Kế toán, kiểm toán - ChapteChapter 6: Time value of money concepts

TIME VALUE OF MONEY CONCEPTSChapter 6© 2013 The McGraw-Hill Companies, Inc.Simple InterestInterest amount = P × i × nAssume 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)Compound Interest Assume we deposit $1,000 in a bank that earns 6% interest compounded annually.What is the balance inour account at theend of three years?Future Value of a Single AmountThe 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 deposit $1,000 for three years that earns 6% interest compounded annually.$1,000.00 × 1.06 = $1,060.00and$1,060.00 × 1.06 = $1,123.60and$1,123.60 × 1.06 = $1,191.02Future Value of a Single AmountWriting in a more efficient way, we can say . . . .$1,191.02 = $1,000 × [1.06]3FV = PV (1 + i)nFutureValueAmount Invested at the Beginning of the PeriodInterestRateNumberof Compounding PeriodsUsing 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 × 1.19102FV = $1,191.02Present Value of a Single AmountInstead 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.Present Value of a Single AmountRemember our equation?FV = PV (1 + i) nWe can solve for PV and get . . . . FV (1 + i)nPV =Present Value of a Single AmountAssume 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 Amounti = .08, n = 5Present Value Factor = .68058$20,000 × .68058 = $13,611.60If you deposit $13,611.60 now, at 8% annual interest, you will have $20,000 at the end of 5 years.FV = PV (1 + i)nFutureValuePresentValueInterestRateNumberof Compounding PeriodsThere are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined.Solving for Other ValuesDetermining 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$1,000 = $1,092 × ?$1,000 ÷ $1,092 = .91575Search the PV of $1 table in row 2 (n=2) for this value.Monetary assets and monetary liabilities are valued at the present value of future cash flows.Accounting Applications of Present Value Techniques—Single Cash AmountMonetary AssetsMoney and claims to receive money, the amount which is fixed or determinableMonetary LiabilitiesObligations to pay amounts of cash, the amount of which is fixed or determinable 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.We impute an appropriate interest rate for a loan of this type to use as the interest rate. No Explicit InterestIASB: “The Conceptual Framework for Financial Reporting”IFRS No. 9: “Financial Instruments”The objective of valuing an asset or liability using present value is to approximate the fair value of that asset or liability.Expected Cash Flow ApproachAn annuity is a series of equal periodic payments for multiple periods.Basic AnnuitiesAn annuity with payments at the end of the period is known as an ordinary annuity.Ordinary AnnuityEnd of year 1$10,000$10,000$10,000$10,0001234TodayEnd of year 2End of year 3End of year 4An annuity with payments at the beginning of the period is known as an annuity due.Annuity DueBeginning of year 1$10,000$10,000$10,000$10,0001234TodayBeginning of year 2Beginning of year 3Beginning of year 4Future Value of an Ordinary AnnuityTo find the future value of an ordinary annuity, multiply the amount of the annuity by the future value of an ordinary annuity factor.Future Value of an Ordinary AnnuityWe 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 Annuity DueTo find the future value of an annuity due, multiply the amount of the annuity by the future value of an annuity due factor.Future Value of an Annuity DueCompute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually.Present Value of an Ordinary AnnuityYou 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?PV1PV2PV3PV4$10,000$10,000$10,000$10,0001234TodayPresent Value of an Ordinary AnnuityIf 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 AnnuityPresent Value of an Ordinary AnnuityCan you find this value in the Present Value of Ordinary Annuity of $1 table?More Efficient Computation $10,000 × 3.16986 = $31,698.60Present 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. $174,680PV of Ordinary Annuity $1Payment $ 20,000.00PV Factor × 8.55948Amount $171,189.60Present Value of an Annuity DueCompute 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 a Deferred AnnuityIn 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 AnnuityOn 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?1/1/1312/31/1312/31/1412/31/1512/31/1612/31/17Present Value?$12,500$12,5001234Present Value of a Deferred AnnuityMore Efficient ComputationCalculate 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.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?1/1/1312/31/1312/31/1412/31/1512/31/1612/31/17Present Value?$12,500$12,5001234Present Value of a Deferred AnnuityOn 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?1/1/1312/31/1312/31/1412/31/1512/31/1612/31/17Present Value?$12,500$12,5001234Solving for Unknown Values in Present Value SituationsIn present value problems involving annuities, there are four variables:Present value of an ordinary annuity or Present value of an annuity dueThe amount of the annuity paymentThe number of periodsThe interest rateIf you know any three of these, the fourth can be determined.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?TodayEnd ofYear 1Present Value $700End ofYear 2End ofYear 3End ofYear 4Solving 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?Accounting Applications of Present Value Techniques—AnnuitiesBecause financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. Long-term BondsLong-term LeasesPension ObligationsValuation of Long-term BondsCalculate the Present Value of the Lump-sum Maturity Payment (Principal Amount)Calculate the Present Value of the Annuity Payments (Interest)On January 1, 2013, Fumatsu Electric issues 10% stated rate bonds with a principal amount 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, 2013. What is the price of the bonds?Valuation of Long-term LeasesCertain long-term leases require the recording of an asset and corresponding liability at the present value of future lease payments.Valuation of Pension ObligationsSome 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. End of Chapter 6© 2013 The McGraw-Hill Companies, Inc.