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**Time Value of Money Concepts**

6 Chapter 6: Time Value of Money Concepts

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**Time Value of Money Interest is the rent paid for the use**

That’s right! A dollar today is more valuable than a dollar to be received in one year. Interest is the rent paid for the use of money over time. The time value of money means that money can be invested today to earn interest and grow to a larger dollar amount in the future. Time value of money concepts are useful in valuing several assets and liabilities. Interest is the amount of money paid or received in excess of the amount borrowed or lent.

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**Explain the difference between simple and compound interest.**

Learning Objectives Explain the difference between simple and compound interest. LO1 Our first learning objective in Chapter 6 is to explain the difference between simple and compound interest.

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**Interest amount = P × i × n**

Simple Interest Interest amount = P × i × n Assume you invest $1,000 at 6% simple interest for 3 years. You would earn $180 interest. ($1,000 × .06 × 3 = $180) (or $60 each year for 3 years) Simple interest is computed by multiplying an initial investment times both the applicable interest rate and the period of time for which the money is borrowed or lent. Assume you invest $1,000 at 6% simple interest for 3 years. You would earn $180 interest.

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Compound Interest Compound interest includes interest not only on the initial investment but also on the accumulated interest in previous periods. Compound interest includes interest not only on the initial investment but also on the accumulated interest in previous periods. Principal Interest

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Compound Interest Assume we will save $1,000 for three years and earn 6% interest compounded annually. Assume we will save $1,000 for three years and earn 6% interest compounded annually. What is the balance in our account at the end of three years? What is the balance in our account at the end of three years?

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Compound Interest Each year we earn interest on the initial investment amount plus any previously earned interest. As a result, at the end of the three years, we have a total of $1,

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**Compute the future value of a single amount.**

Learning Objectives Compute the future value of a single amount. LO2 Our second learning objective in Chapter 6 is to compute the future value of a single amount.

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**Future Value of a Single Amount**

The future value of a single amount is the amount of money that a dollar will grow to at some point in the future. Assume we will save $1,000 for three years and earn 6% interest compounded annually. $1, × = $1,060.00 and $1, × = $1,123.60 $1, × = $1,191.02 The future value of a single amount is the amount of money that a dollar will grow to at some point in the future. Recall our previous example where we assume will save $1,000 for three years and earn 6% interest compounded annually.

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**Future Value of a Single Amount**

Writing in a more efficient way, we can say $1,000 × 1.06 × 1.06 × 1.06 = $1,191.02 or $1,000 × [1.06]3 = $1,191.02 Writing in a more efficient way, we can say $1,000 times 1.06 times 1.06 times 1.06, or even more concise is $1,000 times 1.06 to the third.

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**Future Value of a Single Amount**

$1,000 × [1.06]3 = $1,191.02 We can generalize this as . . . Number of Compounding Periods FV = PV (1 + i)n In fact, the future value of any invested amount can be determined using this concise formula. Another way to find the future value is to use tables that contain the future value of $1 invested for various periods of time and at various interest rates. Table 1 in your textbook is the Future Value of $1 table. Future Value Present Value Interest Rate

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**Future Value of a Single Amount**

Find the Future Value of $1 table in your textbook. Find the Future Value of $1 table in your textbook. Now, find the factor for 6% and 3 periods. Find the factor for 6% and 3 periods.

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**Future Value of a Single Amount**

Find the factor for 6% and 3 periods. Solve our problem like this. . . FV = $1,000 × FV = $1,191.02 To solve our example using the table, we just multiply $1,000 times the table factor of FV $1

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**Compute the present value of a single amount.**

Learning Objectives Compute the present value of a single amount. LO3 Our third learning objective in Chapter 6 is to compute the present value of a single amount.

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**Present Value of a Single Amount**

Instead of asking what is the future value of a current amount, we might want to know what amount we must invest today to accumulate a known future amount. This is a present value question. Present value of a single amount is today’s equivalent to a particular amount in the future. Instead of asking what is the future value of a current amount, we might want to know what amount we must invest today to accumulate a known future amount. This is a present value question. Present value of a single amount is today’s equivalent to a particular amount in the future.

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**Present Value of a Single Amount**

Remember our equation? FV = PV (1 + i) n We can solve for PV and get Using our previous equation for future value, we can solve for present value by dividing the future value by 1 plus the interest rate raised to the number of periods. Another way to solve for the present value is to use the Present Value of $1 table in your textbook. FV (1 + i)n PV =

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**Present Value of a Single Amount**

Hey, it looks familiar! Find the Present Value of $1 table in your textbook. Find the Present Value of $1 table in your textbook. It is Table 2. Does the format look familiar?

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**Present Value of a Single Amount**

Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time. What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually? Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time. What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?

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**Present Value of a Single Amount**

Present Value Factor = $20,000 × = $13,611.60 If you deposit $13, now, at 8% annual interest, you will have $20,000 at the end of 5 years. To solve this question, we multiply the future value of $20,000 by the present value factor for 8% for 5 periods which is If you deposit $13, now, at 8% annual interest, you will have $20,000 at the end of 5 years.

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**Learning Objectives LO4**

Solving for either the interest rate or the number of compounding periods when present value and future value of a single amount are known. LO4 Our fourth learning objective in Chapter 6 is to solve for either the interest rate or the number of compounding periods when present value and future value of a single amount are known.

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**Solving for Other Values**

FV = PV (1 + i)n Number of Compounding Periods Future Value Present Value Interest Rate There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined. There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined by using a little algebra.

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**Determining the Unknown Interest Rate**

Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? a. 3.5% b. 4.0% c. 4.5% d. 5.0% Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to?

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**Determining the Unknown Interest Rate**

Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? a. 3.5% b. 4.0% c. 4.5% d. 5.0% Present Value of $1 Table First, we must determine the present value factor we are looking to find. We divide the present value by the future value and find a factor of Using the present value table, we look for this factor in row 2 since the loan period in this example is for 2 years. This leads us to the answer which is 4.5%. $1,000 = $1,092 × ? $1,000 ÷ $1,092 = Search the PV of $1 table in row 2 (n=2) for this value.

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**Accounting Applications of Present Value Techniques—Single Cash Amount**

Monetary assets and monetary liabilities are valued at the present value of future cash flows. Monetary Assets Monetary Liabilities Monetary assets and monetary liabilities are valued at the present value of future cash flows. Monetary assets include money and claims to receive money, the amount which is fixed or determinable. Monetary liabilities are obligations to pay amounts of cash, the amount of which is fixed or determinable. Examples include notes receivables and notes payable. We value more receivables and payables at the present value of the future cash flows, reflecting an appropriate time value of money. Money and claims to receive money, the amount which is fixed or determinable Obligations to pay amounts of cash, the amount of which is fixed or determinable

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No Explicit Interest Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes. Even though the agreement states it is a noninterest-bearing note, the note does, in fact, include interest. Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes. However, even though the agreement states it is a noninterest-bearing note, the note does, in fact, include interest. (No one will loan you money interest free except, perhaps, your parents!) For these noninterest-bearing notes, we impute an appropriate interest rate for a loan of this type to use as the interest rate. We impute an appropriate interest rate for a loan of this type to use as the interest rate.

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**Expected Cash Flow Approach**

Statement of Financial Accounting Concepts No. 7 “Using Cash Flow Information and Present Value in Accounting Measurements” The objective of valuing an asset or liability using present value is to approximate the fair value of that asset or liability. Statement of Financial Accounting Concepts Number 7, “Using Cash Flow Information and Present Value in Accounting Measurements,” provides a framework for using future cash flows in accounting measurements. The objective of valuing an asset or liability using present value is to approximate the fair value of that asset or liability. This new expected cash flow approach incorporates specific probabilities of cash flows into the analysis. The expected cash flow is multiplied by the risk-free rate of return to arrive at the present value.

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**Explain the difference between an ordinary annuity and an annuity due.**

Learning Objectives Explain the difference between an ordinary annuity and an annuity due. LO5 Our fifth learning objective is to explain the difference between an ordinary annuity and an annuity due.

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**An annuity is a series of equal periodic payments.**

Basic Annuities An annuity is a series of equal periodic payments. An annuity is a series of equal periodic payments. If you make a car payment or a house payment, you likely pay the same amount each month. Both of these are examples of a series of equal periodic payments or an annuity.

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Ordinary Annuity An annuity with payments at the end of the period is known as an ordinary annuity. An annuity with payments at the end of the period is known as an ordinary annuity. End End

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Annuity Due An annuity with payments at the beginning of the period is known as an annuity due. An annuity with payments at the beginning of the period is known as an annuity due. Beginning Beginning Beginning

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**Learning Objectives LO6**

Compute the future value of both an ordinary annuity and an annuity due. LO6 Our sixth learning objective in Chapter 6 is to compute the future value of both an ordinary annuity and an annuity due.

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**Future Value of an Ordinary Annuity**

To find the future value of an ordinary annuity, multiply the amount of a single payment or receipt by the future value of an ordinary annuity factor. To find the future value of an ordinary annuity, multiply the amount of a single payment or receipt by the future value of an ordinary annuity factor found in Table 3 in your textbook.

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**Future Value of an Ordinary Annuity**

We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded annually, on all invested funds. What will be the fund balance at the end of 10 years? Part I We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded annually, on all invested funds. What will be the fund balance at the end of 10 years? Part II The future value of the ordinary annuity is $36,

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**Future Value of an Annuity Due**

To find the future value of an annuity due, multiply the amount of a single payment or receipt by the future value of an ordinary annuity factor. To find the future value of an annuity due, multiply the amount of a single payment or receipt by the future value of an annuity due factor found in Table 5 in your textbook.

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**Future Value of an Annuity Due**

Compute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually. Part I Compute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually. Part II The future value of the annuity due is $46,371.

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**Learning Objectives LO7**

Compute the present value of an ordinary annuity, an annuity due, and a deferred annuity. LO7 Our seventh learning objective in Chapter 6 is to compute the present value of an ordinary annuity, an annuity due, and a deferred annuity.

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**Present Value of an Ordinary Annuity**

You wish to withdraw $10,000 at the end of each of the next 4 years from a bank account that pays 10% interest compounded annually. How much do you need to invest today to meet this goal? First, let’s look at the present value of an ordinary annuity. You wish to withdraw $10,000 at the end of each of the next 4 years from a bank account that pays 10% interest compounded annually. How much do you need to invest today to meet this goal?

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**Present Value of an Ordinary Annuity**

1 2 3 4 Today $10,000 $10,000 $10,000 $10,000 PV1 PV2 PV3 PV4 Here is a graphic that depicts the annuity payments. It may help you to draw a similar graphic when you are working time value of money problems. This graphic illustrates that we are determining the present value of each ordinary annuity.

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**Present Value of an Ordinary Annuity**

Using the Present Value of $1 table, we can find the present value of each ordinary annuity as illustrated. We can then add up the present values to arrive at the present value of the entire annuity stream. So, to answer our question, if you invest $31, today you will be able to withdraw $10,000 at the end of each of the next four years. If you invest $31, today you will be able to withdraw $10,000 at the end of each of the next four years.

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**Present Value of an Ordinary Annuity**

An alternative way to solve this problem is to use the Present Value of Ordinary Annuity of $1 table. We can find the factor on this table at the intersection of the 4th row and the 10% column. Can you find this value in the Present Value of Ordinary Annuity of $1 table? More Efficient Computation $10,000 × = $31,698.60

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**Present Value of an Ordinary Annuity**

How much must a person 65 years old invest today at 8% interest compounded annually to provide for an annuity of $20,000 at the end of each of the next 15 years? a. $153,981 b. $171,190 c. $167,324 d. $174,680 How much must a person 65 years old invest today at 8% interest compounded annually to provide for an annuity of $20,000 at the end of each of the next 15 years?

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**Present Value of an Ordinary Annuity**

How much must a person 65 years old invest today at 8% interest compounded annually to provide for an annuity of $20,000 at the end of each of the next 15 years? a. $153,981 b. $171,190 c. $167,324 d. $174,680 PV of Ordinary Annuity $1 Payment $ 20,000.00 PV Factor × Amount $171,189.60 To solve this problem we use the Present Value of Ordinary Annuity of $1 table and find the factor at the intersection of the row for the 15th period and 8% column. The answer is $171,190 must be invested today at 8% to provide an annuity of $20,000 at the end of each of the next 15 years.

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**Present Value of an Annuity Due**

Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. Now, let’s look at the present value of an annuity due. Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. To solve this problem, we need to find the factor on the Present Value of Annuity Due of $1 at the intersection of the 4th row and the 6% column. The solution is the present value of this annuity due is $36,730.

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**Present Value of a Deferred Annuity**

In a deferred annuity, the first cash flow is expected to occur more than one period after the date of the agreement. In a deferred annuity, the first cash flow is expected to occur more than one period after the date of the agreement.

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**Present Value of a Deferred Annuity**

On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, If you require a 12% return on your investments, how much are you willing to pay for this investment? 1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10 Present Value? $12,500 1 2 3 4 On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, If you require a 12% return on your investments, how much are you willing to pay for this investment? As the graphic illustrates, the annuity is deferred for two periods. One way to solve this problem is to determine the present value of each of the annuities using the factors found at the intersection of the 3rd and 4th periods and 12%.

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**Present Value of a Deferred Annuity**

On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, If you require a 12% return on your investments, how much are you willing to pay for this investment? 1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10 Present Value? $12,500 1 2 3 4 A more efficient computation is as follows: Calculate the PV of the annuity as of the beginning of the annuity period. Discount the single value amount calculated in (1) to its present value as of today. Let’s see how this works for this problem. More Efficient Computation Calculate the PV of the annuity as of the beginning of the annuity period. Discount the single value amount calculated in (1) to its present value as of today.

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**Present Value of a Deferred Annuity**

On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, If you require a 12% return on your investments, how much are you willing to pay for this investment? 1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10 Present Value? $12,500 1 2 3 4 Part I First, let’s calculate the PV of the annuity as of the beginning of the annuity period. Our $12,500 annuity is for 2 periods at 12% so our factor from the Present Value of Ordinary Annuity of $1 table Is The present value of the annuity is $21,126. Part II Next, let’s discount the $21,126 calculated in part 1 to its present value as of today. The present value factor for 2 periods at 12% is The present value of this annuity stream 2 years deferred is $16,841.

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**Learning Objectives LO8**

Solve for unknown values in annuity situations involving present value. LO8 Our eight learning objective for Chapter 6 is to solve for unknown values in annuity situations involving present value.

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**Solving for Unknown Values in Present Value Situations**

In present value problems involving annuities, there are four variables: Present value of an ordinary annuity or Present value of an annuity due The amount of the annuity payment The number of periods The interest rate In present value problems involving annuities, there are four variables: 1. Present value of an ordinary annuity or Present value of an annuity due 2. The amount of the annuity payment 3. The number of periods, and 4. The interest rate. If you know any three of these, the fourth can be determined. If you know any three of these, the fourth can be determined.

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**Solving for Unknown Values in Present Value Situations**

Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made (the annuity amount) to repay the loan in four years? Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made to repay the loan in four years? Today End of Year 1 Present Value $700 End of Year 2 End of Year 3 End of Year 4

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**Solving for Unknown Values in Present Value Situations**

Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made (the annuity amount) to repay the loan in four years? In this problem we know the present value of $700, the interest rate of 8%, and the number of periods of 4. So, we are solving for the amount of the ordinary annuity. Before we can go much further, we need to know the factor from the 4th period and 8% intersection on the Present Value of Ordinary Annuity of $1 table. This factor is To solve this problem we divide the present value by the factor to determine the amount of the annuity is $

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**Learning Objectives LO9**

Briefly describe how the concept of the time value of money is incorporated into the valuation of bonds, long-term leases, and pension obligations. LO9 Our ninth learning objective in Chapter 6 is to briefly describe how the concept of the time value of money is incorporated into the valuation of bonds, long-term leases, and pension obligations.

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**Accounting Applications of Present Value Techniques—Annuities**

Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. Long-term Bonds Long-term Leases Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. Some common examples include long-term bonds, long-term leases, and pension obligations. Pension Obligations

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**Valuation of Long-term Bonds**

Calculate the Present Value of the Lump-sum Maturity Payment (Face Value) On January 1, 2006, Fumatsu Electric issues 10% stated rate bonds with a face value of $1 million. The bonds mature in 5 years. The market rate of interest for similar issues was 12%. Interest is paid semiannually beginning on June 30, What is the price of the bonds? Calculate the Present Value of the Annuity Payments (Interest) Part I When determining the present value of bonds, we must consider two cash flow streams: 1. Calculate the Present Value of the Lump-sum Maturity Payment (Face Value) 2. Calculate the Present Value of the Annuity Payments (Interest). On January 1, 2006, Fumatsu Electric issues 10% stated rate bonds with a face value of $1 million. The bonds mature in 5 years. The market rate of interest for similar issues was 12%. Interest is paid semiannually beginning on June 30, What is the price of the bonds? Part II To determine the present value of the face value of bonds, we use the Present Value of $1 table. To determine the present value of the interest payments, we use the Present Value of Ordinary Annuity of $1 table. By adding together these two present value amounts, we arrive at the present value, or selling price, of the bonds of $926,405.

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**Valuation of Long-term Leases**

Certain long-term leases require the recording of an asset and corresponding liability at the present value of future lease payments. Certain long-term leases require the recording of an asset and corresponding liability at the present value of future lease payments.

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**Valuation of Pension Obligations**

Some pension plans create obligations during employees’ service periods that must be paid during their retirement periods. The amounts contributed during the employment period are determined using present value computations of the estimate of the future amount to be paid during retirement. Some pension plans create obligations during employees’ service periods that must be paid during their retirement periods. The amounts contributed during the employment period are determined using present value computations of the estimate of the future amount to be paid during retirement.

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End of Chapter 6 The end of Chapter 6.

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McGraw-Hill © 2004 The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Introduction to Valuation: The Time Value of Money Chapter 4.

McGraw-Hill © 2004 The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Introduction to Valuation: The Time Value of Money Chapter 4.

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