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Time Value of Money (TVM): A Primer

• 16 Jun 2022

Would you rather receive $1,000 today or the promise that you’ll receive it one year from now? At first glance, this may seem like a trick question; in both instances, you receive the same amount of money.

Yet, if you answered the former, you made the correct choice. Why does receiving $1,000 now provide more value than in the future?

This concept is called the time value of money (TVM), and it’s central to financial accounting and business decision-making. Here’s a primer on what TVM is, how to calculate it, and why it matters.

Access your free e-book today.

What Is the Time Value of Money?

The time value of money (TVM) is a core financial principle that states a sum of money is worth more now than in the future.

In the online course Financial Accounting , Harvard Business School Professor V.G. Narayanan presents three reasons why this is true:

• Opportunity cost: Money you have today can be invested and accrue interest, increasing its value.
• Inflation: Your money may buy less in the future than it does today.
• Uncertainty: Something could happen to the money before you’re scheduled to receive it. Until you have it, it’s not a given.

Essentially, a sum of money’s value depends on how long you must wait to use it; the sooner you can use it, the more valuable it is.

When time is the only differentiating factor, the money you receive sooner will always be more valuable. Yet, sometimes, there are other factors at play. For instance, what’s more valuable: $1,000 today or $2,000 one year from now?

TVM calculations “translate” all future cash to its present value. This way, you can directly compare its values and make financially informed decisions.

“Cash flows expressed in different time periods are analogous to cash flows expressed in different currencies,” Narayanan says in Financial Accounting. “To add or subtract cash flows of different currencies, we first have to convert them to the same currency. Likewise, cash flows of different time periods can be added and subtracted only if we convert them first into the same period.”

Related: 8 Financial Accounting Skills for Business Success

How to Calculate TVM

How you calculate TVM depends on which value you have and which you want to solve for. If you know the money’s present value (for instance, the amount you deposited into your savings account today), you can use the following formula to find its future value after accruing interest:

FV = PV x [ 1 + (i / n) ] (n x t)

Alternatively, if you know the money’s future value (for instance, a sum that’s expected three years from now), you can use the following version of the formula to solve for its present value:

PV = FV / [ 1 + (i / n) ] (n x t)

In the TVM formula:

• FV = cash’s future value
• PV = cash’s present value
• i = interest rate (when calculating future value) or discount rate (when calculating present value)
• n = number of compounding periods per year
• t = number of years

Calculating TVM Manually: An Example

Imagine you’re a key decision-maker in your organization and two projects are proposed:

• Project A is predicted to bring in $2 million in one year.
• Project B is predicted to bring in $2 million in two years.

Before running the calculation, you know that the time value of money states the $2 million brought in by Project A is worth more than the $2 million brought in by Project B, simply because Project A’s earnings are predicted to happen sooner.

To prove it, here’s the calculation to compare the present value of both projects’ predicted earnings, using an assumed four percent discount rate:

PV = 2,000,000 / [ 1 + (.04 / 1) ] (1 x 1)

PV = 2,000,000 / [ 1 + .04 ] 1

PV = 2,000,000 / 1.04

PV = $1,923,076.92

PV = 2,000,000 / [ 1 + (.04 / 1) ] (1 x 2)

PV = 2,000,000 / [ 1 + .04 ] 2

PV = 2,000,000 / 1.04 2

PV = 2,000,000 / 1.0816

PV = $1,849,112.43

In this example, the present value of Project A’s returns is greater than Project B’s because Project A’s will be received one year sooner. In that year, you could invest the $2 million in other revenue-generating activities, put it into a savings account to accrue interest, or pay expenses without risk.

Now, imagine there’s a third project to consider: Project C, which is predicted to bring in $3 million in two years. This adds another variable into the mix: When sums of money aren’t the same, how much weight does timeliness carry?

PV = 3,000,000 / [ 1 + (.04 / 1) ] (1 x 2)

PV = 3,000,000 / [ 1 + .04 ] 2

PV = 3,000,000 / 1.04 2

PV = 3,000,000 / 1.0816

PV = $2,773,668.64

In this case, Project C’s present value is greater than Project A’s, despite Project C having a longer timeline. In this case, you’d be wise to choose Project C.

Calculating TVM in Excel

While the aforementioned example was calculated manually, you can use a formula in Microsoft Excel, Google Sheets, or other data processing software to calculate TVM. Use the following formula to calculate a future sum’s present value:

=PV(rate,nper,pmt,FV,type)

In this formula:

• Rate refers to the interest rate or discount rate for the period. This is “i” in the manual formula.
• Nper refers to the number of payment periods for a given cash flow. This is “t” in the manual formula.
• Pmt or FV refers to the payment or cash flow to be discounted. This is “FV” in the manual formula. You don’t need to include values for both pmt and FV.
• Type refers to when the payment is received. If it’s received at the beginning of the period, use 0. If it’s received at the end of the period, use 1.

It’s important to note that this formula assumes payments are equal over the total number of periods (nper).

Here’s the calculation for Project A’s present value using Excel:

Why Is TVM Important?

Even if you don’t need to use the TVM formula in your daily work, understanding it can help guide decisions about which projects or initiatives to pursue.

“Applying the concept of time value of money to projections of free cash flows provides us with a way of determining what the value of a specific project or business really is,” Narayanan says in Financial Accounting.

As in the previous examples, you can use the TVM formula to calculate predicted returns’ present values for multiple projects. Those present values can then be compared to determine which will provide the most value to your organization.

Additionally, investors use TVM to assess businesses’ present values based on projected future returns, which helps them decide which investment opportunities to prioritize and pursue. If you’re an entrepreneur seeking venture capital funding, keep this in mind. The quicker you provide returns to investors, the higher cash’s present value, and the higher the likelihood they’ll choose to invest in your company over others.

You now know the basics of TVM and can use it to make financially informed decisions. If this piqued your interest, consider taking an online course like Financial Accounting to build your skills and learn more about TVM and other financial levers that impact an organization’s financial health .

Do you want to take your career to the next level? Explore Financial Accounting —one of three online courses comprising our Credential of Readiness (CORe) program —which can teach you the key financial topics you need to understand business performance and potential. Not sure which course is right for you? Download our free flowchart .

About the Author

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Dr. Kevin Bracker; Dr. Fang Lin; and jpursley

Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it. – Albert Einstein

Chapter Learning Objectives

After completing this chapter, students should be able to

• Explain the concepts of future value, present value, annuities, and discount rates
• Solve for the future value, present value, payment, interest rate or number of periods using the 5-key approach on a financial calculator
• Work with annual, semi-annual, quarterly, monthly, biweekly, weekly, or daily periods
• Solve for the present value of a perpetuity
• Solve for the present value or future value of an uneven cash flow stream
• Solve for the interest rate implied by an uneven cash flow stream
• Explain, calculate, and compare investments based on the effective annual rate
• Perform complex time value of money calculations (problems where multiple steps are required in order to reach the final solution)

The Power of Compound Interest

The quote at the start of the chapter is often attributed to Albert Einstein (despite some controversy as to the accuracy of that attribution). However, the validity of the statement itself has merit. Positive returns on investments over long periods of time are central to making money work for you as the power of compounding allows for geometric growth. Consider the following table (before long, you’ll be able to verify these calculations) of someone saving $250 per month for various times at various rates of return. Note that an individual who is 25 would have about 40 years until a standard retirement at age 65 and, assuming their employer offers a 50% match on retirement savings plans such as a 401(k), a total contribution of $250 each month would only be $2000 per year out of pocket before taxes.

Table: Future Value of $250 per month investment

Take a moment to review the table above. Note that at 5 years out, the rate of return makes some difference, but not a dramatic difference. By 15 years out, an individual would have 2.5 times as much at the 15% rate of return as the 5% rate of return. By 30 years out, the 10% rate of return is 2.7 times as much as the 5% rate of return and the 15% rate of return has accumulated 8.3 times the wealth. By 40 years out, an individual has invested $120,000 into her retirement savings (40 years at $3000 per year – with the potential for some of that $120,000 coming from the employer). The power of compounding has generated about $261,500 at 5%, nearly $1.5 million at 10%, and over $7.5 million at 15%. This example illustrates how powerful time and return are as tools for building wealth. Now it is time to show you how to do these and other time value of money calculations.

Future Value

When you put your money in a savings account (or invest it in some fashion), you earn a certain return (sometimes called interest) in order to compensate you. Because of this, a dollar today is not worth the same amount as a dollar sometime in the future. Since you earn money on the dollar invested (or saved) today, you will have more than a dollar at some later future point (making a dollar today worth more than the same dollar received later). The specific amount that you will have at the future date is referred to as a Future Value.

Consider if you had $100 today and were able to earn 12% per year by putting that money in a savings account at XYZ bank. How much would you have in one year? Two years? Three years? At first, you might think that you would have $112 in one year, $124 in two years and $136 in three years as you would earn $12 per year in interest. However, this is WRONG! It ignores the concept of compounding. After one year, you would indeed have $112. However, during the second year you earn 12% interest on the full $112 instead of only the $100 you started with. Therefore, you will earn $13.44 (=112×0.12) in interest in the second year and have $125.44 in two years. During the third year, you will earn $15.05 (=125.44×0.12) in interest and have $140.49 in three years. Therefore, the Future Value of $100 for three years at 12% is $140.49. In other words, $100 today is equivalent to $140.49 received three years from now assuming that you can earn 12% interest annually.

Solving for Future Value

We have three ways to solve for the FV: formula, financial table, and financial calculator.

Method 1: Using a Formula to Find the FV

The first is directly with a formula. Under this method, we use the following formula:

[latex]FV=PV(1+k)^n[/latex]

FV is the future value (in year n) for which we are trying to solve PV is the present value (how much we have today) k is the rate of return we are earning (also referred to as the interest rate, required return, growth rate, or discount rate) n is the number of years which we will be saving (or investing) the money.

Method 2: Using a Table to Find the FV

The second method is to use Financial Tables, in Appendix A. Financial tables are cumbersome and don’t allow us as much flexibility as other methods, so they will not be covered in this text.

Method 3: Using a Financial Calculator to Find the FV

The third method (and the method focused on here) is to use the financial calculator or spreadsheet. Each financial calculator follows the same basic ideas, but the specifics are different for each brand of calculator. The steps below are for the HP10BII, TI-BAII+ and TI-83/84. If this is the first time using your financial calculator, see the detailed instructions  Setting up Your Financial Calculator , in Appendix B.   Please pause here to read that and set up your financial calculator before proceeding.

Calculator Steps to Compute FV:

Note: The order of steps 1-4 is not important. The FV answer will appear as a negative number, ignore the negative sign for now. For the TI-83/84 calculators your P/Y and C/Y on the onscreen display should both be 1 for now.

Example: Finding FV using the Financial Calculator

Find the Future Value of $350 invested for 25 years at 9.5% per year.

Step 1: 25 N Step 2: 9.5 I/YR Step 3: 350 PV Step 4: 0 PMT Step 5: FV⇒

You should get a solution of $3383.93.

In other words, if we invest $350 today and let it compound at 9.5% per year for 25 years, we will have $3383.93 at the end of the 25th year.

Technically, you will get a value of -3383.93. The negative sign is an important aspect of financial calculators. The calculator is looking for the solution that balances both parties of a transaction. Here, since the $350 starting value was positive, the calculator assumes that this amount is being received today. If an individual receives $350, that individual needs to pay back $3383.93. Positive values represent cash inflows and negative values represent cash outflows. In a problem like this, it is not essential. However, later in the chapter, we will introduce problems where the cash flow direction is essential. Specifically, whenever there are nonzero values for two or three of the cash flows (PV, PMT, and/or FV), cash flow direction matters. In those cases, figure out if the cash flow is coming to you (available at that moment to spend) or the cash flow is going away from you (set aside into a savings plan). If the cash flow is coming to you, it is positive. If it is going away from you, it is negative. If we applied that logic in this example, the $350 PV would actually be -350. However, this would not change the value of the FV other than to make it positive.

Present Value

The flip side of Future Value is Present Value. Future value tells us how much a certain amount of money will be worth at some future date assuming a certain rate of return. However, what if we know how much we are supposed to get at some point in the future and want to know what it is worth to us today? Now we must find the Present Value. Assume we are offered an opportunity to receive $200 at the end of two years (call it investment A). How much is this opportunity worth to us today assuming we could earn 8% by placing our money in a savings account (that has risk similar to investment A)? To answer this, we must ask how much we would need to place in a savings account today in order to have $200 at the end of the two years.

[latex]FV=PV(1+k)^n[/latex] [latex]200=PV(1.08)^2[/latex] [latex]\frac{200}{(1.08)^2}=PV[/latex] [latex]\$171.47=PV[/latex]

If we had $171.47 today and placed it in a savings account earning 8%, we would have $200 in two years (the same as through investment A). Assuming that investment A had the same degree of risk as our savings account, then we would buy investment A if it was available for less than $171.47 and put our money in the savings account if investment A cost more than $171.47. We could say that the present value of investment A is $171.47.

Solving for Present Value

We have three ways to solve for the PV: formula, financial table, and financial calculator.

Method 1: Using a Formula to Find the PV

[latex]PV=\frac{FV}{(1+k)^n}[/latex]

FV is the future value (in year n) that we plan to receive PV is the present value (how much it is worth to us today) k is the rate of return we can earn elsewhere (also referred to as the compound rate, required return, or discount rate) n is the number of years which we will have to wait before receiving the money.

Method 2: Using a Table to Find the PV

The second method is to use financial tables and will not be covered in this text.

Method 3: Using a Financial Calculator to Find the PV

The third method is to use the financial calculator (or spreadsheet). Each financial calculator follows the same basic ideas, but the specifics are different for each brand of calculator. The steps below are for the HP10BII, TI-BAII+ and TI-83/84.

Calculator Steps to Compute PV

Note: The order of steps 1-4 is not important. The PV answer will appear as a negative number, ignore the negative sign for now.

Example: Finding PV using the Financial Calculator

Find the Present Value of $5000 received 15 years from today with a 9.5% discount rate.

Step 1: 15 N Step 2: 9.5 I/YR Step 3: 0 PMT Step 4: 5000 FV Step 5: PV⇒

You should get a solution of $1281.62

In other words, if we are offered the opportunity to receive $5000 at the end of 15 years that is equivalent to receiving $1281.62 today.

The examples previously discussed are for situations where we have a specific amount today and want to know what it is worth at some point in the future (FV) or when we plan to receive a certain amount at some point in the future and want to know what it is worth today (PV). These are referred to as lump sum situations because there is only one cash flow that we are discounting or compounding.

Timelines: Let us pause here for a moment to introduce an important tool used in time value of money – timelines. Timelines provide an aid that helps us better visualize what the cash flow stream looks like. Consider an annuity that pays $2000 per year for 4 years with an 8% discount rate. We can illustrate this on a timeline as follows:

Note that the hashmarks represent the end of the time increment and the space between the hashmarks represent the time increment itself. In other words, the year 1 hashmark represents the end of year 1 where the annuity makes its first $2000 payment. Some students find timelines very helpful and use them for most time value of money problems while others use them less frequently. However, when we get to the section on complex time value of money problems later in this chapter, most students will find timelines quite beneficial.

Solving for Present Value of an Annuity

We have three ways to solve for the PV of an annuity: formula, financial table, and financial calculator.

Method 1: Using a Formula to Find the PV of an Annuity

[latex]PVA=PMT\Big(\frac{1-\frac{1}{(1+k)^n}}{k}\Big)[/latex]

PVA is the present value of the anticipated cash flow stream (annuity) PMT is the annuity payment (how much we receive or save each period) k is the rate of return we can earn elsewhere (also referred to as the compound rate, required return, or discount rate) n is the number of periods which we will have to wait before receiving the money.

Method 2: Using a Table to Find the PV of an Annuity

Method 3: using a financial calculator to find the pv of an annuity.

The third method is to use the financial calculator (or spreadsheet) which is what we will focus on. Let’s walk through an example with the financial calculator. An investment that pays $100 at the end of each year for 4 years is an annuity (note that a clue for annuities is to look for the word “each’ or “every” to indicate that the same cash flow is being repeated multiple times). If we wanted to know what that investment is worth to us today and we had a 10% discount rate, we would be finding the present value of that annuity.

Calculator Steps to Compute PV of an Annuity

You should get a solution of $316.99

Solving for Future Value of an Annuity

As with the other TVM calculations we have encountered, there are 3 basic methods to solve for the FV of an annuity: formula, financial table, and financial calculator.

Method 1: Using a Formula to Find the FV of an Annuity

[latex]FVA=PMT\Big(\frac{(1+k)^n-1}{k}\Big)[/latex]

FVA is the future value that our cash flow stream will grow to at the end of n periods PMT is the annuity payment (how much we receive or save each period) k is the rate of return we can earn elsewhere (also referred to as the compound rate, required return, or discount rate) n is the number of periods which we will have to wait before receiving the money.

Method 2: Using a Table to Find the FV of an Annuity

The second method is to use financial tables . These tables are included in Appendix A and will not be covered in this text.

Method 3: Using a Financial Calculator to Find the FV of an Annuity

The third method is the financial calculator (or spreadsheet) approach. Let’s walk through an example using the financial calculator to solve for the future value of an annuity. We want to save $1000 per year (at the end of each year) for 10 years at 12%. How much will this be worth at the end of the 10th year?

Calculator Steps to Compute FV of an Annuity

Note: The order of steps 1-4 is not important. The FV answer will appear as a negative number, ignore the negative sign for now.

You should get a solution of $17,548.74

Note: Ordinary annuities (both present value and future value) assume that cash flows will arrive at the end of each period. Occasionally, you might encounter an annuity due (which means that cash flows arrive at the BEGINNING of each period). It is easy to adjust for this when using a financial calculator by changing the calculator from END of period cash flows to BEGINNING of period cash flows. This process is described in Setting up Your Financial Calculator  in Appendix B (for the TI-83/84, it is just part of the onscreen display in the TVM_Solver).

Solving for PMT, I/YR, or N

Sometimes you may need to find something other than the present value or future value. For instance, you may want to know how much you have to save per year to reach a certain future value (or how much you must earn as a rate of return or how many years it will take). If you are using a financial calculator, these are relatively easy. For example, assume you have $2000 saved already and want to save another $5000 per year to accumulate $80,000 after 10 years. What rate of return must you earn?

Calculator Steps for the Solution

Solution = 8.83%

Reminder: Either the PMT must be negative and the FV positive or the PMT positive and the FV negative. It doesn’t matter which way you do it, but one must be negative and the other positive.

Solving for N and PMT is done along similar lines.

Perpetuities

A Perpetuity is an annuity that lasts forever. While it is difficult to imagine a situation where an individual could buy a cash flow stream that will pay a fixed amount per year through infinity, perpetuities can be useful tools when dealing with long, constant cash flow streams. Consider someone wanting to fund a scholarship or plan for retirement where she is not sure how long she’ll live. A perpetuity can provide a reasonable approximation in either of those situations.

How much would a perpetuity of $100 be worth assuming a discount rate of 10%? Remember this is $100 per year forever. It would seem that this would be worth an infinite amount. However, consider what would happen if you had $1000 today and could put it in the bank to earn 10% interest. You would receive $100 per year and never touch the principal. You would essentially be buying a $100 perpetuity (assuming the bank didn’t change the interest rate). Therefore, a perpetuity has a finite value. The formula for finding the present value of a perpetuity is as follows:

[latex]PV=\frac{PMT}{k}[/latex]

Note: When using this formula, always plug in k as a decimal so that 10% is 0.10

Uneven Cash Flow Streams

Sometimes you will encounter a situation where you have more than one payment, but it is not the same each year. Remember that an annuity requires the payment to be the same each year. If you have multiple cash flows, but they are not the same, you have an uneven cash flow stream. In order to solve a problem like this, treat it as a series of single cash flows (or possibly a series of smaller annuities).

Net Present Value of an Uneven Cash Flow Stream

Consider the following example: you have an investment project that will pay the following cash flows:

Year 1 $1000 Year 2 $500 Year 3 $2000 Year 4 $2000

The discount rate is 15%. Find the Present Value.

Calculator Steps to Compute PV of an Uneven Cash Flow Stream

Solution $3706.18

Note for HP10BII+: The Nj key is used to tell the calculator the number of times that the same cash flow will be received consecutively. If the cash flow only occurs once (in a row) then we do not need to use the Nj key. However, when we have the same cash flow multiple times in a row (such as the $2000 for two years), we can use the Nj key to tell the calculator that this $2000 will occur in two consecutive years.

Note for TI-BAII+: The F screen that appears after you enter a cash flow and down arrow is used to tell the calculator the number of times we have that same cash flow consecutively. If the cash flow only occurs once (in a row) then we do not F screen and just down arrow past it. However, when we have the same cash flow multiple times in a row (such as the 2000 for two years), we use the F screen to tell this to the calculator. The calculator does not have a F screen after the initial cash flow, so we do not need the double down arrow after entering the initial CF.

The above calculator methods are referred to as your Cash Flow Register or Cash Flow Worksheet. It is essential that you always clear all/clear work before entering any cash flows. If you do not do this you will be adding cash flows to a previous problem instead of starting a new problem. The TI-83/84 does not utilize this type of register and does not need to be cleared.

Future Value of an Uneven Cash Flow Stream

The NPV function gives you the present value. You may alternatively want to know how much you will have at the END of the time period (solve for the future value). If this is the case, you start by solving for the NPV. Once you have that, use the 5-key approach to bring that present value forward to the end of the time horizon. For example, if we wanted to know what the above cash flow stream was worth at the END of the fourth year, we would start by solving for the NPV and get the same $3706.18 we calculated earlier. Then, we would go to our 5-key and solve for the future value as follows:

Step 1: 4 N Step 2: 15 I/YR Step 3: 3706.18 PV Step 4: 0 PMT Step 5: Solve for FV⇒ $6482.13

When calculating the PV of an uneven cash flow stream, it should always be less than the sum of the cash flows. When calculating the FV of an uneven cash flow stream, it should always be more than the sum of the cash flows. Also, many financial calculators allow you to solve directly for the future value of an uneven cash flow stream. To see if yours does this, consult your user manual or ask your instructor.

Finding the discount rate of an Uneven Cash Flow Stream

We can also find the discount rate (I/Y) if we have uneven cash flows. Consider the following example: We have an investment project that will pay the following cash flows:

Year 1 $1000 Year 2 $500 Year 3 $2000

If the present value of this investment is $3000, what is the discount rate?

Calculator Steps to Compute I/Y of an Uneven Cash Flow Stream

Solution 7.06%

Note for HP10BII+: The IRR/YR is not the same key as you used for the I/YR, but it serves a similar role — finding the discount rate (or rate of return) for a cash flow stream. The difference is that they I/YR key only works with single cash flows or annuities while the IRR/YR key works with uneven cash flows.

Note for TI-BAII+: The IRR is not the same key as you used for the I/Y, but it serves a similar role — finding the discount rate (or rate of return) for a cash flow stream. The difference is that the I/Y key only works with single cash flows or annuities while the IRR key works with uneven cash flows.

CF0 will always be negative when calculating IRR. If you end up with an error message when calculating the IRR, one of the first things you should do is make sure that your CF0 was a negative value.

Non-Annual Compounding

The more frequently interest is compounded, the greater the effective yield on our savings. Many banks use non-annual compounding periods (monthly, daily, etc). In order to make comparisons, we must find the effective annual yield. This tells us how much we are earning on an annual basis.

Using a Formula to Find the Effective Annual Yield

The formula for effective annual yield is as follows:

[latex]k_{eff}=\Big(1+\frac{k_{nom}}{m}\Big)^m-1[/latex]

k eff is the effective annual yield k nom is the nominal or stated yield m is the number of compounding periods per year

For example, what is the effective interest rate of 8% compounded daily?

[latex]k_{eff}=\Big(1+\frac{0.08}{365}\Big)^{365}-1[/latex]

Note: Be careful not to round when you take .08/365 or you will end up with significant error after compounding it 365 times.

Using a Calculator to Find the Effective Annual Yield

As an alternative, you could use your financial calculator to find the effective interest rate. Again, using 8% compounded daily.

Calculator Steps to Find the Effective Annual Yield

Solution 8.33%.

Note for HP-10BII+: You have changed your payments per year when doing this calculation. If you go back to another TVM problem, be sure to reset your payments per year to one.

Example: Solve a Problem Involving Non-Annual Compounding

We could also look at non-annual compounding with loans or investments. For example, consider a mortgage loan. You are borrowing $80,000 at an 8% rate with monthly payments for 30 years (note that non-annual annuities and lump sums work best with calculators), what is your monthly payment?

Step 1: Convert your calculator to monthly payments by entering 12 P/YR Step 2: -80000 PV Step 3: 8 I/YR Step 4: 360 N (30 years at 12 months per year) Step 5: 0 FV Step 6: PMT

Solution = $587.01 per month

Be VERY careful if you change your payments per year to change it back to 1 P/YR when you are done. Also, each calculator is slightly different in how it sets the periods per year. Be sure to review the Setting up Your Financial Calculator  in Appendix B for calculator specific instructions.

Return to Future Value Tables

Remember the table of future values that we used to start the chapter? We said that the value of $250 set aside every month for 40 years at 10% would be $1,581,019.90. We also suggested that by the end of this chapter, you would be able to do that calculation on your own. Well, now you can.

Step 1: Convert your calculator to monthly payments by entering 12 P/YR Step 2: 0 PV Step 3: 10 I/YR Step 4: 480 N (40 years at 12 months per year) Step 5: 250 PMT Step 6: FV

Solution = $1,581,019.90

Complex Time Value of Money Problems

Everything above this point completes your “Time Value of Money Toolbox.” All the examples to this point have been straight-forward situations. However, sometimes we have what we refer to as complex time value of money problems where there are multiple issues that need addressed within one problem. One of the most common examples of this would be a retirement problem where you have X dollars available today, want to be able to withdraw a certain cash flow stream at retirement throughout your retirement years and want to find out how much you need to save each month until retirement between now and the day you retire to achieve your goal. In order to solve a problem like this, you need to visualize (a time line is very helpful) what information you have and what you are missing (that you need to solve for). You will often need to break this down into multiple steps.

Example: Solve a Complex Time Value of Money Problem

Consider a situation where you are saving for retirement. You currently have $40,000 saved and would like to save an additional $75 per week for the next 30 years. You estimate that when you retire (30 years from today), you want to be able to withdraw $750 each week for the next 20 years and have $200,000 left over at the end of the 20-year retirement period. Assuming you earn 5% during retirement, what rate of return must you earn during the next 30 years to meet your goal?

One way to approach this is to start with a timeline. Note that each period is one week and there are 52 weeks per year. This means that we will have 1560 periods until retirement (1560 = 30×52) and another 1040 periods until the end of retirement (1040 = 20×52). This provides a total of 2600 periods for the entire 50 year time (2600 = 1560 + 1040). Once we’ve created the timeline, we can split it into two timelines. Timeline one will begin today and go to retirement (period 1560) and timeline 2 will begin at retirement (also period 1560) and go to the end of the retirement time frame (period 2600). Here, the timelines help us visualize the information that we know and what we need to find out (specifically our rate of return we must earn over the first 30 years).

Now we can start the calculations. To start, you need to figure out how much you will require at the end of the 30 years. This is the amount you want to have when you retire.

Step 1: Solve for how much you need at retirement.

Set your calculator to 52 periods per year to reflect weekly withdraws during retirement

Set your N to 1040 (52 periods per year for 20 years = 1040 weekly periods)

Set your PMT to 750 (to reflect the weekly withdraw)

Set your FV to 200,000 (to reflect the amount left over)

Set your I/YR to 5 (for your rate of return during retirement)

Solve for PV ⇒$566,527.38

Note – your PMT and FV need to be the same sign. You can make them both positive or both negative, but they are both flowing in the same direction so must be the same sign.

Step 2: Now that you know how much you need when you retire ($566,527.38), you can calculate what rate of return you need to earn over the next 30 years to get there.

Keep your calculator set to 52 periods per year as you are making weekly contributions

Set your N to 1560 (52 periods per year for 30 years = 1560 weekly periods)

Set your PV to -40,000 (to reflect the initial $40,000 contribution)

Set your PMT to -75 (to reflect your weekly $75 contribution)

Set your FV to 566,527.38 (to reflect how much you need at retirement)

Solve for I/YR ⇒5.98%

Note – your PV and PMT both need to be the same sign. Again, you can make them positive or negative, but they are both flowing in the same direction. The FV needs to be the opposite sign. The easiest way to think of this is that you are giving up the $40,000 today and the $75 per week in order to get back the $566,527.38 30 years from today.

Key Takeaways

Time value of money is one of the most powerful and most important concepts in finance. It essentially is as simple as recognizing that because we can earn a return on our money, the value of money changes depending on when it is received or spent. One dollar today is worth more than one dollar received next year. The value of the dollar initially is referred to as a present value while the value of the dollar at a later point in time is referred to as the future value. Compound interest implies that money will grow exponentially over time instead of linearly. This means that relatively small increases in rates of return or time horizons have more power to increase wealth. After completing this chapter, you should be comfortable performing many calculations to see exactly how time value of money can work for you.

Explain why $1 received today is worth more than $1 received one year from today.

What do we mean when we refer to an annuity? How is an annuity different from an annuity due?

What is the relationship between present value and future value?

How do we determine the appropriate discount rate to use when finding present value?

Why is compounding on a monthly basis better for us than compounding on an annual basis?

Determine the answer to each of the following questions.

1a. Find the Future Value of $2500 invested today at 11% for 10 years. 1b. Find the Future Value of $2500 invested today at 11% for 30 years. 1c. Find the Present Value of $6000 received 10 years from today if the discount rate is 5%. 1d. Find the Present Value of $6000 received 10 years from today if the discount rate is 10%. 1e. Find the Future Value of $3000 per year (at the end of each year) invested at 6% for 30 years. 1f. Find the Future Value of $3000 per year (at the end of each year) invested at 12% for 30 years. 1g. Find the Present Value of $4000 per year (at the end of each year) if the discount rate is 15% for 20 years. 1h. Find the Present Value of $4000 per year (at the end of each year) if the discount rate is 15% for 40 years.

Find the interest rates implied by each of the following:

2a. You borrow $1500 today and promise to repay the loan by making a single payment of $2114.00 in 5 years. 2b. You invest $500 today and receive a promise of receiving back $193.50 for each of the next 4 years.

If $2000 is invested today at a 12% nominal interest rate, how much will it be worth in 15 years if interest is compounded

3a. Annually 3b. Quarterly 3c. Monthly 3d. Daily (365-days per year)

How long will it take your money to triple given the following interest rates?

4a. 5% 4b. 10% 4c. 15%

After graduating from college you make it big — all because of your success in business finance. You decide to endow a scholarship for needy finance students that will provide $5000 per year indefinitely, beginning 1 year from now. How much must be deposited today to fund the scholarship under the following conditions.

5a. The interest rate is 10% 5b. The interest rate is 10% and the first payment is made 6 years from today instead of 1 year from today.

Find the present value of the following cash flow stream if the discount rate is 12%:

Years 1-10 $4000 per year Years 11-15 $6000 per year Years 16-20 $8000 per year

Find the value of the following cash flow stream at the end of year 30 if the rate of return is 8.75%:

Years 1-5 $3000 per year Year 6 $7500 Years 7-15 $9000 per year Years 16-30 $12,000 per year

Find the effective annual rate of interest for a nominal rate of 9% compounded

8a. Annually 8b. Quarterly 8c. Monthly 8d. Daily (365 days per year)

Your firm has a retirement plan that matches all contributions on a one-to-two basis. That is, if you contribute $3000 per year, the company will add $1500 to make it $4500. The firm guarantees a 9% return on your investment. Alternatively, you can “do-it-yourself” and you think you can earn 12% on your money by doing it this way. The first contribution will be made 1 year from today. At that time, and every year thereafter, you will put $3000 into the retirement account. If you want to retire in 25 years, which way are you better off?

Jen is planning for retirement. She plans to work for 32 more years. She currently has $15,000 saved and, for the next 15 years, she can save $6,000 at the end of each year. Fifteen years from now, she wants to buy a weekend vacation home that she estimates will require her to withdraw $100,000. How much will she have to save in years 16 through 32 so that she has exactly $750,000 saved when she retires? Assume she can earn 9% throughout the 32-year period.

You are a recent college graduate and want to start saving for retirement. You plan to save $2000 per year for the next 15 years. After that you will stop contributing and just allow your savings to accumulate for another 20 years. Your twin brother would rather wait awhile before he starts saving. He is not going to put away anything for the next ten years, then he will start making contributions at the end of each year for the final 25 years. You both anticipate earning a 9.5% rate of return on your investments. How much must your brother put away at the end of each year to have the same amount of money for retirement as you?

You are considering purchasing a new home. The house you are looking at costs $120,000 and you plan to make a 10% down payment. You checked with a bank and they have two mortgage loan options for you. The first is a 15-year mortgage at 6.25%. The second is a 30-year mortgage at 6.50%.

12a. What are your monthly payments for each loan? 12b. What is the total you will pay over the life of the loan for each loan? 12c. After one year you get a job transfer and have to sell the house. What is the payoff value of your remaining loan balance (hint: find PV of remaining payments)? 12d. Over the first year, how much did you pay in principal and how much did you pay in interest?

Solutions to CH 3 Exercises

Student resources.

Table: Future Value of $250 per month investment, in Appendix B

Financial Tables, in Appendix A

Setting up Your Financial Calculator, in Appendix B

TVM 5-Key Approach Guided Tutorial with HP10BII+, in Appendix B

TVM 5-Key Approach Guided Tutorial with TI-BAII+, in Appendix B

TVM 5-Key Approach Guided Tutorial with TI-83 or TI-84, in Appendix B

Attributions

Image: Mixed from  Godkänd Grön Handskrivning by  Anthony Poynton is licensed under CC0 1.0

Business Finance Essentials Copyright © 2018 by Dr. Kevin Bracker; Dr. Fang Lin; and jpursley is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.

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Case 4: time value of money — concepts and applications. a case study: the general motors company defined benefit pension plan.

• Ivan E. Brick  and 
• Mark W. Guthner

Rutgers Business School, USA

Search for more papers by this author

Introduction

General Motors Company Defined Benefit Pension Plan

Time Value of Money

Recommended

Understanding the Time Value of Money: Key to Making Smart Financial Decisions

Which holds more value: $100 today or $100 a year from now? Does it depend on whether you’re making or receiving the payment? Your answer might hinge on your understanding of ‘the time value of money.’ So, let’s explore this financial concept and understand why is the time value of money important for making savvy monetary decisions. Throughout this exploration, we’ll refer to some time value of money examples and use a time value of money calculator to illustrate our principles clearly.

Understanding the Time Value of Money

Consider this situation: you can take a lump sum of money now or wait a year to collect it. Most people would choose the money now because they have immediate needs or wants that the money can fulfill. But what if the future sum is significantly larger? At what point would your decision shift?

This intuitive understanding that $1 today is worth more than $1 tomorrow, and even more than $1 a year from now, is the essence of the time value of money concept. The time value of money importance is seen in situations like installment loans, such as mortgages or car payments. It’s also crucial for interest-bearing accounts like an IRA. If you decide to invest in real estate, you’ll need to be proficient with these concepts to accurately calculate the time value of money for your cash flows and principal.

How to Calculate the Time Value of Money?

To calculate the time value of money, you need to know and understand a few terms.

• Present value. This is the sum of money you have today.
• Future value. This is the sum of money you will have at some later time.
• A discount rate is the percentage rate used to determine the present value of the future amount. It can often be approximated at the interest rate.

Risk and opportunity cost – two main factors when considering the discount rate

• Risk: The more risk you take on, the higher return you will expect.

Time value of money real-life example , if you put $100 in a bank, you may be willing to accept a $5 return on an investment after a year. This is because the risk that the bank will not repay you is low. If you lend the same $100 to a stranger, you may require a $20 return on investment instead. The person is a stranger. You do not know if they will or will not repay you. To take that level of risk, you require them to pay you $20 extra to use your money.

• Opportunity Cost: The opportunity cost is the cost of the benefit lost by choosing one option over the other(s). If you have the money available right now, you can invest it immediately or apply it elsewhere. Let’s say you choose to apply it somewhere else. The opportunity cost is the value of the interest you could earn while the money is invested.

Time value of money calculations

Given the present value of some money and the discount rate, you can find the future amount using

Future Value = Present Value x (1+Discount Rate)

Let’s say you know how much you want to make, and you know the discount rate you’ll get. If you want to know how much money you’ll need for the initial investment, use

Present Value = Future Value ÷ (1+ Discount Rate)

Let’s use an example to drive the point home. You have $1000 today that you can invest for a year at a 7% discount rate (the interest rate). The value of that $1000 one year from now is

Future Value = $1000 x (1 + 0.07) = $1000 x 1.07 = $1070.

To have $1000 today, you must invest a year ago. Your future value is now $1000; you would use the same discount rate. At that time, your present value would have been

Present Value = $1000 ÷ 1.07 = $934.58.

What if you wanted to project the value of your money beyond a year? For the future value of your $1000, you use

Future Value = Present Value x (1 + Discount Rate) (number of time periods)

So the future value of your $1000 after 5 years, assuming a 7% discount rate per year, would be

Future Value = $1000 x (1 + 0.07) 5 = $1000 x 1.40255= $1,402.55.

Similarly, you can rearrange the formula if you want the initial investment needed to earn $1000 in 5 years. Assuming the same interest rate, your future value will be $1000, and your present value will be

Present Value = $1000 ÷ (1 +0.07) 5 = $1000 ÷ 1.40255 = $712.99.

You would have had to invest $712.99 five years ago at a 7% interest rate to have $1000 today.

Time Value of Money Real-Life Examples

We can use the time value of money in everyday money decisions. Take, for example, the following situations:

Scenario 1 You’ve finally won the lottery! 

The lottery commission lets you choose how you would like to be paid. You can receive $1000 per week for life or an immediate lump sum settlement of $1.5 million. What would be the best option? There is no straightforward answer to this situation. Your answer would depend on a few factors that are specific to your life situation, such as:

• Your age and your life expectancy. If your life expectancy is short, you may not get the full value of your winnings at $1000 weekly.
• What current investment opportunities are available to you? Taking the lump sum but having nothing to invest it in may not be worthwhile.
• The stability of the organization making the payments. Will the lottery commission be around “for life”?

Once you develop the discount rate, you can use the following calculators to help calculate the better payout for you.

Future Value of Present Sum

(Hint: Enter amounts in the Beginning Balance and Periodic Contribution boxes.)

(Hint: Enter amounts in the Periodic Contribution and Ending Balance boxes.)

(Hint: Enter amounts in the Beginning Balance and Ending Balance boxes.)

(Hint: Check this box to show the ending balance in today's dollars.)

(Hint: Check this box to inflate the periodic contributions.)

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By using this calculator you agree to terms and conditions. These calculators are designed to be informational and educational tools only, and when used alone, do not constitute investment or financial advice. We strongly recommend that you seek the advice of a financial services professional before making any type of investment or deciding on your financial matters. This model is provided as a rough approximation of future financial performance. The results presented by this calculator are hypothetical and may not reflect the actual growth of your own investments. We can't take into account potential lender fees, payoff schedule can be longer than in the estimation. Financialfreedom and its affiliates are not responsible for the consequences of any decisions or actions taken in reliance upon or as a result of the information provided by these tools. Financialfreedom is not responsible for any human or mechanical errors or omissions.

Future Value of Cash Flows Calculator

Present Value (PV):
Rate (R):
Payments (PMT):
Periods (q):
Inflation Rate:
Future Value (FV) of Cash Flow:

Scenario 2 You are receiving a payout which is worth $100 today

If taken later, this same payout will be worth $110. When making your decision, consider the following:

• Where is the interest coming from?
• Where can you invest that $100 today, and how much would it be a year from now?

TMV Calculators to help you decide:

Present Value of a Future Sum

Present Value (PV):
Rate (R):
Payments (PMT):
Periods (q):
Inflation Rate:
Future Value (FV):

Present Value of Cash Flows

Scenario 3 you’re going to get an extra $1,000 on your tax refund.

You can do many things with that money, but you’ve narrowed it down to two choices. Should you invest the $1000 for the next 20 years or use it to pay your mortgage today? When you make your decision, you should think about the following:

• Your current and future mortgage rates.
• The investment opportunities for this money.

Potentially you can decide to invest this money into a stable bond. You can use this calculator to decide how much this annuity is worth.

Annuity Future Value Calculator

Present Value (PV):
Rate (R):
Payments (PMT):
Periods (q):
Inflation Rate:
Future Value (FV) of Ordinary Annuity:

Make Better Financial Decisions With Financial Freedom Guru

The time value of money concept is fundamental to making wise financial decisions, especially when deciding between two or more financial options. Should you take the money now, or would it be more beneficial to collect it later?

The answer relies on numerous factors tied to your circumstances. Irrespective of the option you favour, understanding the definition of the time value of money enables you to comprehend just how much is at stake. With tools like a time value of money calculator, you can also gain quantitative insights into your decision, further underscoring the importance of the time value of money.

Last Updated: June 23, 2023

Time Value of Money Video Explanation

I started a digital marketing agency Romanz Media Group Inc. 12 years ago. Running my own business quickly taught me the importance of cash flow. Making sales was not enough, I had to have money in the bank to pay the vendors, staff and personal bills.

During those early stages of the company I learned how to get creative with debt and to save on interest cost. I paid for everything I could with a credit card to both get more points and to extend the payment date by 25 days (credit card grace period). I then utilized a 0% balance transfer offers to rotate this debt.

I learned a lot during this process and made a lot of mistakes. My key lesson is that the most important part of being financially independent is how much I managed to save, rather than how much I earned. Staying disciplined with savings and tracking spending is not easy and I tried many different methods to stay on track.

FinancialFreedom.Guru is a side project where I and my staff are trying to share the practical knowledge on how to understand finances and to build wealth.

Hi there, How to calculate future value in excel?

To calculate the Future Value in Excel, go to the Formulas tab in Excel and select “Financial”. In the sub-menu, choose “FV”. You will need to specify the following values: • “rate” is the interest rate, • “nper” is the number of payment periods, • “pmt” is the payment per period, • “pv” is the present value (if not specified, = 0) • “type” shows when the payment is due – at the beginning of the month (the value is “1”), or at the end of the month (the value is “0”). Insert the right numbers and click “OK.”

Time value of money really matters. A long time ago I had a friend who taught me to always have a hundred dollars tucked away. She called it walking around money. I still believe in this simple system to never using a credit card. That’s my philosophy

Whatever works for you. Time Value of Money concept would say that this $100 is losing value while it is your pocket, so it is better to invest to at least fight the inflation.

Can’t but agree with you. I try to persuade my wife that when she keeps the money in her saving accounts, she is actually losing, on average, 3% per year

Thank you for the bright explanation. My conclusion – the only thing you’ll never get back is time so invest as early as possible )

Time is the most valuable commodity, so invest in whatever gives you most time like health and automation!

Good article. I really do appreciate your posts, they help to teach a number of random people on the internet on how to finance their money. Thank you 

Thank you for reading and I am glad you found it useful!

It’s exactly the case when the sooner the better. The sooner you start investing your money the better it is for you, I mean the sooner the compound effect happens, minus 2 % inflation each year

Yeah it’s very hard to beat compound returns, so even though you don’t have as much money to save when you are younger – you can still come up on top due the time of the compound returns

The best thing about investing is that it is beneficial for you due to compound interest as the time pass

For sure, it’s also important to understand how to properly calculate your returns and compare them to each other fairly. This is where time value of money concept comes in

It is the most basic and important concept that each person must be taught- but not everyone cares

Agreed, it is very intuitive concept and very important for so many decisions like paying off the mortgage early for example, but most people were never taught it.

Nice summary. This stuff can become complicated once you factor in many cash inflows and outflows at various periods

It for sure does. Luckily there are easy functions in Excel, so you can carefully plot all of your cash flows and bring them all to the same present value

Time is money, and nowadays it is quite rare that one has them both at the same time

It takes investment in yourself and systems, so you can create assets that will give you money without spending time

This is a good explanation of the concept. The key moments are what rate you can earn on the money plus what level of risk you have to put the money at to earn that rate 

And you also have to consider opportunity cost, inflation and risk-free rate. Working out the proper discount rate is actually the key to making accurate financial models

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Time Value of Money Determinations and Their Applications

• First Online: 24 March 2023

Cite this chapter

• John Lee 5 ,
• Jow-Ran Chang 6 ,
• Lie-Jane Kao 7 &
• Cheng-Few Lee 8  

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The concepts of present value, discounting, and compounding are frequently used in most types of financial analysis. This chapter discusses the concepts of the time value of money and the mechanics of using various forms of the present value model. These ideas provide a foundation that is used throughout this book.

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Let x = 1/(1 + r). Then

From which Eq. ( 18.6 ) follows.

2 The restriction of our analysis to two periods is convenient for graphical exposition. However, the same conclusions follow when this restriction is dropped.

In the next chapter, we will discuss the valuation of the financial instruments.

Feldstein, M. and L. Summers. “Inflation and the Taxation of Capital Income in the Corporate Sector,” National Tax Journal (December 1979, pp. 445–47).

Google Scholar  

French, K., R. Ruback, and W. Schwert. “Effects of Nominal Contracting on Stock Returns,” Journal of Political Economy 91 (February 1983, pp. 70–96).

Tobin, J. and W. C. Brainard. “Asset Markets and the Cost of Capital,” Economic Progress, Private Values, and Public Policy: Essays in Honor of William Fellner, B. Balassa and R. Nelson, eds. (Amsterdam: North-Holland, 1977).

Van Horne, J. and W. Glassmire. “The Impact of Unanticipated Changes in Inflation on the Value of Common Stocks,” Journal of Finance (December 1972, pp. 1083–92).

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Center for PBBEF Research, Morris Plains, NJ, USA

National Tsing Hua University, Hsinchu, Taiwan

Jow-Ran Chang

College of Finance, Takming University of Science and Technology, Taipei City, Taiwan

Lie-Jane Kao

Rutgers School of Business, The State University of New Jersey, North Brunswick, NJ, USA

Cheng-Few Lee

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Corresponding author

Correspondence to John Lee .

Appendix 18A

Three Hypotheses about Inflation and the Firm’s Value

We began this chapter by asking whether you would prefer to receive $1,000 today or $1,000 a year from now. One reason for selecting the first option is that, as a result of inflation, $1,000 will buy less in a year than it does today. In this appendix, we explore the possible effects of inflation on a firm’s value. According to Van Horne and Glassmire ( 1972 ), unanticipated inflation affects the firm in three ways, characterized by the following hypotheses:

Debtor-creditor hypothesis.

Tax-effects hypothesis.

Operating income hypothesis.

The debtor-creditor hypothesis postulates that the impact of unanticipated inflation depends on a firm’s net borrowing position. In periods of high inflation, fixed money amounts borrowed today will be repaid in the future in a currency with lower purchasing power. Thus, while the rate of interest on the loan reflects expected inflation rates over the term of the loan, a higher than anticipated rate of inflation should result in a transfer of wealth from creditors to debtors. Conversely, if the inflation rate turns out to be lower than expected, wealth is transferred from debtors to creditors. Hence, according to the debtor-creditor hypothesis, a higher than anticipated rate of inflation should, all other things being equal, raise the value of firms with heavy borrowings.

The tax-effects hypothesis concerns the influence of inflation on those firms with depreciation and inventory tax shields. Since these shields are based on historical costs, their real values decline with inflation. Hence, unanticipated inflation should lower the value of the firms with such shields. The magnitude of these tax effects could be very high indeed. For example, Feldstein and Summers ( 1979 ) estimated that the use of depreciation and inventory accounting on a historical cost basis raised corporate tax liabilities by $26 billion in 1977.

In principle, the effects of general inflation should only be felt when parties are forced to comply with nominal contracts, the terms of which fail to anticipate inflation. Hence, in theory, wealth transfers caused by general inflation should be due primarily to the debtor-creditor or tax-effects hypothesis discussed above. Apart from these considerations, if all prices move in unison, real profits should not be affected. Nevertheless, there is strong empirical evidence of a negative association between corporate profitability and the general inflation rate. One possible explanation, called the operating income hypothesis , is that high inflation rates lead to restrictive government fiscal and monetary policies, which, in turn, depress the level of business activity, and hence profits. Further, operating income may be adversely affected if prices of inputs, such as labor and materials, react more quickly to inflationary trends than prices of outputs. Viewed in this light, we might expect firms to react differently to inflation, depending on the reaction speed in the markets in which the firms operate.

Van Horne and Glassmire suggest that, of these three effects of unanticipated inflation on the value of the firm, the operating income effect is likely to dominate. Some support for this contention is provided by French et al. ( 1983 ), who find that debtor-creditor effects and tax effects are rather small.

Appendix 18B

Book Value, Replacement Cost, and Tobin’s q

An objective of financial management should be to raise the firm’s net present value. We have not, however, discussed what constitutes a firm’s value.

An accounting measure of value is the total value of all a firm’s assets, including plant and equipment, plus inventory. Generally, in a firm’s accounts, the book values of the assets are reported. However, this is an inappropriate measure for two reasons. First, it takes no account of the growth rate of capital goods prices since the assets were acquired, and second, it does not account for the economic depreciation of those assets. Therefore, in considering a firm’s value, it is preferable to consider current accounting measures that incorporate inflation and depreciation. The relevant measure of accounting value, then, is replacement cost, which is the cost today of purchasing assets of the same vintage as those currently held by the firm.

However, this accounting concept of value is not the one used in financial management, as it does not incorporate the potential for future earnings through the exploitation of productive investment opportunities. If this broader definition is considered, the value of a firm will depend not only on the accounting value of its assets, but also on the ability of management to make productive use of those assets. In finance theory, the relevant concept of values of common stock, preferred stock, and debt, all of which are determined by the financial markets. Footnote 3

The ratio of a firm’s market value to the replacement cost of its assets is known as Tobin’s q , as shown in Tobin and Brainard ( 1977 ). One reason for looking at this relationship is that if the acquisition of new capital adds more to the firm’s value than the cost of acquiring that capital—that is, it has a positive NPV—then shareholders immediately benefit from the acquisition. On the other hand, if the addition of new capital adds less than its cost to market value, shareholders would be better off if the money were distributed to them as dividends. Therefore, the relationship between market value and replacement cost is crucial in financial management decision-making.

Appendix 18C

Continuous Compounding and Continuous Discounting

In this appendix, we will show how continuous compounding and discounting can be theoretically derived. In addition, we also give some examples to show how these two processes to the real world.

Continuous Compounding

In the general calculation of interest, the amount of interest earned plus the principal is

where r = annual interest rate, m = number of compounding periods per year, and T = number of compounding periods ( m ) times the number of years N .

There are three variables: the initial amount of principal invested, the periodic interest rate, and the time period of the investment. If we assume that you invest $100 for 1 year at 10% interest, you will receive the following:

For a given interest rate, the greater frequency with which interest is compounded affects the interest and the time variables of the above equation; the interest per period decreases, but the number of compounding periods increases. The greater the frequency with which interest is compounded, the larger the amount of interest earned. For interest compounded annually, semiannually, quarterly, monthly, weekly, daily, hourly, or continuously, we can see the increase in the amount of interest earned as follows:

Annual	$110	=	\(100\left( {1 + \frac{.10}{1}} \right)1\)
Semiannual	110.25	=	\(100\left( {1 + \frac{.10}{2}} \right)2\)
Quarterly	110.38	=	\(100\left( {1 + \frac{.10}{4}} \right)4\)
Monthly	110.47	=	\(100\left( {1 + \frac{.10}{{12}}} \right)12\)
Weekly	110.51	=	\(100\left( {1 + \frac{.10}{{52}}} \right)52\)
Daily	110.52	=	\(100\left( {1 + \frac{.10}{{365}}} \right)365\)
Hourly	110.52	=	\(100\left( {1 + \frac{.10}{{8760}}} \right)8760\)
Continuously	110.52	=	\(100\left( {e^{.1\left( 1 \right)} } \right) = 100\left( {2.7183} \right)^{.1}\)

In the case of continuous compounding, the term \(\left( {1 + \frac{r}{m}} \right)^{T}\) goes to e rN as m gets infinitely large. To see this, we start with

where T = m(N) and N = number of years.

If we multiply T by r/r , we can rearrange Eq.  18.10 as follows:

Let x = m/r , and substitute this value into Eq. ( 18.11 )

The term (1 + 1/x) x is equal to e as

This says that as the frequency of compounding becomes instantaneous or continuous, Eq.  18.10 can be written as

Figure  18.10 provides graphs of the value of P = I as a function of the frequency of compounding and the number of years. We can see that for low interest rates and shorter periods, the differences between the various compounding methods are very small. However, as either r or N becomes large the difference becomes substantial. In general, as either r or N or both variables become larger, the frequency of compounding will have a greater effect on the amount of interest that is earned.

Graphical relationships between frequency of compounding r and N

Continuous Discounting

As we have seen in this chapter, there is a relationship between calculating future values and present values. Starting from Eq.  18.10 , which calculates future value, we can rearrange to find the present value

As we mentioned earlier, as \(m \to \infty\) we see that the term (1 + r/m ) T goes to e Nr . Rewriting Eq.  18.14

Equation  18.15 tells us that the present value (P 0 ) of a future amount (P + I) is related by the continuous discounting factor e −Nt . Similarly, the present value of an annuity of future flows can be viewed as the integral of Eq.  18.15 over the relevant time period

where F t is the future cash flow received in period t . In fact, F t can be viewed as a continuous cash flow. For most business organizations, it is more realistic to assume that the cast inflows and outflows occur more or less continuously throughout a given time period instead of at the end or beginning of the period as is the case with the discrete formulation of present value.

Appendix 18D: Applications of Excel for Calculating Time Value of Money

In this appendix, we will show how to use Excel to calculate: (i) the future value of a single amount, (ii) the present value of a single amount, (iii) the future value of an ordinary annuity, and (iv) the present value of an ordinary annuity.

Future Value of a Single Amount

Suppose the principal is $1000 today and the interest rate is 5% per year.

The future value of the principal can be calculated as \(FV = PV\left( {1 + r} \right)^{n}\) , where n is the number of years.

Case 1 . Suppose there is only one period, i.e. n = 1. The future value in one year will be \(1000\left( {1 + 5\% } \right)^{1} = 1050\) .

We can use Excel to directly compute it by inputting “=B1*(1+B2),” as presented in Table 18.3

Or we can also use the function in Excel to compute the future value by inputting “=FV(B2,1, ,B1,0)”

There are five options in this function.

Rate : The interest rate per period.

Nper : The number of payment periods.

Pmt : The payment in each period; If “pmt” is omitted, we should include the “pv” argument below.

Pv : The present value. If “pv” is omitted, it is assumed to be 0. Then we should include the “pmt” argument above.

Type : The number 0 or 1 shows when payments are due. If payments are due at the end of the period, Excel sets it as 0; If payments are due at the beginning of the period, Excel sets it as 1.

The FV function gives us the same amount as what we calculate according to the formula except the sign is negative. Actually, the FV function in Excel is to compute the Future value of the principal that one party should pay back to another party. Therefore, Excel adds a negative sign to indicate the amount needed to pay back, as presented in Table 18.4 .

Case 2 . Now suppose there are 4 periods. The future value of $1,000 at the end of the 4th year will be \(1000\left( {1 + 5\% } \right)^{4} = 1215.51\) .

We use two methods to compute the future value and obtain the same result.

First, we calculate it directly according to the formula, as presented in Table 18.5 .

Second, we use the FV function in Excel to calculate it, as presented in Table 18.6 .

The FV function gives us the same amount as what we calculate according to the formula except the sign is negative. Actually, the FV function in Excel is to compute the Future value of the principal that one party should pay back to another party. Therefore, Excel adds a negative sign to indicate the amount needed to pay back.

Present Value of a Single Amount

The present value of the future sum of money can be calculated as \(PV = FV/\left( {1 + r} \right)^{n}\) , where n is the number of years.

Case 1 . Suppose a project will end in one year and it pays $1000 at the end of that year. The interest rate is 5% for one year.

The present value will be \(1000/\left( {1 + 5\% } \right)^{1} = 952.38\) .

We can use Excel to directly compute it by inputting “=B1/(1+B2),” as presented in Table 18.7 .

Or, we can use the FV function which is quite similar to the FV function we used before. The result is presented in Table 18.8 .

Case 2 . Suppose a project will end in four years and it would pay $1000 only at the end of the last year. The interest rate is 5% for one year.

The present value will be \(1000/\left( {1 + 5\% } \right)^{4} = 952.38\) .

We can use Excel to directly compute it by inputting “=B1/(1+B2)^4,” as presented in Table 18.9 .

Or we use the PV formula in Excel by inputting “=PV(B2,4,,B1,0),” as presented in Table 18.10 .

Future Value of an Ordinary Annuity

Annuity is a series of cash flow of a fixed amount for n periods of equal length. It can be divided into Ordinary Annuity (the first payment occurs at the end of period) and Annuity Due (the first payment is at the beginning of the period)

Case 1. Future Value of an ordinary annuity.

The formula is \(FV = \mathop \sum \limits_{k = 1}^{n} PMT\left( {1 + r} \right)^{k - 1} ,where \,PMT \,is\, {\text{the}}\,{\text{payment}}\,{\text{in}}\,{\text{each }}\,{\text{period}}.\)

Suppose a project will pay you $1,000 at the end of each year for 4 years at 5% annual interest, and the following graph shows the process:

We still use two methods to calculate the future value of this ordinary annuity. First, we directly use the formula to compute it and obtain the value of 4310.125. The result is presented in Table 18.11 .

Then we use the FV function in Excel to compute the future value and obtain 4310.125. Hence, the two methods give us the same result, as presented in Table 18.12 .

Case 2. Future Value of an Annuity Due .

The formula is \(FV = \mathop \sum \limits_{k = 1}^{n} PMT\left( {1 + r} \right)^{k} ,where\,PMT\,is\,{\text{the payment in each period}}.\)

Suppose a project will pay you $1,000 at the beginning of each year for 4 years at 5% annual interest, and the following graph shows the process:

First, we directly use the formula to compute it and obtain the future value of 4525.631. The result is presented in Table 18.13 .

Then we use the FV function in Excel to compute the future value and obtain 4525.63. The only difference between calculating annuity due and computing ordinary annuity is to choose “1” in “type term” of the FV function rather than to choose “0”. The two methods give us the same result, as presented in Table 18.14 .

Present Value of an Ordinary Annuity

Case 1. Present Value of an ordinary annuity.

The formula is \(FV = \mathop \sum \limits_{k = 1}^{n} PMT/\left( {1 + r} \right)^{k} ,where\,PMT\,is\,{\text{the payment in each period}}.\)

Suppose a project will pay you $1500 at the end of each year for 4 years at 5% annual interest.

According to this formula, we directly input “=B1/(1+B5)^4+B2/(1+B5)^3+B3/(1+B5)^2+B4/(1+B5)^1” to get the present value of 5318.93, as presented in Table 18.15 .

In addition, we can use the PV function in Excel directly and obtain the same amount as above, as presented in Table 18.16 .

Case 2. Present Value of an annuity due.

The formula is \(PV = \mathop \sum \limits_{k = 0}^{n - 1} PMT/\left( {1 + r} \right)^{k} ,where\,PMT\,is\,{\text{the payment in each period}}.\)

According to this formula, we directly input “=B1/(1+B5)^3+B2/(1+B5)^2+B3/(1+B5)^1+B4/(1+B5)^0” to get the present value of 5584.87, as presented in Table 18.17 .

Similarly, the PV function gives us the same result as presented in Table 18.18 .

Case 3. An annuity that pays forever (Perpetuity).

In Excel, we directly input “=B1/B2” to get PV = 30,000, as presented in Table 18.19 .

Appendix 18E: Tables of Time Value of Money

See Tables 18.20 , 18.21 , 18.22 and 18.23 .

Questions and Problems

Define following terms:

Present values and future value.

Compounding and discounting process.

Discrete versus continuous compounding.

Liquidity preference.

Discuss how the following four tables listed at the end of the book are compiled.

Present value table.

Future value table.

Present value of annuity table.

Compound value of annuity table.

Suppose that $100 is invested today at an annual interest rate of 12% for a period of 10 years. Calculate the total amount received at the end of this term as follows:

Interest compounded annually.

Interest compounded semiannually.

Interest compounded monthly.

Interest compounded continuously.

What is the present value of $1,000 paid at the end of one year if the appropriate interest rate is 15%?

CF 0 is initial outlay on an investment, and CF 1 and CF 2 are the cash flows at the end of the next two years. The notation r is the appropriate interest rate. Answer the following:

What is the formula for the net present value?

Find NPV when CF 0 = -$1,000, CF 1 = $600, CF 2 = $700, and r = 10%.

If the investment is risk-free, what rate is used as a proxy for r?

ABC Company is considering two projects for a new investment, as shown in table below (in dollars). Which is better if ABC uses the NPV rule to select between the projects? Suppose that the interest rate is 12%.

		Year 0	Year 1	Year 2	Year 3	Year 4
Project A	Costs	10,000	0	0	0	0
	Returns	0	0	0	1,000	20,000
Project B	Costs	5,000	5,000	0	0	0
	Returns	0	10,000	5,000	3,000	2,000

Suppose that C dollars is to be received at the end of each of the next N years, and that the annual interest rate is r over the N years.

What is the formula for the present value of the payments?

Calculate the present value of the payments when C = $1,000, r = 10%, and N = 50.

Would you pay $10,000 now ( t = 0) for the annuity of $1,000 to be received every year for the next 50 years?

If $1,000 per year is to be received forever, what is the present value of those cash flow streams?

Mr. Smith is 50 years old and his salary will be $40,000 next year. He thinks his salary will increase at an annual rate of 10% until his retirement at age 60.

If the appropriate interest rate is 8%, what is the present value of these future payments?

If Mr. Smith saves 50% of his salary each year and invests these savings at the annual interest rate of 12%, how much will he save by age 60?

Suppose someone pays you $10 at the beginning of each year for 10 years, expecting that you will pay back a fixed amount of money each year forever commencing at the beginning of Year 11. For a fair deal when annual interest rate is 10% how much should the annual fixed amount of money be?

ZZZ Bank agrees to lend ABC Company $10,000 today in return for the company’s promise to pay back $25,000 five years from today. What annual rate of interest is the bank charging the company?

Which of the following would you choose if the current interest rate is 10%?

$12 at the end of each year for the next ten years.

$10 at the end of each year forever.

$200 at the end of the seventh year.

$50 now and yearly payments decreasing by 50% a year forever.

$5 now and yearly payments increasing by 5% a year forever.

You are given an opportunity to purchase an investment which pays no cash in years 0 through 5, but will pay $150 per year beginning in year 6 and continuing forever. Your required rate of return for this investment is 10%. Assume all cash flows occur at the end of each year.

Show how much you should be willing to pay for the investment at the end of year 5.

How much should you be willing to pay for the investment now?

If you deposit $100 at the end of each year for the next five years, how much will you have in your account at the end of five years if the bank pays 5% interest compounded annually?

If you deposit $100 at the beginning of each year for the next five years, how much will you have in your account at the end of five years if the bank pays 5% interest compounded annually?

If you deposit $200 at the end of each year for the next 10 years and interest is compounded continuously at an annual quoted rate of 5%, how much will you have in your account at the end of 10 years?

Your mother is about to retire. Her firm has given her the option of retiring with a lump sum of $50,000 now or an annuity of $5,200 per year for 20 years. Which is worth more if your mother can earn an annual rate of 6% on similar investments elsewhere?

You borrow $6145 now and agree to pay the loan off over the next ten years in ten equal annual payments, which include principal and 10% annually compounded interest on the unpaid balance. What will your annual payment be?

Ms. Mira Jones plans to deposit a fixed amount at the end of each month so that she can have $1000 once year hence. How much money would she have to save every month if the annual rate of interest is 12%?

You are planning to buy an annuity at the end of five years from now. The annuity will pay $1500 per quarter for the next four years after you buy it ( t = 6 thru 9). How much would you have to pay for this annuity in year 5 if the annual rate of interest is 8%?

Air Control Corporation wants to borrow $22,500. The loan is repayable in 12 equal monthly installments of $2,000. The corporate policy is to pay no more than an annual interest rate of 10%. Should Air Control accept this loan?

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Lee, J., Chang, JR., Kao, LJ., Lee, CF. (2023). Time Value of Money Determinations and Their Applications. In: Essentials of Excel VBA, Python, and R. Springer, Cham. https://doi.org/10.1007/978-3-031-14283-3_18

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