of discount rate or interest rate depends on factors such as risk, inflation, and the type of cash flow being analyzed
  • Real interest rates adjust for inflation, while nominal interest rates do not account for inflation

Calculations and Formulas

  • Present Value (PV) formula: PV=FV(1+r)nPV = \frac{FV}{(1+r)^n}
    • FV = Future Value
    • r = Discount Rate (per period)
    • n = Number of Periods
  • Future Value (FV) formula: FV=PV(1+r)nFV = PV(1+r)^n
    • PV = Present Value
    • r = Interest Rate (per period)
    • n = Number of Periods
  • Net Present Value (NPV) formula: NPV=t=1nCt(1+r)tC0NPV = \sum_{t=1}^{n} \frac{C_t}{(1+r)^t} - C_0
    • CtC_t = Cash Flow at time t
    • r = Discount Rate
    • t = Time of the cash flow
    • C0C_0 = Initial Investment
  • Internal Rate of Return (IRR) is the discount rate that makes the NPV of all cash flows equal to zero
  • Payback Period is the length of time required to recover the initial investment
  • Discounted Payback Period is the length of time required to recover the initial investment, considering the time value of money

Real-World Applications

  • Investment decision-making and capital budgeting
    • Comparing the present value of future cash inflows and outflows to determine the profitability and feasibility of investment projects
  • Retirement planning and saving
    • Estimating the future value of current savings and determining the required savings rate to achieve a desired retirement income
  • Loan and mortgage analysis
    • Calculating the present value of loan payments and comparing loan options with different interest rates and repayment terms
  • Bond pricing and valuation
    • Determining the fair value of a bond based on the present value of its future coupon payments and face value
  • Lease vs. buy decisions
    • Comparing the present value of lease payments and the cost of purchasing an asset outright
  • Inflation and purchasing power
    • Assessing the impact of inflation on the real value of future cash flows and making informed financial decisions

Common Mistakes to Avoid

  • Ignoring the time value of money and treating all cash flows as equal regardless of timing
  • Using nominal interest rates instead of real interest rates when accounting for inflation
  • Failing to consider the opportunity cost of capital when making investment decisions
  • Mismatching the timing of cash flows and discount rates in present value calculations
  • Double-counting or omitting cash flows in the analysis
  • Using the wrong discount rate or interest rate for the specific situation or risk level
  • Neglecting to consider the impact of taxes on cash flows and investment returns
  • Overlooking the sensitivity of results to changes in assumptions, such as discount rates and growth rates


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.