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Chapter 5 Time Value of Money Time value • What is the difference between simple interest and compound interest? https://www.youtube.com/watch?feature=player_embed ded&v=-qgdMTbTJlA Let’s Look at the worksheet • In the worksheet, is there something special that needs to be done for a present value amount? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) $75.00 -$50.00 5.00 Compounding Freq. (m) (P/Y) Solve for PV Solve for FV FV (Continuous Compounding) Solve for Interest Rate $50.00 8.45% 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y Effective Interest Rate PVA PMT for PVA Interest for PVA (per period) FVA PMT for FVA Interest for FVA PV of Perpetuity PV of Growing Annuity PV of Growing Perpetuity 8.45% $ $ (5.00) 0.08 #DIV/0! $ 0.08 #DIV/0! #DIV/0! FV Example • If you were to invest $3,000 today, what would it be worth in 1 year if you can earn 10% on your investments on average? 2 Years? 5 years? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV -$3,000.00 10.00% 1.00 Solve for FV $3,300.00 FV (Continuous Compounding) $3,315.51 Solve for Interest Rate 1 Solve for PV -$3,000.00 10.00% 2.00 Solve for FV $3,630.00 FV (Continuous Compounding) $3,664.21 Solve for Interest Rate 1 Solve for PV -$3,000.00 10.00% 5.00 1 Solve for FV $4,831.53 FV (Continuous Compounding) $4,946.16 Solve for Interest Rate • What does the word value or worth mean to you? PV Example • When you were 10 years old your grandfather tells you that upon college graduation (age 22) he will give you $30,000. Inflation is expected to be 3.5%. What is this amount worth as a 10 year old? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) $30,000.00 3.50% 12.00 1 Solve for PV -$19,853.50 Solve for FV FV (Continuous Compounding) $0.00 Solve for Interest Rate PV example • When you retire at age 65, your retirement fund promises to pay you $150,000 the first year of your retirement. You are now 25. If inflation is 4%, what is this worth to you in today’s money? • If the company can earn 12% on its retirement investments, how much must they put away today to get the above payment? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) $150,000.00 Compounding Freq. (m) (P/Y) Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Complete this for your age. Solve for PV 4.00% 40.00 -$31,243.36 Solve for FV FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 $150,000.00 12.00% 40.00 1 Solve for PV -$1,612.02 Solve for FV FV (Continuous Compounding) $0.00 Solve for Interest Rate Annuities • What are some examples of cash flows that are annuity cash flow streams? • What is the difference between an annuity due and an ordinary annuity? Annuity Example • If you started at age 19 to save 2,000 per year at the end of the year and could average 11% per year in earnings, how much would you have at retirement at age 65? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV 11.00% 46.00 $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y -$2,000.00 Effective Interest Rate 11.00% PVA PMT for PVA Interest for PVA (per period) #NUM! FVA $ 2,192,337.60 • Would it make a difference if you started at the beginning of the year instead of the end? How much? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV 11.00% 46.00 $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity n -$2,000.00 Effective Interest Rate 11.00% PVA PMT for PVA Interest for PVA (per period) #NUM! FVA $ 2,433,494.74 Building Wealth http://www.youtube.com/watch?feature=player_de tailpage&v=23zghpS9034 Annuity Example • If you started saving $1,500 per year on a monthly basis for 18 years toward your child’s college education, how much would you have if you invested and earned 8%? 12%? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV 8.00% 18.00 $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 12 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) y -$125.00 Effective Interest Rate 8.30% PVA PMT for PVA Interest for PVA (per period) #NUM! FVA $ 60,010.77 Solve for PV 12.00% 18.00 $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 12 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y -$125.00 Effective Interest Rate 12.68% PVA PMT for PVA Interest for PVA (per period) #NUM! FVA $ 94,732.58 Annuity Example • You have analyzed your retirement plan and have concluded that you need 3,850,000 at age 62. You are currently 25. If you can invest at 12% on average, how much must you invest monthly to achieve the financial goal? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) $3,850,000.00 12.00% 37.00 Compounding Freq. (m) (P/Y) Solve for PV -$46,427.02 Solve for FV FV (Continuous Compounding) $0.00 Solve for Interest Rate 12 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity Use your age. y Effective Interest Rate 12.68% PVA PMT for PVA $ (469.94) Interest for PVA (per period) #NUM! FVA PMT for FVA $ (469.94) Annuity example • Let’s assume that at age 62 you have saved the amount in the above slide. You think you will live to be 88 years old. During retirement, you plan to earn 8% on your investments. How much can you withdraw every month for the remainder of your life? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Solve for PV -$3,850,000.00 8.00% 26.00 Compounding Freq. (m) (P/Y) Solve for FV $30,605,216.75 FV (Continuous Compounding) $30,817,205.32 Solve for Interest Rate 12 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y Effective Interest Rate PVA PMT for PVA 8.30% $ 29,360.03 With respect to money what do the following people worry about? • College student 19 years old. • Young married person 31 years old with a child age 5. • A stable working married adult age 50, children are out of the house and college. • A senior citizen age 64 recently retired. Annuity Example • You win a prestigious sweepstake award. The company offers you $50,000 per year for the next 30 years or a lump sum. Answer the following questions. • If the company can invest at 7%, how much would they need to invest to pay the promised cash flow stream? This is the amount of the lump sum offer? • If you could invest at 9%, how much could you withdraw for the 30 years, if you invest the lump sum? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Solve for PV 7.00% 30.00 Compounding Freq. (m) (P/Y) $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) y $50,000.00 Effective Interest Rate 7.00% PVA $ (620,452.06) PMT for PVA Interest for PVA (per period) #NUM! FVA Solve for PV -$620,452.00 9.00% 30.00 Compounding Freq. (m) (P/Y) Solve for FV $8,231,957.64 FV (Continuous Compounding) $9,232,159.31 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y Effective Interest Rate PVA PMT for PVA 9.00% $ 60,392.53 Annuity Example • At retirement you want to receive $60,000 per year for 25 years and can earn 13%. How much must you invest to achieve this goal? • If you are concerned about the 3.75% inflation rate, how much must you invest? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV 13.00% 25.00 $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) y $60,000.00 Effective Interest Rate PVA Solve for PV 13.00% 25.00 13.00% $ (439,799.10) $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y $60,000.00 3.75% Effective Interest Rate 13.00% PVA $ (439,799.10) PMT for PVA Interest for PVA (per period) #NUM! FVA PMT for FVA Interest for FVA PV of Perpetuity $ (461,538.46) PV of Growing Annuity $ (571,956.47) Annuity • If you wanted to receive $10,000 per year for ever, how much would you need to invest at 12%? • What kind of cash flow stream is this? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV 12.00% Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity $0.00 y $10,000.00 Effective Interest Rate #DIV/0! 12.00% PVA $ PMT for PVA Interest for PVA (per period) #NUM! FVA PMT for FVA Interest for FVA PV of Perpetuity $ (83,333.33) Change the interest rate to 10%. How much must you invest? How does the perpetuity work? Mixed Cash Flows • A company is planning a project that will provide the following cash flow stream. If they can earn 14% on average, what is the value of this project? 1 2 3 4 5 6 7 80,000 70,000 70,000 50,000 50,000 60,000 75,000 • If the company could reinvest the above cash flow stream at 12%, what would they have at the end of the 7th year? Pds Cash Flow 0 1 2 3 4 5 6 7 8 9 10 11 Pds $80,000 $70,000 $70,000 $50,000 $50,000 $60,000 $75,000 Discount Rate 14.00% Number of Periods PV of Future Cash Flows Net Present Value IRR FV of Cash Flows 7 $284,166.61 $284,167 #NUM! $711,061.24 Cash Flow 0 1 2 3 4 5 6 7 8 9 10 11 $80,000 $70,000 $70,000 $50,000 $50,000 $60,000 $75,000 Discount Rate 12.00% Number of Periods PV of Future Cash Flows Net Present Value IRR FV of Cash Flows 7 $301,528.07 $301,528 #NUM! $666,582.49 Compounding frequency • How do you change the compounding frequency in a time value problem? • What is the Effective interest rate for a 12%, monthly compounded investment? Quarterly? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Solve for PV -$1,000.00 12.00% 1.00 Compounding Freq. (m) (P/Y) Solve for FV $1,120.00 FV (Continuous Compounding) $1,127.50 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) y Effective Interest Rate 12.00% Solve for PV -$1,000.00 12.00% 1.00 Compounding Freq. (m) (P/Y) Solve for FV $1,125.51 FV (Continuous Compounding) $1,127.50 Solve for Interest Rate 4 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) y Effective Interest Rate 12.55% Solve for PV -$1,000.00 12.00% 1.00 Compounding Freq. (m) (P/Y) Solve for FV $1,127.47 FV (Continuous Compounding) $1,127.50 Solve for Interest Rate 365 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) y Effective Interest Rate 12.75% Example • A newly married couple is considering buying a new home. The house of their dreams costs $325,000. They have 10% to put down on the home and can borrow at 3.95%. • How much must they borrow? • What are the monthly payments on a 30 year mortgage? • What is the total cost of the loan for the 360 months? • Loan Analysis worksheet Loan Amount $292,500.00 Pmt per Period $1,388.02 Loan Maturity (yrs) 30 Total AMT Paid $499,687.71 PMT per Year (P/Y) m 12 Total Financing Costs $207,187.71 Annual Interest Rate 3.95% What is a loan amortization table? • How much interest is paid out of the first month’s payment? • When the couple pays their 180th payment, what is the balance? • If they paid an extra 50 payment per month, how much interest would they save? • If they paid biweekly payments, how much would they save? (divide monthly payments by 12) Period PMT 0 1 2 3 180 181 360 Interest PMT $1,388.02 $1,388.02 $1,388.02 $1,388.02 $1,388.02 $1,388.02 Loan Amount $962.81 $961.41 $960.01 $622.30 $619.78 $4.55 Principal Reduction $425.21 $426.61 $428.01 $765.72 $768.24 $1,383.47 Remaining Balance $292,500.00 $292,074.79 $291,648.18 $291,220.17 $188,286.76 $187,518.52 $292,500.00 Pmt per Period $1,388.02 Loan Maturity (yrs) 30 Total AMT Paid $499,687.71 PMT per Year (P/Y) m 12 Total Financing Costs $207,187.71 Impact of Accelerated PMTS Years of Loan Total AMT Paid Interest Saved 28.08 $484,525.24 $15,162.47 Annual Interest Rate Extra Periodic PMT Biweekly impact =PMT/12 Loan Amount 3.95% $50.00 $292,500.00 Pmt per Period $1,388.02 Loan Maturity (yrs) 30 Total AMT Paid $499,687.71 PMT per Year (P/Y) m 12 Total Financing Costs $207,187.71 Impact of Accelerated PMTS Years of Loan Total AMT Paid Interest Saved 25.93 $467,856.90 $31,830.81 Annual Interest Rate Extra Periodic PMT Biweekly impact =PMT/12 3.95% $115.67 Example of returns • When I was born my Grandfather purchased a stock for $25. When I was 25 the stock was worth $75. What did I earn on the investment? • What would I earn at age 65 if the stock was then worth $250? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) $75.00 -$25.00 25.00 Solve for PV Solve for FV FV (Continuous Compounding) $25.00 Solve for Interest Rate 4.49% 1 $250.00 -$25.00 65.00 1 Solve for PV Solve for FV FV (Continuous Compounding) $25.00 Solve for Interest Rate 3.61% The Rule of 72. What is this rule? 0 6 12 18 24 30 3% 6% 12% 2000 2000 2000 4000 4000 8000 8000 16000 32000 16000 64000 128000 4000 36 42 48 256000 8000 32000 512000 Time • http://www.youtube.com/watch? feature=player_detailpage&v=_zp GZfFbW4M Retirement example • You currently earn 50,000 per year and have been able to save $15,000 in a retirement account. You will retire in 35 years at age 60 and inflation is 4%. What will your income need to be in year 1 of retirement to maintain your current lifestyle? PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0 • If you live to 90, how much do you need in your pension fund at age 60 with 8% return. PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • If you wanted your retirement income to keep up with an expected inflation rate of 4.5%, how much would you need? PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • Growth of annuity = 4.5% How much must you invest each month in your retirement plans to get your desired growing retirement income if you can earn a 12% return? PV = -15000 FV = 3539071 N=35 I/Y=12% m=12 PMT for FVA = ? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV -$50,000.00 4.00% 35.00 1 Solve for FV $197,304.45 FV (Continuous Compounding) $202,760.00 Solve for Interest Rate Retirement example • You currently earn 50,000 per year and have been able to save $15,000 in a retirement account. You will retire in 35 years at age 60 and inflation is 4%. What will your income need to be in year 1 of retirement to maintain your current lifestyle? PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0 • If you live to 90, how much do you need in your pension fund at age 60 with 8% return. PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • If you wanted your retirement income to keep up with an expected inflation rate of 4.5%, how much would you need? PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • Growth of annuity = 4.5% How much must you invest each month in your retirement plans to get your desired growing retirement income if you can earn a 12% return? PV = -15000 FV = 3539071 N=35 I/Y=12% m=12 PMT for FVA = ? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV 8.00% 30.00 $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y $197,304.00 Effective Interest Rate PVA 8.00% $ (2,221,205.68) Retirement example • You currently earn 50,000 per year and have been able to save $15,000 in a retirement account. You will retire in 35 years at age 60 and inflation is 4%. What will your income need to be in year 1 of retirement to maintain your current lifestyle? PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0 • If you live to 90, how much do you need in your pension fund at age 60 with 8% return. PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • If you wanted your retirement income to keep up with an expected inflation rate of 4.5%, how much would you need? PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • Growth of annuity = 4.5% How much must you invest each month in your retirement plans to get your desired growing retirement income if you can earn a 12% return? PV = -15000 FV = 3539071 N=35 I/Y=12% m=12 PMT for FVA = ? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) Compounding Freq. (m) (P/Y) Solve for PV 8.00% 30.00 $0.00 Solve for FV $0.00 FV (Continuous Compounding) $0.00 Solve for Interest Rate 1 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y $197,304.00 4.50% Effective Interest Rate 8.00% PVA $ (2,221,205.68) PMT for PVA Interest for PVA (per period) #NUM! FVA PMT for FVA Interest for FVA PV of Perpetuity $ (2,466,300.00) PV of Growing Annuity $ (3,539,071.58) Retirement example • You currently earn 50,000 per year and have been able to save $15,000 in a retirement account. You will retire in 35 years at age 60 and inflation is 4%. What will your income need to be in year 1 of retirement to maintain your current lifestyle? PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0 • If you live to 90, how much do you need in your pension fund at age 60 with 8% return. PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • If you wanted your retirement income to keep up with an expected inflation rate of 4.5%, how much would you need? PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304 • Growth of annuity = 4.5% How much must you invest each month in your retirement plans to get your desired growing retirement income if you can earn a 12% return? PV = -15000 FV = 3539071 N=35 I/Y=12% m=12 PMT for FVA = ? Future Value (FV) Present Value (PV) Annual Interest Rate (I/Y) Time in Years (N) $3,629,071.00 -$15,000.00 12.00% 35.00 Compounding Freq. (m) (P/Y) Solve for PV Solve for FV FV (Continuous Compounding) $1,000,294.97 Solve for Interest Rate 12 Solve for Time Is this an Ordinary Annuity (y/n) Payment (PMT) (A) Growth of an Annuity Growth of a Perpetuity y Effective Interest Rate PVA PMT for PVA $ Interest for PVA (per period) $ FVA PMT for FVA $ 12.68% (411.98) 0.01 (411.98) Retirement example • You currently earn $XXXX per year and have been able to save $XXX in a retirement account. You will retire in X years at age 65 and inflation is 3.1%. What will your income need to be in year 1 of retirement to maintain your current lifestyle? • If you live to 90, how much do you need in your pension fund at age 60 with 8% return? • If you wanted your retirement income to keep up with an expected inflation rate of 3.1%, how much would you need? • How much must you invest each month in your retirement plans to get your desired growing retirement income, if you can earn a 12% return?