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Time Value of Money

Time value of money is the economic concept that money (or capital) received today has a different value than money that will be received in the future.  Understanding and applying the concept of the time value of money is important in deciding which alternative to pursue when the alternatives will generate revenue or incur costs at different times in the future.

This page reviews the concept of time value of money, discusses the analytical process of discounting future payments to present value, and presents several applications of the concept of time value of money (including capitalizing the present value of future payments to determine the value of an asset).  Embedded in this explanation is a discussion of how to determine a discount rate.

Time Value of Money -- The concept

Time value of money is an economic concept that money (or capital) received today has less value than money that will be received in the future. Several reasons account for this difference in value, for example,

• Money received today is a certainty -- you have the cash -- whereas money that is to be received in the future may never be received.  Risk and uncertainty is one reason why money received today is more valuable than money that may be received in the future.
• Money received today can be invested and generating income, that is, not receiving the payment until a later time has an opportunity cost.
• Inflation between now and when the money is received in the future decreases the purchasing power of the future cash; we know that $4 today will allow me to buy at least one gallon of gas; but will $4 five years from now still buy a gallon of gas at that time. Expected inflation explains why money received today has a greater value than an equal amount of money that will be received in the future.
• Money received today can be spent today and the benefit (utility) of having that item today is greater than being able to purchase that item sometime in the future after the future payment is received. The basic principle is that people prefer to consume (place more value on having) something today rather than receiving and consuming it in the future.

Except in extraordinary circumstances, the value of money received today will have greater value than an equal amount of money that will be received in the future.  For example, $10 today is has more value than receiving $10 in the future due to the four factors listed above.

Present value is today’s value of a payment that will be received in the future.

The corollary is that the present value of a future payment is less than the amount of the future payment.  For example, a $10 payment that will be received in the future has a present value of less than $10.

Example.  Due to the four reasons listed above, I may be willing to accept $9.50 today rather than wait to receive $10 a year from now.

Because a payment received in the future has less value than an equal payment today, we often refer to this thought process or analytical process as “discounting the future payment to its present value.”

Time also is a consideration.  A higher present value will be placed on a future payment that will be received sooner than the present value that will be placed on a payment that will be received later.

Example.  A $10 payment that will be received 6 months from now has a higher present value than a $10 payment that will be received one year from now.  The further into the future a payment will be received, the less the value that will be placed on it today.  With the passage of time, the present value of a future payment increases because the length of time until the payment is growing shorter.

Individuals will Place Different Present Values on the Same Future Payment

Everyone will not place the same value on a future payment.  Consider the following example.

• If I had a choice between receiving $10 one year from now and receiving $9.50 today, and I was indifferent between the two options, we know that I consider a $10 payment one year from now to have a present value of $9.50.  Restated, I consider the present value of $10 that I will receive one year from now to be $9.50.
• If a second person readily chooses to receive the $10 payment one year from now (rather than the $9.50 today), we know that this person feels the present value of the $10 one year from now is more than $9.50.
• If a third person prefers to receive the $9.50 today, we know that the third person feels the present value of $10 one year from now is less than $9.50.

Because each of us thinks differently about the four factors listed above, we will not place the same value on a future payment

Can this Difference be explained?

The extent to which a person discounts future payments depends on his or her discount rate; that is, each individual can have a different discount rate because we have different perceptions of risk, opportunity cost, inflation, and preference to consumer now.

• For example, a person concerned about risk, or who has a strong preference to consume now would likely have a high discount rate and place a relatively lower value on future payments. Likewise, someone who is concerned that future inflation will be relatively high or who has a good investment opportunity also would have a high discount rate.

Introduction to Discount Rate

A discount rate is generally stated as an annual percentage rate. For example, someone may reduce the value of a future payment by 7% for each year until the payment is received. The 7% accounts for this person's concern about risk and inflation, the income that could be earned if the cash was available now, and the preference to consume now rather than in the future.

Even though each of us uses a different discount rate in assessing our opportunities or alternatives, we begin to demonstrate our collective assessment of the discount rate in the market place, more specifically, in the financial markets. For example, the interest rate a bank must pay to motivate you to deposit your cash in a savings account (rather than spend the cash today) is an indication of the value you place on consuming today versus consuming in the future.

• The market rate of interest is the beginning point in determining a discount rate. Often the interest rate paid by the U.S. federal government (10-year treasury yield) is used as a measure of the market interest rate where the risk of being paid in the future is as low as possible (this is based on the assumption that the U.S. government will be one of the last entities to go broke even if the world's economy falls apart). For example, an interest rate of 4.64% for 2007 was calculated from data found at http://research.stlouisfed.org/fred2/data/WGS10YR.txt.
• The market rate of interest also is referred to as the nominal interest rate.

Despite individuals’ different assessments of the future, there are some basic concepts about present value that all business persons need to understand.  The process of calculating present value is such a concept.

Calculating Present Value of a Future Payment

In order to compare the value of several payments, each payment or series of payments need to be discounted to present value.

Discounting is the analytical process by which the present value of a future payment is calculated. For example, discounting allows us to calculate that $10 we are promised to receive a year from now is worth $9.50 today.

Example of Calculating Present Value of a Future Payment

The present value of a future payment can be calculated if the amount of the future payment, the time of the future payment, and the discount rate are specified.  For example, an individual who expects to receive $50 twenty-one (21) months from now and who discounts future payments by 6% for each year until the payment is received can calculate the present value of that payment.

• The formula for computing the present value of a future payment is PV = FV/(1+d)^n, where:
  • PV is present value,
  • FV is amount of the future payment,
  • d is the discount rate (expressed as a decimal), and
  • n is the number of time periods until the future payment is received (although it may not be obvious on this web page, n is treated as an exponent to the factor in parenthesis)

  Example. An individual will be paid $50 twenty-one (21) months from now. The individual has an annual discount rate of 6%; that is, the individual feels a payment received one year from now is only worth about 94% of what it would be worth if the payment was received today. Twenty-one months is 1.75 years; since the discount rate is expressed as an annual rate, the time period has to be measured in terms of years. Using the present value formula,

PV = 50/(1+.06)^1.75
PV = $45.15

Also see Calculating Present Value.

Example of Simple Spreadsheet to Calculate Present Value

A simple computer spreadsheet can be used to calculate present value.

Cells B2, B3, and B4 are used to input the values and

cell B5 is the formula B2/(1+B3)^B4 that provides the answer.

Determining the Discount Rate

Setting a discount rate is challenging because it involves making projections about the future, such as, what is the likelihood that the future payment will be received and what will be the rate of inflation. However, almost all individuals are making these projections on a regular basis; we just may not recognize it. Whenever individuals decide to invest $100 in a savings account to earn interest, they are indicating that the $100 plus interest sometime in the future is more valuable to them (that is, it provides them more utility or satisfaction) than being able to spend the $100 today.

• If the payment will be made by an entity that is less reliable than the U.S. government, there is some risk that should be considered in setting the discount rate. For example, if the payer is a small business, the risk premium may justify raising the discount rate by 1.5% for a total discount rate of 6.14% (4.64 + 1.5), assuming the 2007 market rate of interest of 4.64%.
• Example. A small business promises to pay an investor $100 a year; this promise (assuming a discount rate as described in the previous point) has a present value of $94.22 (100/(1+.0614)^1).

Adjusting for Inflation

In some situations, the amount of the future payment may change between the time the promise is made and when the payment is made; perhaps due to inflation. In that case, you may want to use a real discount rate rather than a nominal discount rate as illustrated in the previous example.

• Example. An investor has agreed to contribute capital to a business in exchange for a portion of the business' profit. Currently, the profit for the next year is projected to be $100 but it will likely increase at the same rate as inflation during the coming year. To determine a discount rate in this situation, again consider the market (nominal) rate of interest and the risk adjustment. Assume these are the same as in the previous example, that is, 6.14%.

• The next step is to adjust this market or nominal interest rate to a real interest rate by factoring out (or subtracting) the inflation rate. The consumer price index (CPI) is a measure of inflation.
• By subtracting the annual percentage change in the CPI (e.g. 4.1% in 2007) from the nominal rate (6.14), a discount rate based on a real interest rate (and risk associated with this investment) would be 2.04%.
• If the analyst was not concerned about the risk (1.5%), the focus would be on the 4.64% market rate of interest. It would also be evident that most of that 4.64% is the market accounting for 4.1% inflation. Accordingly, only .54% (4.64 - 4.1) of the market interest rate is due to minimal risk, lost opportunity, and preference to consume now.
• This is a simplified example of determining a discount rate by calculating the real interest rate; due to annual fluctuations in market interest and inflation rates, multiple years of data should be used to determine a discount rate.
• In this example, the investor would place a present value (PV) of $98.00 on $100 (FV) to be received one year in the future (100/(1+.0204)^1).
  • The discount rate in this example (2.04%) accounts for opportunity cost, preference to consumer now, and more substantial level of risk.

• If inflation is expected to impact future payments, but that impact has not been incorporated into the projected payments, use a real discount rate. If projected future payments have been adjusted to reflect the expected impact of inflation, use a nominal discount rate.
  • Example. If the future payment is expected to increase with inflation, the $100 payment from the small business will actually be $104.10 a year from now (assuming 4.1% inflation). The present value of the payment from the small business can be calculated two ways
    • Inflation adjusted payment discounted by a nominal rate adjusted for risk; that is, $104.10/(1.0614)^1 = $98.08.
    • Future payment is NOT adjusted to reflect expected inflation so a real discount rate is used: $100/(1.0204)^1 = $ 98.00.
    • Although the two numbers are not exactly the same, they suggest that either approach may be reasonable.

• If the real discount rate is used (because future payments have not been adjusted for the expected impact of inflation), recognize that the analysis is assuming that inflation will impact the future payments at the same rate as the inflation adjustment between the nominal and real rates, that is, the general rate of inflation. Is this always a valid assumption?
  • If the inflation rate for the payments is expected to exceed the general rate of inflation, the discount rate will be less than the real interest rate.
  • If the inflation rate for the payments is expected to be less than the general rate of inflation, the discount rate will be greater than the real interest rate.

Understanding and applying the concept of the time value of money is important in deciding which alternative to pursue when your alternatives will generate revenue or require payments at different times in the future.

Application -- Discounting Future Earnings to Present Value

Because the revenue from the asset or investment will be received over time, future earnings must be discounted to present value. Also, each future earnings or payment must be estimated or calculated, if they are expected to vary over time, before discounting them to present value.

• Example. An investment is expected to return $100 one year from now, another $112 two years from now, and soon thereafter be sold for $500. Assuming a discount rate of 5%, the present value of the first payment is $95.24 (100/(1+.05)^1), the present value of the second payment is $101.59 (112/(1+.05)^2), the present value of payment for selling the asset is $453.51 (500/(1+.05)^2), and the total present value of this short-term investment is $650.34 (95.24 + 101.59 + 453.51). Accordingly, an investor who imposes a 5% discount rate on future income would be willing to pay no more than $650.34 for this asset.
• This investment is projected to generate $712 of revenue over the next two years.  But the present value of that revenue is only $650.34, assuming a 5% discount rate.

A computer spreadsheet can help illustrate these calculations.

• Cells D2, A4, A5, A6, C4, C5 and C6 are used to input the values.
• Cell D4 is the formula =C4/(1+D$2)^A4; cell D5 is the formula =C5/(1+D$2)^A5; cell D6 is the formula =C6/(1+D$2)^A6; these cells provide the present value of each payment.
• Cell D7 is the formula =sum(D4:D6); it provides the final answer.
• This spreadsheet can readily be modified to accommodate different situations; for example, additional rows can be inserted above row 7 if the investment will provide more than two future payments (before being sold).
• The time period can be expressed to reflect portions of years; for example, 1.75 years would be the appropriate time period if the payment will be received 21 months from now (21 months /12 months).
• Be sure the time period for the discount rate and the time periods until payment are the same; for example, if the discount rate is expressed as an annual rate, the time periods until payment will be the number of years, but if the discount rate is expressed as a monthly rate, the time periods until payment must be expressed as the number of months.

Role of Assumptions

This analytical process requires the analyst (often the business manager) to make numerous assumptions or projections about future events, especially if the future payments will result from a business that purchases inputs from which a product is produced and marketed (such as farmland being used to raise a crop). In that case, assumptions address

• quantity of future production,
• value of future production (implies an assumption about the impact of inflation and demand on the value of the product),
• time of future production,
• type and quantity of future inputs (implies an assumption about production technology),
• cost of future inputs (implies an assumption about the impact of inflation on input costs),
• time when future inputs will be used, and
• a discount rate (probably a real rate, that is, a nominal or market rate adjusted for an anticipated general rate of inflation).

Inflation is mentioned three times in this list – value of product, input costs and discount rate. However, the inflation rate (or price changes) may be different for each of these items; thus a manager may need to consider each of these factors individually.

• If the manger assumes that inflation will equally affect the three factors, present prices for output and inputs, and a real interest rate can be used to complete the analysis.

Alternative Means of Adjusting for Inflation in Calculating Present Value

Another way to address varying inflation rates is to use a different real discount rate for each item in the analysis. 

• A payment that is expected to increase at the same rate as the general rate of inflation can be discounted by a real interest rate (e.g., nominal or market interest rate minus the general rate of inflation). 
• A payment that is expected to increase at a slower rate than the general rate of inflation would be discounted with a higher discount rate (nominal interest rate minus the lower rate of inflation for that item). 
• A payment that is expected to increase at a rate higher than the general rate of inflation would be discounted with a lower discount rate (nominal interest rate minus the expected higher rate of inflation). 

However, this analytical method assumes each factor in the analysis is discounted individually and then the net outcome is calculated (method 2), rather than discounting only the net outcome (method 1). 

Method 1:  PV (TR – TC)
Method 2:  PV(TR) – PV(TC)

A computer spreadsheet is invaluable if the analyst needs to calculate present value of individual items within the analysis (method 2).

Application -- Calculating Present Value of Future Costs

The previous explanation and examples have emphasized calculating the present value future income. However, the same analytical process can be used to determine the present value of future costs. A simple example may be a manager who wants to compare two alternative opportunities for storing the business' product; that is, build a new warehouse or continue to use the current warehouse but recognize that the current warehouse will require extensive remodeling several times during the next decade. In this situation, the manager would list the expected costs for each alternative, the residual value of the building at the end of the analysis period, calculate the present value of these costs and residual value, total the present values, and determine which alternative is lower cost.

• The new building would require a payment at this time to purchase the warehouse, plus cost to operate (e.g., utilities) and maintain the building. The analysis would be more complex if the business had to borrow a portion of the pruchase price of the new warehouse.
• The current building would require more substantial maintenance and remodeling costs over the next 10 years (assuming that is the period of time the manager decides to use as the basis for the decision) as well as operating expenses, such as utilities.

The economic concept, analytical process, and issues of assumptions/expectations and the discount rate are the same as discussed previously.

Application -- Capitalizing Future Earnings to Determine Value of a Long-term Asset

An important decision for a business manager is determining the price to pay for a long-term asset such as a building or a tract of land. Similarly, a manager must be prepared to determine at what price to sell a long-term asset. In both situations, the fundamental question is "how does one determine the value of a long-term asset?"

Capitalization is the thought process or analytical process of discounting expected future payments to their present value and summing those present values to determine their total present value.  This total is the amount an investor would be willing to pay (invest) today in order to receive the payments in the future.

General Rule for Valuing an Asset

The value of an asset is the revenue it will generate for its owner. This revenue includes the income received from the sale of the products produced by the asset, plus the resale value when the asset is disposed of in the future. However, the cost of operating the asset to produce the products must be subtracted from the revenue.

• Example. The current value of farmland reflects 1) the revenue that will be generated in the future by selling the commodities produced on the land, 2) minus the cost of producing those commodities, and 3) plus the land's future selling price.
• For the purpose of this overview discussion, a long term asset means any asset or investment (whether tangible or intangible) that will generate revenue more than one year into the future.

Short-hand Method to Calculate Long-term Asset Value

If the manager anticipates owning the asset for a long-time, such as a tract of land for 40 years, the resale value may not have much of an impact on the present value because the revenue is not expected to be received for such a long time. However, the annual revenue throughout the period of anticipated ownership will have a major impact on the value of the asset. Accordingly, a short-hand method to calculate the value of a long-term asset is

V = R/d

where: V is value of the long-term asset,
R is annual earnings of long-term asset, and
d is a real discount rate.

Note that the resale value of this long term asset is not explicitly considered in the short-hand formula.

• Using a real discount rate implies (assumes) that the annual earnings will change by the general rate of inflation in the economy; this may or may not be an accurate assumption.
• Example. Annual earnings from the asset are expected to be $37 but will change with inflation, and the real discount rate is 3.5%. The value of this long-term asset would be $1,057.

• Cells B1 and B2 are used to input values; cell B3 is the formula +B1/B2 and provides the answer.

This formula (V=R/d), however, assumes no changes in quantity of sales or production technology. It also assumes that inflation will have an equal impact on sale price, input costs, and interest rates. But, the world is not that simple. The remainder of this page describes an expanded methodology that relaxes some of the simplifying assumptions, but it does not reduce the number of assumptions (some would suggest that it significantly increases the number of assumptions). The expanded methodology also increases the number of computations; fortunately computer programs (such as a spreadsheet) can help with these calculations. The following section suggests such a spreadsheet.

Example:  capitalizing the current rental rate for farmland.  See USDA NASS North Dakota Land Values

Source: National Agriculture Statistics Service, USDA.
* Calculated by dividing Average Rental Rate by Average Value of Rented Land

Expanded Method to Calculate Long-term Asset Value

As stated above, the value of a long-term asset is the present value of its future earnings. In the case of an asset that will be used in a business, revenue and operating expenses for each production period need to be considered. Key considerations for farmland, for example, include type and quantity of commodities that will be produced, price or value of the commodities, type and quantity of inputs used, and input costs.

• This is the same information used in preparing an enterprise analysis; thus the business owner who has prepared enterprise budgets can rely on existing information when valuing land (rather than having to gather all new information). The owner needs to be certain, however, that the enterprise analyses are up-to-date.

• The purpose of this present value analysis is to determine the value of the land resource. Therefore in preparing the enterprise analysis, the owner should subtract all costs except return to land; this includes subtracting an opportunity cost for non-land inputs, such as the owner's unpaid labor, management, and return to risk.

• Farmland often is used to raise several crops; for example, rotating crops is a common practice to reduce disease. An owner will need to prepare several enterprise analyses if the land will be used to raise several crops. This can be handled several ways, depending on how the land will be operated (one crop each year or several crops each year).

• Entire tract:  one crop each year but rotating annually, such as, crop A in year 1, crop B in year 2, crop C in year 3, and crop A in year 4, etc.
• Entire tract:  multiple crops each year, such as, crop A on 50% of the land each year, crops B and C each on 25% of the land each year.

• The manager needs to prepare an analysis for each year if conditions are expected to change, such as which commodities will be produced, prices, costs, or technology (which impacts the inputs that will be used). Such a large number of predictions about the future tend to become guesses the further into the future the analysis is extended. At some point in the analysis, the manager may stop trying to project all the details, but instead, rely on simplifying assumptions. Simplifying the approach does not reduce the number of assumptions; it just reduces how many of them are explicitly considered.
• A computer helps assess how changing assumptions can impact the projected asset value.

The spreadsheet calculates the present value of the future earnings. The spreadsheet does not eliminate or even reduce the number of assumptions that need to be made; instead the spreadsheet facilitates making those assumptions explicit.

• Assume a business wants to calculate the value of farmland that will be purchased at this time (year 1) and will be owned for m years. Also assume the land will be used to produce output y and this product requires the farmer to use inputs x1 through xn.

• V (value of the land) = the sum of the present values of E (annual future earnings); restated, the value of an asset equals its total present value of future earnings.
• E = earnings for each future period (i.e., TR-TC); earnings equals sum of total revenue (including resale price of the asset) minus total explicit cost and opportunity cost for all other owned assets (e.g., the farmer's labor) for each period the asset is owned.

• TR = total revenue equals quantity of output times the selling price (Qy * Py)
• TC = total cost equals sum of quantity of each input times the cost of each input ((Qx1 * Cx1) + (Qx2 * Cx2) + … + (Qxn * Cxn))

• Quantity of output (Qy) = projected production of product for each period of ownership
• Price (Py) = projected price for output produced during each period of ownership
• Input quantity (Qx) = projected quantity of each input used in operating the asset and producing the product
• Cx = projected cost of each input used in operating the asset and producing the product

• The analysis needs to consider the inflation rate for input costs (Cx1…Cxn), product price (Py1…Pyn), and the general inflation rate for years 1 through m (recall, the general inflation rate impacts the discount rate).
• The spreadsheet uses enterprise budget/analysis as the basis for the calculations; that is, E = TR - TC. In an expanded form, the annual earnings for years 1 through m could be expressed as
• E1 = (Qy1 * Py1) - ((Qx11 * Cx11) + (Qx21 * Cx21) + … + (Qxn1 * Cxn1))
• E2 = (Qy2 * Py2) - ((Qx12 * Cx12) + (Qx22 * Cx22) + … + (Qxn2 * Cxn2))
• E3 = (Qy3 * Py3) - ((Qx13 * Cx13) + (Qx23 * Cx23) + … + (Qxn3 * Cxn3))
• …
• Em = (Qym * Pym) - ((Qx1m * Cx1m) + (Qx2m * Cx2m) + … + (Qxnm * Cxnm)) + residual value of the asset

• PV = PV(E1) + PV(E2) + PV(E3) + … PV (En)
• This example illustrates method 1, as described in a previous section.  However, this level of detail can be converted to method 2 if the analyst has enough information to explicitly address the inflation rate for individual items.

• The purpose of this expanded form is to illustrate the numerous assumptions embedded in determing the value of an asset such as farm land.
  • In addition to assumptions about future inflation rates (as mentioned previously), the expanded form also illustrates assumptions about changing production technology. For example in year 1, production technology might rely heavily on input x1 and not use input xn.  By year 10, however, production technology may have changed that now input x1 is not being used but input xn is a major input.
  • The quantity of the various inputs that is used each year reflects the technology that is available and being used at that time.
    • Some may argue, however, that any increases in production due to technology should be attributed to the technology and not be capitalized into the value of the land.

• What is the residue value of asset, especially an asset such as farmland?

Present Value is not the equivalent to Cash Flow (Feasibility)

Determining the present value of an asset does NOT assess the feasibility of purchasing farm land, for example. Restated, the analysis described on this page does not assess whether the income from purchasing land will generate enough cash inflow to pay all cash costs when they are owed. Additional analysis is necessary to answer that question.  Time value of money addresses only profitability.

Bottom Line

A manager may never go through all the complexities and details described on this page. But, it is important that the manager recognize all these assumptions embedded in any discussion about the value of land.

Summary of Key Points

• Decision makers need to consider the present value of future revenue if the decision involves receiving payments at different times in the future.
• The value of an asset is the present value of its future earnings.
• Enterprise analysis supports the development of present value calculations.
• The discount rate is generally a real interest rate.

Last Updated November 16, 2010

Email: David.Saxowsky@ndsu.edu

This material is intended for educational purposes only. It is not a substitute for competent professional advice. Seek appropriate advice for answers to your specific questions.

E-Mail:sswandal@ndsuext.nodak.edu
Published by Agricusiness and Applied Economics
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