# The six factors of one dollar are actually fourteen factors.

Every income-property appraiser has received instruction on the six
factors of one dollar.(1) The literature from both The Appraisal
Foundation(2) and the Appraisal Institute(3) includes this material as
required knowledge for all appraisers who evaluate future income flows.
Advanced income-property appraisers who evaluate irregular income
streams are familiar with present and future value techniques. This
article expands on such knowledge.

This article's first purpose is to illustrate that the six factors of one dollar are actually three factors and that the remaining three are in fact reciprocals of the original three. This material is much easier to remember, apply, and teach when it is presented in such a format.

The article's second purpose is to illustrate that the commonly known six factors are actually fourteen factors, which can be written in a simple tabular format that is easy to remember, apply, and teach. The fourteen factors include the two present values used for the J and K factors. In addition, the table introduces the level annual equivalent (LAE) that is the "stabilized income" used in Ellwood and yield analysis. The reciprocals of each are illustrated without a derivation. The additional eight functions are shown with brief illustrations.

SIX FACTORS OF ONE DOLLAR

The appraisal literature has expanded the typical present value factor (PVF) and future value factor (FVF) analysis into six factors that are usually presented in the following order:(4)

* Amount of $1

* Amount of $1 per period

* Sinking fund factor

* Present worth of $1

* Present worth of $1 per period

* Partial payment

Inconsistent semantics may create a barrier to discovering the close relationship between the factors. For example, "partial payment" is known also as "the installment to amortize one dollar" and "the mortgage constant." The meaning of TABULAR DATA OMITTED "worth" can be interpreted as "factor," "value," or "amount." All of these are operators, it should be remembered--correctly called factors--that may be used to estimate a present value (PV) or future value (FV) at a point in time.

Figure 1 illustrates a reorganization of these factors into a format in which the PVF and corresponding FVF are reciprocals. For example, the PVF reversion, section A of Figure 1, is the reciprocal of the FVF reversion, section B, and vice versa. The present value of the sinking fund factor (PVSFF) in section C is the reciprocal of the future value of the sinking fund factor (FVSFF) in section D, and vice versa. Last, the present value Inwood factor (PVIF) in section E is the reciprocal of the present value mortgage constant (PVMC) in section G. By examining Figure 1, analysts can easily see that the mortgage constant is the reciprocal of the Inwood factor, and vice versa.

Figure 1's organization makes one correction to Ellwood's table of six functions.(5) The sinking fund factor, section C, is actually a PVF rather than a FVF as Ellwood shows it. The solution to the formula in section C give a constant current payment that is necessary to derive a known future value. For example, the sinking fund factor for an interest rate of 10% with ten annual payments is .062745. This factor is used to estimate a payment that is a present value, not a future value.

The organization in Figure 1 reveals TABULAR DATA OMITTED that sections A, C, and E are the three basic PVFs from which the remaining three may be derived. Analysts should remember the PVF of the reversion, the sinking fund, and the PVF of the level annuity; from each of these, the FVF of $1, FVF of the level annuity, and the installment to amortize $1 may be derived, respectively.

ADDITIONAL EIGHT FUNCTIONS OF ONE DOLLAR

The additional eight functions are shown in Figure 2 and are discussed in the next sections.

Section H: PVF of payments when the payment changes by a constant dollar amount

The PVF is for the income stream that is projected to change in constant dollar amounts in a manner similar to a sinking fund to reach a projected fixed future value.(6) This change is the pattern used in deriving the straight-line J factor in n (number of periods). The income stream changes in a constant annual dollar amount that is compounded into a projected future value. For example, assuming a 15% yield rate, the PVF used to find the present value of a stream of ten annual payments that increase $1,000 per year is 6.7167. If the initial cash flow were $10,000, the present value would be $67,167.

Section I: FVF of payments when the payment changes by a constant dollar amount

This FVF may be new to some analysts. The usual assignment in yield analysis is to estimate current market value. An appraiser could, however, be asked to evaluate a property based on its future value. The necessary FVF is the reciprocal of the PVF in section H. For example, using the figures in section H, the FVF is (1/6.7167), or .1488826. If the initial cash flow were $10,000, the future value would be $148,883.

Section J: PVF of payments when the payment changes by a constant rate

This PVF is for an income stream that is projected to change by a constant rate.(7) This type of change is the pattern used in deriving the K factor in section L. For example, using a yield rate of 15%, the PVF of a series of ten annual cash flows that start at $10,000 and increase by 10% annually compounded is 7.17734. The present value is $71,773.

Section K: FVF of payments when the payment changes by a constant rate

Although the typical assignment is to estimate current market value, or PV, a client could need an evaluation based on FV. The necessary factor needed, FVF, is the reciprocal of the PVF found in section J. For example, using the figures in section J, the future value factor is (1/7.17734), or .139327. The FV of this income stream is $139,327.

Section L: Present value factor level annual equivalent (PVFLAE) that is the LAE for constant rate changes

The LAE(8) is the annual, constant income stream whose PV is equal to the PV of the actual income. The investor should make no differentiation between the two income streams because the PVs of both are equal. When the PVFLAE shown in section J for constant rate changes is divided by the Inwood factor in section E, the commonly known K factor is derived. An LAE is the result when the first-year NOI is multiplied by this K factor.(9)

For example, using the data in section J, the PVFLAE of 7.17734 is divided by the Inwood factor in section E, which is 5.018769. The resulting K factor of 1.4301 is the exact amount shown in the Akerson Tables.(10) When this amount is multiplied by the first-year NOI of $10,000, the result of $14,301 is the LAE or stabilized income.

Section M: Future value factor level annual equivalent (FVFLAE) of the LAE for constant rate changes

The FVFLAE in section M is the same as in section D. Because the PV of the level annual income stream is identical to the original income stream, the original can be disregarded. The new income is level, which is the same as a constant ordinary annuity. The FVF of this income pattern was covered in section D. For example, the FVF factor from section D for a rate of 15% over ten annual payments is 20.303718. Using the data from section L, the LAE of $14,301 is multiplied by 20.303718, which gives $290,363, representing the FV of this new constant level income stream.

Analysts should use the PV and the FV of the stabilized income streams with caution. Stabilized income appears frequently in mortgage-equity analysis and direct capitalization. Also, the stabilized income stream from an income-producing property can be compared to the stable income from a comparable investment such as a U.S. T-bond. When using these figures it is important to remember that the whole income stream must be received over the life of the project for the PV of the actual income flows to equal the PV of the LAE.

Section N: PVFLAE that is the LAE for constant dollar amount changes

This step derives the straight-line J factor. When the PVF for the income streams in section H that change by constant dollar amounts in a sinking fund pattern are divided by the Inwood factor in section E, the straight-line J factor is derived. When the amount (1 + appreciation |app~ J), is multiplied by the first-year NOI, the LAE is derived for income that changes in constant amounts. For example, using the data from section H, the PVF was 6.7167 and the Inwood factor for a yield of 15% over ten annual payments is 5.018769. The ratio of the two, 1.338316, is the same straight-line J factor. Because app = 1.0 over 10 years, J is equal to:

(1 + app J) = 1.338322

(1 + 1.0 J) = 1.338322

J = .338322

J may be used in mortgage-equity analysis for income that changes in constant dollar amounts. If the first-year NOI was $10,000, the LAE is $13,383.

This straight-line J factor is not the same as the Ellwood J factor. The latter presumes that the income stream changes in a constant dollar amount similar to a sinking fund that compounds to a fixed future value.

Section O: FVFLAE of the LAE for constant dollar amount changes

This factor represents the FV of the LAE found in section N. Because the original income stream has been transformed into a LAE, the FVFLAE becomes the same as in section D. Using the data in section N, the FVFLAE for ten annual payments at 15% is 20.303718. The FV is $13,383 x 20.303718 = $271,725. The same caution must be exercised in the use of the FV found in section O as in the use of the FV found in section M.

ILLUSTRATION OF THE LAE

This section illustrates the meaning of the LAE. Although it is mentioned several times in The Appraisal of Real Estate,(11) its derivation and meaning are not explained.

Consider an income stream that starts at $20,000 and is projected to grow at the constant rate of 5% annually. The investor wants a yield of 15%.(12)

The FV of this income stream is $73,093. The PVF of $1 per period for 5 years at 15% is 3.352155. Dividing $73,093 by 3.352155 gives $21,804, which is the LAE.

The meaning of this concept can be seen when the LAE is compared with the actual income stream:

TABULAR DATA OMITTED

The PV of the LAE and the project income stream that grows at 5% annually are the same, $73,093. The difference between the two annually must sum to O, as the last column shows.

The K factor from the Akerson tables(13) for 15% and 5 years is 1.0902. When this is multiplied by the first-year NOI, the result is $21,804, which is the LAE. Because the PV of both income streams is the same, the investor should make no distinction between the two and can disregard the original income stream.

The income in this example is changing at a constant annual rate. If the come were changing at a constant dollar annual amount, the LAE would be found by multiplying the first-year NOI by (1 + app J) where J is the straight-line J factor. If the income were changing in a sinking fund manner, the J would be the Ellwood J factor.

The FV of the LAE and the actual income stream should be equal. Using the figures above, the FV for a level annuity of $21,804 over 5 years at 15% is $147,011. The FV from the actual income stream is as follows.

1. American Inst. of Real Estate Appraisers, The Appraisal of Real Estate, 9th ed. (Chicago: American Inst. of Real Estate Appraisers, 1987), 491-504.

2. The Appraisal Foundation, Required Topics for the General Appraiser (Washington, D.C.: The Appraisal Foundation, 1990).

3. Appraisal Institute, Body of Knowledge (Chicago, Appraisal Institute, 1992).

4. See for example, Charles Akerson, The Appraiser's Workbook (Chicago: American Inst. of Real Estate Appraisers, 1985); L. W. Ellwood, Ellwood Tables (Cambridge, MA: Ballinger Publishing, 1977), Part II, 17-50; Financial Tables (Chicago: American Inst. of Real Estate Appraisers, 1981), T3-T75.

5. Ellwood, 19.

6. This PVF is shown in The Appraisal of Real Estate, 9th ed., 695 and in Akerson, 153. Its use and meaning are not sufficiently explained in either case. Also, its relationship to the J factors is not illustrated or explained. Additional information on the derivation and meaning of J factors can be found in Anthony Sanders and C. F. Sirmans, "A Comparison of the Ellwood and Simple Weighted-Average Overall Capitalization Rates," The Appraisal Journal (July 1980): 432-438; and Donald R. Epley and Wayne Kelly, "An Examination of the J

Factor, Index Method, and Gordon Constant Growth Model as Methods to Discount Future Cash Flows," unpublished research, 1992.

7. The PVF is shown in The Appraisal of Real Estate, 9th ed., 699, and Akerson, 153. Its use, meaning and relationship to the K factor is not illustrated or explained.

8. The level annual equivalent has been labeled as a "level equivalent" in The Appraisal of Real Estate, 9th ed., 526. Its meaning and use is not illustrated or explained. Fisher developed a stabilized income and labeled it a G factor in Jeffrey E. Fisher, "Ellwood J Factors: A Further Refinement," The Appraisal Journal (January 1979): 65-75.

9. For an illustration, see Appraisal Institute, Capitalization Theory and Techniques (Chicago: Appraisal Institute, 1991), Figure 4.1, p. 4-3.

10. Akerson, T-96.

11. For example, see The Appraisal of Real Estate, 526, 695, and 699; see also Appraisal Institute, Principles of Capitalization, G4 (Chicago: Appraisal Institute, 1990), 71.

12. Capitalization Theory and Techniques, Problem 4.1.

13. Akerson, T-94.

Donald Epley, MAI, SRA, PhD, is director of the Real Estate Program at Mississippi State University. He is a frequent instructor of courses and seminars for the Appraisal Institute. Mr. Epley is a member of the GAB Education Committee and serves as vice-chair of the GAB Curriculum Committee.

G. Wayne Kelly, PhD, is a graduate of the University of Alabama. His areas of interest are capital budgeting and financial markets. He currently is assistant professor of finance and economics at Mississippi State University.

This article's first purpose is to illustrate that the six factors of one dollar are actually three factors and that the remaining three are in fact reciprocals of the original three. This material is much easier to remember, apply, and teach when it is presented in such a format.

The article's second purpose is to illustrate that the commonly known six factors are actually fourteen factors, which can be written in a simple tabular format that is easy to remember, apply, and teach. The fourteen factors include the two present values used for the J and K factors. In addition, the table introduces the level annual equivalent (LAE) that is the "stabilized income" used in Ellwood and yield analysis. The reciprocals of each are illustrated without a derivation. The additional eight functions are shown with brief illustrations.

SIX FACTORS OF ONE DOLLAR

The appraisal literature has expanded the typical present value factor (PVF) and future value factor (FVF) analysis into six factors that are usually presented in the following order:(4)

* Amount of $1

* Amount of $1 per period

* Sinking fund factor

* Present worth of $1

* Present worth of $1 per period

* Partial payment

Inconsistent semantics may create a barrier to discovering the close relationship between the factors. For example, "partial payment" is known also as "the installment to amortize one dollar" and "the mortgage constant." The meaning of TABULAR DATA OMITTED "worth" can be interpreted as "factor," "value," or "amount." All of these are operators, it should be remembered--correctly called factors--that may be used to estimate a present value (PV) or future value (FV) at a point in time.

Figure 1 illustrates a reorganization of these factors into a format in which the PVF and corresponding FVF are reciprocals. For example, the PVF reversion, section A of Figure 1, is the reciprocal of the FVF reversion, section B, and vice versa. The present value of the sinking fund factor (PVSFF) in section C is the reciprocal of the future value of the sinking fund factor (FVSFF) in section D, and vice versa. Last, the present value Inwood factor (PVIF) in section E is the reciprocal of the present value mortgage constant (PVMC) in section G. By examining Figure 1, analysts can easily see that the mortgage constant is the reciprocal of the Inwood factor, and vice versa.

Figure 1's organization makes one correction to Ellwood's table of six functions.(5) The sinking fund factor, section C, is actually a PVF rather than a FVF as Ellwood shows it. The solution to the formula in section C give a constant current payment that is necessary to derive a known future value. For example, the sinking fund factor for an interest rate of 10% with ten annual payments is .062745. This factor is used to estimate a payment that is a present value, not a future value.

The organization in Figure 1 reveals TABULAR DATA OMITTED that sections A, C, and E are the three basic PVFs from which the remaining three may be derived. Analysts should remember the PVF of the reversion, the sinking fund, and the PVF of the level annuity; from each of these, the FVF of $1, FVF of the level annuity, and the installment to amortize $1 may be derived, respectively.

ADDITIONAL EIGHT FUNCTIONS OF ONE DOLLAR

The additional eight functions are shown in Figure 2 and are discussed in the next sections.

Section H: PVF of payments when the payment changes by a constant dollar amount

The PVF is for the income stream that is projected to change in constant dollar amounts in a manner similar to a sinking fund to reach a projected fixed future value.(6) This change is the pattern used in deriving the straight-line J factor in n (number of periods). The income stream changes in a constant annual dollar amount that is compounded into a projected future value. For example, assuming a 15% yield rate, the PVF used to find the present value of a stream of ten annual payments that increase $1,000 per year is 6.7167. If the initial cash flow were $10,000, the present value would be $67,167.

Section I: FVF of payments when the payment changes by a constant dollar amount

This FVF may be new to some analysts. The usual assignment in yield analysis is to estimate current market value. An appraiser could, however, be asked to evaluate a property based on its future value. The necessary FVF is the reciprocal of the PVF in section H. For example, using the figures in section H, the FVF is (1/6.7167), or .1488826. If the initial cash flow were $10,000, the future value would be $148,883.

Section J: PVF of payments when the payment changes by a constant rate

This PVF is for an income stream that is projected to change by a constant rate.(7) This type of change is the pattern used in deriving the K factor in section L. For example, using a yield rate of 15%, the PVF of a series of ten annual cash flows that start at $10,000 and increase by 10% annually compounded is 7.17734. The present value is $71,773.

Section K: FVF of payments when the payment changes by a constant rate

Although the typical assignment is to estimate current market value, or PV, a client could need an evaluation based on FV. The necessary factor needed, FVF, is the reciprocal of the PVF found in section J. For example, using the figures in section J, the future value factor is (1/7.17734), or .139327. The FV of this income stream is $139,327.

Section L: Present value factor level annual equivalent (PVFLAE) that is the LAE for constant rate changes

The LAE(8) is the annual, constant income stream whose PV is equal to the PV of the actual income. The investor should make no differentiation between the two income streams because the PVs of both are equal. When the PVFLAE shown in section J for constant rate changes is divided by the Inwood factor in section E, the commonly known K factor is derived. An LAE is the result when the first-year NOI is multiplied by this K factor.(9)

For example, using the data in section J, the PVFLAE of 7.17734 is divided by the Inwood factor in section E, which is 5.018769. The resulting K factor of 1.4301 is the exact amount shown in the Akerson Tables.(10) When this amount is multiplied by the first-year NOI of $10,000, the result of $14,301 is the LAE or stabilized income.

Section M: Future value factor level annual equivalent (FVFLAE) of the LAE for constant rate changes

The FVFLAE in section M is the same as in section D. Because the PV of the level annual income stream is identical to the original income stream, the original can be disregarded. The new income is level, which is the same as a constant ordinary annuity. The FVF of this income pattern was covered in section D. For example, the FVF factor from section D for a rate of 15% over ten annual payments is 20.303718. Using the data from section L, the LAE of $14,301 is multiplied by 20.303718, which gives $290,363, representing the FV of this new constant level income stream.

Analysts should use the PV and the FV of the stabilized income streams with caution. Stabilized income appears frequently in mortgage-equity analysis and direct capitalization. Also, the stabilized income stream from an income-producing property can be compared to the stable income from a comparable investment such as a U.S. T-bond. When using these figures it is important to remember that the whole income stream must be received over the life of the project for the PV of the actual income flows to equal the PV of the LAE.

Section N: PVFLAE that is the LAE for constant dollar amount changes

This step derives the straight-line J factor. When the PVF for the income streams in section H that change by constant dollar amounts in a sinking fund pattern are divided by the Inwood factor in section E, the straight-line J factor is derived. When the amount (1 + appreciation |app~ J), is multiplied by the first-year NOI, the LAE is derived for income that changes in constant amounts. For example, using the data from section H, the PVF was 6.7167 and the Inwood factor for a yield of 15% over ten annual payments is 5.018769. The ratio of the two, 1.338316, is the same straight-line J factor. Because app = 1.0 over 10 years, J is equal to:

(1 + app J) = 1.338322

(1 + 1.0 J) = 1.338322

J = .338322

J may be used in mortgage-equity analysis for income that changes in constant dollar amounts. If the first-year NOI was $10,000, the LAE is $13,383.

This straight-line J factor is not the same as the Ellwood J factor. The latter presumes that the income stream changes in a constant dollar amount similar to a sinking fund that compounds to a fixed future value.

Section O: FVFLAE of the LAE for constant dollar amount changes

This factor represents the FV of the LAE found in section N. Because the original income stream has been transformed into a LAE, the FVFLAE becomes the same as in section D. Using the data in section N, the FVFLAE for ten annual payments at 15% is 20.303718. The FV is $13,383 x 20.303718 = $271,725. The same caution must be exercised in the use of the FV found in section O as in the use of the FV found in section M.

ILLUSTRATION OF THE LAE

This section illustrates the meaning of the LAE. Although it is mentioned several times in The Appraisal of Real Estate,(11) its derivation and meaning are not explained.

Consider an income stream that starts at $20,000 and is projected to grow at the constant rate of 5% annually. The investor wants a yield of 15%.(12)

The FV of this income stream is $73,093. The PVF of $1 per period for 5 years at 15% is 3.352155. Dividing $73,093 by 3.352155 gives $21,804, which is the LAE.

The meaning of this concept can be seen when the LAE is compared with the actual income stream:

TABULAR DATA OMITTED

The PV of the LAE and the project income stream that grows at 5% annually are the same, $73,093. The difference between the two annually must sum to O, as the last column shows.

The K factor from the Akerson tables(13) for 15% and 5 years is 1.0902. When this is multiplied by the first-year NOI, the result is $21,804, which is the LAE. Because the PV of both income streams is the same, the investor should make no distinction between the two and can disregard the original income stream.

The income in this example is changing at a constant annual rate. If the come were changing at a constant dollar annual amount, the LAE would be found by multiplying the first-year NOI by (1 + app J) where J is the straight-line J factor. If the income were changing in a sinking fund manner, the J would be the Ellwood J factor.

The FV of the LAE and the actual income stream should be equal. Using the figures above, the FV for a level annuity of $21,804 over 5 years at 15% is $147,011. The FV from the actual income stream is as follows.

Income FV of $1 at 15% FV $20,000 x 1.749006 = $ 34,980. $21,000 x 1.52085 = $ 31,938. $22,050 x 1.3225 = $ 29,161. $23,153 x 1.15 = $ 26,626. $24,310 x -- = $ 24,310. Total = $147,015.

1. American Inst. of Real Estate Appraisers, The Appraisal of Real Estate, 9th ed. (Chicago: American Inst. of Real Estate Appraisers, 1987), 491-504.

2. The Appraisal Foundation, Required Topics for the General Appraiser (Washington, D.C.: The Appraisal Foundation, 1990).

3. Appraisal Institute, Body of Knowledge (Chicago, Appraisal Institute, 1992).

4. See for example, Charles Akerson, The Appraiser's Workbook (Chicago: American Inst. of Real Estate Appraisers, 1985); L. W. Ellwood, Ellwood Tables (Cambridge, MA: Ballinger Publishing, 1977), Part II, 17-50; Financial Tables (Chicago: American Inst. of Real Estate Appraisers, 1981), T3-T75.

5. Ellwood, 19.

6. This PVF is shown in The Appraisal of Real Estate, 9th ed., 695 and in Akerson, 153. Its use and meaning are not sufficiently explained in either case. Also, its relationship to the J factors is not illustrated or explained. Additional information on the derivation and meaning of J factors can be found in Anthony Sanders and C. F. Sirmans, "A Comparison of the Ellwood and Simple Weighted-Average Overall Capitalization Rates," The Appraisal Journal (July 1980): 432-438; and Donald R. Epley and Wayne Kelly, "An Examination of the J

Factor, Index Method, and Gordon Constant Growth Model as Methods to Discount Future Cash Flows," unpublished research, 1992.

7. The PVF is shown in The Appraisal of Real Estate, 9th ed., 699, and Akerson, 153. Its use, meaning and relationship to the K factor is not illustrated or explained.

8. The level annual equivalent has been labeled as a "level equivalent" in The Appraisal of Real Estate, 9th ed., 526. Its meaning and use is not illustrated or explained. Fisher developed a stabilized income and labeled it a G factor in Jeffrey E. Fisher, "Ellwood J Factors: A Further Refinement," The Appraisal Journal (January 1979): 65-75.

9. For an illustration, see Appraisal Institute, Capitalization Theory and Techniques (Chicago: Appraisal Institute, 1991), Figure 4.1, p. 4-3.

10. Akerson, T-96.

11. For example, see The Appraisal of Real Estate, 526, 695, and 699; see also Appraisal Institute, Principles of Capitalization, G4 (Chicago: Appraisal Institute, 1990), 71.

12. Capitalization Theory and Techniques, Problem 4.1.

13. Akerson, T-94.

Donald Epley, MAI, SRA, PhD, is director of the Real Estate Program at Mississippi State University. He is a frequent instructor of courses and seminars for the Appraisal Institute. Mr. Epley is a member of the GAB Education Committee and serves as vice-chair of the GAB Curriculum Committee.

G. Wayne Kelly, PhD, is a graduate of the University of Alabama. His areas of interest are capital budgeting and financial markets. He currently is assistant professor of finance and economics at Mississippi State University.

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Title Annotation: | valuation |
---|---|

Author: | Epley, Donald; Kelly, G. Wayne |

Publication: | Appraisal Journal |

Date: | Jul 1, 1993 |

Words: | 2487 |

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