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Time Value of Money (TVM) - the Intuition

Chapter 4. Time Value of Money (TVM) - the Intuition. A cash flow today is worth more than a cash flow in the future since: Individuals prefer present consumption to future consumption. Monetary inflation will cause tomorrow’s dollars to be worth less than today’s.

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Time Value of Money (TVM) - the Intuition

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  1. Chapter 4 Time Value of Money (TVM) - the Intuition A cash flow today is worth more than a cash flow in the future since: • Individuals prefer present consumption to future consumption. • Monetary inflation will cause tomorrow’s dollars to be worth less than today’s. • Any uncertainty associated with future cash flows reduces the value of the cash flow. Jacoby, Stangeland and Wajeeh, 2000

  2. The Time-Value-of-Money • The Basic Time-Value-of-Money Relationship: FVt+T = PVt% (1 + r)T where • r is the interest rate per period • T is the duration of the investment, stated in the compounding time unit • PVt is the value at period t (beginning of the investment) • FVt+T is the value at period t+T (end of the • investment) Jacoby, Stangeland and Wajeeh, 2000

  3. Future Value and Compounding Compounding: How much will $1 invested today at 8% be worth in two years? (The Time Line) Year 0 1 2 $1.08 $1.1664 $1.08 x 1.08 $1 $1 x 1.08 Or: $1 $1.1664 2 Future Value: FV = $1 x 1.08 = $1.1664 2

  4. TVM in your HP 10B Calculator Housekeeping functions: 1. Set to 8 decimal places: 2. Clear previous TVM data: 3. Set payment at Beginning/End of Period: 3. Set # of times interest is calculated (compounded) per year to 1: Yellow = DISP 8 Yellow C C ALL Yellow MAR BEG/END 1 Yellow PMT P/YR Jacoby, Stangeland and Wajeeh, 2000

  5. FV in your HP 10B Calculator First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above FV example) Yellow C C ALL Key in PV (always -ve) 1 +/- PV Key in interest rate 8 I/YR Key in number of periods 2 N Display should show: 1.1664 Compute FV FV Jacoby, Stangeland and Wajeeh, 2000

  6. An Example - Future Value for a Lump Sum Q. Deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest? A. Multiply the $5000 by the future value interest factor: $5000 % (1 + r)T= $5000 % ___________ = $5000 % 1.9738227 = $9869.1135 At 12%, the simple interest is .12 % $5000 = $per year. After 6 years, this is 6 % $600 = $ ; the compound interest is thus: $ - $3600 = $ Jacoby, Stangeland and Wajeeh, 2000

  7. Present Value and Discounting Discounting: How much is $1 that we will receive in two years worth today (r = 8%)? (The Time Line) Year 0 1 2 $0.9259 $1 $1 / 1.08 $0.8573 $0.9259 / 1.08 Or: $0.8573 $1 2 Present Value: PV = $1 / 1.08 = $0.8573 0

  8. PV in your HP 10B Calculator First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above PV example) Yellow C C ALL 1 FV Key in FV Key in interest rate 8 I/YR Key in number of periods 2 N Display should show: -0.85733882 Compute PV PV Jacoby, Stangeland and Wajeeh, 2000

  9. Example 1 - Present Value of a Lump Sum Q. Suppose you need $20,000 in three years to pay your university tuition. If you can earn 8% annual interest on your money, how much do you need to invest today? A. We know the future value ($20,000), the rate (8%), and the number of periods (3).We are looking for the present amount to be invested (present value). We first define the variables: FV3 = r = percent T= years PV0 = ? Set this up as a TVM equation and solve for the present value: $20,000 = PV0% (1.08)3 Solve for PV: PV0 = $20,000/(1.08)3 = $ $15,876.64 invested today at 8% annually, will grow to $20,000 in three years

  10. Example 2 - Present Value of a Lump Sum Q. Suppose you are currently 21 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate $1 million by the time you reach age 65? A. We first define the variables: FV65 = $ r = percent T= 65 - 21 = years PV21 = ? Set this up as a TVM equation and solve for the present value: $1 million = PV21% (1.10)44 Solve for PV: PV21 = $1 million/(1.10)44 = $ If you invest $15,091.13 today at 10% annually, you will have $1 million by the time you reach age 65

  11. How Long is the Wait? If we deposit $5000 today in an account paying 10%, how long do we have to wait for it to grow to $10,000? Solve for T: FVt+T = PVt% (1 + r)T $10000 = $5000 % (1.10)T (1.10)T = 2 T = ln(2) / ln(1.10) = years Jacoby, Stangeland and Wajeeh, 2000

  12. T in your HP 10B Calculator First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above example) Yellow C C ALL 10,000 FV Key in FV Key in PV +/- PV 5,000 Key in interest rate 10 I/YR Display should show: 7.27254090 Compute T N Jacoby, Stangeland and Wajeeh, 2000

  13. An Example - How Long is the Wait? Q. You have $70,000 to invest. You decided that by the time this investment grows to $700,000 you will retire. Assume that you can earn 14 percent annually. How long do you have to wait for your retirement? A. We first define the variables: FV? = $ r = percent PV0 = $ T= ? Set this up as a TVM equation and solve for T: $700,000 = $70,000 % (1.14)T (1.14)T = 10 Solve for T: T = ln(10)/ln(1.14) = years If you invest $70,000 today at 14% annually, you will reach your goal of $700,000 in 17.57 years Jacoby, Stangeland and Wajeeh, 2000

  14. What Rate Is Enough? • Assume the total cost of a University education will be $50,000 when your child enters college in 18 years. • You have $5,000 to invest today. • What rate of interest must you earn on your investment to cover the cost of your child’s education? • Solve for r : FVt+T = PVt% (1 + r)T $50000 = $5000 % (1 + r)18 (1 + r)18 = 10 (1 + r) = 10(1/18) r = = % per year Jacoby, Stangeland and Wajeeh, 2000

  15. Interest Rate (r) in your HP 10B Calculator First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above example) Yellow C C ALL 50,000 FV Key in FV Key in PV +/- PV 5,000 18 Key in T N Display should show: 13.64636664 Compute r I/YR Jacoby, Stangeland and Wajeeh, 2000

  16. An Example - Finding the Interest Rate (r): Q. In December 1937, the market price of an ABC company common stock was $3.37. According to The Financial Post, the price of an ABC company common stock in December 1999 is $7,500. What is the annually compounded rate of increase in the value of the stock? A. Set this up as a TVM problem. Future value = $ Present value = $ T = 1999 - 1937 = years r = ? FV1999 = PV1937% (1 + r)T so, $7,500 = $3.37 % (1 + r)62 (1 + r)62 = $7,500/3.37 = 2,225.52 Solve for r: r = (2,225.52)1/62 - 1 = = % per year Jacoby, Stangeland and Wajeeh, 2000

  17. Net Present Value (NPV) Example for NPV: You can buy a property today for $3 million, and sell it in 3 years for $3.6 million. The annual interest rate is 8%. Qa. Assuming you can earn no rental income on the property, should you buy the property? Aa. The present value of the cash inflow from the sale is: PV0 = $3,600,000/(1.08)3 = $2,857,796.07 Since this is less than the purchase price of $3 million - don’t buy We say that the Net Present Value (NPV) of this investment is negative: NPV = -C0 + PV0(Future CFs) = + = < 0

  18. Example for NPV (continued): Qb. Suppose you can earn $200,000 annual rental income (paid at the end of each year) on the property, should you buy the property now? Ab. The present value of the cash inflow from the sale is: PV0 = [200,000 /1.08] + [200,000 /1.082] + [3,800,000/1.083] = $3,373,215.47 Since this is more than the purchase price of $3 million - buy We say that the Net Present Value (NPV) of this investment is positive: NPV = -C0 + PV0(Future CFs) = -3,000,000+ 3,373,215.47 = > 0 The general formula for calculating NPV: NPV = -C0 + C1/(1+r) + C2/(1+r)2 + ... + CT/(1+r)T

  19. Simplifications • Perpetuity A stream of constant cash flows that lasts forever • Growing perpetuity A stream of cash flows that grows at a constant rate forever • Annuity A stream of constant cash flows that lasts for a fixed number of periods • Growing annuity A stream of cash flows that grows at a constant rate for a fixed number of periods Jacoby, Stangeland and Wajeeh, 2000

  20. Perpetuity • A Perpetuity is a constant stream of cash flows without end. • Simplification: PVt = Ct+1 / r 0 1 2 3 …forever... |---------|--------|---------|--------- (r = 10%) $100 $100 $100 ...forever… PV0 = $100 / 0.1 = $1000 The British consol bond is an example of a perpetuity. Jacoby, Stangeland and Wajeeh, 2000

  21. Examples - Present Value for a Perpetuity Q1. ABC Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $1,000 per year (starting next year) forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy? A1. The most a rational buyer would pay for the promised cash flows is C/r = $1,000/0.13 = $7,692.31 Q2. ABC Life Insurance Co. tells you that the above policy costs $9,000. At what interest rate would this be a fair deal? A2. Again, the present value of a perpetuity equals C/r. Now solve the following equation: $9,000 = C/r = $1,000/r r = = %

  22. Growing Perpetuity • A growing perpetuity is a stream of cash flows that grows at a constant rate forever. • Simplification: PVt = Ct+1 / (r - g) 0 1 2 3 …forever... |---------|---------|---------|--------- (r = 10%) $100 $102 $104.04 … (g = 2%) PV0 = $100 / (0.10 - 0.02) = $1250 Jacoby, Stangeland and Wajeeh, 2000

  23. An Example - Present Value for a Growing Perpetuity Q. Suppose that ABC Life Insurance Co. modifies the policy, such that it will pay you and your heirs $1,000 next year, and then increase each payment by 1% forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy? A. The most a rational buyer would pay for the promised cash flows is C/(r-g) = $1,000/(0.13-0.01) = $ Note: Everything else being equal, the value of the growing perpetuity is always higher than the value of the simple perpetuity, as long as g>0. Jacoby, Stangeland and Wajeeh, 2000

  24. Annuity • An annuity is a stream of constant cash flows that lasts for a fixed number of periods. • Simplification: PVt = Ct+1(1/r){1 - [1 / (1 + r)T]} FVt+T = Ct+1(1/r){[(1 + r)T]- 1} 0 1 2 3 years |----------|---------|---------| (r = 10%) $100 $100 $100 PV0 = 100 (1/0.1){1 - [ 1/(1.13)]} = $248.69 FV3 = 100 (1/0.1){[1.13 ] - 1} = $331 Jacoby, Stangeland and Wajeeh, 2000

  25. PV and FV of Annuity in your HP 10B Calculator First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above PV example) Yellow C C ALL 100 PMT Key in payment Key in interest rate 10 I/YR Key in number of periods 3 N Display should show: -248.68519910 Compute PV PV Display should show: -331.00000000 Compute FV * 0 PV FV * Note: you can calculate FV directly, by following first 3 steps, and replacing PV with FV in the fourth step.

  26. Present Value of an Annuity - Example 1 Q. A local bank advertises the following: “Pay us $100 at the end of every year for the next 10 years. We will pay you (or your beneficiaries) $100, starting at the eleventh year forever.” Is this a good deal if the effective annual interest rate is 8%? A. We need to compare the PV of what you pay with the present value of what you get: - The present value of your annuity payments: PV0 = 100 (1/0.08){1 - [ 1/(1.0810)]} = $ - The present value of the bank’s perpetuity payments at the end of the tenth year (beginning of the eleventh year): PV10 = C11/r = (100/0.08) = $ The present value of the bank’s perpetuity payments today: PV0 = PV10 /(1+r)10 = (100/0.08)/(1.08)10 = = $ Jacoby, Stangeland and Wajeeh, 2000

  27. Present Value of an Annuity - Example 2 Q. You take a $20,000 five-year loan from the bank, carrying a 0.6% monthly interest rate. Assuming that you pay the loan in equal monthly payments, what is your monthly payment on this loan? A. Since payments are made monthly,we have to count our time units in months. We have: T = monthly time periods in five years, with a monthly interest rate of: r = 0.6%, and PV0 = $ With the above data we have: 20,000 = C (1/0.006){1 - [ 1/(1.00660)]} Solving for C, we get a monthly payment of: $397.91. Note: you can easily solve for C in your calculator, by keying: 3) Key in # of payments 1) Key in the PV 20,000 PV 60 N 2) Key in interest rate PMT 4) Compute PMT I/YR 0.6 Display should show: - 397.91389639

  28. Annuity Due • An annuity due is a stream of constant cash flows that is paid at the beginningof each period and lasts for a fixed number of periods (T). • Simplification: PVt = Ct + Ct+1(1/r){1 - [1 / (1 + r)T-1]} FVt+T = Ct(1/r){(1 + r)T+1- (1+r)} 0 1 2 3 years (T = 3) |----------|---------|---------| (r = 10%) $100 $100 $100 PV0 = 100 + 100 (1/0.1){1 - [ 1/(1.12)]} = $273.55 FV3 = 100 (1/0.1){[1.14 ] - 1.1} = $364.10

  29. PV and FV of Annuity Due in your HP 10B Calculator First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above example) Yellow C C ALL Set payment to beginning of period Yellow MAR BEG/END When finished - don’t forget to set your payment to End of period 100 PMT Key in payment Key in interest rate 10 I/YR Key in number of PAYMENTS 3 N Display should show: -273.55371901 Compute PV PV Display should show: -364.10000000 Compute FV 0 PV FV

  30. Growing Annuity • A Growing Annuity is a stream of cash flows that grows at a constant rate over a fixed number of periods. • Simplification for PV: PVt = Ct+1[1/(r-g)]{1 - [(1+g)/(1+r)]T} 0 1 2 3 |---------|----------|---------| (r = 10%) $100 $102 $104.04 (g = 2%) PV0 = 100 [1/(0.10-0.02)]{1 - (1.02/1.10)3} = $253.37 Jacoby, Stangeland and Wajeeh, 2000

  31. An Example - Present Value of a Growing Annuity Q. Suppose that the bank rewords its advertisement to the following: “Pay us $100 next year, and another 9 annual payments such that each payment is 4% lower than the previous payment. We will pay you (or your beneficiaries) $100, starting at the eleventh year forever.” Is this a good deal if if the effective annual interest rate is 8%? A. Again, we need to compare the PV of what you pay with the present value of what you get: - The present value of your annuity payments (note: g = %): PV0 = C1[1/(r-g)]{1- [(1+g)/(1+r)]T} = 100[1/(0.08-(-0.04))]{1-[(1+(-0.04))/(1.08)]10} = [100/0.12]{1-[0.96/1.08]10} = $576.71 - The present value of the bank’s perpetuity payments today: $578.99 (see example above) Jacoby, Stangeland and Wajeeh, 2000

  32. Growing Annuity - Special Cases • A special case - when r < g, we still use the above formula • Example: 0 1 2 3 |------------------|-------------------|------------------| (r = 4%) $100 $100%1.07 $100%1.072 (g = 7%) PV0 = 100 [1/(0.04-0.07)]{1 - (1.07/1.04)3} = $296.86 (-) (-) (+) Jacoby, Stangeland and Wajeeh, 2000

  33. Growing Annuity - Special Cases • A special case - when r = g, we cannot use the above formula • Example-1: 0 1 2 3 |------------------|-------------------|------------------| (r = 5%) $100 $100%1.05 $100%1.052 (g = 5%) • In general, when cashflow starts at time t+1, use:

  34. Growing Annuity - Special Cases • Example-2: 0 1 2 3 |------------------|-------------------|------------------| (r = 5%) $100 $100%1.05 $100%1.052 (g = 5%) • In general, when cashflow starts at time t, use: Jacoby, Stangeland and Wajeeh, 2000

  35. Growing Annuity • Simplification for FV: FVt+T = Ct+1[1/(r-g)]{(1+r)T - (1+g)T} 0 1 2 3 |---------|----------|---------| (r = 10%) $100 $102 $104.04 (g = 2%) FV3 = 100 [1/(0.10-0.02)]{1.103 - 1.023} = $337.24 Jacoby, Stangeland and Wajeeh, 2000

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