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## Time Value of money concepts

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**Chapter 6**Time Value of money concepts**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**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?**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. Future Value of a Single Amount $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**Writing in a more efficient way, we can say . . . .**$1,191.02 = $1,000 × [1.06]3 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 Number of Compounding Periods FV = PV (1 + i)n Future Value Amount Invested at the Beginning of the Period Interest Rate**Instead of asking what is the future value of a current**amount, we might want to know what amount we must invest todayto accumulate a known future amount. This is apresent valuequestion. 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**i = .08, n = 5**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 5 years. Present Value of a Single Amount**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 Unknown Interest 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.**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 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**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. We impute an appropriate interest rate for a loan of this type to use as the interest rate.**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.**Basic Annuities**An annuity is a series of equal periodic payments.**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**Today**1 2 3 4 Annuity Due An annuity with payments at the beginning of the period is known as an 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**To find the future value of an ordinary annuity, multiplythe**amount of the annuity by the future value of an ordinary annuity factor. 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 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**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**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? 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$31,698.60today 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**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 Present Value of an Ordinary Annuity PV of Ordinary Annuity $1 Payment $ 20,000.00 PV Factor × 8.55948 Amount $171,189.60**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 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**On January 1, 2009, you are considering an investment that**will pay $12,500 a year for 2 years beginning on December 31, 2011. 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 1/1/09 12/31/09 12/31/10 12/31/11 12/31/12 12/31/13 1 2 3 4 Present Value of a Deferred Annuity**Present Value?**$12,500 $12,500 1/1/09 12/31/09 12/31/10 12/31/11 12/31/12 12/31/13 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2009, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2011. If you require a 12% return on your investments, how much are you willing to pay for this investment? • 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.**Present Value?**$12,500 $12,500 1/1/09 12/31/09 12/31/10 12/31/11 12/31/12 12/31/13 1 2 3 4 Present Value of a Deferred Annuity On January 1, 2009, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2011. If you require a 12% return on your investments, how much are you willing to pay for this investment?**In present value problems involving annuities, there are**four variables: Solving for Unknown Values in Present Value 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.**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 ofYear 1 End ofYear 2 End ofYear 3 End ofYear 4 Solving for Unknown Values in Present Value Situations**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?**Because financial instruments typically specify equal**periodic payments, these applications quite often involve annuity situations. Accounting Applications of Present Value Techniques—Annuities 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 January 1, 2009, 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, 2009. What is the price of the bonds? 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 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.