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Time value of money concepts, specifically future value and present value, are essential in a variety of accounting situations. These concepts and the related computational procedures are the subjects of this chapter. Present values and future values of single amounts and present values and future values of annuities (series of equal periodic payments) are described separately but shown to be interrelated.
Chapter TIME VALUE OF MONEY CONCEPTS © 2013 The McGraw-Hill Companies, Inc Slide Simple Interest Interest amount = P × i × n Assume you invest $1,000 at 6% simple interest for years You would earn $180 interest ($1,000 × 06 × = $180) (or $60 each year for years) Slide Compound Interest Assume we deposit $1,000 in a bank that earns 6% interest compounded annually What is the balance in our account at the end of three years? Slide 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 deposit $1,000 for three years that earns 6% interest compounded annually $1,000.00 × 1.06 = $1,060.00 and $1,060.00 × 1.06 = $1,123.60 and $1,123.60 × 1.06 = $1,191.02 Slide Future Value of a Single Amount Using Value way, of $1we Table, we .find Writingthe in aFuture more efficient can say the factor for 6% and periods is 1.19102 So, we can solve=our problem like this $1,191.02 $1,000 × [1.06] Number FV = $1,000 × 1.19102 Number of of n FV = PV (1 + i) Compounding Compounding FV = $1,191.02 Periods Periods Future Future Value Value Amount Amount Invested Invested at at the the Beginning Beginning of of the the Period Period Interest Interest Rate Rate Slide 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 Slide Present Value of a Single Amount Remember our equation? FV = PV (1 + i) n We can solve for PV and get PV = FV (1 + i) n Slide Present Value of a Single Amount Assume you plan to buy a new car in 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 years, if you can earn 8% interest compounded annually? Slide Present Value of a Single Amount i = 08, n = Present Value Factor = 68058 $20,000 × 68058 = $13,611.60 If you deposit $13,611.60 now, at 8% annual interest, you will have $20,000 at the end of years Slide 10 Solving for Other Values FV = PV (1 + i)n Future Future Value Value Present Present Value Value Interest Interest Rate Rate Number Number of of Compounding Compounding Periods Periods There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined Slide 25 Present Value of an Ordinary Annuity PV1 PV2 PV3 PV4 Total Annuity $ 10,000 10,000 10,000 10,000 PV of $1 Factor 0.90909 0.82645 0.75131 0.68301 3.16986 Present Value $ 9,090.90 8,264.50 7,513.10 6,830.10 $ 31,698.60 Can you find this value in the Present Value of Ordinary Annuity of $1 table? More Efficient Computation $10,000 × 3.16986 = $31,698.60 Slide 26 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 b c d $153,981 $171,190 $167,324 $174,680 PV of Ordinary Annuity $1 Payment $ 20,000.00 PV Factor × 8.55948 Amount $171,189.60 Slide 27 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 Slide 28 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 Slide 29 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 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? 1/1/13 12/31/13 12/31/14 $12,500 $12,500 12/31/15 12/31/16 12/31/17 Slide 30 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 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? 1/1/13 12/31/13 12/31/14 $12,500 $12,500 12/31/15 12/31/16 12/31/17 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 Slide 31 Present Value of a Deferred Annuity On January 1, 2013, you are considering an investment that will pay $12,500 a year for 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? 1/1/13 12/31/13 12/31/14 $12,500 $12,500 12/31/15 12/31/16 12/31/17 Slide 32 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 If you know any three of these, the fourth can be determined Slide 33 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? Present Value $700 Today End of Year End of Year End of Year End of Year Slide 34 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? Slide 35 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 Pension Obligations Slide 36 Valuation of Long-term Bonds Calculate the Present Value of the Lump-sum Maturity Payment (Principal Amount) Calculate the Present Value of the Annuity Payments (Interest) Cash Flow Face value of the bond Interest (annuity) Price of bonds On January 1, 2013, Fumatsu Electric issues 10% stated rate bonds with a principal amount of $1 million The bonds mature in 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? Table PV of $1 n=10; i=6% PV of Ordinary Annuity of $1 n=10; i=6% Table Value Amount 0.5584 $ 1,000,000 7.3601 Present Value $ 558,400 $ 368,005 926,405 50,000 Slide 37 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 Slide 38 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 End of Chapter © 2013 The McGraw-Hill Companies, Inc ... Beginning of year Beginning of year Beginning of year Beginning of year 4 Slide 18 Future Value of an Ordinary Annuity To find the future value of an ordinary annuity, multiply the amount of 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 Credit-Adjusted Risk-Free Rate of Interest Present Value. .. balance at the end of 10 years? Slide 20 Future Value of an Annuity Due To find the future value of an annuity due, multiply the amount of the annuity by the future value of an annuity due factor