Time Value of Money
Learning Objectives: ■
Understand the concept of and formulas for the time value of money ■
Apply the time value of money to financial analysis
The concept and the technique of time value of money (TVM) is fundamental in financial analysis and financial management. It is applicable to almost every aspect of long-term financial decision making for public organizations, private firms, and families (e.g., with regard to mortgage applications). This module introduces the TVM concept. It also provides typical applications of TVM in public organizations. The concept and techniques of TVM will be used as building blocks for capital budgeting and cost-benefit analysis in later chapters. Some Basic Concepts of the Time Value of Money The concept of the time value of money has to do with the fact that a dollar today is usually worth more than a dollar at a future time. TVM can be understood and demonstrated from at least two perspectives: financial and cultural. Financially, people are concerned about inflation and risk. The purchasing power of money decreases as the general price level of goods and services rises over time (i.e., due to inflation). Moreover, monies to be received in the future are promises that bear a degree of uncertainty, and this is reflected in our daily lives. For example, we normally earn interest when putting money into interest-bearing vehicles such as a savings account in a bank. Over time, the bank balance will grow, and a dollar today will become more than a dollar in the future. Culturally, people tend to prefer to receive benefits sooner rather than later. The desire for instant gratification gives rise to higher valuations of money at present than money to be received in the future. The degree of preference is culture laden and varies among demographic and social groups. This is especially true as people age; future money may not mean much to those who are at the end of their lives. Consequently, most people value money more in the present than in the future, leading to the concept of TVM. The question of how much more value people place on money that they receive at present than on money that they will receive in the future is important in financial decision making, especially for long-term financial planning such as capital budgeting. As a result, a set of terms has been developed to facilitate the operationalization of the TVM concept. This section provides an outline of the key terms and applications. A key element in TVM calculation is interest, which is the amount paid to compensate lenders (savers) for the use of their money for a specified period of time. The interest rate is the percentage rate at which interest is paid or charged for the use of money for a period of time. The interest rate reflects the valuation of the TVM and is often determined in the marketplace, as are bank lending rates for different risk levels and different loan maturities. The interest rate is a fundamental parameter in investment decision making and capital budgeting. A related concept is the number of periods. Periods are evenly spaced intervals of time (not necessarily years, although that is often the case). The length and number of periods relate to the frequency of compounding, which will be discussed further in the next section. Payments refers to monetary transactions. Payments can be positive (i.e., receipts) or negative (i.e., outlays). For instance, when you invest your savings in an investment instrument, you are making a money outlay, meaning a negative payment from your perspective. However, for the receiving institution, this transaction is a money receipt, or a positive payment. An annuity is a series of equally spaced payments made in equal amounts. The term annuity does not necessarily imply annual payments—the time period for payments can be quarterly, monthly, daily, etc. For instance, you probably pay your office or home rent in equal monthly payments. Formulas to Compute Time Value of Money and Simple Examples
FUTURE VALUE
Future value (FV) refers to the value that a given amount of money invested today will have at a specified date in the future. The calculation of FV requires three parameters: the present dollar amount, present value; the interest rate; and the number of periods into the future for which one wishes to calculate FV. The formula to calculate future value can be derived as follows: Suppose you invest $100 for 1 year at a 10% interest rate. At the end of 1 year, you will earn $10 in interest. Added to the $100 original principal, you will have $110 in your account. This could be expressed mathematically as follows: $110 = $100 × (1 + 0.10) If you continue to invest the $110 (i.e., the initial principal plus the interest) next year at the same 10% interest rate, you will get $11 dollars interest. Adding that to the new $110 principal amount, the account will grow to a total of $121. $121 = $110 × (1 + 0.10) = $100 × 1.1 × 1.1 = $100 × 1.12 If you continue to invest the principal plus interest for a third year, the account will grow to more than $133 by the end of the third year: $133.10 = $121 × (1 + 0.10) = $100 × 1.1 × 1.1 × 1.1 = $100 × 1.13 Summarize the equations above and you get the formula in Equation 1 below, which calculates the total future value of your investment at the end of year n: where pv is the present value of your investment, i is the interest rate, and n (nper) is the number of compounding time periods. This is the formula for calculating future value. It has plenty of applications in financial decision making. Here’s one example:
FUTURE VALUE: EXAMPLE 1
Suppose the government is trying to establish a reserve fund for future equipment replacement. It can invest in a fund that pays a 5% interest rate on an annual basis. You would like to know what the value of the reserve fund would be in 5 years if the government were to invest $5,000 today. This question can be solved easily if you apply the above future value formula. Plug the data pv = $5,000, i = 0.05, and n = 5 into equation 1 and you will find the future value of your reserve fund in 5 years: FV = $5,000 × (1 + 0.05)5 = $6,381.41 A more detailed calculation of FV involves a series of cash flows in future time periods: FUTURE VALUE: EXAMPLE 2 Suppose that your organization does not have a lump sum to invest in the reserve fund but can regularly put a small amount of money into it for future equipment replacement. How much would you have if you invested $2,000 at the end of each year for 5 years if the interest rate remains steady at 5%? This type of problem involves equal payments invested at an evenly spaced interval into an interest-bearing account that permits interest to be reinvested at the same rate for compounding. This is an annuity, as defined earlier, and its calculation for future value has been derived and expressed in equation 2 below. where PMT is the amount of payment or receipt, i is the interest rate, and n is the number of time periods. Reexamining the equipment replacement reserve fund example above, using equation 2, the future value of the $2,000 annual investment at the end of each year for 5 years will be as follows: The future-value-of-annuities formula can be rearranged to solve for the payment, given the amount that will be required at a future time. FUTURE VALUE: EXAMPLE 3 Now suppose that you need to replace a delivery truck in 5 years and the cost of the truck is $100,000. Assume also that your organization can invest money and earn a 5% interest rate at the end of each year for 5 years. How much money does the organization need to invest yearly so that it will have enough funds to purchase the truck at the end of the fifth year? The question can be solved using equation 2, with a minor modification. The amount of each yearly payment can be determined by dividing the factor into both sides of the equation. Plug in the interest rate, the future value, and the number of years, and you get the following: This is a bit more involved mathematically. Fortunately, most spreadsheet programs, such as Excel, have ready-made functions for TVM calculation. What you need to do is to click on the Insert function key, select PMT, and input the required parameters: i, n, and FV, in this case. On the Excel formula bar, you should see the following entry: =PMT(0.05,5,,100000). Excel will calculate PMT and put the value $18,097 in the cell where the formula is inserted. Formulas to Compute Present Value and Simple Examples We have discussed the concepts of future value and annuity and developed the formula to compute them. Conversely, we can calculate the present value (PV) of a future amount or future cash flow if we know the future value, the interest rate, and the number of periods. The derivation of present value (equation 3) is simple. PV can be solved by dividing (1 + i)n into both sides of equation 1: This formula can be used to calculate the present value of a future cash flow so that cash flows at different points in time can be compared. Here is an example to show how this works:
PRESENT VALUE: EXAMPLE 1
A donor has promised to make a $1 million contribution in 10 years for the designated purpose of upgrading a building that the city university owns. However, the university has the option to receive a reduced amount of $800,000 if it would like to have the funds now. If the university has the option to invest the money now at a guaranteed 5% interest rate for 10 years, would it be better for the university to ask for the $800,000 now or wait for the $1 million payment in 10 years? This question can be solved by calculating the present value of the future amount and comparing it with the amount offered at present. Plugging $1 million into equation 3, we can calculate the present value of this future cash flow as follows: Given that the present value (i.e., today’s equivalent value of $1 million received in the future) is lower than what has been offered as an alternative, and investing the money will earn a guaranteed 5% interest rate for the next 10 years, it would be wise for the university to accept the $800,000 now.
PRESENT VALUE OF AN ANNUITY
The formula to compute the present value of an annuity is derived similarly to that for the future value of annuity. The present value can be expressed with the following formula (equation 4): Among other applications, the present value of annuities can be used in making purchase decisions when the relative values of different forms of equipment are the main concern.
PRESENT VALUE: EXAMPLE 2 For instance, the health department of a small city is considering purchasing a refrigerator to store its flu vaccine. There are two brands of refrigerators that could serve the health agency’s purposes equally well. The main consideration is therefore the total cost, meaning the initial purchase price and the operating cost. Given that the equipment will last for 5 years, the timing of the cash flows should be taken into consideration. The relevant information for the two refrigerators is presented in Table 6.1. Apparently, Model A is more expensive to buy but less costly to operate and maintain. Model B is just the opposite, meaning it’s less pricey but more expensive to run. The question is, Which one should you buy, taking into consideration the initial cost and operating expenses? Solving this problem requires calculating and summing two present values of costs: the purchase price and the operating costs for the two alternatives. Determining the present value of the operating cost requires calculating the present value of annuities, with the known interest rate of 5%, the different operating costs, and the number of periods being 5 years. Plug these parameters into the annuity equation and add the initial purchase price, and we get the following results: TABLE 6.1 Purchase and Operating Costs of Alternative Refrigerators For Model A: For Model B: Given that both refrigerators function equally well and both will last for 5 years, Model B is more cost-effective and therefore should be suggested for procurement, when considered from a purely economic viewpoint. Net Present Value: Decision Criteria The above examples illustrate financial decision making, taking into consideration the time value of money. However, not all cases are as simple as these. Often both cash inflows and cash outflows occur at different points in time over the course of one project. The cash flows are not necessarily evenly spaced and in equal amounts. Moreover, decision makers are most concerned about comparing the potential net gains or losses from different project or investment alternatives, within the constraints of budgets and larger social and economic considerations. These questions require the calculation of the net present value (NPV) of the projects or programs or investments (the terms are used interchangeably in this module) involved. Net present value (NPV) of a project is defined as the difference between cash inflows and cash outflows, both discounted to their present values. The general formula to calculate NPV is presented in Equation 5. where b = cash inflows, c = cash outflows, i = discount rate, t = time period, and n = number of time periods. NPV measures the net gain or loss of an investment, project, or program. It is used as a criterion in economic decision making. NPV appeals to common sense, because the criteria can be applied as follows: If NPV is greater than zero, then the project creates value; consequently, we should accept the project. If NPV is less than zero, then the project destroys value; we therefore should reject the project. If NPV is zero, then we are indifferent to the project. Let’s consider an example: The city hospital is considering establishing a new and advanced medical lab in an existing facility to provide enhanced diagnostic services. Your initial analysis indicates that this new lab will bring in $3 million of annual operating revenue to the hospital. It costs approximately $1 million per year to maintain and operate. The start-up cost of the lab is $10 million (primarily for equipment acquisition and installation), and the lab will need major updates and services in about 5 years with no residual value for the equipment purchased (for simplicity’s sake in this example). The best bank loan you can get carries an interest rate of 4.987%. The relevant data are presented in Table 6.2. Based on all the available information, would establishing the lab be a good investment from the hospital’s financial viewpoint? This problem can be solved using equation 5. Plugging in all the parameters, you can calculate the present value (PV), which is listed in the last column of Table 6.3. Adding the PV over time (netting the present value of the cash inflow of the cash outflow), the bottom row of the table shows the NPV of the potential investment. Given that the NPV is negative, indicating a net loss if this project were to be undertaken, your recommendation would be not to undertake this project in its present form. It should be noted that the above analysis is based purely on financial considerations. The potential benefits of the new lab to the community at large and to individual patients would be very relevant to a comprehensive and social cost-benefit analysis. Governments by nature should consider social benefits and costs when making project decisions, and this is the focus of Module 17: Cost-Benefit Analysis. TABLE 6.2 Future Cash Flows for New Lab Project (in thousands of dollars) TABLE 6.3 Discounted Cash Flows and NPV for New Lab Project (in thousands of dollars)
Summary
This module introduces fundamental concepts and technical skills related to time value of money (TVM). It then applies these basic concepts to long-term financial decisions in both personal and public organization contexts. The knowledge and skills learned from this module will be used for cost-benefit analysis and capital budgeting, to be discussed in later modules. Discussion Questions 1. Why is the concept of time value of money important to long-term project decisions? 2. Can you give examples of annuities from your own experience?
Assignments 1. You have received $100,000 from your grandparents to be used for your graduate studies. You plan to start a program in 5 years. You can invest the money at 6% interest, compounded annually. How much will you have in 5 years? 2. Mobile Health needs to replace its mobile medical unit to continue providing diagnostic services and preventative medicine to city residents. The van costs $120,000, and a bank is willing to lend the nonprofit the money at a 6% interest rate for 5 years. Payments are due at the end of each month, and interest is compounded monthly. How much should you budget for the monthly payment? 3. The Department of Corrections (DOC) needs to replace two of its prison transport buses. There are two companies that have a track record of making reliable vehicles, and both make a bus that meets the needs of the department. The DOC pays an 8% interest rate on money it borrows. The bus from Company A has a purchase price of $105,000 and a 10-year service life expectancy, and it averages $2,000 per year in maintenance costs. The bus from Company B has a purchase price of $110,000 and a 10-year service life expectancy, and it averages $1,500 per year in maintenance costs. Which bus should the Department of Corrections select? 4. The information technology (IT) department has recently completed a major refurbishment and upgrade of the city’s data center, at a cost of $10,145,825. The chief information officer (CIO) has informed the central budget office that the IT department will need to do the same kind of upgrade in 5 years, and he expects the total cost of that project to be 10% more than this year’s project, accounting for better future technology and future increases in costs. The city does not expect to have enough funding available to pay for the refurbishment in 5 years and must put money away during each of the next 4 years to meet this need. How much money should the city put aside in each of the next 4 years to be able to pay for the data center refurbishment 5 years from now, assuming the city can invest the money at a 4.5% interest rate? 5. The local community center has been notified that it is the beneficiary of an endowment. The community center can accept cash today in the amount of $550,000 or receive $875,000 in 5 years. The community center expects to be able to invest the money at a 5% interest rate if it were to take the money now. Should it take the money now or wait? 6. The municipal golf course has made a budget request to remodel and upgrade the clubhouse. The project will cost $7,200,000, based on reliable estimates. The additional annual operating expenses after the upgrade are expected to be $200,000 per year. An analysis of the experiences of other municipal golf courses reveals an expected revenue increase of $1 million per year after the upgrade. The useful life of the upgraded clubhouse is estimated at 9 years, and the golf course expects to get a loan for the renovations at an interest rate of 4.25%. Should the renovations be approved? Additional Reading Van Horne, J. C., and Wachowicz, J. M. (2008). Fundamentals of financial management (13th ed.). Essex, UK: Pearson Education.