Chapter 2 - Time Value of Money.pdf

PRINCIPLES OF FINANCE Structure of the Course Chapter 1: Introduction to Finance Chapter 2: Time Value of Money Chapter 3: Financial Markets Chapter 4: Risk and Return Chapter 5: Corporate Finance Chapter 2: Time Value of Money 1. The Concept of “Time Value of Money” 2. The Interest Rate 3. Time Value of Money: Single Amount 4. Time Value of Money: A Stream of Cash Flows 5. Applications of Time Value of Money 1. The Concept of Time Value of Money v Concept: Question 1: Which should you prefer – receive $100 today or $100 next year? Question 2: Which should you prefer – to receive $1,000 today or $2,000 ten years from today? Conclusion: “The money has time value” Meaning: A dollar one year from now is worth less than a dollar today. 1. The Concept of Time Value of Money v Reasons: Inflation Uncertainty Opportunity cost of money v Caution: Different cash flows at different points of time are not comparable! 2. The Interest Rate v Definition: Interest Payment: Represents the dollar amount required to pay the interest cost of a loan for the payment period. Interest Rate: Interest rate is the percent rate charged, or paid, by a lender to a borrower for the use of credit. 𝑰𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝑷𝒂𝒚𝒎𝒆𝒏𝒕𝒔 𝑰𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 = 𝑷𝒓𝒊𝒏𝒄𝒊𝒑𝒂𝒍 𝒂𝒎𝒐𝒖𝒏𝒕 2. The Interest Rate v 2 types of Interest rate: Simple Interest rate: - Simple interest is interest that is paid/earned on only the original amount, or principal, borrowed/lent. Þ Simple interest rate is calculated based on the original amount only. - Simple interest is normally used for a single period of less than a year. Exercise 1: You borrow $10,000 for 60 days at 5% simple interest per year (assume a 360-day year). Calculate the interest payment? 2. The Interest Rate v 2 types of Interest rate: Simple Interest rate: Σ Interest Payments = PV * i * n In which: + PV (Present Value): principal, or original amount borrowed (lent) at time period 0 + i: interest rate per time period + n: number of time periods Exercise 2: Assume that you deposit $100 in a savings account paying 8 percent simple interest and keep it there for 10 years. How much interest will you earn at the end of the 10-year period? 2. The Interest Rate v2 types of Interest rate: Compound Interest rate: Compound interest is interest paid/earned on the original principal and all interest accumulated during past periods. Þ Compound interest rate is calculated based on both the original amount and the previously accumulated interest amount. Exercise 3: Assume that you deposit $100 in a savings account paying 8 percent compound interest and keep it there for 10 years. How much interest will you earn at the end of the 10-year period? 2. The Interest Rate v 2 types of Interest rate: The Frequency of Compounding: 2. The Interest Rate v The Frequency of Compounding: APR (Annual Percentage Rate): an interest rate that does not include any consideration of compounding (also known as Nominal/Stated Interest Rate). EAR (Effective Annual Rate): an annual rate of interest when compounding occurs more often than once a year (also known as Annual Percentage Yield – APY). 2. The Interest Rate v The Frequency of Compounding: The relationship between APR and EAR: EAR = (1+APR/m)m -1 In which: + m = frequency of compounding Note: - Annual compounding: EAR = APR - Continuous compounding: EAR = eAPR – 1 Exercise 4: You have a credit card that carries a rate of interest of 18% per year compounded monthly. What is the interest rate compounded annually? That is, if you borrowed $1 with the card, what would you owe at the end of a year? 2. The Interest Rate v The Frequency of Compounding: Note: As the frequency of compounding increases, so does the effective annual rate: If we allow compounding to occur more and more frequent, the compounding period becomes shorter and shorter. Then m, the number of compounding periods increases. Note: EAR = eAPR – 1 is the maximum interest rate for any value of “APR”. 2. The Interest Rate v The Frequency of Compounding: Exercise 5: You plan to invest $2,000 in an individual retirement arrangement today at a stated interest rate of 8 percent, which is expected to apply to all future years. What is the effective annual rate (EAR) if interest is compounded as follows? 1. Annually 2. Semiannually 3. Daily (assume a 365-day year) 4. Continuously 2. The Interest Rate v The Frequency of Compounding: Exercise 6: Jason Spector has shopped around for the best interest rates for his investment of $10,000 over the next year. He has found the following: Which investment offers Jason the highest effective annual rate of return? 2. The Interest Rate v Effect of Inflation: Real vs. Nominal Interest Rates Consider an investment that costs $100 today and will be worth $115.50 in one year. Suppose prices are currently rising by 5% per year. What is the rate of return on this investment? What will your answer be if we measure the rate of return in terms of buying power? - Nominal interest rate: the percentage change in the number of dollars you have. - Real interest rate: the percentage change in how much you can buy with your dollars – the percentage change in your buying power. 2. The Interest Rate v Effect of Inflation: Real vs. Nominal Interest Rates The Fisher Effect: 1 + 𝑖! = (1 + 𝑖" )(1 + 𝜋 # ) Or, approximately: 𝑖! = 𝑖" + 𝜋 # Principle for present value calculation: Discount nominal cash flows at a nominal rate, Or discount real cash flows at a real rate 2. The Interest Rate v Effect of Inflation: Real vs. Nominal Interest Rates Exercise: Suppose you want to withdraw money each year for the next 3 years, and you want each withdrawal to have $25,000 worth of purchasing power as measured in current dollars. You know that the inflation rate is 4 percent per year. What is the present value of each cash flow if the appropriate nominal discount rate is 10%? 3. TVM: Single Amount v Any investment always has 2 types of value: -Future Value (Terminal Value) (FVn): The value on a future date of an investment made today. If you invest $100 today at 10 percent interest per year, in one year you will have $110. -Present Value (PV): The value today (in the present) of the investment. Þ PV = the value today (in the present) of a payment that is promised to be made in the future. At a 5 percent interest rate, the present value of $105 one year from now is $100. 3. TVM: Single Amount v Future Value: Future Value = Initial Payment + Interest - Simple interest: FVn = PV(1+i.n) Exercise 7: You deposit $1000 into a saving account with simple interest rate of 4%/year for 5 years. How much interest will you earn at the end of the 5-year period? How much money is in your account (if you don’t withdraw the interest) 3. TVM: Single Amount v Future Value: Exercise 8: You want to borrow $500 and you ask your friend to lend you this amount, offering to pay him back $548 in 9 months. What is the annual interest rate for this short term loan? A. 20% B. 8.25% C. 12.8% D. 16.5% Exercise 9: What is the future value of a loan of $2800 to be repaid in 8 months at an annual interest of 9%? A. $2968 B. $2926 C. $3178 D. $2884 3. TVM: Single Amount v Future Value: - Compound interest: FVn = PV(1+i)n Assume that the interest rate is 10% p.a, with an investment of $5 at 10% we obtain: 1 Year $5*(1+0.10) $5.5 2 years $5.5*(1+0.10) $6.05 3 years $6.05*(1+0.10) $6.655 4 Years $6.655*(1+0.10) $7.3205 3. TVM: Single Amount v Future Value: Caution: Time (n) & Interest rate (i) must be in same time units Exercise 10: You deposit 1500 USD on a saving account at the rate of 7.5% per year, compounded monthly. What will you receive after 3 years? Exercise 11: You plan to invest $2,000 in an individual retirement arrangement (IRA) today at a stated interest rate of 8%, which is expected to apply to all future years. a. How much will you have in the account at the end of ten years if interest is compounded as daily? semiannually? b. How much greater will your IRA account balance be at the end of ten years if interest is compounded continuously rather than annually? 3. TVM: Single Amount v Future Value: Future Value is higher with higher present value of the payment (PV bigger), The longer time period until payment (n higher), The higher the interest rate (i higher). Future values with $100 initial deposit and 5%, 10%, and 15% compound annual interest rates 3. TVM: Single Amount v Present Value: Reverses the future value calculation: - Simple interest: 𝑭𝑽 𝑷𝑽 = 𝟏 + 𝒓. 𝒏 - Compound interest: 𝑭𝑽 𝑷𝑽 = (𝟏 + 𝒓)𝒏 3. TVM: Single Amount v Present Value: Exercise 12: An Indiana state savings bond can be converted to $100 at maturity six years from purchase. If the state bonds are to be competitive with U.S. savings bonds, which pay 8 percent annual interest (compounded annually), at what price must the state sell its bonds? Assume no cash payments on savings bonds before redemption. Exercise 13: You just won a lottery that promises to pay you $1 million exactly ten years from today. Because the $1 million payment is guaranteed by the state in which you live, opportunities exist to sell the claim today for an immediate lump-sum cash payment. What is the least you will sell your claim for if you could earn 6 percent on similar risk investments during the ten-year period? 3. TVM: Single Amount v Present Value: Present Value is higher with higher future value of the payment (FV bigger), The shorter time period until payment (n lower), The smaller the interest rate (i lower). 4. TVM: A Stream of Cash Flows v Uneven Cash Flows: Future Value: 𝒏 𝑭𝑽𝒏 = : 𝑪𝑭𝒕 (𝟏 + 𝒊)𝒏(𝒕 𝒕&𝟎 - In which: + CFt: cash flow occur at time t (t= 0, 𝑛) 4. TVM: A Stream of Cash Flows v Uneven Cash Flows: Future Value: Exercise 14: Determine the balance at the end of 5 years in an investment account earning 9 percent annual interest, given the following five end-of-year deposits: $400 in year 1, $800 in year 2, $500 in year 3, $400 in year 4, and $300 in year 5. Determine that balance if these deposits are received at the beginning of the period. 4. TVM: A Stream of Cash Flows v Uneven Cash Flows: Present Value: 𝒏 𝑪𝑭𝒕 𝑷𝑽 = : (𝟏 + 𝒓)𝒕 𝒕&𝟎 - In which: + CFt: cash flow occur at time t (t= 0, 𝑛) 4. TVM: A Stream of Cash Flows v Uneven Cash Flows: Present Value: Exercise 15: Shortly after graduation, you receive an inheritance that you use to purchase a small bed-and breakfast inn as an investment (and a weekend escape). Your plan is to sell the inn after five years. The inn is an old mansion, so you know that appliances, furniture, and other equipment will wear out and need to be replaced or repaired on a regular basis. You estimate that these expenses will total $4,000 during year 1, $8,000 during year 2, $5,000 during year 3, $4,000 during year 4, and $3,000 during year 5, the final year of your ownership. For simplicity, assume that the expense payments will be made at the end of each year. Suppose you invest the lump sum in a bank account that pays 9 percent interest. Determine the amount of money you need to put in the account. Determine this amount if the payments are made at the beginning of the period. 4. TVM: A Stream of Cash Flows v Annuity: An annuity is a series of equal payments or receipts occurring over a specified number of periods. Meaning: CF1 = CF2 = CF3 = CF4 =... = CFn = A Ordinary Annuity: Definition: An annuity for which the payments occur at the end of each period. - Future Value: 𝑨 𝑭𝑽(𝑨)𝒏 = (𝒓 + 𝟏)𝒏 −𝟏 𝒓 - Present Value: 𝑨 𝟏 𝑷𝑽 𝑨 = 𝟏− 𝒓 (𝟏 + 𝒓)𝒏 4. TVM: A Stream of Cash Flows v Annuity: Annuity Due Definition: An annuity for which the payments occur at the beginning of each period. - Future Value: 𝑨 (𝒓 + 𝟏) 𝑭𝑽(𝑨)𝒏 = (𝒓 + 𝟏)𝒏 −𝟏 𝒓 - Present Value: 𝑨 (𝒓 + 𝟏) 𝟏 𝑷𝑽 𝑨 = 𝟏− 𝒓 (𝒓 + 𝟏)𝒏 4. TVM: A Stream of Cash Flows v Annuity: Annuity Due Definition: An annuity for which the payments occur at the beginning of each period. - Future Value: 𝑨 (𝒓 + 𝟏) 𝑭𝑽(𝑨)𝒏 = (𝒓 + 𝟏)𝒏 −𝟏 𝒓 - Present Value: 𝑨 (𝒓 + 𝟏) 𝟏 𝑷𝑽 𝑨 = 𝟏− 𝒓 (𝒓 + 𝟏)𝒏 4. TVM: A Stream of Cash Flows v Annuity: Exercise 16: You wish to save money on a regular basis to finance an exotic vacation in fve years. You are confident that, with sacrifice and discipline, you can force yourself to deposit $1,000 annually, at the end of each of the next five years, into a savings account paying 7 percent annual interest. Compute the future value (FV) of this annuity. What will be your answer, if the annuities are made at the beginning of the year? 4. TVM: A Stream of Cash Flows v Annuity: Exercise 17: Braden Company, a producer of plastic toys, was approached by its principal equipment supplier with an intriguing oﬀer for a service contract. The supplier, Extruding Machines Corporation (EMC), oﬀered to take over all of Braden’s equipment repair and servicing for five years in exchange for a one-time payment today. Braden’s managers know their company spends $7,000 at the end of every year on maintenance, so EMC’s service contract would reduce Braden’s cash outﬂows by this $7,000 annually for five years. Because these are equal annual cash benefits, Braden determines what it is willing to pay for the service contract by valuing it as a five-year ordinary annuity with a $7,000 annual cash ﬂow. If Braden requires a minimum return of 8 percent on all its investments, how much is it willing to pay for EMC’s service contract? What will be your answer, If Braden pays its maintenance costs at the start of each year? 4. TVM: A Stream of Cash Flows v Perpetuity: A perpetuity is an ordinary annuity whose payments or receipts continue forever. - Present Value: 𝑨 𝑷𝑽 𝑨 = 𝒓 Exercise 18: Find the present value of the dividend stream associated with a preferred stock issued by the Alpha and Omega Service Company. A&O, as the company is commonly known, promises to pay $10 per year (at the end of each year) on its preferred shares, and security analysts believe that the firm’s business and financial risk merits a required return of 12.5 percent. 4. TVM: A Stream of Cash Flows v Growing Perpetuity: A growing perpetuity is an annuity promising to pay a growing amount at the end of each year forever. - Present Value: 𝑪𝟏 𝑷𝑽 𝑪 = 𝒓−𝒈 In which: + C" : cash flow at time t=1. + r: discount rate + g: annual growth rate of cash flow 4. TVM: A Stream of Cash Flows v Growing Perpetuity: A growing perpetuity is an annuity promising to pay a growing amount at the end of each year forever. - Present Value: 𝑪𝟏 𝑷𝑽 𝑪 = 𝒓−𝒈 In which: + C" : cash flow at time t=1. + r: discount rate + g: annual growth rate of cash flow 4. TVM: A Stream of Cash Flows v Growing Perpetuity: A growing perpetuity is an annuity promising to pay a growing amount at the end of each year forever. - Present Value: 𝑪𝟏 𝑷𝑽 𝑪 = 𝒓−𝒈 In which: + C" : cash flow at time t=1. + r: discount rate + g: annual growth rate of cash flow 4. TVM: A Stream of Cash Flows v Growing Perpetuity: Exercise 19: Gil Bates is a philanthropist who wants to endow a medical foundation with sufficient money to fund ongoing research. Gil is particularly impressed with the research proposal submitted by the Smith Cancer Institute (SCI). The institute requests an endowment sufficient to cover its expenses for medical equipment. Next year, these expenses will total $10 million, and they will grow by 3 percent per year in perpetuity afterward. The institute can earn an 11 percent return on Gil’s contribution. How large must the contribution be to finance the institute’s medical equipment expenditures in perpetuity? 4. TVM: A Stream of Cash Flows v Growing Perpetuity: Exercise 20: Your client is interested in buying a small company that has fairly reliable cash flows in the future. The projected cash flows are as follows. What is the maximum price that your client should pay for this company if the appropriate discount rate for evaluating this type of company is 12%? Year Cash flows 1 20000 2-5 30000 6-Infinity 50000 --> Mixed, devide into 3 cashflows Move back to PV 5. TVM Application: Amortizing a Loan v Loan amortization refers to a situation in which a borrower makes equal periodic payments over time to fully repay a loan. v Loan amortization schedule is used to determine loan amortization payments and the allocation of each payment to interest and principal. money receive today Example: Suppose you borrow $22,000 at 12 percent compound annual interest to be repaid over the next six years. Equal installment payments are required at the end of each year. In addition, these payments must be sufficient in amount to repay the $22,000 together with providing the lender with a 12 percent return. --> Annuity calculation 5. TVM Application: Amortizing a Loan v Step 1: Determine the annual payment: 𝑨 𝟏 𝟐𝟐, 𝟎𝟎𝟎 = 𝟏− 𝟏𝟐% (𝟏 + 𝟏𝟐%)𝟔 ÞA= $5,351 => Annual payments of $5,351 will completely amortize (extinguish) a $22,000 loan in six years. v Step 2: Amortization schedule: 5. TVM Application: Project Appraisal v Net Present Value (NPV): Definition: - Net Present Value (NPV) of a project is the difference between the Present Value of its benefits and the Present Value of its costs. 𝑵𝑷𝑽 = 𝑷𝑽𝑪𝒂𝒔𝒉 𝑰𝒏𝒇𝒍𝒐𝒘𝒔 − 𝑷𝑽𝑪𝒂𝒔𝒉 𝑶𝒖𝒕𝒇𝒍𝒐𝒘𝒔 𝒏 𝑪𝑭𝒕 𝑵𝑷𝑽 = : 𝒕 − 𝑪𝟎 (𝟏 + 𝒓) 𝒕&𝟎 Economic Implications: Net Present Value (NPV) is a measure of how much value is created or added today by undertaking an investment. 5. TVM Application: Project Appraisal v Net Present Value (NPV): Decision Criteria: - For a specific project: § NPV > 0 => The project is profitable => Accept the Project. § NPV < 0 => The project is not profitable => Reject the Project. - For multiple projects: If 02 projects are independent: + Accept all projects that have NPV > 0 + Reject all projects that have NPV < 0 If the projects are mutually exclusive: => Accept the project with highest NPV. 5. TVM Application: Project Appraisal v Net Present Value (NPV): Exercise 21: Briarcliff Stove Company is considering a new product line to supplement its range line. It is anticipated that the new product line will involve cash investment of $700,000 at time 0 and $1.0 million in year 1. After-tax cash inﬂows of $250,000 are expected in year 2, $300,000 in year 3, $350,000 in year 4, and $400,000 each year thereafter through year 10. Though the product line might be viable after year 10, the company prefers to be conservative and end all calculations at that time. If the required rate of return is 15 percent, what is the net present value of the project? Is it acceptable? 5. TVM Application: Project Appraisal v Net Present Value (NPV): Exercise 22: growing perpetuity The Gent Corp. wants to set up a private cemetery business. According to the CFO, business is “looking up.” As a result, the cemetery project will provide a net cash inﬂow of $60,000 for the ﬁrm during the ﬁrst year, and the cash ﬂows are projected to grow at a rate of 6 percent per year forever. The project requires an initial investment of $925,000. --> present value a. If Gent requires a 13 percent return on such undertakings, should the cemetery business be started? b. The company is somewhat unsure about the assumption of a 6 percent growth rate in its cash ﬂows. At what constant growth rate would the company just break even if it still required a 13 percent return on investment? 5. TVM Application: Project Appraisal v Net Present Value (NPV): Advantages: - Help firms calculate the needed discount rate for profits - - Uses free cash flows. - Recognizes the time value of money. - Is consistent with the firm’s goal of shareholder wealth maximization. Disadvantages: - Requires detailed long-term forecasts of a project’s cash flows. + Future Cash flows + Discount rate - No information about the rate of return on the project 5. TVM Application: Project Appraisal v Net Present Value (NPV): Exercise 23: Alpha Corp. is considering investment in the best of two mutually exclusive projects. Project A involves an overhaul of the existing system; it will cost $45,000 and generate cash inflows of$20,000 per year for the next 3 years. Project B involves replacement of the existing system; it will cost $275,000 and generate cash inflows of$60,000 per year for 6 years. Using a 7.45058% cost of capital, calculate each project's NPV, and make a recommendation based on your findings. 5. TVM Application: Project Appraisal v Internal Rate of Return (IRR): Definition: - Internal Rate of Return (IRR) of a project is the discount rate that equates the Present Value of its benefits and the Present Value of its costs. 𝒏 𝑪𝑭𝒕 𝑵𝑷𝑽 = : 𝒕 − 𝑪𝟎 = 𝟎 (𝟏 + 𝑰𝑹𝑹) 𝒕&𝟎 Economic Implications: Internal Rate of Return (IRR) is a measure of the rate of return that the project earns. 5. TVM Application: Project Appraisal v Internal Rate of Return (IRR): Decision Rule: - For a specific project: § IRR > cost of capital => Accept the Project. § NPV < cost of capital => Reject the Project. - For multiple projects: If 02 projects are independent: + Accept all projects that have IRR > cost of capital + Reject all projects that have IRR < cost of capital If the projects are mutually exclusive: => Accept the project with highest IRR. 5. TVM Application: Project Appraisal v Internal Rate of Return (IRR): Exercise 24: Tony DiLorenzo is evaluating an investment opportunity. He is comfortable with the investment’s level of risk. Based on competing investment opportunities, he believes that this investment must earn a minimum compound annual after-tax return of 9% to be acceptable. Tony’s initial investment would be $7,500, and he expects to receive annual after-tax cash flows of $500 per year in each of the first 4 years, followed by $700 per year at the end of years 5 through 8. He plans to sell the investment at the end of year 8 and net $9,000, after taxes. Determine whether the investment should be accepted or not. 5. TVM Application: Project Appraisal v Internal Rate of Return (IRR): Advantages: - Uses free cash flows. - Recognizes the time value of money. - Is, in general, consistent with the firm’s goal of shareholder wealth maximization. Disadvantages: - Requires detailed long-term forecasts of a project’s cash flows. - Possibility of multiple IRRs or no IRR. - Assumes cash flows over the life of the project can be reinvested at the IRR. 5. TVM Application: Project Appraisal v Comparing NPV and IRR techniques: Ø Ranking Conflicts due to Differing Cash flows patterns: Bennett Company, a medium-sized metal fabricator that is currently contemplating two mutually exclusive projects with a required rate of return of 10%. The projected relevant cash flows for the two projects are presented as below: Initial Year Investment 1 2 3 4 Project A -200 80 80 80 80 Project B -200 0 0 0 400 Make the decision based on NPV and IRR criteria. 5. TVM Application: Project Appraisal v Comparing NPV and IRR techniques: Ø Ranking Conflicts due to Differing Project scale: Bennett Company, a medium-sized metal fabricator that is currently contemplating two mutually exclusive projects with a required rate of return of 10%. The projected relevant cash flows for the two projects are presented as below: NPV value shows the incremental value, the exact IF CONFLICT IN RANKING Initial Year ALWAYS GO FOR NPV Investment 1 2 3 4 Project A -100 50 50 50 50 Project B -400 170 170 170 170 Make the decision based on NPV and IRR criteria. IRR: the assumption that firm can reinvest at the same high rate is ridiculous, SHOULD NOT BASE SOLELY ON IRR 5. TVM Application: Asset Valuation v Asset Valuation process - Basic factors determining an asset’s value: -Principle of Asset valuation: “Asset’s expected future cash flows will be discounted back to the present, using the investor’s required rate of return, to determine the fair value that the asset should be purchased or sold” 5. TVM Application: Asset Valuation v Asset Valuation process Valuation Formula: +&"ℎ -'./ +&"ℎ -'./ +&"ℎ -'./ !""#$ 0#12.3 1 0#12.3 2 0#12.3 8 = ! + " + ⋯ + # %&'(# 1#7(21#3 1#7(21#3 1#7(21#3 51 + 9 51 + 9 51 + 9 1&$#.- 1#$(18 1&$#.- 1#$(18 1&$#.- 1#$(18 Exercise 25: --- Chapter 3 You purchased an asset that is expected to provide $5,000 cash flow per year for 4 years. If you have a 6 percent required rate of return, what is the value of the asset for you? PV Fair value = 17325

Chapter 2 - Time Value of Money.pdf

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