This lesson serves as an introduction to the topic and discusses the following: Show
Objectives and Format of Learning SessionThe objectives of this learning session are to:
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 MoneyIt´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.
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:
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.
When money is borrowed, the amount borrowed is called the principal. The consideration paid for the use of money is called 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 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.
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.
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.
In a well-known transaction, Dutch colonists bought Manhattan Island in 1624 for the equivalent of $24.
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:
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.
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.
(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: Solution:
Link to AH 505, page 49 Example 2: Solution:
Link to AH 505, page 28 Example 3: Solution:
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:
FW$1: Meaning and PurposeThe 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:
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.).
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: Solution:
Link to AH 505, page 33 Example 2: Solution:
Link to AH 505, page 29 Example 3: Solution:
Link to AH 505, page 39 Example 4: Solution:
Link to AH 505, page 49 Example 5: Solution:
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:
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: 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: 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: 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
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 single-payment 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
Solution: 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
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:
PW$1: Meaning and PurposeThe 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. Formula for Calculating PW$1 FactorsThe formula for the calculation of the PW$1 factors is: Where:
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: Solution:
Link to AH 505, page 37 Example 2: Solution:
Link to AH 505, page 33 Example 3: Solution:
Link to AH 505, page 29 Example 4: Solution:
Link to AH 505, page 49 Example 5: Solution:
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 single-payment 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:
Solution: Thus: Link to AH 505, page 29
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:
FW$1/P: Meaning and PurposeThe 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.) Formula for Calculating FW$1/PThe formula for the calculation of the FW$1/P factors is Where:
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: Solution:
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.
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: Solution:
Link to AH 505, page 39 Example 3: Solution: After 10 years:
After 30 years:
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 lump-sum 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: Part I: First 10 years
Link to AH 505, page 37 Part II: End of 20 years (final answer)
Link to AH 505, page 41 Example 5: Solution:
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:
SFF: Meaning and PurposeThe 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. Formula for CalculatingSFFThe formula for the calculation of the SFF is Where:
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: Solution:
Link to AH 505, page 37 Example 2: Solution:
Link to AH 505, page 40 Example 3: Solution:
Link to AH 505, page 41 Example 4: Solution:
Link to AH 505, page 33 Example 5: Solution:
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:
PW$1/P: Meaning and PurposeThe 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. Formula for Calculating PW$1/P FactorsThe formula for the calculation of the PW$1/P factors is as follows: Where:
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:
Solution:
Calculate the present value of the 4-year payment plan (alternative 2) using the PW$1/P factor and compare it to the immediate payment of $20,000 (alternative 1).
Link to AH 505, page 41 Example 2: Solution:
Link to AH 505, page 29 Example 3: Solution:
Link to AH 505, page 33 Example 4: Solution:
Link to AH 505, page 41 Example 5:
Solution:
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:
PR: Meaning and PurposeThe 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 AmortizationIf a loan is repaid over its term in equal periodic installments, the loan is fully amortized. In a fully-amortized 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 10-year loan, at an annual rate of 6%, with annual payments. The formula for the calculation of the PR factors is Where:
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: Solution:
Link to AH 505, page 41 Example 2: Solution:
Link to AH 505, page 28 Example 3: Solution:
Link to AH 505, page 33 Example 4: Solution:
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: Solution: The first step is to calculate the payment amount:
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%.
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:
MC: Meaning and PurposeThe MC factor provides the annualized payment amount per $1 of loan amount for a fully-amortized 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 FactorsTo 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: 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:
Example 1: Solution:
Link to AH 505, page 40
Example 2: Solution:
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:
Intra-Year Compounding
Compounding interest more than once a year is called "intra-year compounding". Interest may be compounded on a semi-annual, quarterly, monthly, daily, or even continuous basis. When interest is compounded more than once a year, this affects both future and present-value calculations. With intra-year 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.
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% (one-half 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:
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 odd-numbered pages; factors for monthly compounding are on the even-numbered 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% (one-half 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:
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:
Definition of an AnnuityAn 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:
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 Solution:
Example 2: Conversion to annuity due factor for PW$1/P Solution:
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 Solution: Example 4: Conversion to annuity due for PR Solution: |
22) the time value of money concept:
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What if I told you, you could get bigger, stronger, with healthier muscles and tendons by performing exercises where you don’t move… at all. Dynamic exercises such as your traditional exercises ...
How is the mindset of the Doctrine of Discovery evident in westward expansion
Manifest Destiny, a phrase coined in 1845, is the idea that the United States is destined—by God, its advocates believed—to expand its dominion and spread democracy and capitalism across the ...
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