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.
Compounding More Frequently than Annually
• 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 = $181.67
Compounding More Frequently than Annually
Annually SemiAnnually Quarterly Monthly
PV 100.00$ 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.
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.
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
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the end of each year at 5% interest for
three years.
Using Excel
PMT 100$ k 5.0%n 3FV? 315.25$
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )
Future Value of an Annuity Due
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
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
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the beginning of each year at 5%
interest for three years.
Using Excel
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )
=315.25*(1.05)
PMT 100.00$ k 5.00%n 3FV $315.25FVA? 331.01$
Present Value of an Ordinary Annuity
• Annuity = Equal Annual Series of Cash Flows
• 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
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes
both principal and interest) at the end of each year for
three years at 10% interest?
Using Excel
PMT 2,000$ I 10.0%n 3PV? $4,973.70
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 3, 2000, )
Present Value of an Annuity Due
• Annuity = Equal Annual Series of Cash Flows
• 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?
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
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes both
principal and interest) at the end of each year for three
years at 10% interest?
PMT 2,000$ I 10.0%n 3PV? $5,471.40
Excel Function
=PV (interest, periods, pmt, FV)
=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
assuming a required return of 8%.
Using EXCEL
Year Cash Flow
1 400
2 800
3 500
4 400
5 300
NPV $1,904.76
Excel Function
Entry in Cell B9
is =-
FV(B2,A8,0,NPV
(B2,B4:B8)
A B
1 FUTURE VALUE OF A MIXED STREAM
2 Interest rate, pct/year 8%
3 Year Year-End Cash flow
4 1 P11,500
5 2 P14,000
6 3 P12,900
7 4 P16,000
8 5 P18,000
9 Future Value P83,608.15
Present Value of a Mixed Stream
• A mixed stream of cash flows reflects no particular
pattern
• Find the present value of the following mixed stream
assuming a required return of 9%.
Using Tables
Year Cash Flow PVIF9%,N PV
1 400 0.917 366.80$
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
• Find the present value of the following mixed stream
assuming a required return of 9%.
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
6 Present value P100
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)
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 (eixn)
where “e” has a value of 2.7183.
• 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
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 (eixn)
where “e” has a value of 2.7183.
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
2 Present value P100
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
Determining Deposits Needed to Accumulate a Future Sum
• Calculator Use
Input Function
3000
5
6
FV
N
I
CPT
PMTSolution is 5,321.89
Determining Deposits Needed to Accumulate a Future Sum
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
3 Annual rate of interest 10%
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
3 Annual rate of interest 10%
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.
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
PV 1,000$ FV 5,525$ n 6k? 33.0%
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
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).