Time Value of Money

Time Value of Money

Learning Goals

  • LG 1: Discuss the role of time value in finance, the use of computational tools, and the basic patterns of cash flow.

    • Time value recognizes that receiving money sooner is more advantageous due to potential investment opportunities; a foundational concept in financial decision-making.

    • Managers utilize time-value tools to evaluate and compare cash inflows and outflows occurring at different points in time, ensuring optimal resource allocation.

  • LG 2: Understand the concepts of future value and present value, their calculation for single amounts, and the relationship between them.

    • Future Value (FV): Represents the worth of an investment or asset at a specified date in the future, taking into account the effect of compound interest.

    • Present Value (PV): Indicates the current worth of a future sum of money, discounted back to the present using an appropriate discount rate.

    • Understanding how to calculate future value and present value is essential for comparing today's cash with future cash inflows or outflows, enabling informed financial decisions.

  • LG 3: Find the future value and the present value of both an ordinary annuity and an annuity due, and find the present value of a perpetuity.

    • Annuity: Defined as a sequence of equal cash flows occurring at regular intervals over a defined period.

    • Ordinary Annuity: Specifies that cash flows occur at the end of each designated period.

    • Annuity Due: Specifies that cash flows occur at the beginning of each designated period.

    • Perpetuity: An annuity that continues indefinitely, providing a constant stream of cash flows without termination.

  • LG 4: Calculate both the future value and the present value of a mixed stream of cash flows.

    • Mixed Stream: Encompasses a series of cash flows that vary in amount and do not adhere to a consistent pattern.

  • LG 5: Understand the effect that compounding interest more frequently than annually has on future value and on the effective annual rate of interest.

    • Compounding interest more frequently than annually leads to a higher future value and an increased effective annual interest rate, reflecting the accelerated growth of investment returns.

  • LG 6: Describe the procedures involved in:

    • (1) Determining the necessary deposits to accumulate a desired future sum.

    • (2) Loan amortization, which involves the systematic repayment of a loan over time.

    • (3) Finding interest or growth rates relevant to investment performance.

    • (4) Finding an unknown number of periods required to achieve a financial goal.

Why This Chapter Matters

  • Accounting: Essential for understanding loan amortization schedules, lease payment calculations, and the determination of bond interest rates.

  • Information Systems: Critical for designing robust systems that accurately measure and value organizational cash flows, ensuring data integrity and reliability.

  • Management: Improves the ability to manage cash receipts and disbursements effectively, optimizing liquidity and financial performance.

  • Marketing: Provides justification for new programs and products based on financial merit, linking investment decisions to potential returns.

  • Operations: Facilitates the evaluation of investments in new equipment, streamlined processes, and inventory management, enhancing operational efficiency and cost control.

  • Personal Life: Aids in calculating savings values, estimating amounts needed for future financial goals, and valuing personal cash flow streams, promoting informed financial planning and security.

Pay Me Now or Pay Me Later: Cincinnati's Parking Deal

  • The city of Cincinnati entered into a deal involving the leasing of parking spaces to a private company in exchange for an upfront payment coupled with recurring annual payments.

  • Such deals involve a trade-off between receiving immediate funds and foregoing future income streams, requiring careful financial analysis.

  • Time Value of Money: Comprises analytical tools used to evaluate cash flows occurring at different times, taking into account the principle that money's value changes over time.

    • Due to the potential to earn interest, a dollar received in the future is worth less than a dollar received today, influencing investment decisions and financial strategies.

5.1 The Role of Time Value in Finance

  • Time value of money: It is better to receive money sooner because of its potential to earn interest, leading to greater wealth accumulation over time.

  • Money in hand today can be invested: Providing opportunities to generate returns and increase its value through various investment vehicles.

  • Managers need tools to compare cash inflows and outflows at different times: This allows for informed decision-making regarding investments, projects, and financial strategies.

Future Value Versus Present Value

  • Consider the opportunity to spend $15,000\$15,000 today, which is expected to produce $17,000\$17,000 over 5 years: Cash flows are represented as follows: −$15,000,$3,000,$5,000,$4,000,$3,000,$2,000-\$15,000, \$3,000, \$5,000, \$4,000, \$3,000, \$2,000

  • Time-value-of-money analysis helps managers answer such questions: By quantifying the financial implications of decisions, leading to more effective resource allocation.

  • Two primary ways to compare cash today versus cash in the future:

    • Future Value: Determining what amount in the future is equivalent to $15,000\$15,000 today, considering interest earned.

    • Present Value: Determining what amount today is equivalent to the stream of cash flows coming in the next 5 years, adjusted for the time value of money.

  • Time Line: Serves as a visual aid, depicting cash flows associated with an investment over time, facilitating comprehension and analysis.

Time Line

A horizontal line that marks time zero at the leftmost end and progresses to future periods from left to right, commonly used to illustrate investment cash flows.

  • Managers need to compare cash flows at a single point in time: To facilitate objective assessment and informed decision-making.

  • Future Value Technique: Involves compounding each cash flow to its future value at the end of the investment's duration and summing these values to determine the total future value.

  • Present Value Technique: Utilizes discounting to determine the present value of each cash flow at time zero, summing these values to obtain the total present value.

  • Managers usually adopt the present value approach for investment decisions: Because it provides a clear understanding of the current worth of future cash flows, enabling sound financial planning.

Computational Tools

  • Financial calculators and spreadsheets streamline time-value techniques: Facilitating efficient and accurate analysis, saving time and reducing the risk of errors.

Financial Calculators

  • Financial calculators come equipped with preprogrammed financial routines: Simplifying complex calculations and enhancing user productivity.

  • Important keys include N, I, PV, PMT, FV, and CPT: Representing essential variables used in time-value calculations.

  • Keystrokes are often menu-driven: Providing ease of use and intuitive operation.

  • Conceptual understanding is crucial: Ensuring users comprehend the underlying principles and mechanics of time-value analysis.

Electronic Spreadsheets

  • Spreadsheets boast built-in routines that simplify time-value calculations: Providing a versatile platform for financial analysis and modeling.

  • Solutions identify cell entries for calculating time values: Facilitating transparency and traceability in calculations.

  • Changing input variables automatically changes the solution: Enabling dynamic analysis and sensitivity testing.

Cash Flow Signs

  • Calculators and spreadsheets require accurate entry of cash inflows (positive) and outflows (negative): Ensuring precise and meaningful results.

Basic Patterns of Cash Flow

  • Cash flow, including both inflows and outflows, can be characterized by its general pattern:

    • Single Amount: A lump sum that is either held currently or expected at some future date (e.g., $1,000\$1,000 today or $650\$650 in 10 years).

    • Annuity: A consistent periodic stream of cash flow, such as paying or receiving $800\$800 at the end of each of the next 7 years.

    • Mixed Stream: A series of unequal periodic cash flows that lack a discernible pattern, exemplified by cash flow streams A and B.

5.2 Single Amounts

  • Investing $2,000\$2,000 per year at 5% interest from age 25 to 65 (40 years) would grow to $242,000\$242,000: Illustrating the substantial impact of long-term investing and the time value of money.

Future Value of a Single Amount

  • Need to find the value at some future date of a given amount of money placed on deposit today: By understanding future value, individuals can assess wealth accumulation potential.

  • (Future Value) The value at a specified future date of an amount placed on deposit today and earning interest at a defined rate: Determined through the application of compound interest principles over a set period.

  • Depends on the interest rate and time: Higher interest rates and longer time horizons result in greater future values.

The Concept of Future Value

  • Compound Interest: Interest earned on a given deposit that becomes integrated into the principal at the end of each specified period, facilitating exponential growth.

  • Principal: The base amount of money on which interest is calculated and paid.

  • Annual compounding is the most common type: Offering convenience and standardization in financial calculations.

The Equation for Future Value

  • FVn=PV∗(1+r)nFV_n = PV * (1 + r)^n

  • Where:

    • FVnFV_n = future value at the end of period n

    • PVPV = initial principal, or present value

    • rr = annual rate of interest paid

    • nn = number of periods (typically years)

  • Example:

    • Jane Farber places $800\$800 in a savings account paying 6% interest compounded annually.

    • How much will be in the account at the end of 5 years?

    • FV5=$800∗(1+0.06)5=$1,070.58FV_5 = \$800 * (1 + 0.06)^5 = \$1,070.58

Calculator Use

  • Input $800\$800 as PV (negative), 5 as N, 6 as I, compute FV: Simplifying the calculation process and providing accurate results.

Spreadsheet Use

  • Employ the FV function in Excel: FV(rate, nper, pmt, pv, type).

  • Example:

    • Present value = $800\$800

    • Annual rate of interest = 6%

    • Number of years = 5

    • Future value = $1,070.58\$1,070.58

A Graphical View of Future Value

  • The higher the interest rate, the higher the future value: Demonstrating the positive correlation between interest rates and investment growth.

  • The longer the period, the higher the future value: Underscoring the benefits of long-term investing strategies.

Present Value of a Single Amount

  • Determing the value today of a future amount of money: Essential for evaluating investment opportunities and making informed financial decisions.

  • How much would I have to deposit today at 7% to accumulate $3,000\$3,000 in 5 years?: Present value calculations help determine the initial investment needed to achieve a financial goal.

  • Depends largely on the interest rate and the point in time at which the amount is to be received: Indicating the impact of discount rates and time horizons on present value.

The Concept of Present Value

  • The process of finding present values is often referred to as discounting cash flows: Adjusting future cash flows to their equivalent value today, considering the time value of money.

  • What is the most I would pay now for an opportunity to receive FVnFV_n dollars n periods from today?: The present value represents the maximum amount one should be willing to invest today.

  • This process is actually the inverse of compounding interest: Discounting brings future values back to their present worth.

  • The annual rate of return is referred to as the discount rate, required return, cost of capital, and opportunity cost: Serving as the benchmark for evaluating investment profitability.

The Equation for Present Value

  • PV=FVn(1+r)nPV = \frac{FV_n}{(1 + r)^n}

  • Pam Valenti wishes to find the present value of $1,700\$1,700 that she will receive 8 years from now with an opportunity cost of 8%.

  • PV=$1,700(1+0.08)8=$918.46PV = \frac{\$1,700}{(1 + 0.08)^8} = \$918.46

Calculator Use

  • Using the calculator’s financial functions and the inputs shown, you should find the present value to be $918.46\$918.46: Enabling precise calculations of present value using financial tools.

Spreadsheet Use

  • Excel’s present value function PV(rate,nper,pmt,fv,type) mirrors the future value function.

A Graphical View of Present Value

  • The figure clearly shows that, everything else being equal,

    • the higher the discount rate, the lower the present value, and

    • the longer the period of time, the lower the present value: Demonstrating the inverse relationship between discount rates, time horizons, and present values.

5.3 Annuities

  • How much will you have at the end of 5 years if your employer withholds and invests $1,000\$1,000 of your bonus at the end of each of the next 5 years, guaranteeing you a 9 percent annual rate of return?: Illustrating the application of annuity calculations in wealth accumulation.

  • An annuity is a stream of equal periodic cash flows over a specified time period: Offering structured payments over time.

Types of Annuities

  • Ordinary annuity: Cash flow occurs at the end of each period.

  • Annuity due: Cash flow occurs at the beginning of each period.

Finding the Future Value of an Ordinary Annuity

  • FVn=CF∗((1+r)n−1)rFV_n = CF * \frac{((1 + r)^n - 1)}{r}

  • Example:

    • Fran Abrams wants to determine how much money she will have at the end of 5 years if she chooses annuity A, the ordinary annuity.

    • She will deposit $1,000\$1,000 annually, at the end of each of the next 5 years, into a savings account paying 7% annual interest:

    • FV5=$1,000∗(1+0.07)5−10.07=$5,750.74FV_5 = \$1,000 * \frac{(1 + 0.07)^5 - 1}{0.07} = \$5,750.74

Calculator use

  • Using the calculator inputs shown at the left, you can confirm that the future value of the ordinary annuity equals $5,750.74\$5,750.74

Spreadsheet Use

  • To calculate the future value of an annuity in Excel, we will use the same future value function that we used to calculate the future value of a lump sum, but we will add two new input values.

  • Recall that the future value function’s syntax is FV(rate,nper,pmt,pv,type).

Finding the Present Value of an Ordinary Annuity

  • Quite often in finance, there is a need to find the present value of a stream of cash flows to be received in future periods.

  • Annuitites are a stream of equal periodic cash flows.

  • PVn=(CFr)∗(1−1(1+r)n)PV_n = (\frac{CF}{r}) * (1 - \frac{1}{(1 + r)^n})

  • Braden Company wants to determine the most it should pay to purchase a particular ordinary annuity that consists of cash flows of $700\$700 at the end of each year for 5 years with a minimum return of 8%.

  • PVn=($7000.08)∗(1−1(1+0.08)5)=$2,794.90PV_n = (\frac{\$700}{0.08}) * (1 - \frac{1}{(1 + 0.08)^5}) = \$2,794.90

Calculator Use

  • Using the calculator’s inputs shown at the left, you will find the present value of the ordinary annuity to be $2,794.90\$2,794.90

Spreadsheet Use

  • The present value of the ordinary annuity also can be calculated as shown on the following Excel spreadsheet.

Finding the Future Value of an Annuity Due

  • Remember that the cash flows of an annuity due occur at the start of the period.

  • FVn=CF∗((1+r)n−1)r∗(1+r)FV_n = CF * \frac{((1 + r)^n - 1)}{r} * (1 + r)

Finding the Present Value of an Annuity Due

  • Each annuity due cash flow is discounted back 1 less year than for an ordinary annuity.

  • PVn=(CFr)∗(1−1(1+r)n)∗(1+r)PV_n = (\frac{CF}{r}) * (1 - \frac{1}{(1 + r)^n}) * (1 + r)

  • In Example 5.8 of Braden Company, we found the present value of Braden’s $700\$700, 5-year ordinary annuity discounted at 8% to be $2,794.90\$2,794.90. If we now assume that Braden’s $700\$700 annual cash flow occurs at the start of each year and is thereby an annuity due.

Calculator Use

  • Before using your calculator to find the present value of an annuity due, you must either switch it to BEGIN mode or use the DUE key, depending on the specifics of your calculator.

Spreadsheet Use

  • The present value of the annuity due also can be calculated as shown on the following Excel spreadsheet.

  • The format is PV(rate,nper,pmt,fv,type).

Matter of fact: Getting Your (Annuity) Due

  • Kansas truck driver Donald Damon got the surprise of his life when he learned that he held the winning ticket for the Powerball lottery drawing held November 11, 2009. The advertised lottery jackpot was $96.6\$96.6 million. Damon could have chosen to collect his prize in 30 annual payments of $3,220,000\$3,220,000 (303\$3.22 million 5 \$96.6 million), but instead he elected to accept a lump sum payment of $48,367,329.08\$48,367,329.08, roughly half the stated jackpot total.

Finding the Present Value of a Perpetuity

  • Perpetuity: An annuity with an infinite life, providing continual annual cash flow.

  • PV=CFrPV = \frac{CF}{r}

  • Ross Clark wishes to endow a chair in finance at his alma mater. The university indicated that it requires $200,000\$200,000 per year to support the chair, and the endowment would earn 10% per year.

5.4 Mixed Streams

  • Two types of cash flow streams exist: annuity and mixed stream.

  • Mixed Stream: A stream of unequal periodic cash flows that reflect no particular pattern.

  • Financial managers frequently need to evaluate opportunities that are expected to provide mixed streams of cash flows.

Future Value of a Mixed Stream

  • We determine the future value of each cash flow at the specified future date and then add all the individual future values to find the total future value.

  • Shrell Industries expects to receive the following mixed stream of cash flows over the next 5 years.

    End of year Cash flow

    1 $11,500\$11,500

    2 $14,000\$14,000

    3 $12,900\$12,900

    4 $16,000\$16,000

    5 $18,000\$18,000

  • If Shrell expects to earn 8% on its investments, how much will it accumulate by the end of year 5 if it immediately invests these cash flows when they are received?

Present Value of a Mixed Stream

  • Finding the future value of a mixed stream of cash flows is similar to finding the future value of a mixed stream. We determine the present value of each future amount and then add all the individual present values together to find the total present value.

5.4 Compounding Interest More Frequently Than Annually

  • Interest is often compounded more frequently than once a year.

Semiannual Compounding

  • Two compounding periods within the year.

  • One-half of the stated interest rate is paid twice a year.

Quarterly Compounding

  • Four compounding periods within the year.

  • One-fourth of the stated interest rate is paid four times a year.

A General Equation for Compounding More Frequently Than Annually

  • FVn=PV∗(1+rm)m∗nFV_n = PV * (1 + \frac{r}{m})^{m*n}

  • mm = number of times per year interest is compounded

Using Computational Tools for Compounding More Frequently Than Annually

  • We can simplify the process of doing the calculations by using a calculator or spreadsheet program.

Continuous Compounding

  • In the extreme case, interest can be compounded continuously.

  • In this case, m in Equation 5.8 would approach infinity. Through the use of calculus, we know that as m approaches infinity,

    • FVn=(PV)∗(er∗n)FV_n = (PV) * (e^{r*n})

    • Where e is the exponential function, which has a value of approximately 2.7183.

Nominal and Effective Annual Rates of Interest

  • Need objective comparisons of loan costs or investment returns over different compounding periods.

  • Nominal (Stated) Annual Rate: Contractual annual rate of interest.

  • Effective(True)Annual Rate (EAR)**: Annual rate of interest actually paid or earned.

  • EAR=(1+rm)m−1EAR = (1 + \frac{r}{m})^m - 1

  • For an EAR example related to the “payday loan” business, with discussion of the ethical issues involved, see the Focus on Ethics box.

  • At the consumer level, “truth-in-lending laws” require disclosure on credit card and loan agreements of the annual percentage rate (APR).

  • Nominal annual rate x multiplying the periodic rate x the number of periods

    • =annual

  • “Truth-in-savings laws,” on the other hand, require banks to quote the annual percentage yield (APY) on their savings products.

  • APY = effective annual rate

  • Used to quote the most attractive interest rates: law loan rates and high savings rates.

5.6 Special Applications of Time Value

  • Future value and present value techniques have a number of important applications in finance.

    • (1) Determining deposits needed to accumulate a future sum,

    • (2) Loan amortization,

    • (3) Finding interest or growth rates, and

    • (4) Finding an unknown number of periods. In 1993, the first.

  • Check Into Cash location opened in Cleveland, Tennessee.

Ethics: How Fair Is “Check Into Cash”?

  • The 391 percent mentioned above is an annual nominal rate 3 (365414).

  • Should the weeks rate (15 percent) be compounded to calculate the effective annual interest rate?

  • A payday loan is a small, unsecured, short-term loan ranging from $100\$100 to $1,000\$1,000 (depending on the state) offered by a payday lender such as Check Into Cash.

  • The cost of $100 of overdraft protection is $26.90, a credit card late fee on $100 is $37, and the late/disconnect fee on a $100 utility bill is $46.16.

  • Bankrate .com reports that nonsufficient funds (NSF) fees average $26.90 per occurrence.

Determining Deposits Needed to Accumulate a Future Sum

  • Determining the equal annual end-of-year deposits into an account paying annual interest of 6% to accumulate the $30,000\$30,000 payment at that time.

  • You know the future value ($\$30,000), but we want to solve for the annual cash payment to achieve that goal.

  • CF=FVn((1+r)n−1)rCF = \frac{FV_n}{\frac{((1 + r)^n - 1)}{r}}

  • Solved with a calculator or in excel

  • Example: You want it determine the amount of equal deposits required to accumulate $30,000\$30,000 at the end of 5 years with 6% interest.

Loan Amortization

  • Loan Amortization: The determination of equal periodic loan payments.

  • Loan Amortization Schedule A schedule of equal payments to repay a loan. It shows the allocation of each loan payment to interest and principal.

  • Finding the future payments, over the term of the loan, whose present value at the loan interest rate equals the amount of initial principal.

  • Example = Say you borrow $6,000\$6,000 at 10% and agree to make equal annual end-of-year payments over 4 years

  • CF=PV∗r(1−1(1+r)n)CF = \frac{PV * r}{(1 - \frac{1}{(1 + r)^n})}

  • Example - You want to determine the annual loan amount to full pay off a $6,000\$6,000 10% loan in 4 years

    • =annual

Finding Interest of Growth Rates

  • It is often necessary to calculate the compound annual interest or growth rate of a series of cash flows (rate of change in values).

  • Use Equation 5.1 or equation 5.13

  • r=((FVnPV)1/n−1)r = ((\frac{FV_n}{PV})^{1/n} - 1)

  • If the loan payments occur in equal amounts then refer back to section 5.3

  • Example = Ray Noble purchased an investment 4 years ago for $1,250\$1,250. What compound annual rate of return has Ray earned on this investment now that it is worth $1,520\$1,520?

Finding an Unknown Number of Periods

  • Sometimes it is necessary to calculate the number of time periods needed to generate a given amount of cash flow from an initial amount.

  • Case - Person wants to determine amount of time it will take initial deposit to grow to a specified future amount at a stated interest rate

It is assumed that students will have access to a financial calculator or computer when completing these problems.