1 the time value of money october 3, 2012. 2 learning objectives the “time value of money” and...

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The Time Valueof Money

October 3, 2012

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Learning ObjectivesLearning Objectives

The “time value of money” and its The “time value of money” and its importance to you and business importance to you and business decisionsdecisions

The future value and present The future value and present value of a single amount.value of a single amount.

The future value and present The future value and present value of an annuity.value of an annuity.

The present value of a series of The present value of a series of uneven cash flows.uneven cash flows.

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The Time Value of The Time Value of MoneyMoney Money grows in amount over time Money grows in amount over time

as it earns from investments.as it earns from investments. However, money that is to be However, money that is to be

received at some time in the received at some time in the future is worth less than the same future is worth less than the same dollar amount to be received dollar amount to be received today. Why?today. Why?

Similarly, a debt of a given Similarly, a debt of a given amount to be paid in the future is amount to be paid in the future is less burdensome than that debt less burdensome than that debt to be paid now. Why?to be paid now. Why?

Some Examples

Bought Oakland house for $29,500 in 1969$23,600 mortgage, $175 mo. pymtI bought my house in Los Altos in 1979 for $135,000$40,000 30 yr mortgage, $300 moIn 2009, would still paying $300 mo!House sold for over $1.25 million in 2006Current owner paying $5,500 per monthI now own $935,000 home, no mortgage!Time value of money

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!

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 The Future Value of a Single AmountSingle Amount

Suppose that you have $100 today Suppose that you have $100 today and plan to put it in a bank account and plan to put it in a bank account that earns 8% (k) per year. that earns 8% (k) per year.

How much will you have after 1 year? How much will you have after 1 year? After one year:After one year:

$100 + (.08 x $100) = $100 + $8 = $100 + (.08 x $100) = $100 + $8 = $108$108Or Or

If k = 8%, then 1 + k = 1 + .08 or 1.08 ThenThen, , $100 x (1.08)$100 x (1.08)11 = = $108$108

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After one year:After one year:$100 x (1.08)$100 x (1.08)1 1 = $100 x 1.08 = $108= $100 x 1.08 = $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 1.4693 = $146.93$146.93

After fifteen years:

$100 x (1.08)15 = $100 x 3.1722* = $317.22= $317.22

Equation:

The Future Value of a The Future Value of a Single AmountSingle Amount

Suppose that you have $100 today and plan to Suppose that you have $100 today and plan to put it in a bank account that earns 8% per year. put it in a bank account that earns 8% per year.

How much will you have after 1 year? 5? 15?How much will you have after 1 year? 5? 15?

FV = PV (1 + k)n *Table I, p. A-1Appendix

The Future Value of a Single Amount

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

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Present Value of a Single Present Value of a Single AmountAmount

Value today of an amount to be Value today of an amount to be received or paid in the future.received or paid in the future.

Example:Example: Expect to receive $100 in one year. If can invest at 10%, what is it worth today?

0 1 2

$100PV = 100 (1.10)1

=

PV = FVn x1

(1 + k)n

*Table II, p. A-2, Appendix

$ $100 x .9091* = $90.91

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Present Value of a Single Present Value of a Single AmountAmount

Example:Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today?

=PV = 100

(1+.10)8

0 1 2 3 4 5 6 7 8

$100

PV = FVn x1

(1 + k)n

Value today of an amount to be Value today of an amount to be received or paid in the future.received or paid in the future.

$100 x .4665* = $46.65 *Table II, p. A-2, Appendix

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N I/YR PV PMT FV

Financial Calculator Financial Calculator Solution - PVSolution - PV

- 46.65- 46.65

PV = 100 (1+.10)8 = 46.65Using Formula:

8 10 ?

100

Previous Example:Previous Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today?

Calculator Enter:N = 8I/YR = 10PMT = 0FV = 100CPT PV = ?

0

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AnnuitiesAnnuities

An annuity is a series of An annuity is a series of equalequal cash flows spaced evenly over cash flows spaced evenly over time.time.

For For exampleexample, you pay your , you pay your landlord an annuity since your landlord an annuity since your rentrent is the same amount, paid on is the same amount, paid on the same day of the month for the same day of the month for the entire year.the entire year.

$500 $500 $500 $500 $500

Jan Feb Mar Dec

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Future Value of an Future Value of an AnnuityAnnuity

0 1 2 3

$0 $100 $100 $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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$100(1.08)2 $100(1.08)1

$108.00$116.64$324.64$324.64

You deposit $100 each year (end of year) into a savings account.


0 1 2 3

$0 $100 $100 $100

$100.00

$100(1.08)0

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= 100(3.2464*) = $324.64FVA = PMTx( ) (1+k) - 1

k

n = 100

(1+.08)3 - 1 .08( )

0 1 2 3

$0 $100 $100 $100$100(1.08)2 $100(1.08)1

$108.00$116.64$324.64$324.64

$100.00

$100(1.08)0

*Table III, p. A-3, Appendix

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N I/YR PV PMT FV


Calculator SolutionCalculator Solution

3 8 0 -100 ?

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

0 1 2 3

$0 $100 $100 $100

324.64

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Present Value of an Present Value of an AnnuityAnnuity

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

0 1 2 3

$0 $100 $100 $100

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$100 / (1.08)2

$92.60$85.73$79.38

$100/(1.08)1 $100 / (1.08)3

$257.71$257.71

How much would the following cash flows be worth

to you today if you could earn 8% on your deposits?

0 1 2 3

$0 $100 $100 $100

2020


= 100(2.5771*) = $257.71 PVA = PMTx( )

1(1+k)n1 -

k

$100 / (1.08)2

$92.60$85.73$79.38

$100/(1.08)1 $100 / (1.08)3

$257.71$257.71

0 1 2 3

$0 $100 $100 $100

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

.08= 100

1 - 1 (1.08)3( )

*Table IV, p. A-4, Appendix

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N I/YR PV PMT FV

3 8 ? 100 0


Calculator SolutionCalculator Solution

PV=?

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

0 1 2 3

$0 $100 $100 $100

-257.71

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Annuity DueAnnuity Due An annuity is a series of equal cash An annuity is a series of equal cash

payments spaced evenly over time. payments spaced evenly over time.

Ordinary Annuity:Ordinary Annuity: The cash payments The cash payments occur at the occur at the ENDEND of each time period. of each time period.

Annuity Due:Annuity Due: The cash payments occur The cash payments occur at the at the BEGINNINGBEGINNING of each time period. of each time period.

Lotto is an example of an annuity dueLotto is an example of an annuity due

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Future Value of an Annuity Future Value of an Annuity DueDue

You deposit $100 each year (beginning of year) into a savings account.


0 1 2 3

$100 $100 $100 FVA=?

2424


$100(1.08)2 $100(1.08)1$100(1.08)3

$108

$116.64$125.97

$350.61$350.61

0 1 2 3

$100 $100 $100

You deposit $100 each year (beginning of year) into a savings account.


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$100(1.08)2 $100(1.08)1$100(1.08)3

$108

$116.64$125.97

$350.61$350.61

0 1 2 3

$100 $100 $100


=100(3.2464)(1.08)=$350.61FVA = PMTx( ) (1+k) (1+k)n - 1

k

(1+.08)3 - 1 .08

= 100 (1.08)( )

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 = 0CPT FV = 324.64 (1.08) = 350.61

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Present Value of an Annuity Present Value of an Annuity DueDue

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

PV=?

0 1 2 3

$100 $100 $100

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$100 / (1.08)2

$92.60$85.73

$100/(1.08)1

$278.33$278.33

0 1 2 3

$100 $100 $100

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

$100.00

$100/(1.08)0

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= 100(2.5771)(1.08) = 278.33 PVA = PMTx( )

1(1+k)n1 -

k(1+k)

$100 / (1.08)2

$92.60$85.73

$100/(1.08)1

$278.33$278.33

0 1 2 3

$100 $100 $100

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

.08 = 100

1 - 1 (1.08)3

(1.08)( )

$100.00

$100/(1.08)0

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 = -257.71 (1.08) = -278.33

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Amortized LoansAmortized Loans

A loan that is paid off in equal A loan that is paid off in equal amounts that include principal amounts that include principal as well as interest.as well as interest.

Solving for loan payments Solving for loan payments (PMT).(PMT).

Note: The amount of the loan is Note: The amount of the loan is the present value (PV)the present value (PV)

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N I/YR PV PMT FV

0 1 2 3 4 5

$5,000 $? $? $? $? $?

––1,186.981,186.98

5 6 5,000 ? 0

ENTER:N = 5I/YR = 6PV = 5,000FV = 0CPT PMT = ?

Amortized LoansAmortized Loans You borrow $5,000 from your parents to purchase a You borrow $5,000 from your parents to purchase a

used car. You agree to make payments at the end of used car. You agree to make payments at the end of each year for the next 5 years. If the interest rate on each year for the next 5 years. If the interest rate on this loan is 6%, how much is your annual payment?this loan is 6%, how much is your annual payment?

Compounding more than once per Year

If m = number of compounds, thenN = 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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$20,000 = PMT(40.184782)PVA = PMTx( )

1(1+k)n1 -

kPMT = 497.70

You borrow $20,000 from the bank to purchase a You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the used car. You agree to make payments at the end of each month for the next 4 years. If the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much annual interest rate on this loan is 9%, how much is your monthly payment?is your monthly payment?

= PMT .0075

1 - 1 (1.0075)48

$20,000 ( )

Note: Tables no longer work

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ENTER:N = 48I/YR = .75PV = 20,000FV = 0CPT PMT = ?


N I/YR PV PMT FV

– – 497.70497.70

48 .75 20,000 ? 0

You borrow $20,000 from the bank to purchase a You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the used car. You agree to make payments at the end of each month for the next 4 years. If the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much annual interest rate on this loan is 9%, how much is your monthly payment?is your monthly payment?

Note:

N = 4 * 12 = 48

I/YR = 9/12 = .75

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

PerpetuitiesPerpetuities

PMT k

PVP =

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PerpetuitiesPerpetuities

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

Proof: $62.50 x .08 = $5.00Proof: $62.50 x .08 = $5.00

PMT k

PVP =

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:Example: A share of preferred stock pays a constant dividend of $5 per year. What is the present value if k =8%?

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Solving for kSolving for kExample:Example: A $200 investment has grown to $230 over

two years. What is the ANNUAL return on this investment?

0 1 2

$230$200

FV = PV(1+ k)n

230 = 200(1+ k)2

1.15 = (1+ k)2

1.0724 = 1+ k

1.15 = (1+ k)2

k = .0724 = 7.24%

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N I/YR PV PMT FV

Enter known values: N = 2I/YR = ?PV = -200PMT = 0FV = 230

Solve for: I/YR = ?

2 -200 230?

Solving for k - Solving for k - Calculator SolutionCalculator Solution

Example:Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment?

7.247.24

0

Solving for N

Example: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 = 1.9995, or 2 years

N I/YR PV PMT FV

? 7.24 -200 0 230

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Compounding more than Compounding more than Once per YearOnce per Year

$500 invested at 9% annual interest for $500 invested at 9% annual interest for 2 years. Compute FV.2 years. Compute FV.

$500(1.09)2 = $594.05 Annual

$500(1.045)4 = $596.26 Semi-annual

$500(1.0225)8 = $597.42 Quarterly

$500(1.0075)24 = $598.21 Monthly

$500(1.000246575)730 = $598.60 Daily

Compounding Compounding FrequencyFrequency

1 the time value of money october 3, 2012. 2 learning objectives the “time value of money” and...

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