💳Principles of Finance Unit 7 – Time Value of Money: Single 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 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: $PV = \frac{FV}{(1+r)^n}$, 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: $FV = PV(1+r)^n$, 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: $PV = \frac{FV}{(1+r)^n}$
  • FV = Future Value
  • r = Discount Rate (per period)
  • n = Number of Periods
• Future Value (FV) formula: $FV = PV(1+r)^n$
  • PV = Present Value
  • r = Interest Rate (per period)
  • n = Number of Periods
• Net Present Value (NPV) formula: $NPV = \sum_{t=1}^{n} \frac{C_t}{(1+r)^t} - C_0$
  • $C_t$ = Cash Flow at time t
  • r = Discount Rate
  • t = Time of the cash flow
  • $C_0$ = 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