3.2 Future Value and Present Value
7 min read•august 14, 2024
Time value of money is a crucial concept in finance. It explains why a dollar today is worth more than a dollar tomorrow. and 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
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- The future value formula is , where:
- is the future value
- is the present value
- is the periodic interest rate
- is the number of periods
- The periodic interest rate () 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% ()
- Quarterly compounding with an 8% annual interest rate results in a periodic interest rate of 2% ()
- The number of periods () is determined by multiplying the number of years by the number of compounding periods per year
- For a 5-year investment with , the number of periods would be 60 ()
- A 10-year investment with semi- would have 20 periods ()
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 1,647
- An initial deposit of 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 , where:
- is the present value
- is the future value
- is the periodic
- is the number of periods
- The periodic discount rate () is the rate used to discount future cash flows to their present value, representing the 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% ()
- A 6% annual discount rate with semi-annual discounting translates to a periodic discount rate of 3% ()
- The number of periods () 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 ()
- An 8-year discounting period with annual discounting would have 8 periods ()
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 6,756
- An expected payment of 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 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 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 (NPV) and assess its financial viability
- A project with an initial investment 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 75,000 for 10 years, discounted at a 12% annual rate, results in an NPV of about -$40,346 (negative NPV, reject the project)
- 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 1,013
- A 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 1,629
- A future value of 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 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
Key Terms to Review (18)
Annual compounding: Annual compounding refers to the process of calculating interest on an investment or loan once per year, where the interest earned is added to the principal at the end of each year. This means that each year, the interest calculation is based on the total amount in the account, including any previously earned interest, which leads to exponential growth over time. It is a crucial concept in understanding how both future value and present value are determined in finance.
Annuities: Annuities are financial products that provide a series of payments made at equal intervals, typically used as a way to receive steady income over time. They can be structured as immediate or deferred, depending on when the payments start, and are often utilized in retirement planning and investment strategies. Annuities can help individuals manage longevity risk, which is the risk of outliving one's savings.
Bonds: Bonds are debt securities issued by corporations, governments, or other entities to raise capital. When investors purchase bonds, they are essentially lending money to the issuer in exchange for periodic interest payments and the return of the bond's face value upon maturity. Bonds play a crucial role in personal finance, corporate finance, and public finance as a way to secure funding.
Cash flow projection: Cash flow projection is the process of estimating future financial inflows and outflows over a specific period of time. This tool helps individuals and businesses anticipate their financial position, allowing for better planning and decision-making. Understanding cash flow projections is crucial for analyzing both the future value and present value of investments, as it aids in evaluating potential profitability and liquidity needs.
Compound Interest: Compound interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods. This concept emphasizes that money can grow at a faster rate due to the effect of compounding, where the earnings on an investment generate additional earnings over time. The growth potential of an investment increases significantly as the time period extends, making it crucial for understanding how investments grow and for making informed financial decisions.
Discount Factor: The discount factor is a numerical value used to determine the present value of future cash flows by accounting for the time value of money. It reflects how much a future amount of money is worth today, taking into consideration factors such as interest rates and the time until payment. Essentially, it allows investors and financial analysts to evaluate how much future cash flows are worth in today's terms, enabling better financial decision-making.
Discount Rate: The discount rate is the interest rate used to determine the present value of future cash flows, reflecting the time value of money. It serves as a critical factor in finance, influencing investment decisions, project evaluations, and the valuation of financial instruments by adjusting future earnings back to their value today.
Future Value: Future value refers to the amount of money an investment will grow to over a specified period at a given interest rate. It connects the concept of time value of money to the understanding that money available today can be invested to earn returns, ultimately increasing its worth in the future. This notion is fundamental for making informed financial decisions, evaluating investment opportunities, and assessing how present capital can yield future benefits.
Fv formula: The fv formula, or future value formula, is a mathematical equation used to calculate the value of an investment or cash flow at a specified future date, taking into account a certain interest rate and the number of periods it will be invested. This concept is crucial for understanding how money grows over time due to the effects of interest, making it essential in evaluating investment opportunities and financial planning. The ability to calculate future value enables individuals and businesses to assess the potential worth of their current investments.
Inflation Rate: The inflation rate is the percentage increase in the price level of goods and services in an economy over a specific period, typically measured annually. This rate reflects how much purchasing power has diminished, affecting savings, investments, and overall economic growth. A rising inflation rate can indicate an overheating economy, while a low or negative rate may signal economic stagnation.
Investment valuation: Investment valuation is the process of determining the current worth of an asset or investment based on its expected future cash flows, market conditions, and risk factors. This concept plays a critical role in financial decision-making as it helps investors evaluate potential investments by comparing the present value of expected returns to the initial investment cost. Accurately valuing investments is essential for assessing whether they align with an investor's financial goals and risk tolerance.
Loan amortization: Loan amortization is the process of paying off a debt over time through regular payments that cover both principal and interest. This systematic repayment allows borrowers to gradually reduce their outstanding balance, with early payments typically allocating more towards interest and later payments focusing on reducing the principal. Understanding this concept is crucial for evaluating how loans work in terms of their future value and present value calculations, as well as their practical applications in real-life financial scenarios.
Monthly compounding: Monthly compounding refers to the process of calculating interest on an investment or loan on a monthly basis, rather than annually. This means that interest earned in one month is added to the principal, and in the following month, interest is calculated on this new total. As a result, monthly compounding can lead to a higher future value due to the effect of earning interest on previously accumulated interest.
Net cash flow: Net cash flow is the difference between cash inflows and cash outflows over a specific period. It represents the actual cash that a business generates or uses, providing insight into its financial health and liquidity. Positive net cash flow indicates that a company has enough cash to cover its expenses, invest in growth, and return value to shareholders, while negative net cash flow can signal potential financial trouble.
Net Present Value: Net Present Value (NPV) is a financial metric that calculates the value of a project or investment by determining the difference between the present value of cash inflows and the present value of cash outflows over time. This concept is crucial in assessing the profitability of an investment, as it helps in making informed decisions about allocating resources effectively.
Opportunity Cost: Opportunity cost refers to the value of the next best alternative that is foregone when making a choice. It emphasizes that every decision comes with trade-offs, highlighting the importance of considering what is sacrificed in order to pursue a particular option. Understanding opportunity cost is crucial in financial decisions, especially when evaluating future value and present value, as it allows individuals to assess the true cost of their choices and the potential benefits they might miss out on.
Present Value: Present value (PV) is the current worth of a sum of money that is to be received or paid in the future, discounted back to the present using a specific interest rate. This concept highlights the time value of money, emphasizing that a dollar today holds more value than a dollar in the future due to its potential earning capacity. By understanding present value, individuals and businesses can make informed decisions about investments, financing, and evaluating cash flows over time.
Pv formula: The present value (PV) formula is a financial equation used to determine the current worth of a cash flow or series of cash flows that will be received in the future, discounted back at a specific interest rate. This formula highlights the concept that money today has a different value compared to the same amount in the future due to factors like inflation and opportunity cost. Understanding this formula is essential for evaluating investment opportunities and comparing financial options over time.