Micah Jeffery
October 9, 2023

Abstract

     This paper explores the concept of the time value of money, tracing its historical roots from ancient philosophy to its modern-day application as a foundational principle in financial mathematics. The paper talks about the main equations associated with the time value of money, including Future Value, Present Value, and Net Present Value, showing their significance in accurately assessing the value of investments while considering important factors like taxation, inflation, and risk. The paper also advocates for adjustments in terminology, namely the use of "rate of return" over "interest rate." It also shows the importance of post-tax income in calculations. The usefulness of the time value of money is shown by its wide-ranging applications in personal and corporate finance, inventory management, and tax optimization. This concept remains an indispensable tool for informed decision-making and profit maximization in the dynamic landscape of economics.

Introduction

     The concept of the time value of money is based on the idea that a nominal sum of money has more value now than the same nominal sum of money has in the future. In other words, a dollar today is better than a dollar tomorrow. The reason why this is the case is that if the dollar was invested today, it would grow to be worth more tomorrow in nominal terms. If that dollar was not received until tomorrow, then the interest or growth from the potential investment would be foregone. This missed opportunity for profit is the opportunity cost of a delay in receiving a sum of money.

     Initially, the general concept of the time value of money was explored by Greek philosophers such as Thales and Aristotle, by the Talmud, by figures like Martín de Azpilcueta and Halil MeteSoner, and even by Jesus. In the Parable of the Talents, Jesus tells a story of a master who gave three of his servants each a sum of money while he was away. When the master returned two of the servants returned more than the sum that they were given but the third only returned the principal. The master responds to the third servant by saying “You ought to have invested my money with the bankers, and at my coming I should have received what was my own with interest” and he “cast the worthless servant into the outer darkness. In that place there will be weeping and gnashing of teeth” (English Standard Bible, 2001, Matthew 25:14-30). The money was worth less when the servant returned it than when he received it because of the opportunity cost of the foregone interest. The servant was therefore treated as if he had stolen money because according to the time value of money, he had.

     The time value of money existed as a concept long before the equation was solidified but in 1202, Fibonacci authored the first book to use the time value of money mathematically (Slobodnyak & Sidorov, 2022, pp. 1-2). This concept in the field of financial mathematics has since expanded and assumed the role of one of the fundamental principles of finance.

Equations

     The most basic equation of the time value of money, which deals with determining what a sum of money will be worth in nominal terms after a given period, is typically written as follows:

FV = PV × (1 + i)n

     FV, the Future Value, is the nominal amount that the initial sum will equal after the period of time. PV, the Present Value, is the initial sum of money. i, the Interest Rate, is the expected or known rate of return for a given period. There is an issue with referring to it as the interest rate but that will be addressed later in this paper. The variable n represents the Number of Periods. The same period must be used to determine i as well as n for the equation to work (Dubos Joseph Masson, 2001, pp. 36-37). This equation outputs the future value of a given present value, but it can be rearranged to output the present value of a given future value:

PV=FV/(1+i)n

     The variables are the same in this equation except for i, which is referred to as the discount rate or required rate of return rather than the interest rate but functions in a similar way. The next relevant equation is called the Net Present Value, and it is used to determine the sum profitability of an investment when all future values are discounted to the present:

NPV = Present Value of Cash Inflows – Present Value of Cash Outflows

     Traditionally, if an investment initially costs $100 and then brings in $110 after 10 years, the net value would be $10, and the investment would be mistakenly considered profitable. With NPV however, the cost’s present value would equal $100 while the present value of the revenue, assuming a required rate of return of 5% per year, would equal about $67.53. The NPV would equal -$32.47, which demonstrates how useful this equation is in determining actual value.

To find the present value of an ordinary annuity, a fixed regular payment at the end of each period, the following equation can be used:

PV=PMT×([1-(1+i)n]/i)

     PMT represents the payment amount for each period. The PMT must remain constant each period for an ordinary annuity. If the payment varies in a nonregular way, then the present value of each payment must be calculated separately. Especially for situations that require many calculations, it is expedient to use Excel, a financial calculator, or another similar tool rather than to solve each equation individually.

Rate Adjustments

     While the principle of the time value of money has played a significant role in the financial sector, there are several areas where improvements can be made to enhance clarity and accuracy. The variable i, which is commonly denoted as Interest, should be substituted with the symbol r, to denote Rate of Return, which provides a more comprehensive description of the variable. Secondly, the net income after taxes should be used rather than the net income before taxes. Thirdly, inflation must be considered when calculating the real rate of return. Lastly, an adjustment that should be made, to be able to accurately compare the Present Value or Future Value of two or more investments, is to standardize the rates by adding a risk premium.

     While the variable, i is almost universally referred to as the interest rate, that is not completely accurate. According to Alexander Pierre Faure, “Interest rates are the reward paid by a borrower (debtor) to a lender (creditor) for the use of money for a period, and they are expressed in percentage terms per annum” (2014, p. 1). Although the TVM formulas can and often do use an interest rate for the i variable, it is not limited to simply the yearly payment amount of money borrowed or lent. Rate of growth in the market value of a stock or other asset as well as the expected profit rate from an investment such as a piece of equipment or acquisition of a company are also common inputs for the i variable. The term “interest rate” is used in the context of TVM to represent the rate of return but for the sake of definitional uniformity and accuracy, “rate of return” should be used instead of “interest rate.” The rate of return is the net gain or loss over a specific amount of time as a percentage of the investment’s initial cost which better describes the variable. The variable should also be represented with an r rather than an i to represent “rate of return.”

     Because the tax rate affects the amount that an investment brings in, it should never be ignored. If a $100 investment with a rate of return of 10% increases by $10 and the tax rate is 20% which would be $2, then the after-tax income will be $8. The rate of return should be determined using the expected after-tax amount rather than the before-tax amount. In the previous example, the nominal rate of return should be calculated as 8% rather than 10% because the after-tax amount was $8 and not $10. Because tax rates are based on non-inflation adjusted rates, this is vital to accurately compare investments.

     Inflation, which is the general increase in price of all goods in the economy, has a significant impact on the real rate of return. The nominal rate of return is the stated (or estimated) rate of return that ignores the impact of inflation on the real value of a sum of money. As inflation increases, the purchasing power (i.e., the real value) of the dollar decreases. The real rate of return is the nominal rate of return less the rate of inflation. If the rate of return (adjusted for risk) is 5% but the rate of inflation that year was or is expected to be 3%, then the real rate of return would be 2%. This would make the rate accurate by denoting the amount of growth in value rather than the growth in dollars.

     If two investments have the same expected rate of return but one involves almost no risk while the other requires a substantial amount of risk, then the former is always preferable. If the nominal rate of return is used in the time value of money equation, then two investments of different real values will be shown as the same Present Value. To accurately calculate the Present Value or Future Value of these investments, the rates need to be discounted to the standard of the risk-free rate. The risk-free rate is the rate at which an investment can be made with essentially no chance of loss or variability. The standard example of this rate is a short-term U.S. Treasury bond because since it is backed by the U.S. government, it is typically the closest an investment can get to having no risk at all. If the two investments are a company’s bond and a company’s stock with nominal expected rates of 6% and 7% respectively, if the bond is deemed only slightly riskier than a short-term U.S. Treasury bond and the stock is expected to be relatively volatile, then the nominal rates could be discounted 1% and 3% respectively, to standardize them according to risk. The amount that should be discounted can vary drastically and there is no one formula that can be used to find the correct amount to discount from the rate, but this conservative approach should be utilized to standardize the rates.

     Combining each of these adjustments, the r should be determined by the after-tax nominal growth rate minus the inflation rate and minus the risk discount rate. If an investment of $100 was expected to bring in $10 before a tax of 20% net of any expenses, then the investment would have increased $8 in nominal terms. An increase from $100 to $108 would be an 8% nominal rate of return. If the expected inflation rate was 3% and the risk discount rate was 2%, then the real rate of return would be 3% (8%-3%-2%). Using this rate will ensure that the rates and values are all standardized and adjusted for precise and real values.

Applications

     The concept and formulas of the time value of money have many applications, including for personal and corporate investments, for inventories, and for taxes.

When evaluating investments, people may overlook factors like opportunity cost, tax rates, and inflation, which can significantly impact their profitability. Leveraging the time value of money equations and concepts elucidates the decision-making process, mitigates ill-advised decisions, and maximizes profitability. Given the scenario of two alternative investments, one that requires $100 and returns $15 each year after tax for 15 years and another riskier investment that costs $100 and results in $500 after 20 years after tax. Without making the TVM calculations, it would be incredibly challenging to correctly choose the best option. Given an inflation rate of 3%, a risk-free rate of 5%, and a risk premium of 1% for the 15-year investment, the required rate of return would be 9%. Using this required rate of return in the TVM equation would result in a Net Present Value of $20.91 ($120.91 - $100). If the inflation and risk-free rate stayed the same and the 20-year investment had a risk premium of 3%, the required rate of return would be 11%. The Net Present Value of the 20-year investment would be equal -$37.98 ($62.02-$100). Given these values, the 15-year investment is favorable while the 20-year investment should not be taken.

     Using the principle of the time value of money, as well as estimates of inflation, is important to effectively manage inventory quantities. According to a journal article on how inflation and the time value of money impact inventory ordering decisions, “a finite replenishment inventory system is affected by inflation and time value of money - the difference becomes significant as the inflation rate and the time value of money increase” (Sarker & Pan, 1994, p. 71). When inflation is higher than the discount rate or the rate of return, companies should hold a higher quantity of inventory because it is better to purchase the material or goods while the price is low. Conversely, if inflation is lower than the rate of return, then it is better to hold a smaller amount of inventory. Comparing the discount rate and the rate of inflation can help companies effectively manage inventory quantities in a changing economy.

     According to the IRS, the total amount in individual tax refunds for April 2023 was $236.615 billion (2023). This was the amount of tax withholdings that exceeded the amount of tax due by individuals. Given that a sum of money is worth less in the future than it is now, the $236.615 billion in refunds is worth less than the periodic withholdings. Assuming the average individual’s accessible rate of return is 5%, that paychecks are paid monthly (with each month withholding $19.7 billion in excess of taxes owed), and that the average refund was issued at the end of May, the Future Value of the excess withholdings would equal $247.2 billion. This amount is almost $10.6 billion greater than the amount refunded. From the individual’s standpoint, it is important to minimize one’s tax withholdings to avoid effectively paying more taxes than necessary.

     The time value of money is a fundamental concept with implications in various aspects of personal finance and corporate decision-making. Whether evaluating investment opportunities, managing inventory quantities, or considering tax implications, understanding the impact of inflation and the discount rate is important. As demonstrated through the example of investment choices, TVM calculations can make the difference between a wise financial decision and a costly one. In the context of inventory management, the relationship between inflation and the time value of money is important to understand for optimizing inventory quantities to adapt to changing economic conditions. Lastly, in the realm of taxation, the concept of TVM highlights the need to minimize tax withholdings to maximize the value of one's money. Therefore, a firm grasp of the time value of money is an invaluable tool for individuals and organizations alike, enabling them to make informed financial choices and ultimately achieve their financial goals.

Conclusion

     In conclusion, the concept of the time value of money is deeply ingrained in the world of finance, dating back to ancient philosophies and evolving into a fundamental principle of modern financial mathematics. Its equations, such as the Future Value, Present Value, and Net Present Value, provide invaluable tools for assessing the true worth of investments, taking into account factors like taxation, inflation, and risk. By making adjustments, such as replacing "interest rate" with "rate of return" and considering post-tax income, this concept becomes even more precise and applicable. With applications spanning personal and corporate finance, inventory management, and tax optimization, the time value of money remains an indispensable asset for informed decision-making and maximizing profitability in an ever-changing economic landscape.

References

Dubos Joseph Masson. (2001). Essentials of Cash Management (7th ed., pp. 36–38). Association for Financial Professionals.

English Standard Bible. (2001). ESV Online. https://esv.literalword.com/

Faure, A. P. (2014). Interest rates 1: What are interest rates? SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2542083

IRS. (2023, April 21). Filing season statistics for week ending April 21, 2023. Www.irs.gov. https://www.irs.gov/newsroom/filing-season-statistics-for-week-ending-april-21-2023

Sarker, B. R., & Pan, H. (1994). Effects of inflation and the time value of money on order quantity and allowable shortage. International Journal of Production Economics, 34(1), 65–72. ScienceDirect. https://doi.org/10.1016/0925-5273(94)90047-7

Slobodnyak, I., & Sidorov, A. (2022). Time value of money application for the asymmetric distribution of payments and facts of economic life. Journal of Risk and Financial Management, 15(12), 573. https://doi.org/10.3390/jrfm15120573