# The Fundamental Theory of Time Value of Money

Time value of money (TVM) essentially means that money has different values based on when the money is received. In essence, money today is worth more than money in the future by the simple fact that money today can be used to earn interest in the future. Within financial accounting, time value of money needs to be factored in due to the need to value financial resources such as assets and liabilities. For financial statements, the statements must factor in time value of money because the financial statements need to reflect the current value of those resources. Time value of money calculations are used for valuing bonds, leases, pensions, long-term assets, stock-based compensation, consolidations, and disclosures. Not all of these topics will be covered as they are generally more advanced than the scope of this textbook.

Within financial accounting, there are two major concepts: present value and future value. Present value means the value of an asset when all future cash inflows have been factored in to the present by using the time value of money. The concept of future value is also important. Future value is the value of an asset at a specific future date after factoring in all cash inflows and outflows and adjusting for the time value of money.

In this chapter, we will study all of the various calculations needed to calculate the time value of money on various types of cash flows. We will study both present value as well as future value calculations. Within those topics we will cover single-sum calculations as well as annuity calculations. An annuity is a series of payments generally paid in fixed intervals (monthly, quarterly, annually etc.) while a single-sum is made in one lump-sum payment.

# Interest

Interest is a cash payment from the borrower to a lender for the use of borrowing money. In essence, interest is the cost of borrowing money from a lender. When a loan is repaid, the borrower will be paying a combination of principal and interest with each payment. For example, imagine that you borrow \$5,000 from a bank to purchase a high-end bicycle and you agree to pay 5% interest. You also agree to repay the loan in with a single payment in one year from the date of the loan. After one year, your payment will be \$5,250 which consists of \$5,000 principal and \$250 of interest. Interest is always expressed in terms of percentages which makes comparisons easier.

Many factors will affect the rate of interest. These factors could be general market conditions and credit worthiness of the borrower. Credit worthiness is the likelihood that the borrower will not be able to repay the loan in full. When a borrower is unable to repay the loan in full it is called defaulting on the loan. There is a general relationship between credit risk and interest rate charged by lenders. Lenders are willing to offer lower interest rates to borrowers with good credit ratings. Borrowers with poor credit ratings will be charged higher interest or be refused the loan.

When computing interest, we need to understand the three variables in the calculation:

Principal: The original amount borrowed

Interest rate: The percentage rate charged to the borrower as a cost of borrowing the money.

Time periods: The amount of time the loan is to be repaid, expressed in a standardized unit of time (days, weeks, months, years etc.).

The amount of interest paid is increased when the principle is larger, the interest rate is higher or the amount of time the loan is outstanding is longer.

# Simple Interest and Compound Interest

Simple interest computes interest only on the principal while compound interest, on the other hand, computes interest based on the principal and the accumulated interest. For example, if \$5,000 is borrowed at 5% per year with 12 monthly payments then the monthly payment would be \$437.50 (\$416.67 principal and \$20.83 interest) based on simple interest. The total amount repaid would be \$5,250. The reason is because the 5% interest is only being calculated on the \$5,000 even though a little fraction of interest has accumulated each month. If we compute the same amount using compound interest then the total amount repaid would be \$5,255.81, with an additional \$5.81 being repaid compared to simple interest. So where did the extra \$5.81 come from? The extra interest comes from interest being calculated on the accumulated interest. This extra amount might not seem like much but when compound interest is utilized over long period of time then the amount of extra interest earned can result in extravagant sums of money.

# Simple Interest

As mentioned before, simple interest calculates interest based only on the principal amount. Simple interest shows us the growth of the principal amount over a period of time. The simple interest formula is as follows:

Interest = principal X interest X number of periods (time)

Where principal is the initial amount in a monetary unit (dollars, euros, yen etc.), interest is the percentage rate of interest and number of periods is number of intervals that interest will be calculated on the principal.

For example, using our original example from above related to the bicycle, we can calculate simple interest on that transaction as follows:

Interest = \$5,000 X .05 X 1

Interest = \$250

The principal is \$5,000 at 5% interest paid in one lump sum after one year. The 5% interest is converted to a decimal of .05. We can also do a similar calculation for months or days. For example, let's say the loan was repaid over 8 months. The calculation would be as follows:

Interest = \$5,000 X .05 X 8/12

Interest = \$166.67

As you can see, we convert our period into a fraction of 8/12. We can also do some simple analysis using this formula. For example, let's assume we wanted to figure out how much interest we were going to be paying daily for the bike. We could figure that out as follows:

Interest = \$5,000 X .05 X 1/365

Interest = \$.68

Therefore, we can conclude that the bike costs us 68 cents per day to borrow.

# Compound Interest

Compound interest is calculated based on the principle plus any accumulated interest that has not been paid to the borrower. The best way to think about it is if you had \$25,000 in a savings account. Every month, the bank will deposit the monthly interest into the account and next month's interest payment will be calculated based on the principle plus the interest you were paid last month. The end result is that after each month, the next month's interest payment will be slightly higher as interest is being paid on previous interest. The illustration below demonstrates the calculation of compound interest:

As seen above, we can see the amount of interest being paid each month gradually increasing. The amount being paid above \$1,500 is being earned based on interest that was paid in previous months. Also note that no additional funds have been deposited into the account other than the original \$25,000 deposit in January. We can notice the difference between simple and compound interest by recalculating based on simple interest.

The primary difference is seen in the difference in the ending balances. The higher ending balance in the compound interest calculation reflects how much interest has been earned on interest paid in previous months.

Within business, simple interest is only used for short-term decisions whereas compound interest will be used for long-term decisions.

# The Fundamental Theory of Present Value and Future Value

Essential to accounting is the need to determine present value or future value. This type of calculation will allow us to factor in future cash flows and the impact of interest on those cash flows. We can use the time value of money calculations to calculate either a present value or future value.

Present Value: The value of a future cash flow stream that is discounted to a single value if it were received in a single lump sum. The present value is concerned with the value of the cash flow stream on the day of the valuation. The discount interest rate is considered to be compounding.

Future Value: The value of a cash flow stream that is discounted to a single value as if it were to be received in a single lump sum at a future date. The future value is concerned with the value of the cash flow stream at a future date. The discount interest rate is considered to be compounding.

Essential to both types of calculations is the rate of interest and the compounding periods. The rate of interest is the interest that is to be factored into the future value or present value calculation. The rate of interest needs to be adjusted for the time period of compounding periods. For example, if the compounding period is monthly then the interest rate needs to be the amount of interest that would be accumulated during that time period. The compounding period is the frequency that the interest is calculated. Common compounding periods are annually, semiannually, quarterly and monthly.

To help assist with these calculations we can use either a table of interest and compounding factors or use a more modern method of using a calculator. The type of calculator can be a standard calculator or a business calculator. We can also use calculators that are specifically designed for calculating time value of money calculations. The specific TVM calculators are generally easier to use. In this textbook we will discuss both methods to show you how each is used.

Before we begin we will learn the basics of present value and future value. We will learn how to use the tables and calculators after we have mastered the fundamentals. By learning the fundamentals first will allow you to understand the material better.

# Present Value

Present value calculations consist of a single sum (one payment) and annuities (multiple payments). We will cover the single sum calculations first since they are easiest and then proceed to learn about the annuity calculations. Remember, the present value is converting the cash flow stream into the value at the date of the calculation which is the current date.

### Present Value of Single Sums

The process of converting a given future value amount is performed by using the process of discounting. Discounting is when we apply the impact of interest on a future value to convert it to a present value.

As learned earlier, we must have three variables in all time value calculations, which are:

1) Principal (for present value calculations this amount is the future value of the single sum)

2) Interest

3) Time

Once we have identified the relevant information for each variable, our first step is to calculate the present value factor. Once we have calculated the present value factor we will simply multiply the future value by the present value factor. The result of the calculation is the present value. As an important reminder, when converting a future value to the present value the present value is always smaller than the future value due to the impact of interest.

Example: You are to be given \$5,000 at the end of three years. The interest rate is 5% per year. What is the present value?

Our first step is to calculate the present value factor. We use the following formula to do the calculation:

Present Value Factor = 1/(1 + I)n

where I is the interest rate and n is the number of periods.

Using the data given in our example, we can calculate the present value factor as follows:

Present Value Factor = 1/(1+.05)3

Present Value Factor = 1/(1.05)3

Present Value Factor = 1/1.157625 or 0.8638375985314761

Now that we have simply multiply the \$5,000 by 0.8638375985314761 as follows:

Present Value = \$5,000 x 0.8638375985314761 = \$4,319.19 (rounded)

The \$4,319.19 means you would need that amount at 5% interest over 3 years to earn \$5,000.

### Methods to Simplify Present Value Calculations

If we think about the calculation we just performed we quickly realize that the present value factor is a relatively simple calculation. The time periods are usually standardized and common interest rates are used frequently. To help speed up time value calculations, we can use a chart that contains all of the common present value calculations. Then the process of calculating the present value factor is simply a matter of locating the correct factor in the chart. Below is a typical present value chart for a single-sum:

 Periods 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 1 0.99010 0.98522 0.98039 0.97561 0.97087 0.96618 0.96154 0.95694 0.95238 0.94787 0.94340 2 0.98030 0.97066 0.96117 0.95181 0.94260 0.93351 0.92456 0.91573 0.90703 0.89845 0.89000 3 0.97059 0.95632 0.94232 0.92860 0.91514 0.90194 0.88900 0.87630 0.86384 0.85161 0.83962 4 0.96098 0.94218 0.92385 0.90595 0.88849 0.87144 0.85480 0.83856 0.82270 0.80722 0.79209 5 0.95147 0.92826 0.90573 0.88385 0.86261 0.84197 0.82193 0.80245 0.78353 0.76513 0.74726

There are time value factor charts for both present value and future value as well as single-sum and annuities. The most common factor charts are the following: present value of single-sum, future value of single-sum, present value of an ordinary annuity, future value of an ordinary annuity, present value of an annuity due and future value of an annuity due. We will briefly cover each of these charts in the subsequent sections. The following are links to the six factor charts:

Another method we have within our reach is the use of a calculator. Nearly all business calculators have the ability to calculate time value of money. In this case we will cover a basic calculation using a standard business calculator. The most common business calculators are the Texas Instruments BA II Plus, the HP 10BII+ and the HP 12CP. A variety of other business calculators are also available. Below is the Texas Instruments BA II Plus:

We also have access to internet based or computer based applications which have been precisely programmed to calculate time value of money. The internet and application based calculators are generally more user friendly and allow more detailed information to be provided (charts, graphs, tables, payment schedules). We will briefly cover how to use a standard internet-based time value of money calculator using Wolfram Alpha. We teach all three methods and let you decide which method works best for you. We recommend learning how to use all three methods since you might encounter a variety of situations where one or two of the methods is not available to you, for example, on a standardized exam.

### Example of Using the Simplified Methods for Present Value Calculations

In this example we will take our original data that was used in the previous example and then recompute the example using the simplified methods of factor charts, business calculators and our online calculator.

Example: You are to be given \$5,000 at the end of three years. The interest rate is 5% per year. What is the present value?

Before we begin, we need to understand exactly what this question is asking us to do and to determine which key data points we are given. In this example we are asked to calculate the present value of a single sum when we are given the future value, the time periods and the interest rate.

Factor charts: To begin using the factor charts for this example, we need to use the correct chart. As you know, we have six possible factor charts to select from. In this case we need the present value of one chart.

 Periods 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 1 0.99010 0.98522 0.98039 0.97561 0.97087 0.96618 0.96154 0.95694 0.95238 0.94787 0.94340 2 0.98030 0.97066 0.96117 0.95181 0.94260 0.93351 0.92456 0.91573 0.90703 0.89845 0.89000 3 0.97059 0.95632 0.94232 0.92860 0.91514 0.90194 0.88900 0.87630 0.86384 0.85161 0.83962 4 0.96098 0.94218 0.92385 0.90595 0.88849 0.87144 0.85480 0.83856 0.82270 0.80722 0.79209 5 0.95147 0.92826 0.90573 0.88385 0.86261 0.84197 0.82193 0.80245 0.78353 0.76513 0.74726

As you can see from the above, we simply locate the number of time periods on the left which is three. Then we locate the interest rate of 5% along the top and then locate our factor which is 0.86384.

Once we have located the factor we simply multiply the factor by our single sum of \$5,000 as seen below:

\$5,000 X .86384 = \$4,319.20

If we reference our result to the original calculation we performed manually we can see that we got the same result with the exception of the one cent rounding.

In our examples of using a business calculator we will be using the Texas Instruments BAII Plus due to it being the most commonly used business calculator. Other calculators have similar buttons so if you can figure out how to use the BAII plus you can probably also figure out how to use your calculator.

Online Calculator:

In our examples of using an online calculator we will be using the online website wolframalpha.com. Out of all of our methods the online calculator is by far the easiest method to use so it will probably become your preferred method of calculating TVM.

Using our example, we will go to the front page of wolframalpha.com and type "present value of 5000". We are presented the following calculator to use which contains several variables we can change.

At this point, we simply need to input our variables into the calculator. We need to change the interest rate to 5%, change interest periods to 3 and change the compounding frequency to annual. Selecting the "also include compounding frequency" will allow you to change the compounding frequency. Once we have that input we simply press enter to perform the calculation. The result is shown below:

See how simple that was? The results page also contains additional information which might be helpful.

So far we have covered present value of single sums. In the next section we will cover present values of ordinary annuities and annuity due.

### Present Value of Ordinary Annuities

An ordinary annuity is a series of equal payments where the payment is due at the end of the time period, for example, at the end of the month. This is contrast to an annuity due where the payment is due at the beginning of the time period, for example, at the beginning of the month.

To calculate the present value of an ordinary annuity, we will use the following formula:

### Simplified Methods for Calculating Present Value of Ordinary Annuities

We will now cover how to use the three simplified methods we cleared about previously: present value factor charts, business calculators and online calculators.

Present Value of an Ordinary Annuity Factor Chart:

Using the data in our previous example, we can locate the present value factor from the following factor chart. Remember to always use the present value of an ordinary annuity chart:

Online Calculator: