Time value money_ppt

The Time Valueof Money

Learning ObjectivesLearning Objectives

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

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.

The Time Value of MoneyThe Time Value of Money

Money grows over time when it earns interest.

Therefore, money that is to be received at some time in the future is worth less than the same dollar amount to be received today.

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

Link to FinanCenter

http://www.financenter.com/

The Future Value of a Single AmountThe 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 years?

15 years? After one year:

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

OR:

$100 (1.08)1 = $108

The Future Value of a Single AmountThe 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 (1.08)1 = $108

FV = PV (1 + k)n

After five years:

$100 (1.08)5 = $146.93$146.93

After fifteen years:

$100 (1.08)15 = $317.22= $317.22

Equation:

$1000

900

800

700

600

400

500

300

200

0

Year0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

k = 8%

k = 4%

k = 0%

The Future Value of a Single AmountThe Future Value of a Single AmountGraphical PresentationGraphical PresentationDifferent Interest RatesDifferent Interest Rates

Present Value of a Single AmountPresent Value of a Single AmountValue today of an amount to be 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

= 90.90

PV = FVn x1

(1 + k)n

Present Value of a Single AmountPresent Value of a Single Amount

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

= 46.65PV = 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 received or paid in the future.

$100

90

80

70

60

40

50

30

20

0

Year0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

k = 10%

Present Value of a Single AmountPresent Value of a Single AmountGraphical PresentationGraphical Presentation

k = 5%

k = 0%

N I/YR PV PMT FV

Financial Calculator Solution - PVFinancial Calculator Solution - 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 = 10FV = 100CPT PV = ?

Financial Calculator Solution - FVFinancial Calculator Solution - FV

Previous Example:Previous Example: You invest $200 at 10%. How much is it worth after 5 years?

Using Formula: FV = $200 (1.10)5 = $322.10

N I/YR PV PMT FV

Financial Calculator Solution - FVFinancial Calculator Solution - FV

322.10322.10

5 10 -200 ?

Calculator Enter:N = 5I/YR = 10PV = -200CPT FV = ?

Using Formula: FV = $200 (1.10)5 = $322.10

Previous Example:Previous Example: You invest $200 at 10%. How much is it worth after 5 years?

AnnuitiesAnnuities

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 $500 $500 $500 $500

Jan Feb Mar Dec

Future Value of an AnnuityFuture Value of an Annuity

0 1 2 3

$0 $100 $100 $100

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?


$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



= 100(3.2464) = 324.64FVA = PMT( ) (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

N I/YR PV PMT FV

Future Value of an AnnuityFuture Value of an AnnuityCalculator SolutionCalculator Solution

3 8 -100 ?

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

0 1 2 3

$0 $100 $100 $100

324.64

Present Value of an AnnuityPresent 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 1 2 3

$0 $100 $100 $100

Present Value of an AnnuityPresent Value of an Annuity

$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

Present Value of an AnnuityPresent Value of an Annuity

= 100(2.5771) = 257.71 PVA = PMT( )

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


.08= 100

1 - 1 (1.08)3( )

N I/YR PV PMT FV

3 8 ? 100

Present Value of an AnnuityPresent Value of an AnnuityCalculator SolutionCalculator Solution

PV=?

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

0 1 2 3

$0 $100 $100 $100

-257.71

AnnuitiesAnnuities

An annuity is a series of equal cash payments spaced evenly over time.

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

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

Future Value of an Annuity DueFuture Value of an Annuity Due

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


0 1 2 3

$100 $100 $100 FVA=?


$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.



$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 = PMT( ) (1+k) (1+k)n - 1

k

(1+.08)3 - 1 .08

= 100 (1.08)( )

Present Value of an Annuity DuePresent 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?

PV=?

0 1 2 3

$100 $100 $100

Present Value of an Annuity DuePresent Value of an Annuity Due

$100 / (1.08)2

$92.60$85.73

$100/(1.08)1

$278.33$278.33

0 1 2 3

$100 $100 $100


$100.00

$100/(1.08)0

Present Value of an Annuity DuePresent Value of an Annuity Due

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

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


.08 = 100

1 - 1 (1.08)3

(1.08)( )

$100.00

$100/(1.08)0

Amortized LoansAmortized Loans

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

Solving for loan payments.

N I/YR PV PMT FV

0 1 2 3 4 5

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

––1,186.981,186.98

5 6 5,000 ?

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

Amortized LoansAmortized Loans 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?


$20,000 = PMT(40.184782)PVA = PMT( )

1(1+k)n1 -

kPMT = 497.70

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 ( )

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


N I/YR PV PMT FV

– – 497.70497.70

48 .75 20,000 ?

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?

Note:

N = 4 * 12 = 48

I/YR = 9/12 = .75

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

PerpetuitiesPerpetuities

PMT k

PVP =

Example: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

PMT k

PVP =


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

Solving for kSolving for k

Example: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%

N I/YR PV PMT FV

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

Solve for: PMT. = ?

2 -200 230?

Solving for k - Calculator SolutionSolving for k - Calculator Solution

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

7.247.24

Compounding more than Once per YearCompounding more than Once per Year

$500 invested at 9% annual interest for 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 Frequency Frequency

Continuous CompoundingContinuous Compounding

Compounding frequency is infinitely large.Compounding period is infinitely small.

FV = PV x ekn

Example:Example: $500 invested at 9% annual interest for 2 years with continuous compounding.

FV = $500 x e.09 x 2 = $598.61

Date post:	20-Jun-2015
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