💰Intro to Finance Unit 3 – Time Value of Money

Key Concepts

• Time value of money (TVM) fundamental principle in finance that money available now is worth more than an identical sum in the future
• Present value (PV) current worth of a future sum of money or stream of cash flows given a specified rate of return
• Future value (FV) value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today
• Discounting process of finding the present value of a future cash flow
  • Applies the concept of time value of money
  • Determines the value of a future cash flow in today's dollars
• Compounding process of calculating the future value of an investment based on a given interest rate and time period
• Annuity series of equal payments or receipts that occur at evenly spaced intervals over a fixed period of time
• Perpetuity annuity that has no end, or a stream of cash payments that continues forever
• Interest rate percentage charged or paid for the use of money, typically expressed as an annual percentage rate (APR)

Time Value Basics

• Money has a time value because of the potential 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 earning capacity
• The time value of money is affected by inflation, which erodes the purchasing power of money over time
• Opportunity cost represents the potential benefits an individual or business misses out on when choosing one alternative over another
• Risk and uncertainty also play a role in the time value of money, as future cash flows are not guaranteed
• The time value of money is an important concept in making investment decisions and assessing the viability of projects
• Ignoring the time value of money can lead to suboptimal financial decisions and inaccurate valuations

Present Value Calculations

• Present value (PV) is the current worth of a future sum of money or stream of cash flows given a specified rate of return
• The basic formula for calculating present value is: $PV = FV / (1 + r)^n$
  • FV = Future value r = Interest rate per period n = Number of periods
• Present value calculations are used to determine the value of future cash flows in today's dollars
• The discount rate used in present value calculations represents the required rate of return or the opportunity cost of capital
• A higher discount rate will result in a lower present value, while a lower discount rate will result in a higher present value
• Net present value (NPV) is the sum of the present values of all cash inflows and outflows of an investment or project
  • A positive NPV indicates that a project is expected to be profitable and should be accepted
  • A negative NPV suggests that a project should be rejected

Future Value Calculations

• 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
• The basic formula for calculating future value is: $FV = PV * (1 + r)^n$
  • PV = Present value r = Interest rate per period n = Number of periods
• Future value calculations are used to determine the growth of an investment over time
• The interest rate and the number of compounding periods have a significant impact on the future value
• Continuous compounding assumes that interest is compounded continuously, rather than at discrete intervals
  • The formula for future value with continuous compounding is: $FV = PV * e^{rt}$, where e is the mathematical constant (approximately 2.71828)
• Rule of 72 a quick way to estimate the number of years required to double an investment, calculated by dividing 72 by the annual interest rate

Annuities and Cash Flows

• An annuity is a series of equal payments or receipts that occur at evenly spaced intervals over a fixed period of time
• Ordinary annuity assumes that cash flows occur at the end of each period
  • The formula for the present value of an ordinary annuity is: $PV = PMT * [(1 - (1 + r)^{-n}) / r]$
  • PMT = Payment amount per period r = Interest rate per period n = Number of periods
• Annuity due assumes that cash flows occur at the beginning of each period
  • The formula for the present value of an annuity due is: $PV = PMT * [(1 - (1 + r)^{-n}) / r] * (1 + r)$
• Perpetuity is an annuity that has no end, or a stream of cash payments that continues forever
  • The formula for the present value of a perpetuity is: $PV = PMT / r$
• Uneven cash flows can be analyzed by calculating the present value of each individual cash flow and then summing them together

Interest Rates and Compounding

• Interest rate is the percentage charged or paid for the use of money, typically expressed as an annual percentage rate (APR)
• Simple interest is calculated only on the principal amount, and not on accumulated interest
  • The formula for simple interest is: $I = P * r * t$
  • I = Interest earned P = Principal amount r = Annual interest rate t = Time (in years)
• Compound interest is calculated on the initial principal and the accumulated interest from previous periods
  • The formula for compound interest is: $A = P * (1 + r/n)^{nt}$
  • A = Final amount P = Principal amount r = Annual interest rate n = Number of times interest is compounded per year t = Number of years
• Effective annual rate (EAR) is the actual annual rate of return, taking into account the effect of compounding
  • The formula for EAR is: $EAR = (1 + r/n)^n - 1$
• Annual percentage yield (APY) is the effective annual rate of return, taking into account the effect of compounding and any fees

Real-World Applications

• Retirement planning: TVM concepts help individuals determine how much they need to save to achieve their desired retirement income
• Loan amortization: Banks and lenders use TVM to calculate the periodic payments required to pay off a loan over a specified term
• Capital budgeting: Companies use NPV and other TVM techniques to evaluate the profitability and feasibility of investment projects
• Mortgage payments: TVM is used to calculate monthly mortgage payments based on the loan amount, interest rate, and loan term
• Savings and investment growth: TVM helps individuals understand how their savings and investments can grow over time due to compounding interest
• Lease vs. buy decisions: TVM can be used to compare the costs and benefits of leasing an asset versus buying it outright
• Inflation adjustments: TVM concepts are used to adjust future cash flows for the impact of inflation to determine their real value

Common Pitfalls and Tips

• Forgetting to consider the time value of money can lead to incorrect financial decisions
• Ensure that all cash flows are discounted to the same point in time (usually the present) for accurate comparisons
• Be consistent with the use of interest rates (e.g., annual, semi-annual, or monthly) throughout your calculations
• Double-check the inputs (interest rate, number of periods, etc.) in your TVM calculations to avoid errors
• Consider the impact of taxes on your investment returns and cash flows
• Be aware of the limitations of TVM techniques, such as the assumption of constant interest rates and cash flows
• Conduct sensitivity analyses to understand how changes in key variables (e.g., interest rates) affect the outcome of your TVM calculations
• Use technology (financial calculators or spreadsheets) to perform complex TVM calculations accurately and efficiently