# Discounted Cash Flow and the Time Value of Money

How to Discount Cash Flow, Calculate PV, FV and Net Present Value

Business Encyclopedia ISBN 978-1929500109 © 2019 Solution Matrix Ltd All Rights Reserved

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

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

The **Discounted cash flow** **concept (DCF)** is an application of the time value of money principle—the idea that money that will be received or paid at some time in the future has less value, today, than an equal amount collected 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 know 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 flows in or out at the designated time in the future.

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 less than the corresponding future value, except for cash flow events occurring today, in which case
**PV**=**FV**). - The further into the future before a cash flow event occurs, the more discounting lowers the present value below its future value.
- the total of all of the cash flow present values for the cash flow series, across a timespan extending into the future, is the
**net present value**(**NPV**) of a**cash flow stream**.

DCF is center stage when decision makers and investors evaluate potential investments, action proposals, or purchases. Looking forward in time, the analyst projects cash inflows and outflows (cash flow streams) the investor can expect from each of these. Other things being equal, analysts will recommend the action or investment with the more substantial cash flow NPV, as the "better business decision."

### Explaining Time Value of Money in Context

Sections below describe and illustrate Discounted Cash Flow and other Time Value of Money terms in context with similar ideas from business analysis, banking, and finance, focusing on four themes:

- First, defining Time Value of Money in Finance, and why this value is real and measurable.
- Second, the meaning of Time Value and Discounting concepts, such as Present Value, Future Value, and Discount Rate.
- Third, example calculations showing how to discount future values to present values in cash flow streams, and how to calculate Net Present Value (NPV).
- Fourth, the role of DCF and NPV in Business Case Analysis and Business Planning.

## Contents

- What is the meaning of "discounted cash flow" and the "time value of money?"
- Is "time value of money" real value? Why Is Time Value important in Finance?
- Defining present value, future value, and net present value.
- What is the mathematical basis for calculating DCF and NPV?
- How do analysts choose the discount (interest) rate for DCF analysis?
- How do business people use DCF and NPV for comparing competing investment proposals?
- What are the roles of DCF and NPV 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 when they state the time value of money concept like this:

Having thehas a value that is tangible, measurable, and real.useof money for some 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.**

Funds you have now could (in principle) be invested now, and gain return or interest between the present and the future time. You cannot use now funds that you will not have until a "future" time.**Risk**.

Funds you have now are not at risk, but funds arriving in the future are uncertain. A well-known proverb states this principle more colorfully: "A bird in the hand is worth two in the bush."**Inflation**.

A sum you have today will very likely buy more than an equal amount you will not have until years in future. The buying power of money decreases over time due to inflation.

## Defining Present and Future Value, 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, *discounts *to a value 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 value today (its 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 six 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 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 arrive 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 the 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 to create DCF. 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

**Future Value**

Definition Formula

**FV = PV ( 1 + i )^{ n}**

**FV** = Future Value

**PV** = Present Value

** i** = Interest (discount) rate

**= Period numbe**

*n**r*

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 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 a "future value" and "interest rate" for input, as shown.

Now, the formula starts in the future and looks *backward* in time, toward 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 compound, 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:

Year-end discounting for a stream of ** n** fuutre cash llows (FVs)

### 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 cases 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, discounting works as though all cash flow occurs on the last day of each period. When periods are one year in length, of course, analysts call this 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 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 more substantial discount effect) than the mid-period approach. Period-end calculations are more severe than mid-period versions because they impact 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. The period-end approach divides each FV value by a more substantial discount factor than do mid-period calculations.

Some analysts prefer to describe the difference between approaches by saying the "period-end" discounting is the more conservative approach. - 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 can change as time passes. Mid-period discounting comes closer, they say, to applying the discounting effect precisely when cash flows.

### Formulas for Mid-Period Discounting

The methods below show NPV calculations for the mid-year approach (at panel top) and for discounting with periods other than one year (at mid-panel).

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

For working examples of these formulas and guidance for spreadsheet implementation, see the spreadsheet-based ebook Financial Metrics Pro.

## Choosing a 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 discount rate or another as the most appropriate rate for the organization.

- In private industry, many companies use their own cost of capital (or a 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 come with less risk. The higher "discount rate" is 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 $200 over the next five years making for a 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 gains 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 the 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 a 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, to compare different uses for funds.
- The business case covers a long period (two or more years).
- Inflows and outflows change differently over time. For example, the most substantial "inflows" come at a different time from the most substantial "outflows."
- Two or more alternative cases for comparison differ concerning cash flow timing within the analysis period.

For live spreadsheet examples of discounted cash flow calculations and more in-depth coverage of DCF usage, see the Excel-based ebook Financial Metrics Pro Financial Metrics Pro.