CHAPTER 06: TIME VALUE OF MONEY

CHAPTER 06: TIME VALUE OF MONEY Learning Objectives (Focus is on Present Values NOT Future Values) LO6-1 Explain the difference between simple and compound interest. LO6-3 Compute the present value of a single amount. LO6-4 Solve for either the interest rate or the number of compounding periods when present value and future valueof a single amount are known. LO6-5 Explain the difference between an ordinary annuity and an annuity due situation. LO6-7 Compute the present value of an ordinary annuity, an annuity due, and a deferred annuity. LO6-8 Solve for unknown values in annuity situations involving present value. NOT COVERED (Future Values) LO6-2 Compute the future value of a single amount. LO6-6 Compute the future value of both an ordinary annuity and an annuity due. LO6-9 Briefly describehow the concept of the time value of money is incorporated into the valuation of bonds, long-term leases, and pension obligations.

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)

Assume we deposit $1,000 in a bank that earns 6% interest compounded annually. Compound Interest What is the balance in our account at the end of three years?

Compound Interest

Writing in a more efficient way, we can say . . . . $1,191.02 = $1,000 × [1.06]3 Future Value of a Single Amount Number of Compounding Periods FV = PV × (1 + i)n Future Value Amount Invested at the Beginning of the Period Interest Rate

Future Value of a Single Amount Using the Future Value of $1 Table, we find the factor for 6% and 3 periods is 1.19102. So, we can solve our problem like this. . . FV = $1,000 × 1.19102 FV = $1,191.02

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 apresent value question. Present value of a single amount is today’s equivalent to a particular amount in the future. Present Value of a Single Amount

Remember our equation? FV = PV × (1 + i) n We can solve for PV and get . . . . FV (1 + i)n PV = 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? Present Value of a Single Amount

Present Value of a Single Amount i = .08, n = 5 Present Value Factor = .68058 $20,000 × .68058 = $13,611.60 Present Value of $1 Table If you deposit $13,611.60 now, at8% annual interest, you will have$20,000 at the end of 5 years.

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.

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% Determining the UnknownInterest Rate Present Value of $1 Table $1,000 = $1,092 × ? $1,000 ÷ $1,092 = .91575 Search the PV of $1 table in row 2 (n=2) for this value.

Basic Annuities An annuity is a seriesof equal periodic payments. Period 1 Period 2 Period 3 Period 4 $10,000 $10,000 $10,000 $10,000

Today 1 2 3 4 Ordinary Annuity An annuity with payments at the end of the period is known as an ordinary annuity. $10,000 $10,000 $10,000 $10,000 End of year 1 End of year 2 End of year 3 End of year 4

An annuity with payments at the beginning of the period is known as an annuity due. Today 1 2 3 4 Annuity Due $10,000 $10,000 $10,000 $10,000 Beginning of year 1 Beginning of year 2 Beginning of year 3 Beginning of year 4

You wish to withdraw $10,000 at the endof 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? Present Value of an Ordinary Annuity

Today 1 2 3 4 Present Value of an Ordinary Annuity $10,000 $10,000 $10,000 $10,000 PV1 PV2 PV3 PV4

If you invest $31,698.60 today you will be able to withdraw $10,000 at the end of each of the next four years. Present Value of an Ordinary Annuity

Can you find this value in the Present Value of Ordinary Annuity of $1 table? Present Value of an Ordinary Annuity More Efficient Computation$10,000 × 3.16986 = $31,698.60

Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. Present Value of an Annuity Due

In present value problems involvingannuities, there are four variables: Solving for Unknown Values inPresent Value of Annuity Situations 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 If you know any three of these,the fourth can be determined.

Present Value $700 Today End ofYear 1 End ofYear 2 End ofYear 3 End ofYear 4 Solving for Unknown Valuesin 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?

Solving for Unknown Valuesin 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?

NOT COVERED ==

Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes. Accounting Applications of Present Value Techniques—Single Cash AmountNOT COVERED Even though the agreement states it is a noninterest-bearing note, the note does, in fact, include interest. We impute an appropriate interest rate for noninterest-bearing notes.

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. The present value factor is obtained using the company’s credit-adjusted risk-free rate of interest.

To find the future value of an ordinaryannuity, multiplythe amount of the annuity by the future value of an ordinary annuity factor. Future Value of an Ordinary Annuity (NOT COVERED)

We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded interest annually, on all invested funds. What will be the fund balance at the end of 10 years? Future Value of an Ordinary Annuity

To find the future value of an annuity due, multiplythe amount of the annuity by the future value of an annuity due factor. Future Value of an Annuity Due (NOT COVERED)

Compute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually. Future Value of an Annuity Due

In a deferred annuity, the first cash flow is expected to occur more than one period after the date of the agreement. Present Value of a Deferred Annuity (NOT COVERED)

Present Value? $12,500 $12,500 1/1/13 12/31/13 12/31/14 12/31/15 12/31/16 12/31/17 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2015. If you require a 12% return on your investments, how much are you willing to pay for this investment?

Present Value? $12,500 $12,500 1/1/13 12/31/13 12/31/14 12/31/15 12/31/16 12/31/17 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2015. If you require a 12% return on your investments, how much are you willing to pay for this investment? • More Efficient Computation • Calculate the present value 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.

Present Value? $12,500 $12,500 1/1/13 12/31/13 12/31/14 12/31/15 12/31/16 12/31/17 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2015. If you require a 12% return on your investments, how much are you willing to pay for this investment?

Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. Accounting Applications of Present Value Techniques—Annuities (NOT COVERED) Long-term Bonds Long-term Leases Pension Obligations

Valuation of Long-term Bonds Calculate the Present Value of the Lump-sum Maturity Payment (Face Value) On June 30, 2013, Ebsen Electric issued 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 December 31, 2013. What was the price of the bond issue? Calculate the Present Value of the Annuity Payments (Interest)

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.

Valuation of Long-term Leases On January 1, 2013, Todd Furniture Company signed a 20-year lease for a new retail showroom. The lease agreement calls for annual payments of $25,000 for 20 years beginning on January 1, 2013. The appropriate rate of interest for this long-term lease is 8%. Calculate the value of the asset acquired and the liability assumed by Todd (the present value of an annuity due at 8% for 20 years).

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.

Valuation of Pension Obligations On January 1, 2013, Todd Furniture Company hired a new sales manager for the new showroom. The sales manager is expected to work 30 years before retirement on December 31, 2042. Annual retirement benefits will be paid at the end of each year of retirement, a period that is expected to be 25 years. The sales manager will earn $2,500 in annual retirement benefits for the first year worked, 2013. How much must Todd Furniture contribute to the company pension fund in 2013 to provide for $2,500 in annual pension benefits for 25 years that are expected to begin in 30 years? Todd Furniture’s pension fund is expected to earn 5% interest.

Valuation of Pension Obligations This is a two-part calculation. The first part requires the computation of the present value of a 25-year ordinary annuity of $2,500 as of December 31, 2042. Next we calculate the present value of the December 31, 2042 amount. This second present value is the amount Todd Furniture will contribute in 2013 to fund the retirement benefit earned by the sales manager in 2013.

End of Chapter 6

CHAPTER 06: TIME VALUE OF MONEY