Time value of money is a crucial concept in finance. It explains why a dollar today is worth more than a dollar tomorrow. Future value and present value calculations help us understand how money grows over time and what future cash flows are worth today.

These concepts are essential for financial decision-making. They allow us to compare investments, plan for retirement, and evaluate projects. By mastering future and present value calculations, we can make smarter choices about saving, investing, and spending money.

Future Value of a Single Sum

Calculating Future Value

• The future value formula is $FV = PV (1 + r)^n$, where:
  • $FV$ is the future value
  • $PV$ is the present value
  • $r$ is the periodic interest rate
  • $n$ is the number of periods
• The periodic interest rate ($r$) is calculated by dividing the annual interest rate by the number of compounding periods per year
  • If the annual interest rate is 6% and compounding occurs monthly, the periodic interest rate would be 0.5% ($6% / 12$)
  • Quarterly compounding with an 8% annual interest rate results in a periodic interest rate of 2% ($8% / 4$)
• The number of periods ($n$) is determined by multiplying the number of years by the number of compounding periods per year
  • For a 5-year investment with monthly compounding, the number of periods would be 60 ($5 12$)
  • A 10-year investment with semi-annual compounding would have 20 periods ($10 2$)

Interpreting Future Value

• The future value represents the amount to which a present sum of money will grow over a specified period, given a certain interest rate and compounding frequency
  • A $1,000 investment earning 5% annually, compounded monthly for 10 years, will grow to approximately $1,647
  • An initial deposit of $5,000 earning 3% annually, compounded quarterly for 7 years, will result in a future value of about $6,142
• The future value takes into account the time value of money, recognizing that money available now is worth more than the same amount in the future due to its earning potential
  • Assuming a positive interest rate, the future value will always be greater than the present value
  • The difference between the future value and the present value represents the total interest earned over the investment period

Present Value of a Single Sum

Calculating Present Value

• The present value formula is $PV = FV / (1 + r)^n$, where:
  • $PV$ is the present value
  • $FV$ is the future value
  • $r$ is the periodic discount rate
  • $n$ is the number of periods
• The periodic discount rate ($r$) is the rate used to discount future cash flows to their present value, representing the opportunity cost of capital or the required rate of return on an investment
  • An annual discount rate of 10% with monthly discounting results in a periodic discount rate of approximately 0.833% ($10% / 12$)
  • A 6% annual discount rate with semi-annual discounting translates to a periodic discount rate of 3% ($6% / 2$)
• The number of periods ($n$) follows the same principle as in the future value calculation, determined by multiplying the number of years by the number of discounting periods per year
  • For a 4-year discounting period with quarterly discounting, the number of periods would be 16 ($4 4$)
  • An 8-year discounting period with annual discounting would have 8 periods ($8 1$)

Interpreting Present Value

• The present value represents the current worth of a future sum of money, discounted at a specific rate over a given period
  • A future cash flow of $10,000 received in 5 years, discounted at an annual rate of 8% compounded monthly, has a present value of approximately $6,756
  • An expected payment of $50,000 in 12 years, discounted at a 5% annual rate compounded semi-annually, is worth about $27,920 today
• The present value accounts for the time value of money by recognizing that future cash flows are worth less than their nominal value due to the opportunity cost of waiting to receive them
  • Assuming a positive discount rate, the present value will always be less than the future value
  • The difference between the future value and the present value represents the total discount applied over the discounting period

Applying Future and Present Value

Time Value of Money Problems

• Time value of money (TVM) problems involve determining the future value or present value of cash flows, given the interest rate, compounding frequency, and time horizon
  • Calculating the future value of a series of periodic investments (annuity) to determine the ending balance of a savings account
  • Determining the present value of a stream of future rental income to assess the value of a real estate investment
• Solving TVM problems requires identifying the known variables (PV, FV, r, or n) and using the appropriate formula to solve for the unknown variable
  • To find the interest rate (r) needed to grow a $10,000 investment to $15,000 in 6 years with annual compounding, use the future value formula and solve for r
  • Calculating the number of years (n) it will take for a $5,000 initial deposit to reach $20,000, assuming an 8% annual interest rate compounded quarterly, involves using the future value formula and solving for n

Financial Applications

• Retirement planning often involves calculating the future value of regular savings contributions or the present value of a desired retirement income stream
  • Estimating the future value of monthly contributions of $500 over 30 years, earning a 7% annual return compounded monthly, to determine the retirement savings balance
  • Calculating the present value of a desired annual retirement income of $60,000 for 20 years, discounted at a 5% rate compounded annually, to determine the required savings at retirement
• Capital budgeting decisions require comparing the present value of expected future cash flows from a project to its initial investment to determine its net present value (NPV) and assess its financial viability
  • A project with an initial investment of $100,000 and expected annual cash inflows of $30,000 for 5 years, discounted at a 10% annual rate, has an NPV of approximately $18,421 (positive NPV, accept the project)
  • An investment opportunity requiring $500,000 upfront and generating annual cash flows of $75,000 for 10 years, discounted at a 12% annual rate, results in an NPV of about -$40,346 (negative NPV, reject the project)
• Loan amortization schedules use present value calculations to determine the periodic payments required to fully repay a loan, given the loan amount, interest rate, and repayment term
  • A $200,000 mortgage with a 4.5% annual interest rate and a 30-year repayment term would require monthly payments of approximately $1,013
  • A $50,000 car loan with a 6% annual interest rate and a 5-year repayment term would have monthly payments of about $966

Future Value vs Present Value

Inverse Relationship

• Future value and present value are inverse concepts: a present value can be grown to a future value, and a future value can be discounted back to a present value
  • A present value of $1,000 invested at a 5% annual interest rate, compounded annually for 10 years, will grow to a future value of approximately $1,629
  • A future value of $10,000 to be received in 7 years, discounted at an 8% annual rate compounded semi-annually, has a present value of about $5,835
• The relationship between future value and present value is determined by the interest rate (or discount rate) and the time horizon
  • A higher interest rate will result in a larger future value for a given present value, while a higher discount rate will lead to a smaller present value for a given future value
  • A longer time horizon will increase the difference between the present value and future value, as compound interest has more time to accrue (for future value) or discounting has a more significant effect (for present value)

Financial Decision-Making

• In financial decision-making, comparing the present value of future cash flows to their required investment allows investors to assess the profitability and feasibility of different opportunities
  • If the present value of a project's expected cash flows exceeds its initial investment (positive NPV), the project is considered financially viable and should be accepted
  • When evaluating multiple investment opportunities, the one with the highest positive NPV should be chosen, assuming similar risk levels
• The choice between receiving money now (present value) or in the future (future value) depends on factors such as the investor's required rate of return, opportunity costs, and time preferences
  • An investor with a high required rate of return may prefer receiving money now, as the future value of that money at their desired rate would be significantly higher
  • If an investor has immediate cash needs or attractive alternative investment opportunities, they may opt for the present value instead of waiting for a future payoff
  • Individuals with longer investment horizons and lower immediate cash needs may be more willing to defer gratification and invest for a higher future value