Time Value of Money Concepts Copyright © 2007 by The McGraw-Hill Companies, Inc All rights reserved 6-2 Time Value of Money Interest is the rent paid for the use of money over time That’s right! A dollar today is more valuable than a dollar to be received in one year 6-3 Learning Objectives Explain the difference between simple and compound interest 6-4 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) 6-5 Compound Interest Compound interest includes interest not only on the initial investment but also on the accumulated interest in previous periods Principal Interest 6-6 Compound Interest 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? 6-7 Compound Interest Original balance First year interest Balance, end of year $ 1,000.00 60.00 $ 1,060.00 Balance, beginning of year Second year interest Balance, end of year $ 1,060.00 63.60 $ 1,123.60 Balance, beginning of year Third year interest Balance, end of year $ 1,123.60 67.42 $ 1,191.02 6-8 Learning Objectives Compute the future value of a single amount 6-9 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 $1,000.00 × 1.06 = $1,060.00 and $1,060.00 × 1.06 = $1,123.60 and $1,123.60 × 1.06 = $1,191.02 6-10 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 6-11 Future Value of a Single Amount $1,000 × [1.06]3 = $1,191.02 We can generalize this as FV = PV (1 + i)n Future Value Present Value Number of Compounding Periods Interest Rate 6-12 Future Value of a Single Amount Find the Future Value of $1 table in your textbook Find the factor for 6% and periods 6-13 Future Value of a Single Amount Find the factor for 6% and periods Solve our problem like this FV = $1,000 × 1.19102 FV = $1,191.02 FV $1 6-14 Learning Objectives Compute the present value of a single amount 6-15 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 6-16 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 6-17 Present Value of a Single Amount Find the Present Value of $1 table in your textbook Hey, it looks familiar! 6-18 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? 6-19 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 6-20 Learning Objectives Solving for either the interest rate or the number of compounding periods when present value and future value of a single amount are known 6-21 Solving for Other Values FV = PV (1 + i)n Future Value Present Value Interest Rate Number of Compounding 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 6-22 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% 6-23 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% Present Value of $1 Table c 4.5% $1,000 = $1,092 × ? d 5.0% $1,000 ÷ $1,092 = 91575 Search the PV of $1 table in row (n=2) for this value Accounting Applications of Present Value Techniques—Single Cash Amount 6-24 Monetary assets and monetary liabilities are valued at the present value of future cash flows Monetary Assets Monetary Liabilities 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 6-25 No Explicit Interest Some notes 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 6-26 Expected Cash Flow Approach Statement of Financial Accounting Concepts No “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 Expected Cash Flow × Risk-Free Rate of Interest Present Value 6-27 Learning Objectives Explain the difference between an ordinary annuity and an annuity due 6-28 Basic Annuities An annuity is a series of equal periodic payments 6-29 Ordinary Annuity An annuity with payments at the end of the period is known as an ordinary annuity End End 6-30 Annuity Due An annuity with payments at the beginning of the period is known as an annuity due Beginning Beginning Beginning 10 6-31 Learning Objectives Compute the future value of both an ordinary annuity and an annuity due 6-32 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 6-33 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? Am ount of annuity $ 2,500.00 $ 36,216.50 Future value of ordinary annuity of $1 (i = 8%, n = 10) Future value × 14.4866 11 6-34 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 6-35 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 Am ount of annuity $ 10,000 FV of annuity due of $1 (i=6%, n=4) Future value × 4.63710 $ 46,371 6-36 Learning Objectives Compute the present value of an ordinary annuity, an annuity due, and a deferred annuity 12 6-37 Present Value of an Ordinary Annuity You wish to withdraw $10,000 at the end of each of the next years from a bank account that pays 10% interest compounded annually How much you need to invest today to meet this goal? 6-38 Present Value of an Ordinary Annuity Today $10,000 $10,000 $10,000 $10,000 PV1 PV2 PV3 PV4 6-39 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 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 13 6-40 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 6-41 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 6-42 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 14 6-43 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 Am ount of annuity $ 10,000 PV of annuity due of $1 × 3.67301 (i=6%, n=4) Present value of annuity $ 36,730 6-44 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 6-45 Present Value of a Deferred Annuity On January 1, 2006, you are considering an investment that will pay $12,500 a year for years beginning on December 31, 2008 If you require a 12% return on your investments, how much are you willing to pay for this investment? Present Value? 1/1/06 $12,500 12/31/06 Payment $ 12,500 12,500 12/31/07 $12,500 12/31/08 12/31/09 PV of $1 i = 12% 0.71178 0.63552 $ $ 12/31/10 PV 8,897 7,944 16,841 n 15 6-46 Present Value of a Deferred Annuity On January 1, 2006, you are considering an investment that will pay $12,500 a year for years beginning on December 31, 2008 If you require a 12% return on your investments, how much are you willing to pay for this investment? Present Value? 1/1/06 $12,500 12/31/06 12/31/07 $12,500 12/31/08 12/31/09 12/31/10 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 6-47 Present Value of a Deferred Annuity On January 1, 2006, you are considering an investment that will pay $12,500 a year for years beginning on December 31, 2008 If you require a 12% return on your investments, how much are you willing to pay for this investment? Present Value? 1/1/06 Payment $ 12,500 $12,500 12/31/06 PV of Ordinary Annuity of $1 n=2, i = 12% 1.69005 $ 12/31/07 PV 21,126 $12,500 12/31/08 12/31/09 Future Value $ 21,126 12/31/10 PV of $1 n=2, i = 12% 0.79719 $ PV 16,841 6-48 Learning Objectives Solve for unknown values in annuity situations involving present value 16 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 6-49 The interest rate If you know any three of these, the fourth can be determined Solving for Unknown Values in Present Value Situations 6-50 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 Solving for Unknown Values in Present Value Situations 6-51 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? Pres ent value $ 700.00 PV of ordinary annuity of $1 (i=8%, n=4) Annuity am ount ÷ 3.31213 $ 211.34 17 6-52 Learning Objectives Briefly describe how the concept of the time value of money is incorporated into the valuation of bonds, long-term leases, and pension obligations Accounting Applications of Present Value Techniques—Annuities 6-53 Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations Long-term Bonds Long-term Leases Pension Obligations 6-54 Valuation of Long-term Bonds Calculate the Present Value of the Lump-sum Maturity Payment (Face Value) Calculate the Present Value of the Annuity Payments (Interest) Cash Flow Face value of the bond Inte rest (annuity) Price of bonds On January 1, 2006, Fumatsu Electric issues 10% stated rate bonds with a face value 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, 2006 W hat 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 0.5584 $ 1,000,000 7.3601 Pre sent Va lue Amount $ 558,400 $ 368,005 926,405 50,000 18 6-55 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 6-56 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 6-57 End of Chapter 19 ... 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 6-35 Future Value of. .. Present value of an ordinary annuity or Present value of an annuity due The amount of the annuity payment The number of periods 6-49 The interest rate If you know any three of these, the fourth... Valuation of Long-term Bonds Calculate the Present Value of the Lump-sum Maturity Payment (Face Value) Calculate the Present Value of the Annuity Payments (Interest) Cash Flow Face value of the bond