This lesson serves as an introduction to the topic and discusses the following:  The concept of the time value of money,
 Timelines for cash flows,
 Simple versus compound interest, and
 Use of Assessors´ Handbook Section 505, Capitalization Formulas and Tables.
The objectives of this learning session are to:  Provide an understanding of the time value of money.
 Explain the compound interest functions presented in Assessors' Handbook Section 505, Capitalization Formulas and Tables, and their relationship to appraisal.
 Demonstrate how to use the factor tables and each compound interest function in Assessors' Handbook Section 505, to estimate the market value of property for property tax purposes.
In this learning session, instruction is provided through structured reading and illustrated examples. To assist in the attainment of the learning objectives, examples are incorporated within the lessons to illustrate the concept being discussed. Problems available at the end of lessons are to be worked by participants to ensure comprehension of concepts and calculations discussed in the lesson. A final examination is also available at the conclusion of this session, in order that certified property tax appraisers employed by a California County Assessor’s Office or the Board of Equalization may demonstrate their overall comprehension of the subject matter, attest to their participation in the learning session and receive training credit for completion of the session. The Concept of the Time Value of Money It´s intuitive to most people that a dollar today is preferable to a dollar to be received in the future. (Think about $1,000,000 today compared to $1,000,000 to be received 5 years from today. Which would you rather have?) A dollar today can be invested to accumulate to more than a dollar in the future, which also makes a future dollar worth less than a dollar today. Hence, money has a time value. More generally, the time value of money is the relationship between the value of a payment at one point in time and its value at another point in time as determined by the mathematics of compound interest. Because of the time value of money, payments made at different points in time cannot be directly compared. The compound interest functions—the mathematics of the time value of money – allow us to bring the payments to the same point in time for comparison purposes. Time value of money calculations have wide application in finance, real estate, and personal financial decisions. An understanding of them is essential in the field of valuation. There are several situations in real estate where dollars at different points in time are compared:  An investor buys a property today for a certain amount of money in order to receive expected income from the property in the future (the purchase price today is compared to the expected future income stream).
 A lender makes a loan today in exchange for a promise by the borrower to pay scheduled amounts in the future (the loan amount is compared to the promised payments).
 A property owner wants to prepare for a future estimated expense by setting aside a given amount of money each month or year (the future expense is compared to the money set aside each year).
 An investor expects that a property will be worth a certain amount in the future and wants to estimate the equivalent amount today (the expected future value is compared to a present value).
The process by which payments (or cash flows) are moved forward in time is called compounding. The process by which payments (or cash flows) are moved backward in time is called discounting. All time value of money calculations involve either compounding or discounting — that is, moving amounts either forward or backward in time. A series of cash flows can be graphically represented using a cash flow timeline. A timeline depicts the timing and amount of the cash flows. For example, the following timeline depicts cash inflows of $100 to be received at the end of each of the next 5 years: Cash flows in a timeline are often labeled positive or negative. By convention, positive cash flows correspond to cash inflows; negative cash flows correspond to cash outflows. Whether a cash flow is an inflow or an outflow depends on perspective (i.e., as a borrower or lender). The borrower´s inflow is the lender´s outflow, and vice versa. In using timelines, and in solving time value of money problems, one should adopt the perspective of either borrower or lender and stay with that perspective throughout the problem. Consider a simple time value of money problem. In making a purchase you are given two payment alternatives:  Pay $400 immediately.
 Pay five installments of $100 each at the end of each of the next five years.
As depicted on a cash flow timeline: In deciding which alternative is better, we can´t simply add up the five payments of $100 and compare this sum ($500) with alternative 1 ($400 today). To do so would ignore the time value of money because the two alternatives involve payments at different times. Instead, we must determine the value today (at time 0) of the five future payments of $100 of alternative 2 and compare this to $400, which is the value today of alternative 1. As we shall see, determining the value today (the present value) of the five payments under alternative 2 involves calculating the present value of those payments at a given rate of interest. Time value of money calculations allow us to solve problems such as the one above and many others. When money is borrowed, the amount borrowed is called the principal. The consideration paid for the use of money is called interest.  The rate of interest can be thought of as a price per period for the use of money.
 From the perspective of the lender, interest is earned; from the perspective of the borrower, interest is paid.
Simple Interest Simple interest refers to the situation in which interest is calculated on the original principal amount only. With simple interest, the base on which interest is calculated does not change, and the amount of interest earned each period also does not change. Compound Interest Compound interest refers to the situation in which interest is calculated on the original principal and the accumulated interest. With compound interest, interest is calculated on a base that increases each period, and the amount of interest earned also increases with each period. Application of Simple Interest Suppose someone invests $100 for 50 years and receives 5% per year in simple interest. To calculate simple interest, multiply the beginning balance by the rate 0.05 × $100 = $5. The growth in the investment is depicted in the table below: With simple interest, each year´s interest is based on the original principal amount only. Application of Compound Interest Assume the same investment of $100 for 50 years, but at compound interest: With compound interest, interest is earned on both the original principal and accumulated interest. Interest is earned on interest. In the preceding example, with simple interest, the accumulated amount after 50 years is only $350. With compound interest, the accumulated amount is $1,147. As the term increases, the difference between the final amount with compound interest versus simple interest becomes more and more dramatic. "Miracle of Compound Interest" In a wellknown transaction, Dutch colonists bought Manhattan Island in 1624 for the equivalent of $24.  This seems like a steal, but if the seller had deposited the $24 and earned an annual rate of 6%, the future compound amount would have been about $141 billion in the year 2010.
 This is roughly equal to the total assessed value of all land and improvements in the City and County of San Francisco in the year 2010.
 Over the same time period (386 years), the future value of $24 at simple interest of 6% would have been only $580.
From a lender´s (or investor´s) perspective, compound interest is a good thing; the lender earns interest on interest from the borrower. Conversely, from a borrower´s perspective, compound interest is not so good. The borrower, in effect, pays interest on interest throughout the term of the loan. Compound Interest Functions Six compound interest functions are used to solve time value of money problems. Not surprisingly, all of the functions are based on compound, not simple, interest. Each compound interest function is defined by a formula, which is the basis for calculating the compound interest factors for that function. Each formula requires a periodic interest rate and the number of periods Most time value of money problems involve the use of only one compound interest function (or factor), but some require the use of two or more. Understanding the compound interest functions, and how the factors derived from them are used to solve time value of money problems, is the heart of this subject matter. Each compound interest formula, and the factors derived from it, involves three variables:  An interest rate,
 A term (number of periods), and
 A compounding interval (how frequently interest is compounded).
In essence, using a compound interest factor does one of two things:  Adds compound interest to a present value to arrive at a future value.
 Subtracts compound interest from a future value to arrive at a present value.
Published tables of compound interest factors are used to solve time value of money problems. It´s easier to refer to a table of factors than to calculate the desired factor from one of the formulas each time you need it. Time value of money problems can also be solved using a financial calculator or spreadsheet software. Essentially, the software calculates the necessary factor and processes the calculation. We approach the subject by first showing how compound interest factors are derived from each of the formulas, then showing how the factors are used to solve various time value of money problems. This approach provides a fundamental understanding of the material and a good basis for later using financial calculators and spreadsheet applications to solve time value of money problems. The six compound interest functions are listed below; the following table briefly describes each function and gives an example of how it might be used.  Future Worth of $1 (FW$1)
 Present Worth of $1 (PW$1)
 Future Worth of $1 Per Period (FW$1/P)
 Sinking Fund Factor (SFF)
 Present Worth of $1 Per Period (PW$1/P)
 Periodic Repayment (PR)
Review the above table as an introduction to the compound interest functions. Note that the first two compound interest functions (FW$1 and PW$1) deal with a single amount, or payment, while the remaining four deal with a series of payments (an annuity). Although all of the compound interest functions have appraisal applications, two are of particular importance because of their central role in the income approach – the present worth of $1 (PW$1) and the present worth of $1 per period (PW$1/P). Assessors´ Handbook Section 505 (AH 505), Capitalization Formulas and Tables, contains a set of compound interest factors for use by property tax appraisers.  AH 505 begins with introductory material (pages 111) that describes the compound interest functions, shows their formulas, and demonstrates some sample problems. We will cover this material in the presentation.
 The remainder of AH 505 (pages 12109) contains tables of compound interest factors over a range of interest rates for the six compound interest functions.
 AH 505 contains compound interest factors at interest rates from 1 to 25%, at 1/2 % intervals, for time periods up to 50 years. If a problem requires a term longer than 50 years, the tables can be extended using certain calculations.
 For each interest rate, there is a separate page for either annual or monthly compounding. The frequency of compounding (such as on a monthly basis versus annual) directly affects the present and future values in discounting – see separate lesson on Frequency of Compounding (Lesson 9).
Briefly look through AH 505 (Use browser's back button to return to lessons). To use the AH 505 tables, follow these steps:  Locate the page at the desired interest rate, selecting the page for either annual or monthly compounding (a common error is using the annual page when the monthly page is required, or vice versa).
 Find the desired term (number of periods, months or years) by going down the far left side of the page.
 Go across to the proper column to find the factor for the desired compound interest function.
(For solving the examples, we have recreated portions of the AH505 tables. We also provide a link to the actual pages in AH 505. If you go to an actual page, use the zoom feature for the best view.) Example 1: Find the FW$1 factor at 10% for 10 years, given annual compounding. Solution: In AH 505:  Search for the page at 10%, annual compounding, (step 1 above); the correct page is page 49.
 Go down 10 years and across to column 1 (FW$1) (steps 2 and 3); the correct factor is 2.593742.
Link to AH 505, page 49 Example 2: Find the PW$1 factor at 5% for 5 years, given monthly compounding. Solution: In AH 505:  Search for the page at 5%, monthly compounding; the correct page is page 28.
 Go down 5 years and across to column 4 (PW$1); the correct factor is 0.779205
Link to AH 505, page 28 Example 3: Find the PW$1/P factor at 12.5% for 8 years, given annual compounding. Solution: In AH 505:  Search for the page at 12.5%, annual compounding; the correct page is page 59.
 Go down 8 years and across to column 5 (PW$1/P); the correct factor is 4.882045
Link to AH 505, page 59
Page 2 This lesson discusses the Future Worth of $1 (FW$1); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. This lesson:  Explains the FW$1 function's meaning and purpose,
 Provides the formula for FW$1 factors,
 Contains practical examples of how to apply the FW$1 factor,
 Explains the Rule of 72, and
 Shows how to calculate the future value of multiple payments.
FW$1: Meaning and Purpose The FW$1 is the amount to which $1 will grow at periodic interest rate i after n periods, assuming the payment of $1 occurs at the beginning of the first period. The FW$1 is used to compound a single present amount to its future amount. The FW$1 factors are in column 1 of AH 505. The future worth of 1 factor (FW$1) is based on the premise that $1 deposited at the beginning of a period earns interest during the period and becomes part of the principal at the beginning of the next period. This continues for the number of periods in the problem. The formula for the calculation of the FW$1 factors is FW$1 = (1 + i)n Where:  FW$1 = Future Worth of $1 Factor
 i = Periodic Interest Rate, often expressed as an annual percentage rate
 n = Number of Periods, often expressed in years
All of the other compound interest formulas published in AH 505 are derived from the basic compounding expression in the FW$1 factor, (1 + i)n. As we will see, this mathematical expression is the basic building block of all the other compound interest formulas. The periodic interest rate, i, must match the compounding period, n (this holds for all compound interest functions). For example, if n is stated in years, indicating annual compounding, i must be stated as an annual rate; if n is stated in months, indicating monthly compounding, i must be stated as a monthly rate. For now, we will assume annual compounding, so our periods, n, will be in years and the periodic interest rate, i, will be the annual percentage rate. Later, we will introduce the concept of more than one compounding period per year (monthly, quarterly, etc.). In order to calculate the annual FW$1 factor for 4 years at an annual interest rate of 6%, use the formula below:  FW$1 = (1 + i)n
 FW$1 = (1 + 0.06)4
 FW$1 = (1.06)4
 FW$1 = 1.262477
Viewed on a timeline: On the timeline, the initial deposit of $1 is shown as negative because from the point of view of a depositer it would be a cash outflow. The future value is shown as positive because it would be a cash inflow. The depositor gives up money now in order to receive money later. To locate the FW$1 factor in AH 505, go to page 33 of AH 505. Go down 4 years and across to column 1. The FW$1 factor is 1.262477. Link to AH 505, page 33 In most problems, we don’t want the FW$1; we want the future worth of some other amount that has been deposited or invested. To put it another way, we want to use the FW$1 factor to solve a TVM problem. When working problems, we will use the notation shown below. Don’t worry too much about the notation now. Using it will become easier as we work problems throughout the lessons. Example 1: You deposit $2,000 today at an annual interest rate of 6%. How much will you have at the end of 10 years, assuming annual compounding? Solution:  FV = PV × FW$1 (6%, 10 yrs, annual)
 FV = $2,000 × 1.790848
 FV = $3,582
 Find the annual FW$1 factor (annual compounding) for 6% at a term of 10 years. In AH 505, page 33, go down 10 years and across to column 1 to find the correct factor of 1.790848.
 The future value of $3,582 is equal to the present value of $2,000 multiplied by the factor.
Link to AH 505, page 33 Example 2: The Jones family places $100,000 in an investment that will provide an annual rate of return of 5%. What will the investment be worth in 2 years? Solution:  FV = PV × FW$1 (5%, 2 yrs, annual)
 FV = $100,000 × 1.102500
 FV = $110,250
 Find the annual FW$1 factor (annual compounding) for 5% at a term of 2 years. In AH 505, page 29, go down 2 years and across to column 1 to find the correct factor of 1.102500.
 The future value of $110,250 is equal to the present value of $100,000 multiplied by the factor.
Link to AH 505, page 29 Example 3: You have $450,000 to invest and can earn an annual interest rate of 7.50%. How much will your investment be worth in 10 years? Solution:  FV = PV × FW$1 (7.50%, 10 yrs, annual)
 FV = $450,000 × 2.061032
 FV = $927,464
 Find the annual FW$1 factor (annual compounding) for 7.50% at a term of 10 years. In AH 505, page 39, go down 10 years and across to column 1 to find the correct factor of 2.061032.
 The future value of $927,464 is equal to the present value of $450,000 multiplied by the factor.
Link to AH 505, page 39 Example 4: A given product costs $500 today. The cost of the product is expected to rise at an annual rate of 10%. How much will the product cost in 5 years? Solution:  FV = PV × FW$1 (10%, 5 yrs, annual)
 FV = $500 × 1.610510
 FV = $805
 Find the annual FW$1 factor (annual compounding) for 10% at a term of 5 years. In AH 505, page 49, go down 5 years and across to column 1 to find the correct factor of 1.610510.
 The future value of $805 is equal to the present value of $500 multiplied by the factor.
Link to AH 505, page 49 Example 5: Approximately how long does it take a given amount to grow to 10 times its original amount, given an annual interest rate of 5% with annual compounding? Solution:  Use the compound interest tables to estimate the answer by inspecting the tables.
 On the annual page for 5% (AH 505, page 29), inspect the FW$1 column (column 1) to find a factor that is approximately equal to 10 (FV ÷ PV = 10 ÷ 1).
 The answer is slightly greater than 47 years (the factor for 47 years is 9.905971).
Link to AH 505, page 29 The Rule of 72 is a rule of thumb that is closely related to the FW$1 factor. The rule assumes annual compounding. The Rule of 72 can be used to estimate either of the following:  The number of years it would take for an amount to double at a given annual interest rate, or
 The annual interest rate, if an amount has doubled in a given number of years.
The formula for the Rule of 72 is: Or, transposing: (Note: When using the Rule of 72, the annual interest rates are stated as percentages, not as decimals.) The smaller the difference between the factors of 72 (i.e., the number of years and the annual interest rate) the more accurate the estimate. For example, when the factors are 9 and 8, the estimate is more accurate than when the factors are 36 and 2. Example 1: You deposit $1,000 in an account that pays an annual interest rate of 6%. Approximately how long will it take this deposit to grow to $2,000? Solution: The estimate may be confirmed using the compound interest tables in AH 505, page 33, column 1. At 12 years, the FW$1 factor is approximately equal to 2, indicating a doubling (the actual factor is 2.012196). Link to AH 505, page 33 Example 2: Eight years ago you received a small inheritance that you deposited in a savings account. The amount has now doubled. What compound annual interest rate have you earned over the past 8 years? Solution: To confirm the estimate, search in AH 505 for the annual rate at which the FW$1 factor for 8 years is approximately equal to 2, indicating a doubling. In AH 505, page 45, column 1, the FW$1 factor at 8 years is approximately equal to 2 (the actual factor is 1.992563). Link to AH 505, page 45 Example 3: You’ve purchased a house that you think will double in value in 10 years. At what annual compound rate will the property have appreciated? Solution: At an annual rate of 7.00% the FW$1 factor for 10 years is 1.967151 (AH 505, page 37). Link to AH 505, page 37 At an annual rate of 7.50% the FW$1 factor for 10 years is 2.061032 (AH 505, page 39). Link to AH 505, page 39 Interpolating, the annual rate at which the FW$1 factor is 2, is somewhere between 7.00 and 7.50%, approximately 7.2%. We have calculated the future value of single amounts or payments, using the FW$1 factor. Many problems involve more than one payment, making it necessary to calculate the future value of multiple payments–that is, the future value of a stream of payments. Determining the future value of multiple payments is a straightforward extension of the singlepayment situation. When we calculated the future value of a single amount or payment, we multiplied the payment by the appropriate FW$1 factor. This compounded the payment to its future value. If there is more than one payment, we must multiply each payment by the appropriate FW$1 factor and add up all of the future values. The sum of the future values is the total future value of the stream of payments. Example 1 You plan to make the following three deposits into a savings account:  $10,000 at the end of the first year
 $15,000 at the end of the second year
 $20,000 at the end of the third year
At an annual interest rate of 5%, how much will you have in the account at the end of the third year (i.e., what is the total future value of all three payments)? Solution: Calculate the future value of each payment as of the end of year 3 using the appropriate FW$1 factor (AH 505, page 29, column 1) and add those future values. This sum is the future value of all three payments at the end of 3 years. Thus: The first payment is compounded forward for two periods (years); the second payment for one period (year); and the final payment, which itself at the end of year 3, requires no compounding. Link to AH 505, page 29 Viewed on a Timeline: On the timeline, the deposits are shown as negative because from the perspective of the depositor they represent cash outflows, and the resulting future values are shown as positive because they represent cash inflows at the end of year 3.
Page 3 This lesson discusses the Present Worth of $1 (PW$1); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The Lesson:  Explains the function’s meaning and purpose,
 Provides the formula for the calculation of PW$1 factor,
 Shows how to calculate the present value of multiple payments, and
 Contains practical examples of how to apply the PW$1 factor.
PW$1: Meaning and Purpose The PW$1 factor is the amount that must be deposited today to grow to $1 in the future, given periodic interest rate i and n periods. The PW$1 factor is used to discount a single future amount to its present amount. The PW$1 factors are in column 4 of AH 505. The PW$1 factor can be thought of as the opposite of the FW$1 factor; mathematically, the PW$1 and FW$1 factors are reciprocals: Whereas the FW$1, discussed in Lesson 1, provides the future value of a single present amount, the PW$1 provides the present value of a single future amount. The value of the PW$1 factor will always be less than $1, explicitly demonstrating that a dollar to be received in the future is worth less than a dollar today. The formula for the calculation of the PW$1 factors is: Where:  PW$1 = Present Worth of $1 Factor
 i = Periodic Interest Rate, often expressed as an annual percentage rate
 n = Number of Periods, often expressed in years
In order to calculate the PW$1 factor for 4 years at an annual interest rate of 6%, use the formula below: Viewed on a timeline: A depositor would be willing to give up $0.792094 today (shown as negative on the timeline) in order to receive $1 at the end of 4 years (shown as positive). To locate the factor, go to page 33 of AH 505, go down 4 years, and then across to column 4. The correct PW$1 factor is 0.792094. Link to AH 505, page 33 Example 1: How much must be deposited today in order to have $15,000 at the end of 10 years, assuming an annual interest rate of 7% with annual compounding? Solution:  PV = FV × PW$1 (7%, 10 yrs, annual)
 PV = $15,000 × 0.508349
 PV = $7,625
 Find the annual PW$1 factor (annual compounding) for 7% at a term of 10 years. In AH 505, page 37, go down 10 years and across to column 4 to find the correct factor of 0.508349.
 The present value of $7,625 is equal to the future value of $15,000 multiplied by the factor.
Link to AH 505, page 37 Example 2: Someone promises to pay you $25,000 in 5 years. Given an annual interest rate of 6% with annual compounding, how much should you pay for this promise today? Solution:  PV = FV × PW$1 (6%, 5 yrs, annual)
 PV = $25,000 × 0.747258
 PV = $18,681
 Find the annual PW$1 factor (annual compounding) for 6% at a term of 5 years. In AH 505, page 33, go down 5 years and across to column 4 to find the correct factor of 0.747258.
 The present value of $18,681 is equal to the future value of $25,000 multiplied by the factor.
Link to AH 505, page 33 Example 3: If you want to have $10,000 after 3 years, and you can invest at an annual rate of 5% compounded annually, how much should you invest today? Solution:  PV = FV × PW$1 (5%, 3 yrs, annual)
 PV = $10,000 × 0.863838
 PV = $8,638
 Find the annual PW$1 factor (annual compounding) for 5% at a term of 3 years. In AH 505, page 29, go down 3 years and across to column 4 to find the correct factor of 0.863838.
 The present value of $8,638 is equal to the future value of $10,000 multiplied by the factor.
Link to AH 505, page 29 Example 4: Ten years from now, you will need to make a lumpsum payment of $500,000. Assuming annual compounding, how much should you invest today in order to cover the future payment? The annual interest rate is 10%. Solution:  PV = FV × PW$1 (10%, 10 yrs, annual)
 PV = $500,000 × .385543
 PV = $192,772
 Find the annual PW$1 factor (annual compounding) for 10% at a term of 10 years. In AH 505, page 49, go down 10 years and across to column 4 to find the correct factor of 0.385543.
 The present value of $192,772 is equal to the future value of $500,000 multiplied by the factor.
Link to AH 505, page 49 Example 5: Acme Enterprises promises to pay the holders of its most recent bond issue $1,000 per bond at the end of 25 years (there are no annual or semi–annual interest payments; this is called a "zero coupon" bond). If the annual interest rate is 8.50%, assuming annual compounding, how much should each bond sell for when issued? Solution:  PV = FV × PW$1 (8.50%, 25 yrs, annual)
 PV = $1,000 × 0.130094
 PV = $130.09
 Find the annual PW$1 factor (annual compounding) for 8.50% at a term of 25 years. In AH 505, page 43, go down 25 years and across to column 4 to find the correct factor of 0.130094.
 The present value of $130.09 is equal to the future value of $1,000 multiplied by the factor. The bond should sell for $130.09.
Link to AH 505, page 43 We have calculated the present value of single amounts or payments, using the PW$1 factors. Many problems involve more than one payment, making it necessary to calculate the present value of multiple payments–that is, the present value of a stream of payments. Determining the present value of multiple payments is a straightforward extension of the singlepayment situation. When we calculated the present value of a single future payment, we multiplied the future payment by the appropriate PW$1 factor. This discounted the future payment to its present value. If there is more than one future payment, multiple each payment by the appropriate PW$1 factor and add the present values. The sum of the present values is the total present value of the stream of future payments. Example 1: Consider the following 3 payments:  $10,000 at the end of the first year
 $15,000 at the end of the second year
 $20,000 at the end of the third year
At an annual interest rate of 5%, what is the total present value of the 3 payments? Solution: Calculate the present value of each payment using the PW$1 factors and add those present values. The sum is the present value of all 3 payments. Thus: Link to AH 505, page 29 Viewed on a timeline: A person would be willing to pay $40,406 now (shown as negative on the timeline) in order to receive the three future payments of $10,000, and $15,000, and $20,000 (shown as positive).
Page 4 This lesson discusses the Future Worth of $1 per Period (FW$1/P); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:  Explains the function’s meaning and purpose,
 Provides the formula for the calculation of FW$1/P factor, and
 Contains practical examples of how to apply the FW$1/P factor.
FW$1/P: Meaning and Purpose The FW$1/P factor is the amount to which a series of periodic payments of $1 will compound at periodic interest rate i over n periods, assuming payments occur at the end of each period. The FW$1/P factor is used to compound a series of periodic equal payments to their future value. The FW$1/P factors are in column 2 of AH 505. FW$1/P factors are applicable to ordinary annuity problems. An annuity may be defined as a series of periodic payments, usually equal in amount, and payable at the end of the period. (See Lesson 10 for further discussion of annuities.) The formula for the calculation of the FW$1/P factors is Where:  FW$1/P = Future Worth of $1 per Period Factor
 i = Periodic Interest Rate, often expressed as an annual percentage rate
 n = Number of Periods, often expressed in years
In order to calculate the annual FW$1/P factor for 4 years at an annual interest rate of 6%, use the formula below: Viewed on a timeline: On the timeline, the deposits of $1 are shown as negative because from the point of view of a depositor they would be cash outflows. The future values are shown as positive because they would be cash inflows. The depositor gives up money at the end of each year in order to receive money at the end of year 4. To locate the FW$1/P factor, go to page 33 of AH 505, go down 4 years and across to column 2. The correct factor is 4.374616. Link to AH 505, page 33 Example 1: What is the future value of 3 payments of $1,000 with the payments made at the end of each of the next 3 years? (Assume an annual interest rate of 10%.) Solution:  FV = PMT × FW$1/P (10%, 3 yrs, annual)
 FV = $1,000 × 3.310000
 FV = $3,310
 Find the annual FW$1/P factor (annual compounding) for 10% and a term of 3 years. In AH 505, page 49, go down 3 years and across to column 2 to find the correct factor of 3.310000.
 The future value of $3,310 is equal to the periodic payment of $1,000 multiplied by the factor.
Link to AH 505, page 49 Viewed on a timeline: The problem could have been solved by using the FW$1 factor applicable to each payment, but it would have taken 4 calculations.  Endofyear 1 payment FW = $1,000 × 1.210000* = $1,210
Endofyear 2 payment FW = $1,000 × 1.100000** = $1,100 Endofyear 3 payment FW = $1,000 × 1.000000 (no compounding) = $1,000 Total value at end of year 3 = $3,310 * 1.210000 = FW$1 (10%, 2 years, annual) (AH 505, page 49, column 1) ** 1.100000 = FW$1 (10%, 1 year, annual) (AH 505, page 49, column 1) Using the FW$1/P annuity factor simplifies the calculation. Annuity factors are essentially shortcuts that can be used when cash flows or payments are equal and at regular intervals. Example 2: You deposit $13,000 at the end of each year for 23 years. If the account earns an annual rate of 7.50%, compounded annually, how much will be in the account after 23 years? Solution:  FV = PMT × FW$1/P (7.50%, 23 yrs, annual)
 FV = $13,000 × 57.027895
 FV = $741,363
 Find the annual FW$1/P factor (annual compounding) for 7.50% and a term of 23 years. In AH 505, page 39, go down 23 years and across to column 2 to find the correct factor of 57.027895.
 The future value of $741,363 is equal to the payment of $13,000 multiplied by the factor.
Link to AH 505, page 39 Example 3: Mr. Foresight deposits $1,500 at the end of each month into a retirement account that returns an annual rate of 6%, compounded monthly. How much will he have after 10 years? After 30 years? Solution: After 10 years:  FV = PMT × FW$1/P (6%, 10 yrs, monthly)
 FV = $1,500 × 163.879347
 FV = $245,819
After 30 years:  FV = PMT × FW$1/P (6%, 30 yrs, monthly)
 FV = $1,500 × 1,004.515042
 FV = $1,506,773
 Find the monthly FW$1/P factors (monthly compounding) for 6% at terms of 10 and 30 years, respectively. In AH 505, go to page 32, column 2, to find the correct factors of 163.879347 and 1,004.515042.
 The future values of $245,819 and $1,506,773, respectively, are equal to the payment of $1,500 multiplied by the appropriate factors. What a difference 20 years makes.
Link to AH 505, page 32 Example 4: Mrs. Foresight invests $20,000 in a 401k account at the end of each year for 10 years, earning an annual rate of 7%, compounded annually. At the end of 10 years, she invests the lumpsum balance for another 10 years, earning an annual rate of 8%, compounded annually. How much will Mrs. Foresight have at the end of 20 years? (Hint: This problem combines the FW$1/P and the FW$1) Solution: This is a twopart problem. Part I: First 10 years  FV = PMT × FW$1/P (7%, 10 yrs, annual)
 FV = $20,000 × 13.816448
 FV = $276,329
 Find the correct FW$1/P factor (AH 505, page 37, column 2) to calculate the future value of the payments after the first 10 years; the correct factor is 13.816448.
 After the first 10 years the future value is $276,329.
Link to AH 505, page 37 Part II: End of 20 years (final answer)  FV = PMT × FW$1/P (8%, 10 yrs, annual)
 FV = $276,329 × 2.158925
 FV = $596,574
 Use the future value calculated in Part I ($276,329) as the present value in the next calculation.
 Find the FW$1 factor (AH 505, page 41, column 1); the correct factor is 2.158925.
 Calculate the final future value (at the end of 20 years) using the formula:
Link to AH 505, page 41 Example 5: You want to save $8,000 to buy a car. You will deposit $185.71 at the end of every month. Your first deposit will be a month from today. If your account pays an annual interest rate of 12%, compounded monthly, approximately how many months will it take to save $8,000? Solution:
 In AH 505, page 56, column 2, find the FW$1/P factor that is closest to 43.077917; the correct factor is 43.076878.
 In order to save $8,000, you will need to deposit $185.71 for approximately 3 years or 36 months.
Link to AH 505, page 56
Page 5 This lesson discusses the Sinking Fund Factor (SFF); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:  Explains the function’s meaning and purpose,
 Provides the formula for the calculation of the SFFs, and
 Shows practical examples of how to apply the SFF.
SFF: Meaning and Purpose The SFF is the equal periodic payment that must be made at the end of each of n periods at periodic interest rate i, such that the payments compound to $1 at the end of the last period. The SFF is typically used to determine how much must be set aside each period in order to meet a future monetary obligation. The factors for the sinking fund are in column 3 of AH 505. The SFF can be thought of as the “opposite” of the FW$1/P factor; mathematically, the SFF and the FW$1/P factor are reciprocals: Conceptually, the FW$1/P factor provides the future amount to which periodic payments of $1 will compound, while the SFF provides the equal periodic payments that will compound to a future value of $1. The formula for the calculation of the SFF is Where:  SFF = Sinking Fund Factor
 i = Periodic Interest Rate, often expressed as an annual percentage rate
 n = Number of Periods, often expressed in years
In order to calculate the SFF for 4 years at an annual interest rate of 6%, use the formula below: The table below shows how the sinking fund payments of 0.228591 per year compound to $1 at the end of 4 years. The payments are at the end of each year, so the beginning balance in year 1 is 0. Viewed on a timeline: To locate the SFF, go to page 33 of AH 505, go down 4 years and across to column 3. The correct factor is 0.228591. Link to AH 505, page 33 Example 1: A company has just issued bonds with a face value of $150 million that are due and payable in 10 years. How much should the company deposit at the end of each year in order to retire the bond issue at the end of year 10, assuming the company can earn an annual interest rate of 7% on its deposits? Solution:  PMT = FV × SFF (7%, 10 yrs, annual)
 PMT = $150,000,000 × 0.072378
 PMT = $10,856,700 (annual deposit required)
 Find the annual SFF (annual compounding) for 7% and a term of 10 years. In AH 505, page 37, go down 10 years and across to column 3 to find the correct SFF of 0.072378.
 The required annual deposit, $10,856,700, is equal to the future value (the required amount at the end of year 10 multiplied by the SFF).
Link to AH 505, page 37 Example 2: When you retire in 25 years, you would like to have $500,000 in your 401k retirement account. If you can earn an annual rate of 8%, how much should you deposit at the end of each month in order to reach your goal? Solution:  PMT = FV × SFF (8%, 25 yrs, monthly)
 PMT = $500,000 × 0.001051
 PMT = $525.50 (monthly deposit required)
 Find the monthly SFF (monthly compounding) for 8% and a term of 25 years. In AH 505, page 40, go down 25 years and across to column 3 to find the correct SFF of 0.001051.
 The required monthly deposit, $525.50, is equal to the future value (amount desired upon retirement) multiplied by the SFF.
Link to AH 505, page 40 Example 3: In a balloon payment loan, only interest payments are made during the term of the loan; all of the principal is repaid at the end of the term. Suppose that you must repay a balloon loan in the amount of $1,000,000 that will be due 10 years from today. At an annual interest rate of 8%, how much should you deposit at the end of each year to fund the balloon payment? Solution:  PMT = FV × SFF (8%, 10 yrs, annual)
 PMT = $1,000,000 × 0.069029
 PMT = $69,029 (annual deposit required)
 Find the annual SFF (annual compounding) for 8% and a term of 10 years. In AH 505, page 41, go down 10 years and across to column 3 to find the correct SFF of 0.069029.
 The required annual deposit, $69,029, is equal to the future value (in this case the amount of the balloon payment) multiplied by the SFF.
Link to AH 505, page 41 Example 4: You own a small apartment building and five years from now, you expect to replace the roof at an estimated cost of $50,000. How much should you set aside at the end of each year to fund the future roof replacement, given an annual interest rate of 6%? Solution:  PMT = FV × SFF (6%, 5 yrs, annual)
 PMT = $50,000 × 0.177396
 PMT = $8,869.80 (annual deposit required)
 Find the annual SFF (annual compounding) for 6% and a term of 5 years. In AH 505, page 33, go down 5 years and across to column 3 to find the correct SFF of 0.177396.
 The required annual deposit, $8,869.80, is equal to the future value (in this case the future cost of the new roof) multiplied by the SFF.
Link to AH 505, page 33 Example 5: A borrower has a $200,000 balloon payment due in 10 years. To ensure that he can make the future payment, he plans to make equal annual deposits at the end of each year into an account that earns an annual interest rate of 4%. How much should he deposit at the end of each year? Solution:  PMT = FV × SFF (4%, 10 yrs, annual)
 PMT = $200,000 × 0.083291
 PMT = $16,658.20 (annual deposit required)
 Find the annual SFF (annual compounding) for 4% and a term of 10 years. In AH 505, page 25, go down 10 years and across to column 3 to find the correct SFF of 0.083291.
 The required annual deposit, $16,658.20, is equal to the future value (in this case the amount of the balloon payment) multiplied by the SFF.
Link to AH 505, page 25
Page 6 This lesson discusses the Present Worth of $1 Per Period (PW$1/P); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:  Explains the function’s meaning and purpose,
 Provides the formula for the calculation of PW$1/P factors, and
 Shows practical examples of how to apply the PW$1/P factor.
PW$1/P: Meaning and Purpose The PW$1/P is the present value of a series of future periodic payments of $1, discounted at periodic interest rate i over n periods, assuming the payments occur at the end of each period. The PW$1/P is typically used to discount a future level income stream to its present value. Another way to conceptualize the PW$1/P is the amount that must be deposited today to fund withdrawals of $1 at the end of each of the n periods at periodic interest rate i, assuming a periodic rate i can be earned on the outstanding balance. This compound interest function, together with the PW$1, is the basis of yield capitalization and its primary variant, discounted cash flow analysis. The PW$1/P factors are in column 5 of AH 505. The formula for the calculation of the PW$1/P factors is as follows: Where:  PW$1/P = Present Worth of $1 Per Period Factor
 i = Periodic Interest Rate, often expressed as an annual percentage rate
 n = Number of Periods, often expressed in years
In order to calculate the PW$1/P factor for 4 years at an annual interest rate of 6%, use the formula below: Viewed on a timeline: On the timeline, the initial deposit of $3.465106 is shown as negative because from the point of view of a depositor it would be a cash outflow. The future values of $1 at the end of each year are shown as positive because they would be cash inflows. To locate the PW$1/P factor, go to page 33 of AH 505, go down 4 years and across to column 5. The correct factor is 3.465106. Link to AH 505, page 33 Example 1: You’ve admired your neighbor’s vintage car for years, and he’s finally agreed to sell it to you. He offers you the following payment alternatives:  Pay $20,000 now, or
 Pay $6,000 at the end of each of the next 4 years with an annual interest rate of 8%
Which is the better alternative? Solution:  PV = PMT × PW$1/P (8%, 4 years, annual)
 PV = $6,000 × 3.312127
 PV = $19,873
Calculate the present value of the 4year payment plan (alternative 2) using the PW$1/P factor and compare it to the immediate payment of $20,000 (alternative 1).  Find the annual PW$1/P factor (annual compounding) for 8% at a term of 4 years. In AH 505, page 41, go down 4 years and across to column 5 to find the correct factor of 3.312127.
 The present value of $19,873 is equal to the periodic payment of $6,000 multiplied by the factor.
 You want to select the payment alternative with the lowest cost in presentvalue terms. Because the present value of the four payments ($19,873) is less than the immediate payment of $20,000 (no discounting of the immediate payment is required), the fourpayment alternative is preferable after adjusting for the time value of money.
Link to AH 505, page 41 Example 2: You will receive annual payments of $10,000 at the end of each year for the next 15 years with an annual interest rate of 5%. What is the present value of this stream of payments? Solution:  PV = PMT × PW$1/P (5%, 15 years, annual)
 PV = $10,000 × 10.379658
 PV = $103,796.58
 Find the annual PW$1/P factor (annual compounding) for 5% and 15 years. In AH 505, page 29, go down 15 years and across to column 5 to find the correct factor of 10.379658.
 The present value of $103,796.58 is equal to the periodic payment of $10,000 multiplied by the factor.
Link to AH 505, page 29 Example 3: At the end of each year following your retirement, you want to withdraw $20,000 from your 401k retirement account. You expect to live for 20 years after you retire. Assuming that you can earn an annual interest rate of 6%, what balance will you need in your retirement account to fund your planned withdrawals? Solution:  PV = PMT × PW$1/P (6%, 20 years, annual)
 PV = $20,000 × 11.469921
 PV = $229,398
 Find the annual PW$1/P factor (annual compounding) for 6% and 20 years. In AH 505, page 33, go down 20 years and across to column 5 to find the correct factor of 11.469921.
 The present value of $229,398 is equal to the annual payment of $20,000 multiplied by the PW$1/P factor. When you retire, the balance in your account must equal the present value of the 20 years of planned future withdrawals.
Link to AH 505, page 33 Example 4: Mr. Fortunate has won the $64 million California lottery. He will receive 20 annual payments of $3,200,000, with the first payment to be received immediately. Acme Investment Company is offering Mr. Fortunate $30,000,000 for the right to receive his 20 payments. If the annual interest rate is 8% with annual compounding, should he accept the offer? Solution:  PV = PMT × PW$1/P (8%, 19 years, annual)
 PV = $3,200,000 × 9.603599
 PV = $30,731,517
 Total PV = $30,731,517 + $3,200,000 = $33,931,517
 Use the PW$1/P factor for 19 years to discount the future 19 payments of $3,200,000 (AH 505, page 41, column 5).
 Add the initial payment of $3,200,000 (this occurs immediately and is not discounted) to calculate the total present value of the promised payments.
 Acme is offering $30,000,000 for a stream of cash flows valued at $33,931,517 (assuming an 8% discount rate). Mr. Fortunate should decline Acme’s offer.
Link to AH 505, page 41 Example 5: The subscription to your favorite magazine is about to expire. The magazine company offers you three renewal options:  Pay $100 now for a fouryear subscription.
 Pay $32 per year at the end of each year for four years.
 Pay $54 today and another $54 two years from today.
Assuming you want to receive the magazine for at least four more years, if the annual interest rate is 10%, which option is the best deal? Solution:  PV = PMT × PW$1/P (10%, 4 years, annual)
 PV = $32 × 3.169865
 PV = $101.44

 PV = FV × PW$1 (10%, 2 years, annual)
 PV = $54.00 × 0.826446
 PV = $44.63
 Total PV = $54.00 + $44.63 = $98.63
 Determine the present value of each renewal option and select the option with the lowest present value.
 The present value of option 1 is $100; payment is immediate and no discounting is required.
 The present value of option 2 is calculated using the PW$1/P factor (AH 505, page 49, column 5).
 The present value of option 3 is the initial payment of $54 (no discounting required) plus the present value of the second payment of $54 discounted for 2 years using the PW$1 factor (AH 505, page 49, column 4). Option 3 has the lowest present value and is the best deal.
Link to AH 505, page 49 Link to AH 505, page 49
Page 7 This lesson discusses the Periodic Repayment (PR), one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:  Explains the function’s meaning and purpose,
 Discusses the process of loan amortization,
 Provides the formula for the calculation of PR factors, and
 Contains practical examples of how to apply the PR factor.
PR: Meaning and Purpose The PR is the payment amount, at periodic interest rate i and number of periods n, in which the present worth of the payments is equal to $1, assuming payments occur at the end of each period. The PR is also called the loan amortization factor or loan payment factor, because the factor provides the payment amount per dollar of loan amount for a fully amortized loan. The PR factors are in column 6 of AH 505. The PR can be thought of as the “opposite” of the PW$1/P which was discussed in Lesson 6; mathematically, the PR and the PW$1/P factors are reciprocals as shown below: Conceptually, the PW$1/P factor provides the present value of a future series of periodic payments of $1, whereas the PR factor provides the equal periodic payments the present value of which is $1. Loan Amortization If a loan is repaid over its term in equal periodic installments, the loan is fully amortized. In a fullyamortized loan, each payment is part interest and part repayment of principal. Over the term of a fully amortized loan, the principal amount is entirely repaid. From the standpoint of the lender, a loan is an investment. In an amortized loan, the portion of the payment that is interest provides the lender a return on the investment, and the portion of the payment that is principal repayment provides the lender a return of the investment. An amortization schedule shows the distribution of loan payments between principal and interest throughout the entire term of a loan. Amortization schedules are useful because interest and principal repayment may be treated differently for income tax purposes and it is necessary to keep track of the separate amounts for each. The loan amortization schedule below shows an amortization schedule for a 10year loan, at an annual rate of 6%, with annual payments. The formula for the calculation of the PR factors is Where:  PR = Periodic Repayment Factor
 i = Periodic Interest Rate, often expressed as an annual percentage rate
 n = Number of Periods, often expressed in years
In order to calculate the PR factor for 4 years at an annual interest rate of 6%, use the formula below: Viewed on a timeline: On the timeline, the four payments are negative because from a borrower´s perspective they would be cash outflows. The amount borrowed, $1, is positive because from the borrower´s perspective it would be a cash inflow. To locate the PR factor in AH 505, go to page 33 of AH 505. Go down 4 years and across to column 6. The PR factor is 0.288591. Link to AH 505, page 33 Example 1: You have just borrowed $50,000, to be repaid in equal annual installments at the end of each of the next 20 years. The annual interest rate is 8%. What is the amount of each annual payment? Solution:  PMT = PV × PR (8%, 20 yrs, annual)
 PMT = $50,000 × 0.101852
 PMT = $5,092.60
 Find the annual PR factor (annual compounding) for 8% at a term of 20 years. In AH 505, page 41, go down 20 years and across to column 6 to find the correct factor of 0.101852.
 The annual payment of $5,092.60 is the loan amount of $50,000 multiplied by the PR factor.
Link to AH 505, page 41 Example 2: You borrow $200,000 to buy a house, using a fullyamortizing mortgage with monthly payments for 30 years at an annual interest rate of 5%. What is your monthly payment of principal and interest? Solution:  PMT = PV × PR (5%, 30 yrs, annual)
 PMT = $200,000 × 0.005368
 PMT = $1,073.60
 Find the monthly PR factor (monthly compounding) for 5% at a term of 30 years. In AH 505, page 28, go down 30 years and across to column 6 to find the correct factor of 0.005368.
 The monthly payment of $1,073.60 is equal to the loan amount multiplied by the monthly PR factor.
Link to AH 505, page 28 Example 3: John borrows $75,000 at an annual rate of 6%, repayable in equal annual payments at the end of each of the next 10 years. How much of John’s first payment is principal and how much is interest? Solution:  PMT = PV × PR (6%, 10 yrs, annual)
 PMT = $75,000 × 0.135868
 PMT = $10,190
 Find the annual PR factor (annual compounding) for 6% at a term of 10 years. In AH 505, page 33, go down 10 years and across to column 6 to find the correct factor of 0.135868.
 The annual payment of $10,190 is the loan amount multiplied by the annual PR factor. The payment is divided between interest and principal repayment.
 The outstanding loan balance for the first year is $75,000, so the interest amount for the first year is $75,000 × 0.06 = $4,500.
 The amount of principal repayment is $10,190 (annual payment)  $4,500 (interest amount) = $5,690.
Link to AH 505, page 33 Example 4: A friend just about ready to retire has $400,000 in his 401k retirement account. If he can earn an annual rate of 4% on the account and wishes to exhaust the fund over 20 years with equal annual withdrawals, how much can he withdraw at the end of each year? Solution:  PMT = PV × PR (4%, 20 yrs, annual)
 PMT = $400,000 × 0.073582
 PMT = $29,433
 Find the annual PR factor (annual compounding) for 4% at a term of 20 years. In AH 505, page 25, go down 20 years and across to column 6 to find the correct factor of 0.073582.
 The annual payment of $29,433 is the $400,000 balance in the retirement account multiplied by the annual PR factor.
 The friend could withdraw $29,433 at the end of each year for 20 years, assuming he could earn an annual rate of 4% on the account balance. After 20 years the account would be empty.
Link to AH 505, page 25 The primary use of the PR factor is to provide the amount of the periodic payment necessary to retire a given loan amount. But you can also use it to provide the amount of periodic payment that a given amount will support, assuming an annual interest rate and term, as in this example. Example 5: You take out a $100,000 mortgage loan at an annual rate of 6% with monthly payments for 30 years. You plan to sell the property after 12 years. At that time, what will be the outstanding balance (i.e., remaining principal) on the loan? Solution: The first step is to calculate the payment amount:  PMT = PV × PR (6%, 30 yrs, monthly)
 PMT = $100,000 × 0.005996
 PMT = $599.60
 Find the monthly PR factor (monthly compounding) for 6% at a term of 30 years. In AH 505, page 32, go down 30 years and across to column 6 to find the correct monthly factor of 0.005996.
 The monthly payment of $599.60 is the loan amount of $100,000 multiplied by the monthly PR factor.
Link to AH 505, page 32 The remaining balance of an amortizing loan is the present value of the loan’s remaining payments discounted at the loan’s contract rate of interest. The second step is to discount the remaining 18 years of monthly payments using the PW$1/P factor at 6%.  PV = PMT × PW$1/P (6%, 18 yrs, monthly)
 PV = $599.60 × 131.897876
 PV = $79,085.97
 Find the monthly PW$1/P factor (monthly compounding) for 6% for 18 years. In AH 505, page 32, go down 18 years and across to column 5 to find the correct monthly PW$1/P factor of 131.897876.
 The remaining loan balance of $79,085.97 is the payment amount of $599.60 multiplied by the PW$1/P factor.
Link to AH 505, page 32
Page 8 This lesson discusses the Mortgage Constant (MC), which is listed in the monthly tables of Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:  Explains the meaning and purpose of the MC,
 Explains how to find MC factors in AH 505 and calculate MC factors, and
 Contains practical examples of how to apply the MC factor.
MC: Meaning and Purpose The MC factor provides the annualized payment amount per $1 of loan amount for a fullyamortized loan with monthly compounding and payments. Mathematically, the MC factor is simply the monthly PR factor multiplied by 12. The MC factor is also known as the annualized mortgage constant or constant annual percent. The MC factors are in column 7 of the monthly pages of AH 505. Calculating MC Factors To locate the MC factor for a term of 30 years at an annual interest rate of 6%, go to page 32 of AH 505, go down 30 years and across to column 7. The MC factor is 0.0719461. MC factors are found in Column 7 of the monthly tables only. Link to AH 505, page 32 You can confirm that the MC factor is the monthly periodic repayment factor multiplied by 12: 0.005996 × 12 = 0.071952 (small difference due to rounding) This means that for every $1 of loan amount, the annual total of the 12 monthly payments will be $0.071952 (or $0.072). Or, stating it another way, the sum of the 12 monthly payments will be equal to 7.1952% (or 7.2%) of the loan amount. We could have calculated the MC factor by first calculating the monthly PR factor and then multiplying it by 12 (note that both i and n must be expressed in months, not years) using the formula below:  MC = PR × 12
 MC = 0.00599551 × 12
 MC = 0.0719461
Example 1: A buyer takes out a mortgage loan for $250,000 at an annual rate of 8% with monthly payments for 30 years. What percentage of the original loan amount will she pay on an annualized basis? Solution:  The problem simply asks for the MC factor, which we can look up directly in AH 505.
 Go to AH 505, page 40, column 7, 30 years, to find the MC factor of 0.0880517. This is equivalent to 8.80517%
Link to AH 505, page 40 One can confirm the answer by calculating the monthly payment, multiplying it by 12, and dividing this product by the original loan amount (difference between factor and table and calculation due to rounding):  PMT = PV × PR (8%, 30 years, annual)
 PMT = $250,000 × 0.007338
 PMT = $1,834.50
 MC = (PMT × 12) ÷ PV
 MC = ($1,834.50 × 12) ÷ $250,000
 MC = $22,014 ÷ $250,000
 MC = 0.088056, or 8.8056%
Example 2: In the band of investment method for deriving an overall capitalization rate (RO), the rate is a weighted average of the equity dividend rate (RE) and the mortgage constant (MC), with the weightings based on the respective proportions of equity and debt. The current equity dividend rate is 10% and a loan can be obtained at an annual interest rate of 6% with monthly payments for 30 years at a loantovalue ratio of 75%. Calculate an overall capitalization rate using the band of investment. Solution:  Find the MC factor in AH 505, page 32, column 7, for 30 years. The correct MC factor is 0.0719461.
 Use the band of investment method to estimate the overall rate (RO) using the calculation shown in the table below.
 The estimated overall capitalization rate (RO) is 7.90%.
Link to AH 505, page 32
Page 9 This lesson discusses the frequency of compounding and its affect on the present and future values using the compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:  Explains compounding frequency and intrayear compounding,
 Demonstrated calculation of FW$1 and PW$1 factors given monthly compounding, and
 Concludes with generalizations with respect to frequency of compounding and future and present value.
IntraYear Compounding Up to this point, we generally have assumed that interest was calculated at the end of each year, based on the principal balance at the beginning of the year and the annual interest rate. That is, we have assumed that interest was compounded (or discounted) on an annual basis, and in solving problems we have used the annual compounding pages in AH 505. Compounding interest more than once a year is called "intrayear compounding". Interest may be compounded on a semiannual, quarterly, monthly, daily, or even continuous basis. When interest is compounded more than once a year, this affects both future and presentvalue calculations. With intrayear compounding, the periodic interest rate, instead of being the stated annual rate, becomes the stated annual rate divided by the number of compounding periods per year. The number of periods, instead of being the number of years, becomes the number of compounding periods per year multiplied by the number of years. As shown in the following table: With monthly compounding, for example, the stated annual interest rate is divided by 12 to find the periodic (monthly) rate, and the number of years is multiplied by 12 to determine the number of (monthly) periods. In lesson 2, we calculated the annual FW$1 factor at a stated annual rate of 6% for 4 years with annual compounding. The resulting factor was 1.262477. Now let’s calculate the FW$1 for an annual rate of 6% for 4 years, but with monthly compounding. In this case, the periodic monthly rate is 0.5% (onehalf of one percent per month, 6% ÷ 12), and the number of monthly compounding periods is 48 (12 periods/year × 4 years). In order to calculate the FW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, use the formula below:  FW$1 = (1 + i)n
 FW$1 = (1 + 0.5%)48
 FW$1 = (1 + 0.005)48
 FW$1 = (1.005)48
 FW$1 = 1.270489
The FW$1 factor with monthly compounding, 1.270489, is slightly greater than the factor with annual compounding, 1.262477. If we had invested $100 at an annual rate of 6% with monthly compounding we would have ended up with $127.05 four years later; with annual compounding we would have ended up with $126.25. AH 505 contains separate sets of compound interest factors for annual and monthly compounding. Factors for annual compounding are on the oddnumbered pages; factors for monthly compounding are on the evennumbered pages.The FW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, is in AH 505, page 32 (monthly page). Link to AH 505, page 32 In lesson 3, we calculated the PW$1 factor at an annual rate of 6% for 4 years with annual compounding. The resulting factor was 0.792094. Let’s calculate the PW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding. In this case, the periodic monthly rate is 0.5% (onehalf of one percent per month, 6% ÷ 12), and the number of monthly compounding periods is 48 (12 periods/year × 4 years). In order to calculate the PW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, use the formula below: The PW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, can be found in AH 505, page 32. The amount of the factor is 0.787098. Link to AH 505, page 32 The following two generalizations can be made with respect to frequency of compounding and future and present values:  When interest is compounded more than once a year, a future value will always be higher than it would have been with annual compounding, all else being equal.
 When interest is compounded more than once a year, a present value will always be lower than it would have been with annual compounding, all else being equal.
Thus, with our examples for the FW$1 and the PW$1:  Given FW$1, at a rate of 6%, for a term of 4 years: 1.270489 (compounded monthly) > 1.262477 (compounded annually)
 Given PW$1, at a rate of 6%, for a term of 4 years: 0.787098 (compounded monthly < 0.792094 (compounded annually)
We would have obtained similar results with FW$1/P and PW$1/P, respectively. Most appraisal problems involve annual payments and require the use of annual factors. Monthly factors are also useful because most mortgage loans are based on monthly payments, and it is often necessary to make mortgage calculations as part of an appraisal problem. For other compounding periods, the factors for which are not included in AH 505, the appraiser can calculate the desired factor from the appropriate compound interest formula. As noted, AH 505 contains factors for annual and monthly compounding only.
Page 10 This lesson discusses annuities in the context of the compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:  Defines an annuity and two types of annuity,
 Explains how to convert an ordinary annuity factor into the corresponding annuity due factor, and
 Contains examples of converting annuity factors.
Definition of an Annuity An annuity is a series of equal cash flows, or payments, made at regular intervals (e.g., monthly or annually). The payments must be equal, and the interval between payments must be regular. The following compound interest functions in AH 505 involve annuities: There are two types of annuities:  Ordinary Annuity
 Annuity Due
An ordinary annuity is an annuity in which the cash flows, or payments, occur at the end of the period. An ordinary annuity of cash inflows of $100 per year for 5 years can be represented like this: The cash flows occur at the end of years 1 through 5. And the first cash flow occurs at the end of year 1. Most appraisal problems involve ordinary annuities; that is payments are assumed to occur at the end of the period. All of the formulas and factors in AH 505 pertain to ordinary annuities only. An annuity due is an annuity in which the cash flows, or payments, occur at the beginning of the period. An annuity due is also called an annuity in arrears. An annuity due of cash inflows of $100 per year for 5 years can be represented like this: The cash flows occur at the beginning of years 1 through 5. And the first cash flow occurs at time 0 (now). As noted, most appraisal problems assume that payments occur at the end of the period (ordinary annuity). But if payments occur at the beginning of the period (annuity due), an ordinary annuity factor in AH 505 can be converted to its corresponding annuity due factor with a relatively simple calculation. Although financial calculators and spreadsheet software make it even easier to convert from an ordinary annuity to an annuity due, it is useful to understand how to "manually" convert the ordinary annuity factors in AH 505 to annuity due factors. Conversion of ordinary annuity factor to annuity due factor for FW$1/P or PW$1/P: To determine the Future Worth of $1 Per Period (FW$1/P) or Present Worth of $1 Per Period (PW$1/P) factor for an annuity due, refer to the corresponding factor in AH 505 for an ordinary annuity and multiply it by a factor of (1 + the periodic interest rate). The periodic rate will differ depending on the compounding interval in the problem. For example, with annual compounding, the periodic rate would be the same as the annual rate; with monthly compounding the periodic rate would be the annual rate divided by 12. Example 1: Conversion to annuity due factor for FW$1/P Calculate the FW$1/P factor for 4 years at an annual interest rate of 6% with annual compounding, assuming payments occur at the beginning of each year. Solution: With annual compounding, the periodic rate equals the annual rate (6 percent, or 0.06).  Annuity Due Factor = Ordinary Annuity Factor (in AH 505, page 33) × (1 + periodic rate)
 Annuity Due Factor = 4.374616 × (1 + 0.06)
 Annuity Due Factor = 4.637093
Example 2: Conversion to annuity due factor for PW$1/P Calculate the PW$1/P factor for 4 years at an annual interest rate of 6% with monthly compounding, assuming payments occur at the beginning of each month. Solution: With monthly compounding, the periodic rate is 6% ÷ 12 = onehalf of one percent per month, or 0.06 ÷ 12 = 0.005.  Annuity Due Factor = Ordinary Annuity Factor (in AH 505, page 32) × (1 + periodic rate)
 Annuity Due Factor = 42.580318 × (1 + 0.005)
 Annuity Due Factor = 42.793220
Conversion of ordinary annuity factor to annuity due factor for SFF or PR: To determine the Sinking Fund Factor (SFF) or Periodic Repayment (PR) Factor for an annuity due, refer to the corresponding factor in AH 505 for an ordinary annuity and divide it by a factor of (1+ the periodic interest rate). Be sure to divide, not multiply, when converting the SFF and PR factors. Note: the periodic rate will differ depending on the compounding interval in the problem. Example 3: Conversion to annuity due for SFF Calculate the SFF for 4 years at an annual interest rate of 6% with annual compounding, assuming payments occur at the beginning of each year. Solution: Example 4: Conversion to annuity due for PR Calculate the PR factor for 4 years at an annual interest rate of 6% with monthly compounding, assuming payments occur at the beginning of each month (monthly annuity due factor). Solution: 