# Discounted Cash Flow, Net Present Value, Time Value of Money

Definitions, Meaning Explained, Usage, Calculated Examples

Business Encyclopedia ISBN 978-1929500109

Updated 2018-02-19 © 2017 Solution Matrix Ltd

*Time value of money concepts are the cornerstone of modern finance.*

## What are Discounted Cash Flow and the Time Value of Money?

**Discounted cash flow** **DCF** is an application of the **time value of money** concept—the idea that money to be received or paid at some time in the future has less *value*, today, than an equal amount actually received or paid today.

The DCF calculation finds the value appropriate today—the **present value**—for the future cash flow. The term "discounting" applies because the DCF present value is always lower than the cash flow **future value.**

In modern finance, time value of money concepts play a central role in decision support and planning. When investment projections or business case results extend more than a year into the future, professionals trained in finance usually want to see cash flows presented in two forms, with discounting and without discounting. Financial specialists, that is, want to see the time value of money impact on long-term projections.

### Two Central Terms: Present Value and Future Value

In discounted cash flow analysis DCF, two time value of money terms are central:

**Present value**(**PV)**is what the future cash flow is worth*today.***Future value**(**FV**) is the value that actually flows in or out at the future time.

A $100 cash inflow that will arrive two years from now could, for example, have a present value today of about $95, while its future value is by definition $100.

- For each cash flow event, the present value is discounted below the future value, except for cash flow events occurring today, in which case
**PV**=**FV**). - The longer the time period before an actual cash flow event occurs, the greater the present value of future money is discounted below its future value.
- The total discounted value (present value) for a series of cash flow events across a time period extending into the future is the
**net present value**(**NPV**) of a**cash flow stream**.

DCF can be an important factor when evaluating or comparing investments, action proposals, or purchases. Looking forward in time, the analyst projects cash inflows and outflows (cash flow stream) expected to follow from each of these. Other things being equal, the action or investment with the larger cash flow NPV is recommended as the better decision.

## Contents

- What are discounted cash flow and the time value of money?
- Is "time value" of money tangible real value? Is Time Value important in Finance?
- Defining present value, future value, and net present value.
- What are the mathematics in calculating DCF and NPV?
- How is the discount (interest) rate chosen for discounted cash flow analysis?
- How are DCF and NPV used in business for comparing competing investment proposals?
- When and where are DCF and NPV used in business case analysis?

## Related Topics

- Defining, explaining, and measuring cash flow: See the article Cash Flow.
- Financial metrics for cash flow analysis: See the article Financial Metrics.

## Is Time Value of Money Real Value?

Is Time Value Important in Finance?

When first hearing the definition of discounted cash flow, many people understandably react with comments like these: "It sounds like fiction" or "The time value of money cannot refer to real value because DCF does not measure real cash flow" or "It's an interesting calculation, but there's no tangible value involved."

However, business professionals recognize that the results of discounting calculations do represent real tangible value, readily seen if the time value of money concept is stated like this:

Having thehas value that is tangible, measurable, and real.useof money for a specific period of time

Discounted cash flow (DCF) is one application of this concept, while interest paid for a loan is another. With DCF, the discounting lowers the present value PV of future funds below the future value FV of the funds for at least three reasons:

**Opportunity**. Money you have now could (in principle) be invested now, and gain return or interest between now and the future time. Money you will not have until a future time cannot be used now.**Risk**. Money you have now is not at risk. Money expected in the future is less certain. A well known proverb states this principle more colorfully: "A bird in hand is worth two in the bush."**Inflation**: A sum you have today will very likely buy more than an equal sum you will not have until years in future. Inflation over time reduces the buying power of money.

## Defining Present Value, Future Value, and Net Present Value

What Do PV, FV, and NPV Mean?

What future money is worth today is called its present value (PV) and what it will be worth in the future when it finally arrives is called not surprisingly its future value (FV).

The right to receive a $100 payment one year from now (the future value) might be worth to us

today$95 (its present value).

Present value in other words, is *discounted *below future value.

### Present Values for a Cash Flow Stream

When the analysis concerns a series of cash inflows or outflows coming at different future times, the series is called a **cash flow stream**. Each future cash flow has its own value today (its own present value). The sum of these present values is the net present value for the cash flow stream.

Consider an investment today of $100, that brings net gains of $100 each year for 6 years. The future values and present values of these cash flow events might look like this:

One 6-Year cash flow stream, as Future Values (FV), and Present Values (PV) at two different discount rates.

All three sets of bars represent the same investment cash flow stream.

- Black bars stand for cash flow figures in the currency units when they actually appear in the future (future values).
- Green and Blue bars are values of the same cash flows
*now*, in present value terms. - The net values in the legend show that after five years, the net cash flow expected is $500, but the
**Net present value**(**NPV)**today is discounted to something less.

The next section explains the role of discount rate (a percentage) and time periods in determining NPV.

### Interest Rates and Time Periods in Discounting

The size of the discounting effect depends on two things: the amount of time between now and each future payment (the number of discounting periods) and an interest rate called the discount rate. The example shows that:

- As the number of discounting periods between now and the cash arrival increases, the present value decreases.
- As the discount rate (interest rate) in the present value calculations increases, the present value decreases.

Whether you will or will not calculate present values yourself, your ability to use and interpret NPV / DCF figures will benefit from a simple understanding of the way that interest rates and discounting periods work together in discounting. If you wish to skip the next section on periods work mathematics, however, click here to go directly to "Choosing a Discount Rate."

## Mathematics in Discounting Calculations

Examples Calculating FV, PV and NPV

Many if not most business people outside of finance, are unfamiliar with time value of money terms and calculations. The subject becomes approachable, however, if the explanation begins by noting that DCF mathematics are very closely related to a subject that is familiar to most people: calculations for interest growth and compounding.

Remember briefly how interest calculations work. The FV formula looks into the future and might ask, for instance: What is the future value (FV) in one year, of $100 invested today (the PV), at an annual interest rate of 5%?

### FV_{1} = $100 ( 1 + 0.05)^{1} = $105

When the FV is more than one period into the future, as most people know, interest compounding takes place. Interest earned in earlier periods begins to earn interest on itself, in addition to interest on the original PV. Compound interest growth is delivered by the exponent in the FV formula, showing the number of periods. What is the future value in five years of $100 invested today at an annual interest rate of 5%?.

### FV_{5} = $100 ( 1 + 0.05)^{5} = $128

The same formula can be rearranged to deliver a present value given at the future value and interest rate for input, as shown at left.

Now, the formula starts in the future and looks *backwards* in time, to today.

The formula now asks: What is the value today of a $100 payment arriving in one year, using a discount rate of 5%?

### PV_{1} = ($100) / (1.0 + 0.05)^{1 }

= $100 / (1.05)

= $95

You should be able to see why PV will decrease if we either (a) increase the interest rate, or (b) increase the number of periods before the FV arrives. What is the present value of $100 we will receive in 5 years, using a 5% discount rate?

### PV_{5 } = $100 / (1.0 +0.05)^{5}

= $100 / (1.276)

= $75.13

When the FV is more than one period into the future, as most people know, interest compounding takes place. Interest earned in earlier periods begins to earn interest on itself, in addition to interest on the original PV. Compound interest growth is delivered by the exponent in the FV formula, showing the number of periods. What is the future value in five years of $100 invested today at an annual interest rate of 5%?.

### FV_{5} = $100 ( 1 + 0.05)^{5}

= $128

When discounting is applied to a series of cash flow events, a cash flow stream, as illustrated in the graph example above, net present value for the stream is the sum of PVs for each FV:

### Should You Use Mid-Period (or Mid-Year) Discounting? What Difference Does it Make?

Finally, note two commonly used variations on the examples shown thus far. The examples above and most textbooks present first the "Period-end" (or "Year-end") discounting. Period-end discounting is the more frequently used DCF approach. The approach, moreover, usually turns up as the default approach for spreadsheet and calculator DCF functions.

**Period-End Discounting**

With the **period-end** approach, all discounting for a period is applied as though all cash flow occurs on the last day of the period. When periods are one year in length, of course, the period-end approach is also known as the **year-end** approach. With year-end discounting, all of the period's cash flow is assumed to occur on day 365 of the year.

**Mid-Period Discounting**

Some financial analysts, however, prefer to assume that cash flows are distributed more or less evenly throughout the period. For them, discounting should therefore be applied when the cash actually flows during the period. Calculating present values this way is mathematically equivalent to saying that all cash flow occurs at mid-period. For this reason, this approach is called **mid-period** discounting. And, of course, the name mid-year discounting applies when periods are one year in length.

### What Are the Differences Between the Two Approaches?

- Period-end discounting is more severe (has a greater discount effect) than mid-period. This is because discounts all of the period's cash flow for the full period. You can see how this works mathematically from the formulas in the next section. Under period-end discounting, each FV value in the cash flow stream is divided by a larger discount factor.

Some analysts prefer to describe this difference by saying the period-end approach is more**conservative**. - Those preferring the other approach say that discounting mid-period is more
**accurate.**

Remember that the discount rate recognizes the values of opportunity, risk, and inflation—values that change continuously as time passes. Mid-period discounting comes closer, they say, to applying the discounting effect precisely when cash actually flows.

### Formulas for Mid-Period Discounting

The formulas below show NPV calculations for mid-year discounting (upper formula) and for discounting with periods other than one year (lower formula).

In any case, the business analyst will want to find out which of the above discount methods is preferred by the organization's financial specialists, and why, and follow their practice (unless there is justification for doing otherwise).

Working examples of these formulas, along with guidance for spreadsheet implementation and good-practice usage are available in the spreadsheet-based tool Financial Metrics Pro.

## Choosing Discount (Interest) Rate for Cash Flow Analysis

The analyst will also want to find out from the organization's financial specialists which discount rate the organization uses for discounted cash flow analysis. Financial officers who have been with an organization for some time, usually develop good reasons for choosing one rate or another as the most appropriate rate for the organization.

- In private industry, many companies use their own cost of capital (or weighted average cost of capital) as the preferred discount rate.
- Government organizations typically prescribe a discount rate for use in the organization's planning and decision support calculations. In the United States, for instance, the Office of Management and Budget (OMB) publishes a quarterly circular with prescribed discount rates for Federal Government use.
- Financial officers may use a higher discount rate for investments or decisions viewed as risky, and a lower discount rate when expected returns from a proposed action are seen as less risky. The higher rate is viewed as a hedge against risk, because it puts relatively more emphasis (weight) on near-term returns compared to distant future returns.

## Comparing Competing Investment Proposals with DCF and NPV

Consider two competing investments in computer equipment. Each calls for an initial cash outlay of $100, and each returns a total a $200 over the next 5 years making net gain of $100. But the timing of the returns is different, as shown in the table below (Case Alpha and Case Beta), and therefore the present value of each year’s return is different. The sum of each investment’s present values is called the discounted cash flow(DCF) or net present value (NPV). Using a 10% discount rate again, we find:

Timing | CASE Alpha | CASE Beta | ||
---|---|---|---|---|

Net Cash Flow | Present Value | Net Cash Flow | Present Value | |

Now | – $100.00 | – $100.00 | – $100.00 | – $100.00 |

Year 1 | $60.00 | $54.54 | $20.00 | $18.18 |

Year 2 | $60.00 | $49.59 | $20.00 | $16.52 |

Year 3 | $40.00 | $30.05 | $40.00 | $30.05 |

Year 4 | $20.00 | $13.70 | $60.00 | $41.10 |

Year 5 | $20.00 | $12.42 | $60.00 | $37.27 |

Total | Net CF _{A} = $100.00 | NPV_{A} = $60.30 | Net CF_{B} = $100.00 | NPV_{B} = $43.12 |

Comparing the two investments, the larger early returns in Case Alpha lead to a better net present value (NPV) than the later large return in Case Beta. Note especially the *Total* line for each present value column in the table. This total is the net present value (NPV) of each cash flow stream." When choosing alternative investments or actions, other things being equal, the one with the higher NPV is the better investment.

## What Roles Do DCF and NPV Play in Business Case Analysis?

In brief, an NPV / DCF view of the cash flow stream should probably appear with a business case summary when:

- The business case deals with an "investment" scenario of any kind, in which different uses for money are being compared.
- The business case covers long periods of time (two or more years).
- Inflows and outflows change differently over time (e.g., the largest inflows come at a different time from the largest outflows).
- Two or more alternative cases are being compared and they differ with respect to cash flow timing within the analysis period.

For a working spreadsheet example of discounted cash flow calculations more in-depth coverage of discounted cash flow usage, please see Financial Metrics Pro.