FINANCIAL MANAGEMENT PPT BY FINMAN Time value of money official

Principles of Managerial Finance

Time Value of Money

MBA 656 FINANCIAL MANAGEMENT CYCLE 1BY: MARY ROSE HABAGAT GELITA COLON

WHY THIS TOPIC MATTERS TO YOU

IN PROFESSIONAL LIFE:

ACCOUNTING:

You need to understand time-value-of-money calculations to account for certain transactions such as loan amortization, lease payments, and bond interest rates.

INFORMATION SYSTEM:You need to understand time-value-of-money calculations to design systems that accurately measure and value the firm’s cash flows.

MANAGEMENT:You need to understand time-value-of-money calculations so that you can manage cash receipts and disbursements in a way that will enable the firm to receive the greatest value from its cash flows.

MARKETING

You need to understand time value of money because funding for new programs and products must be justified financially using time-value-of-money techniques.

OPERATIONSYou need to understand time value of money because the value of investments in new equipment, in new processes, and in inventory will be affected by the time value of money.

IN YOUR PERSONAL LIFE

Time value techniques are widely used in personal financial planning. You can use them to calculate the value of savings at given future dates and to estimate the amount you need now to accumulate a given amount at a future date.

You also can apply them to value lump-sum amounts or streams of periodic cash flows and to the interest rate or amount of time needed to achieve a given financial goal.

Learning Objectives• Discuss the role of time value in finance and the use

of computational aids used to simplify its application.

• Understand the concept of future value, its calculation

for a single amount, and the effects of compounding

interest more frequently than annually.

• Find the future value of an ordinary annuity and an

annuity due and compare these two types of annuities.

• Understand the concept of present value, its

calculation for a single amount, and its relationship to

future value.

Learning Objectives

• Calculate the present value of a mixed stream of cash

flows, an annuity, a mixed stream with an embedded

annuity, and a perpetuity.

• Describe the procedures involved in:

– determining deposits to accumulate a future sum,

– loan amortization, and

– finding interest or growth rates

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that

are spread out over time.

• Time value of money allows comparison of cash flows

from different periods.

Question?

Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in one year, or one that would return

$500,000 after two years?

Answer!

It depends on the interest rate!

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that

are spread out over time.

• Time value of money allows comparison of cash flows

from different periods.

Present Value and Future Value

PRESENT VALUE

• Is the cash on hand today

• It is the amount you need today in to reach a future value

• PRESENT VALUE TECHNIQUE uses discounting to find its present valueof each cash flow at time zero and then sums these values to find the investment’s value today

FUTURE VALUE

• Is cash you will receive at a given future date

• It is the amount you will receive in the future from your cash on hand

• FUTURE VALUE TECHNIQUE uses compounding to find future value of each cash flow at the end of the investment’s life and then sums these values to find the investment’s future value

ILLUSTRATION

Simple Interest

• Year 1: 5% of $100 = $5 + $100 = $105

• Year 2: 5% of $100 = $5 + $105 = $110

• Year 3: 5% of $100 = $5 + $110 = $115

• Year 4: 5% of $100 = $5 + $115 = $120

• Year 5: 5% of $100 = $5 + $120 = $125

With simple interest, you don’t earn interest on interest.

Compound Interest

• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00

• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25

• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76

• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55

• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

With compound interest, a depositor earns interest on interest!

Computational Aids

• Use the Equations

• Use the Financial Tables

• Use Financial Calculators

• Use Spreadsheets

Computational Aids

Future value interest factor or present value interest factor

Computational Aids

Time Value Terms

• PV0 = present value or beginning amount

• k = interest rate

• FVn = future value at end of “n” periods

• n = number of compounding periods

• A = an annuity (series of equal payments or

receipts)

Four Basic Models

• FVn = PV0(1+k)n = PV(FVIFk,n)

• PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)

• FVAn = A (1+k)n - 1 = A(FVIFAk,n) k

• PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n)

k

BASIC PATTERNS OF CASH FLOW

• SINGLE AMOUNT: a lump sum amount either currently held or expected at some future date

• ANNUITY: a level periodic stream of cash flow

• MIXED STREAM: a stream of unequal cash flows that reflect no particular pattern

Future Value Example

You deposit $2,000 today at 6%

interest. How much will you have in 5

years?

$2,000 x (1.06)5 = $2,000 x FVIF6%,5

$2,000 x 1.3382 = $2,676.40

Algebraically and Using FVIF Tables

Future Value Example

You deposit $2,000 today at 6%

interest. How much will you have in 5

years?

Using Excel

PV 2,000$ k 6.00%n 5FV? $2,676

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.06, 5, , 2000)

Compounding More Frequently than Annually

• Compounding more frequently than once a year

results in a higher effective interest rate because you

are earning on interest on interest more frequently.

• As a result, the effective interest rate is greater than

the nominal (annual) interest rate.

• Furthermore, the effective rate of interest will increase

the more frequently interest is compounded.


• For example, what would be the difference in future

value if I deposit $100 for 5 years and earn 12%

annual interest compounded (a) annually, (b)

semiannually, (c) quarterly, an (d) monthly?

Annually: 100 x (1 + .12)5 = $176.23

Semiannually: 100 x (1 + .06)10 = $179.09

Quarterly: 100 x (1 + .03)20 = $180.61

Monthly: 100 x (1 + .01)60 = $181.67


Annually SemiAnnually Quarterly Monthly

PV 100.00$ 100.00$ 100.00$ 100.00$

k 12.0% 0.06 0.03 0.01

n 5 10 20 60

FV $176.23 $179.08 $180.61 $181.67

On Excel

Continuous Compounding• With continuous compounding the number of

compounding periods per year approaches infinity.

• Through the use of calculus, the equation thus

becomes:

FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

• Continuing with the previous example, find the Future

value of the $100 deposit after 5 years if interest is

compounded continuously.




becomes:

FVn (continuous compounding) = PV x (ekxn)


FVn = 100 x (2.7183).12x5 = $182.22

Present Value Example

How much must you deposit today in order to

have $2,000 in 5 years if you can earn 6%

interest on your deposit?

$2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5

$2,000 x 0.74758 = $1,494.52

Algebraically and Using PVIF Tables

Present Value Example

How much must you deposit today in order to

have $2,000 in 5 years if you can earn 6%

interest on your deposit?

FV 2,000$ k 6.00%n 5PV? $1,495

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.06, 5, , 2000)

Using Excel

Annuities• Annuities are equally-spaced cash flows of equal size.

• Annuities can be either inflows or outflows.

• An ordinary (deferred) annuity has cash flows that

occur at the end of each period.

• An annuity due has cash flows that occur at the

beginning of each period.

• An annuity due will always be greater than an

otherwise equivalent ordinary annuity because interest

will compound for an additional period.

Annuities

Future Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the end of each year at 5% interest for

three years.

FVA = 100(FVIFA,5%,3) = $315.25

Year 1 $100 deposited at end of year = $100.00

Year 2 $100 x .05 = $5.00 + $100 + $100 = $205.00

Year 3 $205 x .05 = $10.25 + $205 + $100 = $315.25

Using the FVIFA Tables

Future Value of an Ordinary Annuity



deposit $100 at the end of each year at 5% interest for

three years.

Using Excel

PMT 100$ k 5.0%n 3FV? 315.25$

Excel Function


=FV (.06, 5,100, )

Future Value of an Annuity Due



deposit $100 at the beginning of each year at 5%

interest for three years.

FVA = 100(FVIFA,5%,3)(1+k) = $330.96

Using the FVIFA Tables

FVA = 100(3.152)(1.05) = $330.96

Future Value of an Annuity Due



deposit $100 at the beginning of each year at 5%

interest for three years.

Using Excel

Excel Function


=FV (.06, 5,100, )

=315.25*(1.05)

PMT 100.00$ k 5.00%n 3FV $315.25FVA? 331.01$

Present Value of an Ordinary Annuity


• Example: How much could you borrow if you could

afford annual payments of $2,000 (which includes

both principal and interest) at the end of each year for

three years at 10% interest?

PVA = 2,000(PVIFA,10%,3) = $4,973.70

Using PVIFA Tables

Present Value of an Ordinary Annuity






Using Excel

PMT 2,000$ I 10.0%n 3PV? $4,973.70

Excel Function


=PV (.10, 3, 2000, )

Present Value of an Annuity Due






Using PVIFA Tables

PVA = 2000(PVIFA,10%,3)(1+k) = $5,471.40

PVA = 2000(2.487)(1.1) = $5,471.40

Present Value of an Annuity DueUsing Excel



afford annual payments of $2,000 (which includes both

principal and interest) at the end of each year for three

years at 10% interest?

PMT 2,000$ I 10.0%n 3PV? $5,471.40

Excel Function


=PV (.10, 3, 2000, )

Present Value of a Perpetuity

• A perpetuity is a special kind of annuity.

• With a perpetuity, the periodic annuity or cash flow

stream continues forever.

PV = Annuity/k

• For example, how much would I have to deposit today

in order to withdraw $1,000 each year forever if I can

earn 8% on my deposit?

PV = $1,000/.08 = $12,500

Future Value of a Mixed Stream

• A mixed stream of cash flows reflects no particular

pattern

• Find the future value of the following mixed stream

assuming a required return of 8%.

Using Tables

Year Cash Flow PVIF9%,N PV

1 400 0.917 366.80$

2 800 0.842 673.60$

3 500 0.772 386.00$

4 400 0.708 283.20$

5 300 0.650 195.00$

PV 1,904.60$

Year Cashflow (1) No. of years earning int. (n)

(2)

FVIF (3)

Future Value [(1)x(3)] (4)

1 P11,500 5-1 = 4 1.360 P15,640

2 14,000 5-2 = 3 1.260 17,640

3 12,900 5-3 = 2 1.166 15,041

4 16,000 5-4 = 1 1.080 17,280

5 18,000 5-5 = 0 1.000 18,000

Fixed value of mixed stream P83,601.40

Future Value of a Mixed Stream

• Find the present value of the following mixed stream


Using EXCEL

Year Cash Flow

1 400

2 800

3 500

4 400

5 300

NPV $1,904.76

Excel Function

Entry in Cell B9

is =-

FV(B2,A8,0,NPV

(B2,B4:B8)

A B

1 FUTURE VALUE OF A MIXED STREAM

2 Interest rate, pct/year 8%

3 Year Year-End Cash flow

4 1 P11,500

5 2 P14,000

6 3 P12,900

7 4 P16,000

8 5 P18,000

9 Future Value P83,608.15

Present Value of a Mixed Stream

• A mixed stream of cash flows reflects no particular

pattern



Using Tables

Year Cash Flow PVIF9%,N PV

1 400 0.917 366.80$

2 800 0.842 673.60$

3 500 0.772 386.00$

4 400 0.708 283.20$

5 300 0.650 195.00$

PV 1,904.60$

Present Value of a Mixed Stream



Using EXCEL

Year Cash Flow

1 400

2 800

3 500

4 400

5 300

NPV $1,904.76

Excel Function

Entry in Cell B9 is

=NPV(B2,B4:B8)

A B

1 PRESENT VALUE OF A MIXED STREAM OF CASH FLOWS

2 Interest rate, pct/year 9%

3 Year Year-End Cash Flow

4 1 P400

5 2 P800

6 3 P500

7 4 P400

8 5 P300

9 Present Value P1,904.76

Compounding Interest More Frequently Than Annually

• Interest is often compounded more frequently than once a year. Savings institutions compound interest semi-annually, quarterly, monthly, weekly, daily, or even continuously.

SEMIANNUAL COMPOUNDING of interest involves two compounding periods within the year. Instead of the stated interest rate being paid once a year, one-half of the stated interest rate is paid twice a year.

QUARTERLY COMPOUNDING of interest involves four compounding periods within the year. One-fourth of the stated interest rate is paid four times a year.

Example:

Future Value from Investing P100 at 8% Interest Compounded Semiannually over 24 Months (2 Years)

Period Beginning Principal (1)

Future Value interest factor (2)

Future value at end of period [(1)x(2)]

(3)

6 months P100.00 1.04 P104.00

12 months 104.00 1.04 108.16

18 months 108.16 1.04 112.49

24 months 112.49 1.04 116.99

Example:

Future Value from Investing P100 at 8% Interest Compounded Quarterly over 24 Months (2 Years)

Period Beginning Principal (1)

Future Value interest factor (2)

Future value at end of period [(1)x(2)]

(3)

3 months P100.00 1.02 P102.00

6 months 102.00 1.02 104.04

9 months 104.04 1.02 106.12

12 months 106.12 1.02 108.24

15 months 108.24 1.02 110.41

18 months 110.40 1.02 112.62

21 months 112.61 1.02 114.87

24 months 114.86 1.02 117.17

Future Value at the End of Years 1 and 2 from Investing P100 at 8% Interest, Given Various Compounding Periods

Compounding Period

End of Year Annual Semiannual Quarterly

1 P108.00 P108.16 P108.24

2 116.64 116.99 117.17

Example:

As shown, the more frequently interest is compounded, the greater the amount of money accumulated. This is true for any interest rate for any period of time.

• FVIFi,n = (1+i/m)mxn

• The basic equation for future value can no w be rewritten as

FVIFi,n = (1+i/m)mxn

USING COMPUTATIONAL TOOLS FOR COMPOUNDING MORE FREQUENTLY

THAN ANNUALLY

• Semiannual Quarterly Input Function

100

4

4

PV

N

I

CPT

FV

Solution is 116.99

Input Function

100

8

2

PV

N

I

CPT

FVSolution is 117.17

Spreadsheet Use

A B

1 FUTURE VALUE OF A SINGLE AMOUNT WITH SEMIANNUAL AND QUARTERLY COMPOUNDING

2 Present value P100

3 Interest rate, pct per year compounded semiannually 8%

4 Number of years 2

5 Future value with semiannual compounding P116.99


7 Interest rate, pct per year compounded quarterly 8%

8 Number of years 2

9 Future value with quarterly compounding P117.17

Entry in cell B5 is = FV(B3/2,B4*2,0)Entry in cell B9 is = FV(B7/4,B8*4,0,-B2,0)




becomes:

FVn (continuous compounding) = PV x (eixn)


• Continuing with the previous example, To find the value at the

end f 2 years of Fred Moreno’s P100 deposit in an account

paying 8% annual interest compounded continuously




becomes:

FVn (continuous compounding) = PV x (eixn)


Continuous Compounding• CALCULATOR USE

Input Function

0.16 2nd

1.1735

x

=100

Solution is 117.35

Continuous Compounding• Spreadsheet Use

A B

1 FUTURE VALUE OF SINGLE AMOUNT WITH CONTINOUS COMPOUNDING


3 Annual rate of interest, compounded continously

8%

4 Number of years 2

5 Future value with continuous compounding P117.35

Entry in Cell B5 is =B2*EXP(B3*B4)

Nominal & Effective Rates

• The nominal interest rate is the stated or contractual

rate of interest charged by a lender or promised by a

borrower.

• The effective interest rate is the rate actually paid or

earned.

• In general, the effective rate > nominal rate whenever

compounding occurs more than once per year

EAR = (1 + i/m) m -1

Nominal & Effective Rates

• For example, what is the effective rate of interest on

your credit card if the nominal rate is 18% per year,

compounded monthly?

EAR = (1 + .18/12) 12 -1

EAR = 19.56%

Special Applications of Time Value

Future value and present value techniques have a number of important applications in finance. We’ll study four of them in this section:

1.Determining deposits needed to accumulate a future sum.

2.Loan amortization

3.Finding interest or growth rates, and

4.Finding an unknown number of periods

Determining Deposits Needed to Accumulate a Future Sum

Supposed you want to buy a house 5 years from now, and you estimate that an initial down payment of P30,000 will be required at that time. To accumulate the P30,000, you will wish to make equal annual end-of-year deposits into an account paying annual interest of 6 percent.

FVAn = PMT X (FVIFAi,n)

PMT = FVAn

FVIFAi,n

FVIFAi,n) = 1x[ (1+i)n – 1] i


• Calculator Use

Input Function

3000

5

6

FV

N

I

CPT

PMTSolution is 5,321.89


A B

1 ANNUAL DEPOSITS NEEDED TO ACCUMULATE A FUTURE SUM

2 Future value P30,000

3 Number of years 5

4 Annual rate of interest 6%

5 Annual deposit P5,321.89

Entry in Cell B5 is =-PMT(B4,B3,0,B2).

Spreadsheet Use

Table Use: Use Table A-3

Loan Amortization The term loan amortization refers to the

determination of equal periodic loan payments.

Lenders use a loan amortization schedule to determine these payment amounts and the allocation of each payment to interest and principal.

Amortizing a loan actually involves creating an annuity out of a present amount.

Loan AmortizationYou borrow P6000 at 10 percent and agree to make equal annual end of year payments over 4 years.

PVAn = PMT X (FVIFAi,n)

PMT = PVAn

PVIFAi,n

PVIFAi,n = 1x[ 1 - 1 ] (1+i)n

Loan Amortization• Calculator Use

Input Function

6000

4

10

PV

N

I

CPT

PMT

Solution is 1,892.82

Loan Amortization

Loan Amortization

A B

1 ANNUAL PAYMENT TO REPAY A LOAN

2 Loan Principal (present value) P6,000


4 Number of years 4

5 Annual payment P1,892.82

Entry cell B5 is = -PMT(B3,B4,B2)

Loan AmortizationA B C D E

1

2 Data: Loan Principal

P6000


4 Number of years 4

5 Annual Payments

6 Year Total To interest To Principal Year-End Principal

7 0 6,000

8 1 1892.82 600.00 1,292.82 4,707.18

9 2 1892.82 470.72 1,422.11 3,285.07

10 3 1892.82 328.51 1,564.32 1,720.75

11 4 1892.82 172.07 1,720.75 0

Key Cell Entries

Cell B8:=-PMT($D$3,$D$4,$D$2),copy t B9;B11

Cell C8:=-CUMIPMT($D$3,$D$4,$D$2,A8,A8,0), copy to C9:C11

CellD8:=-CUMPRINC($D$3,$D$4,$D$2,A8,A8,0),copy to D9:D11

Cell E8:=E7-D8,copy to E9:E11

Loan Amortization

• Use Table A-4

Determining Interest or Growth Rates

• At times, it may be desirable to determine the

compound interest rate or growth rate implied by a

series of cash flows.

• For example, you invested $1,000 in a mutual fund in

1994 which grew as shown in the table below?1994 1,000$ 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525

It is first important to notethat although there are 7

years show, there are only 6time periods between the

initial deposit and the final value.







PV 1,000$ FV 5,525$ n 6k? 33.0%







Excel Function

=Rate(periods, pmt, PV, FV)

=Rate(6, ,1000, 5525)

Finding an unknown Number of Periods

• Ann Bates wishes to determine the number of years it will take for her initial P1000 deposit, earning 8% annual interest, to grow to equal P2,500. Simply stated, at an 8% annual rate of interest, how many years, n will it take for Ann’s P1000,PV, to grow to P2,500,FV?

• Table Use:• We begin by dividing the amount deposited in the

earliest year by the amount received in the latest year. This will result to present value interest factor

• Use Table A-2

Finding an Unknown Number of Periods

A B

1 YEARS FOR A PRESENT VALUE TO GROW TO A SPECIFIED FUTURE VALUE

2 Present value (deposit) P1000

3 Annual Rate of Interest, compounded annually 8%

4 Future value 2,500

5 Number of years 11.91

Entry in Cell B5 is =NPER(B3,0,B2,-B4).

Date post:	15-Jul-2015
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FINANCIAL MANAGEMENT PPT BY FINMAN Time value of money official

Economy & Finance