1 / 57

Chapter 4

Chapter 4. Chapter Outline Future Values and Compound Interest Present Values Multiple Cash Flows Level Cash Flows: Perpetuities and Annuities Inflation and the Time Value of Money Effective Annual Interest Rates. The Time Value of Money. Introduction. Money Problems …

elu
Download Presentation

Chapter 4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 4 • Chapter Outline • Future Values and Compound Interest • Present Values • Multiple Cash Flows • Level Cash Flows: Perpetuities and Annuities • Inflation and the Time Value of Money • Effective Annual Interest Rates The Time Value of Money Semih Yildirim ADMS 3530

  2. Introduction • Money Problems … • As a person or a financial manager you will often have to compare cash payments which occur at different dates: Cash flows now, versus cash flows later. • To make optimal decisions, you must understand the relationship between a dollar received (paid) today and a dollar received (paid) in the future. • Two basic types of cash flow problems: • Present Value (PV) problems. • Future Value (FV) problems. Semih Yildirim ADMS 3530

  3. Introduction • Present Value (PV) Problems • PV problems involve calculating the value today of future cash flow(s). • For example: • Interest rates are 7%. If I need to have $100,000 saved in 10 years, how much money must I put aside todayto create that cash flow? • Interest rates are 12%. If I need to create an income of $5,000 per year for 10 years, how much money must I put aside todayto create that cash flow? Semih Yildirim ADMS 3530

  4. Introduction • Future Value (FV) Problems • FV problems involve calculating the value an investment will grow to after earning interest. • For example: • Interest rates are 5%. If I invest $1,000 today, how much willitbeworth in 8 years? • Interest rates are 10%. If I open an account and invest $2,500 per year, how much willitbeworth in 12 years? Semih Yildirim ADMS 3530

  5. Future Values • Compound Interest vs Simple Interest • There are two types of interest you may receive: • Compound interest. • Simple interest. 1. Simple Interest • Simple interest means that interest is earned only on your original investment: • No interest is earned on the interest. • Example: Assume interest rates are 6%. • You invest $100 in an account paying simple interest. How much will the account be worth in 5 years? Semih Yildirim ADMS 3530

  6. Period (t) 0 1 2 3 4 5 Future Values • Simple Interest • You earn interest only on the amount invested. • Therefore you would earn: • $100 x 6% = $6.00 per year for 5 years. Answer – you would have $130 after 5 years: Balance in your account: $100 $106 $112 $118 $124 $130 Semih Yildirim ADMS 3530

  7. Future Values 2. Compound Interest • Compound interest means that interest is earned on interest. • The result: the income you earn would be higher than it would be with simple interest. Why? • Because you earn interest on both the original investment and the interest earned in previous years. • Try the example again using compound interest: • Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years? Semih Yildirim ADMS 3530

  8. Period (t) 0 1 2 3 4 5 Future Values • Compound Interest • You earn interest on your interest: • $100 x 6% = $6.00 the first year. • $106 x 6% = $6.36 the second year. • $112.36 x 6% = $6.74 the third year … etc. After 5 years you would have $133.82 : Balance in your account: $100 $106 $112.36 $119.10 $126.25 $133.82 Semih Yildirim ADMS 3530

  9. Future Values • Formula for Calculating FV FV = Investment x (1 + r)t • Where r is the interest rate and t is the number of periods. • Try the example again using the formula above: • Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years? FV = $100 x (1 + 0.06)5 = $100 x 1.3382 = $133.82 Semih Yildirim ADMS 3530

  10. Why Compound Interest? Future Value (Dollars) Semih Yildirim ADMS 3530

  11. Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2000, compounded at 8%. FYI - The value of Manhattan Island land is well below this figure. Semih Yildirim ADMS 3530

  12. Present Value • Assume interest rates are 10%. • You have just won a lottery and must choose between the following two options: • Receive $1,000,000 today. • Receive $1,000,000 five years from now. Which option gives you the biggest “winnings” • The above example demonstrates a basic financial principle: • A dollar received today is worth more than a dollar received tomorrow. • The key question is: • How much less valuable is a dollar received tomorrow as versus a dollar received today? • That question is answered by using the interest rate (also known as the discountrate) to calculate the PV of the second option. Semih Yildirim ADMS 3530

  13. Present Value • Formula for Calculating PV PV = Future Value x 1/(1 + r)t • You have been offered $1 million five years from now. Interest rates are 10%. • What is that worth to you in today’s dollars? PV = $1.0 million x 1/ (1 + 0.10)5 = $1.0 million x 0.620921 = $620,921 Semih Yildirim ADMS 3530

  14. The equivalent of $620,921 now $1 million now Present Value • More Money Problems … • Thus, you could have $1 million today. • Or you could have the second option, which equates to $620,921 in today’s dollars. vs • You knew before that the first option was better, but now you can calculate exactly how much better off you are: • $379,079 better off! Semih Yildirim ADMS 3530

  15. Present Value vs Future Value • PV and FV are related! • Have you noticed that $620,921 becomes $1 million (and that $1 million requires $620,921) if you have a time period of 5 years and a discount rate of 10%? PV at 10% $1,000,000 $620,921 FV at 10% Semih Yildirim ADMS 3530

  16. Present Value vs Future Value • PV and FV are related! • $620,921 invested for 5 years at 10% grows to $1 million. • Or, working it in reverse, if rates are 10%, and you need $1 million in 5 years, you must put aside $620,921 right now. FV = PV x (1 + r)t = $620,921 x (1 + 0.10)5 = $620,921 x 1.61051 = $1 million PV = FV x 1/(1 + r)t = $1 million x 1/ (1 + 0.10)5 = $1 million x 0.620921 = $620,921 Semih Yildirim ADMS 3530

  17. Present Value vs Future Value • PV and FV are related! • To calculate the FV of money which is available now (PV) to be invested for t years at an interest rate r, multiply the PV by (1+r)t.To calculate the PV of a future payment, runtheprocessinreverse and divide the FV by (1+r)t. • (1+r)t is called the future value factor. • r is called the discountrate • Finding the PV is often called discounting Semih Yildirim ADMS 3530

  18. Present Value vs Future Value • Two Key Principles for Financial Calculations • Think of the example in which we compared receiving $1 million today against $1 million received 5 years from now. • You should see from that example that: • A dollar received today is worth more than a dollar received tomorrow. Lesson: The value of cash flows received at different times can never be directly compared. You must first discount all cash flows to a common date and then compare them. Semih Yildirim ADMS 3530

  19. Present Value vs Future Value • Finding the Unknown … FV = PV x (1 + r)t PV = FV x 1/(1 + r)t The FV and PV formulashave many applications. Note that the variables usedin these two equations are: FV PV r t Given any three variables in the equation, you can always solve for the remaining variable! Semih Yildirim ADMS 3530

  20. Present Value vs Future Value Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? Semih Yildirim ADMS 3530

  21. Present Value vs Future Value Semih Yildirim ADMS 3530

  22. Present Value vs Future Value Semih Yildirim ADMS 3530

  23. Multiple Cash Flows • Future Value Calculations • So far, we have looked at problems involving only a single cash flow. • This is unrealistic – most business investments will involve multiple cash flows over time. • We need a method for coping with such streams of cash flows! • To calculate the FV of multiple cash flow stream, you need to calculate the FV of each CF and then add up these FVs. • To calculate the PV of multiple cash flow stream, you need to calculate the PV of each CF and then add up these PVs Semih Yildirim ADMS 3530

  24. $1,200 $1,400 $1,000 0 1 2 3 FV of Multiple Cash Flows Doing Future Value Calculations • Assume interest rates are 8%. You make 3 deposits to your bank account: $1,200 today, $1,400 one year later, $1,000 two years later. • How much money will you have in your account 3 years from now? • Calculate what each cash flow will be worth at the specified future date and add up these future values. FV in Year 3: $1,080.00 = $1,000 x 1.08 $1,632.96 = $1,400 x (1.08)2 $1,511.65 = $1,200 x (1.08)3 $4,224.61 Semih Yildirim ADMS 3530

  25. PV of Multiple Cash Flows EXAMPLE • Assume interest rates are 8%. • You wish to buy a car making three installments: • $8,000 today • $4,000 one year later. • $4,000 two years later. • How much money would you have to place in an account today to generate this stream of cash flows? • If Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments, which do you prefer? Semih Yildirim ADMS 3530

  26. $8,000 $4,000 $4,000 0 1 2 Multiple Cash Flows • Present Value Calculations • You would need to place $15,133.06 in an account today to generate the desired cash flows: PV today: $8,000.00 $4,000 / (1.08) = $3,703.30 $4,000 / (1.08)2 = $3,429.36 $15,133.06 Semih Yildirim ADMS 3530

  27. Example: FV & PV of Mixed Stream(4% Interest Rate) Compounding - $12,166.5 FV$6,413.8 $3,509.6 $5,624.3 $4,326.4 $3,120.0 -$10,000 $3,000 $5,000 $4,000 $3,000 $2,000.0 0 1 2 3 4 5 End of Year $2,884.6 PV$5,271.7 $4,622.8 $3,556.0 $2,564.4 $1,643.9 Discounting Semih Yildirim ADMS 3530

  28. Multiple Cash Flows • Special Situations • In the previous examples, we worked with multiple cash flows of different sizes. • Sometimes we have a situation in which a seriesofequal cash flows is involved: • Any sequence of equally spaced, level cash flows is called an Annuity. • If the payment Stream lasts forever, it is called a perpetuity. Semih Yildirim ADMS 3530

  29. PV of an Annuity: the Long Method In previous examples, we have worked with multiple cash flows of different sizes. • Suppose we now need to calculate the PV of a stream of level future cash flows. • We could use the same procedure as before: • Calculate what each cash flow would be worth today, i.e. get its PV. • Add up these present values. • EXAMPLE • Assume interest rates are 10%. You wish to buy a car making three installments of $4000, first installment will be a year from today • How much money would you have to place in an account today to generate this stream of cash flows? Semih Yildirim ADMS 3530

  30. $4,000 $4,000 4,000 0 1 2 3 PV of an Annuity: the Long Method • You would need to place $9,947.41 in an account today to generate the desired cash flows: PV today: $4,000 / (1.10) = $3,636.36 $4,000 / (1.10)2 = $3,305.79 $4,000 / (1.10)3 = $3,005.26 $9,947.41 Semih Yildirim ADMS 3530

  31. PV of Annuity Formula C = cash payment r = interest rate t = Number of periods cash payment is received PVannuity = $4,000 x [1/0.10 – 1/(0.10 (1 + 0.10)3)] = $4,000 x 2.48685 = $9,947.41 • Using the PV of an annuity calculation, we get the same answer as before: • Put aside $9,947.41 to fund the cash flows. or Semih Yildirim ADMS 3530

  32. Calculating the FV of an Annuity • Suppose interest rates are 10% and you decide to save $4,000 per year for 20 years. How much will you have saved for your retirement? • This is a FV problem. We could use the same procedure as we used for multiple cash flows of different sizes: • Calculate what each cash flow would be worth in, 20 years, i.e. get its FV and Add up these future values. Can you see the problem with using this method? • Calculating the FV this way would mean working out the FV for 20 separate cash flows ...  • Is there an easier way? Semih Yildirim ADMS 3530

  33. FV of Annuity Formula C = cash payment r = interest rate t = Number of periods cash payment is received FVannuity = $4,000 x [ ((1 + 0.10)20 – 1) / 0.10 ] = $4,000 x 57.27499949 = $229,100 • Using the FV of an annuity calculation, we see that you will have $229,100 in your account when you retire in 20 years. Semih Yildirim ADMS 3530

  34. Examples of Special Annuities Semih Yildirim ADMS 3530

  35. Annuities Due Semih Yildirim ADMS 3530

  36. Annuities Due Semih Yildirim ADMS 3530

  37. Annuities Due Semih Yildirim ADMS 3530

  38. Examples of Special Annuities Semih Yildirim ADMS 3530

  39. Annuities with missing CFs Semih Yildirim ADMS 3530

  40. Annuities with missing CFs Semih Yildirim ADMS 3530

  41. C C Cash Payment Cash Payment r PV Interest rate Present Value PV of Perpetuity • Perpetuities • The PV of a perpetuity is calculated by dividing the cash payment by the interest rate: = PV of a perpetuity = • The interest rate on a perpetuity is calculated by dividing the cash payment by the PV: = Interest rate on a perpetuity = Semih Yildirim ADMS 3530

  42. C Cash Payment $1,000 r Interest rate 4% PV of Perpetuity • Example • What would you pay to own a guaranteed income of $1,000 per year to be received forever, if interest rates are 4%? = PV of a perpetuity = = = $25,000 Semih Yildirim ADMS 3530

  43. Growing Cash Flows • Cash Flows Growing at a Constant Rate • What if the cash flows in a financial problem are not equal, but are instead growing at a constant rate? • For example: • Assume the discount rate is 8%. You are thinking of buying a condo which generates $12,000 per year in net cash flow inperpetuity. These cash flows grow at 3% per year. What is the maximum price youshould pay for this condo? Semih Yildirim ADMS 3530

  44. C $12,000 Cash Payment r-g 0.08 – 0.03 Discount Rate – Growth Rate Growing Cash Flows • Valuing Growing Perpetuities • The PV of a growingperpetuity is calculated by dividing the cash payment by the discount rate less the growth rate: = PV of a perpetuity = = = $240,000 Semih Yildirim ADMS 3530

  45. Growing Cash Flows • Cash Flows Growing at a Constant Rate • In the previous problem we assumed that the cash flows grew at a constant rate forever. • It may be more reasonable to assume a constant growth rate for a limitedtimeperiod. • For example: • Assume the condo in the previous problem generates $12,000 per year in net cash flow for 20 years. These cash flows grow at 3% per year. Now, what is the maximum price youshould pay for this condo? Semih Yildirim ADMS 3530

  46. Growing Cash Flows • Valuing Finite Growing Cash Flows • The PV of cash flows which grow at a constant rate for a limited time period (T) is calculated by: PV of cash flow = = $240,000 * 0.6125 = = $147,000 Semih Yildirim ADMS 3530

  47. Inflation, Real and Nominal Rates • Inflation - Rate at which prices as a whole are increasing. • Nominal Interest Rate - Rate at which money invested grows. • Real Interest Rate - Rate at which the purchasing power of an investment increases. The relation between nominal and real rates and the (expected) future inflation rate is approximation formula Semih Yildirim ADMS 3530

  48. Inflation, Real and Nominal Rates Example If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate? Semih Yildirim ADMS 3530

  49. Frequency of Compounding General Formula: FVt= PV0(1 + [r/m])mt t: Number of Years m: Compounding Periods per Year r: Annual Interest Rate FVt,m: FV at the end of Year t PV0: PV of the Cash Flow today Impact of Frequency Julie Miller has $1,000 to invest for 2 years at an annual interest rate of 12%. Annual FV2 = 1,000(1+ [.12/1])(1)(2) = 1,254.40 Semi FV2 = 1,000(1+ [.12/2])(2)(2) = 1,262.48 Qrtly FV2 = 1,000(1+ [.12/4])(4)(2) = 1,266.77 Monthly FV2= 1,000(1+ [.12/12])(12)(2) = 1,269.73 Daily FV2= 1,000(1+[.12/365])(365)(2)= 1,271.20 Semih Yildirim ADMS 3530

  50. Effective Interest Rates • EAR vs APR • So far, we have used annual interest rates applied to annual cash flows. • But interest can be applied daily, weekly, monthly, semi-annually – or for any other convenient time period. • The simplest way to deal with this situation is to convert to annual rates. • There are two ways to convert a periodical interest rate to an annual basis: • Calculate the Effective Annual Rate (EAR). • Calculate the Annual Percentage Rate (APR). Semih Yildirim ADMS 3530

More Related