# Time Value of Money

### Overview of Present and Future Value Concepts

Time value of money is the idea that money possessed today is worth more than the same dollar amount received in the future. All borrowers and lenders are impacted by the time value of money.

Return on an investment over time is known as interest, while the actual baseline amount of an investment associated with a loan or a borrowing is known as the principal. Lenders loan money with the expectation of earning interest on the loan while also ensuring the return of their principal. Borrowers expect to receive funds to use in their enterprise but likewise expect to pay back the principal plus the interest that accrues over time. The interest rate is typically stated as a percentage of the principal that reflects the cost of borrowing money, expressed in annual terms. For example, a $1,000 loan that carries a 6% stated rate of interest is expected to accrue$60 of interest each year the loan remains outstanding, or 6% of the principal amount. Therefore, after one year the economic value of the investment would be $1,060: the original principal of$1,000 plus $60 of interest. If such an investment remained outstanding for another year, the total value would be$1,123.60 $\left(\1\rm{,}060\:\times\:1.06\,=\,\1\rm{,}123.60\right)$. This effect, the repetitive calculation of interest earnings, means that an original fixed sum of money invested across time can earn interest many times over, each time interest is calculated. This is known as compounding. It can be valuable to own an investment that continues to compound on itself over a long period of time.

A lump sum of money or a fixed regular payment stream paid in the same amount every period for a period of time, known as an annuity, can be measured at a present value or a future value. Present value is the amount a sum of money is worth now, in the present. Future value is what a sum of money will be worth at a future point in time, given the effects of interest. If the rate of interest and compounding frequency are known, present value and future value can be calculated. A common practical use for the time value of money is estimating funds needed for an individual to retire. If the amount the future retiree plans to invest now is known and they can estimate the rate of interest or earnings on their investment, they will know how much money they will have to use after retirement.

### Present Value of a Single Amount

The present value of a single amount can be derived if future value (FV) is known, the interest rate is stated in annual terms (r), and the number of periods to which the rate will apply (n) is known. As present values are always less than the future value, this is called discounting to the present value.

To begin working with time value of money calculations, it is important to first distinguish the information available and the information needed to be calculated. If the amount in hand now or due in the future is a single sum rather than a repeated cash flow, it's called a lump sum. If the amount in question is a fixed, repeated cash flow made every period, it is an annuity. If the current amount is known, then present value is known, and calculations can be done to determine future value. If the future value is known, then calculations can be performed to determine present value.

The present value of a lump sum can be computed using future value (FV), the stated interest rate (r), and the number of periods the lump sum will compound.
${\rm{FV}}={\rm{PV}}\times\left(1+ r\right)^n$
For example, Sara needs to have $40,000 to replace her car next year. She can earn 6% interest on her investment and have just one year until she needs to purchase a new car.$40,000 is the future value, 6% is the interest rate, and 1 is the number of periods.
\begin{aligned}\40\rm{,}000&=\rm{PV}\times(1+0.06)^{1}\\\40\rm{,}000&=\rm{PV}\times(1.06)^{1}\\\40\rm{,}000&=\rm{PV}\times1.06\\\40\rm{,}000/1.06&=\rm{PV}\\\rm{PV}&=\37\rm{,}736\end{aligned}
Sara would have to invest $37,736 today and earn 6% on her investment for 1 year in order to have the$40,000 she needs to buy her car one year from today. In using the time value of money, it is always important to see if the answer achieved is reasonable. Since a sum of money increases in value over time, it makes sense that a sum that is worth $40,000 a year from now will be worth slightly less at present:$37,736. This can also be proven by taking the present value of $37,736 and multiplying by the interest rate of 6%. This equals$2,264 in accrued interest for one period. When added $(\37,736+\2,264)$ the result is $40,000. This example was for just one year. However, the time value of money formula works for multiple periods. Assume Sara already has a relatively new car and plans to wait 10 years to buy a new one. She again wants to have$40,000 set aside for the car and can earn 6% on her investment, which is her expected return each year.
${\rm{FV}}={\rm{PV}}\times\left(1+r\right)^n$
\begin{aligned}\40\rm{,}000&=\rm{PV}\times(1+0.06)^{10}\\\40\rm{,}000&=\rm{PV}\times(1.06)^{10}\\\40\rm{,}000&=\rm{PV}\times1.790848\\\40\rm{,}000/1.790848&=\rm{PV}\\\rm{PV}&=\22\rm{,}336\end{aligned}

Given that she now has 10 years for her money to compound, Sara only needs to invest $22,336 today in order to have the$40,000 she needs for her new car 10 years from today.

A table, rather than a calculator, can be used to solve time value of money problems. Using the one period example, find the interest rate r of 6% and the period n of 1. A table provides a factor of 0.943. By multiplying 0.943 by the future value of $40,000, present value can be calculated. There are some minor differences in calculated answers and the table answers because of rounding. $\40\rm{,}000\times0.943=\37\rm{,}720$ #### Present Value Table - Part 1 Using the multiple period example, the process can be repeated. A table provides a factor of 0.558 when a period of 10 and an r of 6% are used. So,$40,000 10 years from now is worth less in terms of present value than $40,000 one year from now. $\40\rm{,}000\times.558=\22\rm{,}320$ #### Present Value Table - Part 2 The same answer can also be derived using a future value of$1 table.
${\rm{FV}}={\rm{PV}}\times\left(1+r\right)^n$

The $40,000 future value, which had an $\rm n=1$, is divided by a future value of$1 factor of 1.06. The result is $37,736. Similarly, the$40,000 future value, which had an $\rm n=10$, is divided by a factor of 1.79085. The result is $22,336. #### Future Value of$1 Table - Part 2

Keep in mind that, depending on how many decimals are included in the present-value or future value table used, calculations may not match 100%. For example, if a present value is calculated using a present-value table with 5 decimals, and an attempt is made to replicate this result using a future value table rounded to 2 decimals, the results may be off by a few dollars. This difference is due to the two tables not using the same level of accuracy. Therefore, if tables are used, a consistent amount of decimals should also be used.

### Future Value of a Single Amount

The future value of a single amount can be derived if present value (PV) is known, the interest rate is known (r), and the number of periods to which the rate will apply (n) is known. As money we have today can be invested to generate a return, we refer to this as compounding or growing across time.
If the rate of interest is known, the future value of an amount today can be calculated after just one period. For example, Robert has 100,000 in hand today and can invest his money for one year at an 8% rate of interest. ${\rm{FV}}={\rm{PV}}\times\left(1+r\right)^n$ \begin{aligned}\rm{FV}&=\100\rm{,}000\times(1+0.08)^1\\\rm{FV}&=\100\rm{,}000\times(1.08)^1\\\rm{FV}&=\100\rm{,}000\times1.08\\\rm{FV}&=\108\rm{,}000\end{aligned} To derive the future value of a present amount, after multiple periods, it is possible to find the answer if the rate of interest is known. Assume Robert has100,000 in hand today and can invest his money for 20 years at an 8% rate of interest.
${\rm{FV}}={\rm{PV}}\times\left(1+r\right)^n$
\begin{aligned}\rm{FV}&=\100\rm{,}000\times(1+0.08)^{20}\\\rm{FV}&=\100\rm{,}000\times(1.08)^{20}\\\rm{FV}&=\100\rm{,}000\times(4.660957)\\\rm{FV}&=\466\rm{,}096\end{aligned}
A table can be used, rather than a calculator, to solve this problem. The same facts as the two previous examples can be used. Using the one period example, look for the interest rate r of 8% and the period n of 1. A table provides a factor of 0.926. By dividing the present value of $100,000 by 0.926, a future value can be calculated. There are some minor differences in calculated answers and the table answers because of rounding. $\100\rm{,}000/0.926=\107\rm{,}992$ #### Present Value Table for Future Value - Part 1 Using the multiple period example, look for the interest rate r of 8% and the period n of 20. If a table provides a factor of 0.215, the future value can be calculated by dividing the present value of$100,000 by 0.215. Again, a logical or expected result should be kept in mind. It is easy to calculate without thinking about whether the answer makes sense. For such a long time period, one would expect a significant increase in value, which is exactly what we see here, as $100,000 invested for 20 years at an 8% rate would grow to$465,116.
$\rm{PMT}=\100\rm{,}000/0.215=\465\rm{,}116$

This calculation can also be done using a future value of $1 table. #### Future Value of$1 Table: Future Value - Part 1

The same method in choosing factors is used. However, using the future value of $1 table, one can multiply the present value of a lump sum by the future value of$1 factor. The $100,000 present value, which had an $n=1$, is multiplied by a factor of 1.08. The result is$108,000.

#### Future Value of $1 Table: Future Value - Part 2 Similarly, the$100,000 present value, which had an $n=20$, is multiplied by a factor of 4.66096. The result is $466,096. ### Present and Future Value of an Ordinary Annuity An annuity is a financial arrangement that pays out a fixed payment stream over a set period of time. Both the present and future value of annuities can be calculated. Oftentimes, receipts or payments are made on a regular basis. An annuity is a fixed, regular cash flow that recurs over time. An ordinary annuity is an annuity payment that assumes the periodic payment occurs at the end of a time period. Lottery winners are frequently given the choice of a single lump sum payment option or an annuity option. Obviously, personal characteristics affect the decision, but so does the time value of money. Suppose Tom wins the lottery and is offered 40 annual payments of$100,000 or a lump sum of $1,800,000 payable immediately. Using the time value of money formula, Tom can determine the present value of the lump sum and the annuity and see which one is worth more. The formula for the present value of an annuity is: ${\rm{PV}}=\frac{\rm{PMT}}i\ \times\left(1-\frac1{\left(1+i\right)^n}\right)$ FV, PV, n, and i must be defined. A new parameter for use with annuities is PMT, or the annuity payment that recurs on an ongoing basis. In Tom's case, the PMT is$100,000, the number of periods is 40, and a market rate of interest for similar investments of 5% is assumed. The market rate of interest is the going or comparative interest rate for similar investments or loans.
\begin{aligned}{\rm{PV}}_n&=\frac{\100\rm{,}000}{i}\times\left(1-\frac1{\left(1+i\right)^n}\right)\\{\rm{PV}}_n&=\2\rm{,}000\rm{,}000\times(1-(1/7.039988712))\\{\rm{PV}}_n&=\2\rm{,}000\rm{,}000\times(1-0.142045682)\\{\rm{PV}}_n&=\2\rm{,}000\rm{,}000\times(0.857954)\\{\rm{PV}}_n&=\1\rm{,}715\rm{,}909\end{aligned}