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The Time Value of Money Chapter 8 October 3, 2012

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Learning Objectives The “time value of money” and its importance to you and business decisions The future value and present value of a single amount. The future value and present value of an annuity. The present value of a series of uneven cash flows.

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The Time Value of Money Money grows in amount over time as it earns from investments. However, money that is to be received at some time in the future is worth less than the same dollar amount to be received today. Why? Similarly, a debt of a given amount to be paid in the future is less burdensome than that debt to be paid now. Why?

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**Some Examples Bought Oakland house for $29,500 in 1969**

$23,600 mortgage, $175 mo. pymt I bought my house in Los Altos in 1979 for $135,000 $40, yr mortgage, $300 mo In 2009, would still paying $300 mo! House sold for over $1.25 million in 2006 Current owner paying $5,500 per month I now own $935,000 home, no mortgage! Time value of money

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**Indians – Manhattan Island**

In 1624, Indians got $24 for Manhattan island People think they were “taken” If invested at 8%, compounded annually, today they would have $223,166,200,000,000 (trillion) If compounded semiannually, $396 trillion If compounded quarterly, $534 trillion You could buy Manhattan Island today for around $500 billion They could pay off the nat’l debt/buy back US! Time value of money!

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**16 year old saves for retirement!**

Earns $2,000 per year for 6 years/stops Reinvests at 10% per year At 21 years old, she is worth $15,431 At age 65, with no add’l investment, if she just lets it ride, she will be worth $1,022,535 If she waits just one more year to get started, she would be worth only $929,578 She loses $92,957! (final years earnings) So start saving now! You’ll never miss it.

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**The Future Value of a Single Amount**

Suppose that you have $100 today and plan to put it in a bank account that earns 8% (k) per year. How much will you have after 1 year? After one year: $100 + (.08 x $100) = $100 + $8 = $108 Or If k = 8%, then 1 + k = or Then, $100 x (1.08)1 = $108

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**The Future Value of a Single Amount**

Suppose that you have $100 today and plan to put it in a bank account that earns 8% per year. How much will you have after 1 year? 5? 15? After one year: $100 x (1.08)1 = $100 x = $108 After five years: $100 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 = $146.93 $100 x (1.08)5 = $100 x = $146.93 After fifteen years: $100 x (1.08)15 = $100 x * = $317.22 Equation: FV = PV (1 + k)n *Table I, p. A-1 Appendix

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**The Future Value of a Single Amount**

Calculator solution: N = 15 I/Y = 8 PV = -$100 PMT = 0 Compute (CPT) FV = $317.22

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**Present Value of a Single Amount**

Value today of an amount to be received or paid in the future. PV = FVn x 1 (1 + k)n *Table II, p. A-2, Appendix Example: Expect to receive $100 in one year. If can invest at 10%, what is it worth today? PV = 100 (1.10)1 = $100 $100 x .9091* = $90.91 $

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**Present Value of a Single Amount**

Value today of an amount to be received or paid in the future. PV = FVn x 1 (1 + k)n Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today? = PV = 100 (1+.10)8 $100 $100 x .4665* = $46.65 *Table II, p. A-2, Appendix

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**Financial Calculator Solution - PV**

Previous Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today? PV = 100 (1+.10)8 = 46.65 Using Formula: N I/YR PV PMT FV Calculator Enter: N = 8 I/YR = 10 PMT = 0 FV = 100 CPT PV = ? ? 100

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Annuities An annuity is a series of equal cash flows spaced evenly over time. For example, you pay your landlord an annuity since your rent is the same amount, paid on the same day of the month for the entire year. $500 Jan Feb Mar Dec

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**Future Value of an Annuity**

$0 $100 You deposit $100 each year (end of year) into a savings account (saving up for an IPad). How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

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**Future Value of an Annuity**

$0 $100 $100(1.08)2 $100(1.08)1 $100(1.08)0 $100.00 $108.00 $116.64 $324.64 You deposit $100 each year (end of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

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**Future Value of an Annuity**

$0 $100 $100 $100 $100(1.08)2 $100(1.08)1 $108.00 $116.64 $324.64 $100.00 $100(1.08)0 How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? = 100 (1+.08)3 - 1 .08 ( ) FVA = PMTx( ) (1+k) - 1 k n = 100(3.2464*) = $324.64 *Table III, p. A-3, Appendix

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**Future Value of an Annuity Calculator Solution**

$0 $100 N I/YR PV PMT FV Enter: N = 3 I/YR = 8 PV = 0 PMT = -100 CPT FV = ? 324.64 ?

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**Present Value of an Annuity**

How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $0 $100

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**Present Value of an Annuity**

How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100/(1.08)1 $100 / (1.08)2 $100 / (1.08)3 $0 $100 $92.60 $85.73 $79.38 $257.71

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**Present Value of an Annuity**

How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100 / (1.08)2 $92.60 $85.73 $79.38 $100/(1.08)1 $100 / (1.08)3 $257.71 $0 $100 .08 = 100 1 - 1 (1.08)3 ( ) PVA = PMTx( ) 1 (1+k)n 1 - k = 100(2.5771*) = $257.71 *Table IV, p. A-4, Appendix

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**Present Value of an Annuity Calculator Solution**

$0 $100 PV=? N I/YR PV PMT FV Enter: N = 3 I/YR = 8 PMT = 100 FV = 0 CPT PV = ? ?

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Annuity Due An annuity is a series of equal cash payments spaced evenly over time. Ordinary Annuity: The cash payments occur at the END of each time period. Annuity Due: The cash payments occur at the BEGINNING of each time period. Lotto is an example of an annuity due

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**Future Value of an Annuity Due**

$100 FVA=? You deposit $100 each year (beginning of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

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**Future Value of an Annuity Due**

$100 $100(1.08)3 $100(1.08)2 $100(1.08)1 $108 $116.64 $125.97 $350.61 You deposit $100 each year (beginning of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

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**Future Value of an Annuity Due**

$100(1.08)2 $100(1.08)1 $100(1.08)3 $108 $116.64 $125.97 $350.61 $100 How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? (1+.08)3 - 1 .08 = 100 (1.08) ( ) FVA = PMTx( ) (1+k) (1+k)n - 1 k =100(3.2464)(1.08)=$350.61

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**Calculator solution to annuity due**

Same as regular annuity, except Multiply your answer by (1 + k) to account for the additional year of compounding or discounting Future value of an annuity due: n = 3, i/y = 8%, pmt = -100, PV = 0 CPT FV = (1.08) =

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**Present Value of an Annuity Due**

How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100 PV=?

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**Present Value of an Annuity Due**

$100 How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100/(1.08)0 $100/(1.08)1 $100 / (1.08)2 $100.00 $92.60 $85.73 $278.33

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**Present Value of an Annuity Due**

How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100 $100/(1.08)0 $100/(1.08)1 $100 / (1.08)2 $100.00 $92.60 $85.73 $278.33 .08 = 100 1 - 1 (1.08)3 (1.08) ( ) PVA = PMTx( ) 1 (1+k)n 1 - k (1+k) = 100(2.5771)(1.08) =

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**Calculator solution to annuity due**

Same as regular annuity, except Multiply your answer by (1 + k) to account for the additional year of compounding or discounting Present value of an annuity due: N = 3, i/y = 8%, PMT = 100, FV = 0, CPT PV = (1.08) =

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Amortized Loans A loan that is paid off in equal amounts that include principal as well as interest. Solving for loan payments (PMT). Note: The amount of the loan is the present value (PV)

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**Amortized Loans $5,000 $? ENTER: N = 5 I/YR = 6 PV = 5,000 FV = 0**

You borrow $5,000 from your parents to purchase a used car. You agree to make payments at the end of each year for the next 5 years. If the interest rate on this loan is 6%, how much is your annual payment? $5,000 $? N I/YR PV PMT FV ENTER: N = 5 I/YR = 6 PV = 5,000 FV = 0 CPT PMT = ? –1,186.98 ,000 ?

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**Compounding more than once per Year**

If m = number of compounds, then N = n x m and K = k / m Annual i.e. N = 4 K = 12% Semi-annual N = 4 x 2 = 8 K = 12% / 2 = 6% Quarterly N = 4 x 4 = 16 K = 12% / 4 = 3% Monthly N = 4 x 12 = 48 K = 12% / 12 = 1%

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**Amortized Loans ( ) 1 1 - (1.0075)48 $20,000 = PMT .0075 1 -**

You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment? = PMT .0075 1 - 1 (1.0075)48 $20,000 ( ) PVA = PMTx( ) 1 (1+k)n 1 - k $20,000 = PMT( ) PMT = Note: Tables no longer work

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**Amortized Loans ENTER: N = 48 I/YR = .75 PV = 20,000 FV = 0**

You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment? ENTER: N = 48 I/YR = .75 PV = 20,000 FV = 0 CPT PMT = ? N I/YR PV PMT FV – Note: N = 4 * 12 = 48 I/YR = 9/12 = .75 ,000 ? 0

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Perpetuities A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity. PMT k PVP =

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**Perpetuities If k = 8%: PVP = $5/.08 = $62.50**

A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity (i.e., retirement payments) Example: A share of preferred stock pays a constant dividend of $5 per year. What is the present value if k =8%? PMT k PVP = If k = 8%: PVP = $5/.08 = $62.50 Proof: $62.50 x .08 = $5.00

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**Solving for k FV = PV(1+ k)n**

Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment? $230 $200 230 = 200(1+ k)2 1.15 = (1+ k)2 FV = PV(1+ k)n 1.15 = (1+ k)2 = 1+ k k = = 7.24%

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**Solving for k - Calculator Solution**

Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment? Enter known values: N = 2 I/YR = ? PV = -200 PMT = 0 FV = 230 Solve for: I/YR = ? N I/YR PV PMT FV 7.24 2 -200 230 ?

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Solving for N Example: A $200 investment has grown to $230. If the ANNUAL return on this investment is 7.24%, how long would it take? Enter known values: N = ? I/YR = 7.24 PV = -200 PMT = 0 FV = 230 N = , or 2 years N I/YR PV PMT FV ?

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**Compounding more than Once per Year**

$500 invested at 9% annual interest for 2 years. Compute FV. Compounding Frequency $500(1.09)2 = $ Annual $500(1.045)4 = $ Semi-annual $500(1.0225)8 = $ Quarterly $500(1.0075)24 = $ Monthly $500( )730 = $ Daily

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