the time value of money

1

The Time Value of MoneyIntroduction to Time Value of Money, TVMFuture Value, FVLump-sum amountAnnuityUneven cash flow

Present Value, PVLump-sum amountAnnuityUneven cash flow

FV and PV ComparisonSolving for r and nIntra-year Interest CompoundingAmortization

2

Time Value of Money

Why is it important to understand and apply time value of money concepts?

What is the difference between a present value amount and a future value amount?

What is an annuity?

What is the difference between the Annual Percentage Rate and the Effective Annual Rate?

What is an amortized loan?

How is the return on an investment determined?

3

The Time Value of Money

Time value of money is considered the most important concept in finance

Mathematics of finance

“Nuts & Bolts” of financial analysis—apply of TVM concepts to determine value

Interest = Rate of return = r = i = k = Y

4

The Time Value of Money

“$1 received today is more valuable than $1 received in one year.” Why?Because if you have the opportunity to earn a positive return, investing the $1 today will cause it to grow to greater than $1 in one year. For example, $1 invested at 5 percent will grow to $1.05 in one year because 5¢ of interest will be earned.

5

Future Value and Present ValueFuture Value (FV)—determine to what amount an investment will grow over a particular time periodre-invested interest (earned in previous periods)

earns interestcompounding—interest compounds or grows the

investment

Present value (PV)—determine the current value of an amount that will be paid, or received, at some time in the futurePV is the future amount restated in current dollars;

future interest has not been earned, thus it is not included in the PV

discounting—deflate, or discount, the future amount by future interest that can be earned (“deinterest” the FV)

6

Lump-Sum Amounts, Annuities, and Uneven Cash Flow Streams

Lump-sum amount—a single amount invested (received) today or in the future; growth in value is the result of interest onlyAnnuity—equal payments made (received) at equal intervals; growth in value is the result of additional payments as well as interestordinary annuity—end of period paymentsannuity due—beginning of period payments

Uneven Cash Flows—payments that are not all equal that are generally made (received) at equal intervals; growth in value is the result of additional payments as well as interest

7

Cash Flow Time Lines

Helps you to visualize the timing of the cash flows associated with a particular situation

Constructing a cash flow time line is easy:

Time 0 1 2 3 4

Cash Flows -500

r = 10%

FVn = ?

8

Approaches to TVM SolutionsTime line solutionSolve using a cash flow time line

Equation (numerical) solutionUse equations to solve the problem

Financial calculator solutionFinancial calculators are programmed to solve time

value of money problems using the numerical solution

Spreadsheet solutionSpreadsheets contain functions that can be used to

solve time value of money problems using the numerical solution

Interest tablesObsolete

9

Future Value

Determine to what amount an investment will grow over a particular time period if it is invested at a positive rate of return.CompoundingLump-sum amountAnnuityUneven cash flow stream

10

Future Value, FV, of a Lump-Sum Amount

Example: If you invest $500 today at 10%, what will the investment be worth in four years if interest is paid annually?

Time 0 1 2 3 4

Cash Flows -500

r = 10%

FVn = ?

11

Future Value

Graphically, these computations are: 0 1 2 3 4

10%

× 1.10 × 1.10 × 1.10-500

End of year amount

= 550.00 = 605.00 = 665.50 = 732.05

× 1.10

FV4 = 500(1.10 x 1.10 x 1.10 x 1.10)

= 500(1.10)4

= 732.05

12

Future Value

The future value of an amount invested today for n years, FVn, can be found using the following equation:

FVn = PV(1 + r)n = PV(interest multiple)

FVn = future value in period nPV = present, or current, value

r = interest rate per periodn = number of periods interest is earned

13

Equation (Numerical) Solution

Determined by applying the appropriate equation:

FVn = PV x (1 + r)n

In our example: PV = $500, r = 10.0%, n = 4

FVn = 500 x (1 + r)n


FVn = 500 x (1.10)n


FV4 = 500 x (1.10)4


FV4 = 500 x (1.10)4 = 732.05

14

Financial Calculator Solution


N I/Y PV PMT FV


4


4 10


4 10 -500


4 10 -500 0


4 10 -500 0 ?


4 10 -500 0 ?

732.05

15

Future Value of an Annuity

Annuity—a series of equal payments that are made at equal intervalsOrdinary annuity—end of the period

Annuity due—beginning of the period

The future value of an annuity, FVA, can be computed by solving for the future value of a lump-sum amount

16

Future Value of an Annuity, FVA

-100

Time

Cash Flows

0 1 2 3

-100

7%

-100x (1.07)0

100.00x (1.07)1

107.00x (1.07)2

114.49 FVA = 321.49

FVA = 100(1.07)2 + 100(1.07)1 + 100(1.07)0 = 321.49FVA = 100(1.07)2 + 100(1.07)1 + 100(1.07)0 = 321.49 = 100[(1.07)2 + (1.07)1 + (1.07)2] = 100(3.2149) = 321.49

17

FVA—Equation (Numerical) Solution

r1-r)(1

PMT r)(1PMT FVAn1n

0

tn

In our example: PMT = $100, r = 7%, n = 3

321.49 )100(3.2149

0.07

1-(1.07)100 FVA

3

n

18

FVA—Annuity Due

Annuity due is an annuity with cash flows that occur at the beginning of the period.

When compared to an ordinary annuity, which has end-of-period cash flows, the cash flows of an annuity due earn one additional period of interest.

19

FVA—Annuity Due

100.00x (1.07)1

107.00x (1.07)2

Time

Cash Flows

0 1 2 37%

114.49FVA = 321.49

114.49

107.00 x (1.07)

114.49x (1.07)

122.50FVA(DUE) =

-100 -100 -100-100 -100 -100 x (1.07)

x (1.07)0

343.99

20

FVA(DUE)—Equation (Numerical) Solution


343.99 )100(3.4399

1.07 0.07

1-(1.07)100 FVA(DUE)

3

n

r)(1

r1-r)(1 n

FVA(DUE)n = PMT

r1-r)(1 n

FVA = PMT

21

FVA—Financial Calculator Solution

N I/Y PV PMT FV

321.49

In our example: n = 3, r = 7%, PMT = $100 In our example: n = 3, r = 7%, PMT = $100

3

In our example: n = 3, r = 7%, PMT = $100

3 7.0


3 7.0 0


3 7.0 0 -100


3 7.0 0 -100 ?

FVA =

3 7.0 0 -100 ? N I/Y PV PMT FV

343.99

BEGIN

FVA(DUE) =

Solutions: Future value computations

FV of $25,000 lump-sum amount: N = 5, I/Y = 5,

PV = -25,000, PMT = 0, FV = ?

FV of $5,499.40 annuity: N = 5, I/Y = 5, PV = 0,

PMT = -5,499.40, FV = ?

FV of $25,000 lump-sum amount: N = 5, I/Y = 5,

PV = -25,000, PMT = 0, FV = 31,907

FV of $5,499.40 annuity: N = 5, I/Y = 5, PV = 0,

PMT = -5,499.40, FV = 31,907

23

Uneven Cash Flow Streams

Uneven cash flow stream—cash flows that are not all the same (equal)

Simplifying techniques (that is, using a single equation) used to compute FVA cannot be used

24

FV—Uneven Cash Flow Streams

-600 -200

4%

-400

1(1.04)

0 1 2 3

200.00

2(1.04)416.00

0(1.04)

648.96_______

1,264.96

012 200(1.04) 400(1.04) 600(1.04) FV

25

FV of Uneven Cash Flow Streams—Equation (Numerical) Solution

tn

1tt

nn

22

11n

r) (1CF

r) (1CF r) (1CF r) (1CF FV

1,264.96

)200(1.0000 )400(1.0400 )600(1.0816

200(1.04) 400(1.04) 600(1.04) FV 012n

26

FV of Uneven CF Streams—Calculator Solution

Input the cash flows, find the present value, PV, and then compute the future value, FV, of PV.

Discussed in the next section.

27

Present Value

Determine the current value of an amount that will be paid, or received, at a particular time in the future.

Finding the present value (PV), or discounting, an amount to be received (paid) in the future is the reverse of compounding, or determining the future value of an amount invested today.

We find the PV by “de-interesting” the FV.

28

Present Value—Lump-Sum Amount

What is the PV of $800 to be received in four years if your opportunity cost is 8 percent?

Stated differently: How much would you be willing to pay today for an investment that will pay $800 in four years if you have the opportunity to invest at 8 percent per year?

29

Present Value—Lump-Sum Amount

8%

PV = ?Cash Flows 800

0 1 2 3 4Time

30

Present Value—Equation (Numerical) Solution

Remember that FV is computed as follows:

FV = PV x (1 + r)n

FV = PV x (1 + r)n

In our example, FV4 = 800, n = 4, r = 8.0%

800 = PV x (1 + r)n


800 = PV x (1.08)n


800 = PV x (1.08)4


Remember that FV is computed as follows:

800 = PV x (1.08)4

800 = PV x 1.36049PV = 800/1.36049 = 588.02PV = 800/1.36049

31

Present Value—Equation (Numerical) Solution

nr)(1

1FVPV

588.02 3) 800(0.7350

4(1.08)

1 800

PV Equation:

In our example: FV = $800, r = 8.0%, and N = 4

32

Present Value—Time Line Solution

Graphically, this computation is:

0 1 2 3 48%

588.02

End of year amount

635.07 685.87 740.74 800.00

08.1

1

08.1

1

08.1

1

08.1

1

33

PV Lump-Sum Amount—Financial Calculator Solution

In our example: FV = $800, r = 8.0%, n = 4

N I/Y PV PMT FV


4


4 8.0


4 8.0 0 800


4 8.0 0


4 8.0 ? 0 800

-588.02

34

Present Value of an Annuity, PVA

100 100

7%

100

1(1.07)

1

0 1 2 3

93.46 2(1.07)

1

87.34 3(1.07)

1

81.63 81.63262.43= PVA

321 (1.07)

1100

(1.07)

1100

(1.07)

1100

)100(2.6243 (1.07)

1

(1.07)

1

(1.07)

1100

321

262.43

35

PVA—Equation (Numerical) Solution

r

PMT r)(1

PMT PVA nr)(1

n

1t

1 - 11


262.43 )100(2.6243

0.7

1100 PVA

3(1.07)

1 -

36

PVA—Annuity Due

Annuity due is an annuity with cash flows that occur at the beginning of the period.

37

PVA—Annuity Due

100 100

7%

100

0 1 2 3

93.46

87.34

81.63 262.43 =

PVA

100100

7%

100

0 1 2 3

100 1(1.07)

1

2(1.07)

1

3(1.07)

1

1(1.07)

1

2(1.07)

1

3(1.07)

1

100100

7%

100

0 1 2 3

100 (1.07)

(1.07)

(1.07)

100.00

93.46

87.34

280.8

0

(DUE)

38

PVA(DUE)—Equation (Numerical) Solution


280.80 )100(2.8080

1.07 0.7

1100 PVA(DUE)

3(1.07)3

1 -

r

1 nr)(1

1 -

PVA = PMT r

1 nr)(1

1 -

PVA(DUE)n = PMT x (1 + r)

39

PVA—Financial Calculator Solution

N I/Y PV PMT FV

-262.43

In our example: n = 3, r = 7%, PMT = $100 3 7.0 100 0

In our example: n = 3, r = 7%, PMT = $100 3 7.0 ? 100 0

= PVA


-280.80

N I/Y PV PMT FV

3 7.0 100 0

BEGIN

3 7.0 100 0BEGIN

3 7.0 ? 100 0

= PVA(DUE)

Calculator Solution:Calculator Solution:

N = 5, I/Y = 5, PMT = -5,499.40, FV = 0, PV = 25,000

000,25

)54595.4(40.499,5)05.1(x05.0

)05.1(

11

40.499,5PVA5

Calculator Solution:

N = 5, I/Y = 5, PMT = -5,499.40, FV = 0, PV = ?

Numerical Solution:

41

Uneven Cash Flow Streams

Uneven cash flow stream—cash flows that are not all the same (equal)

Simplifying techniques (that is, using a single equation) used to compute PVA cannot be used

42

Present Value of an Uneven Cash Flow Stream

600 200

4%

400

1(1.04)

1

0 1 2 3

576.92

2(1.04)

1

369.82 3(1.04)

1

177.80 177.80

1,124.54

321 (1.04)

1200

(1.04)

1400

(1.04)

1600

43

PV of Uneven Cash Flows— Equation (Numerical) Solution

t

n

1t

nn2211

r)(11

CF

r)(11

CF r)(1

1 CF

r)(11

CF PV

1,124.54

(1.04)

1200

(1.04)1

400 (1.04)

1600 321

44

PV of Uneven Cash Flows—Financial Calculator Solution

Use the cash flow (CF) register (see calculator instructions)

Input CFs in the order they occur—that is, first input CF1, then input CF2, and so on

CF0—most calculators require you to input a value for before entering any other cash flows

Enter the value for I NPV = PV of uneven cash flows

45

PV of Uneven Cash Flows—Financial Calculator Solution

CF0 = 0

CF1 =600

CF2 =400

CF3 =200

r = 4%

Compute NPV = 1,124.54

46

FV of Uneven CF Streams—Calculator Solution

Input the cash flows, find the present value, PV, and then compute the future value, FV, of PV

In our example, PV = $1,124.54, so the future value is:

FV = $1,124.54(1.04)3 = $1,264.95

Calculator solution:

CF0 = 0

CF1 = 2 million

CF2 = 4 million

CF3 = 5 million

Numerical solution:

PV = ($2 million)/(1.06)1 + ($4 million)/(1.06)2

+ ($5 million)/(1.06)3

= ($2 million)(0.943396) + ($4 million)(0.889996) +($5 million)(0.839619)

= 1.8868 + 3.5600 + 4.1981 = 9.6449

I = 6

NPV = 9.6448

48

Comparison of PVA, FVA, and Lump-Sum Amount

PMT = $100; r = 7%; n = 3

FVA = $321.49 PVA = $262.43

100 100

7%

100

0 1 2 3

PVA = 262.43 FVA = 321.49FV = 262.43 x (1.07)3 = 321.49

PV = 321.49/(1.07)3 = 262.43

A B

C

49

PVA, FVA, and Lump-Sum Amount

PMT = $100; r = 7%; n = 3; PVA = $262.43

1 $262.43 $18.37 $280.80 $100.002 180.80 12.66 193.46 100.003 93.46 6.54 100.00 100.00

PMT = $100; r = 7%; n = 3; PVA = $262.43

Beginning Interest Ending Payment/Year Balance @ 7% Balance Withdrawal

FVA = 321.49

50

Solving for Time (n) and Interest Rates (r)—Lump Sums

The computations for lump-sum amounts included four variables: n, r, PV, and FV.

If three of the four variables are known, then we can solve for the unknown variable—e.g., if n, PV, and FV are known, we can solve for r.

51

Solving for Interest Rates, r—Lump-Sum Amount

If $200 that was invested three years ago is now worth $245, what rate of return (r) did the investment earn?

-200

Time

Cash Flows

0 1 2 3

245

r = ?

52

Solving for r for a Lump-Sum—Equation (Numerical) Solution

FV = PV (1+r)n

245 =200(1+r)3

200245 3r)(1

1/3

200245 r)(1

1.0 -2.45 r 0.333

= 7.0%

53

Solving for r for a Lump-Sum—Financial Calculator Solution

In our example: PV = $200, FV = $245, n = 3

N I/Y PV PMT FV


-200


-200 245


3 -200 245


3 -200 0 245


3 ? -200 0 2457.00

54

Solving for Number of Years, n—Lump-Sum Amount

If $712 is invested at 6 percent, how long will it take to grow to $848?

-712 848

r = 6% n = ?

Time

Cash Flows

0 1 2 …

55

Solving for n for a Lump-Sum—Equation (Numerical) Solution

712848 n(1.06)

n712(1.06) 848

nr)PV(1 FV

56

Solving for n for a Lump-Sum—Financial Calculator Solution

In our example: PV = $712, FV = $848, r = 6%

N I/Y PV PMT FV


-712


-712 848


6.0 -712 848


6.0 -712 0 848


? 6.0 -712 0 848

3.00

57

Solving for Time (n) and Interest Rates (r)—Annuities

The computations for annuities included four variables: n, r, PMT, and PVA or FVA.

If three of the four variables are known, then we can solve for the unknown variable—e.g., if n, PVA (or FVA), and PMT are known, we can solve for r.

58

Solving for Interest Rates, r, for Annuities

The current value of an investment that will pay $300 each year for three years is $817. What rate of return (r) will the investment earn?

300

0 1 2 3

PVA = -817

r = ?

300 300

59

Solving for r for Annuities—Equation (Numerical) Solution

r

1300 817

r

1PMT PVA

3

n

r)(1

r)(1

1

1

-

-

To solve, use a trial-and-error process

Solution = 5.0%

60

Solving for r for Annuities—Financial Calculator Solution

In our example: PMT = $300, PVA = $817, n = 3

N I/Y PV PMT FV


3 -817 300 0

5.00


3 ? -817 300 0

61

Solving for Number of Years, n, for Annuities

If $480 is invested each year at 8 percent, how long will take to grow to $2,816?

-480

FVA = 2,816

r = 8% n = ?

Time

Cash Flows

0 1 2 …-480 -480

62

Solving for n for Annuities—Equation (Numerical) Solution

0.08(1.08)

480 2,816

rr)(1

PMT FVA

n

n

1 -

1 -

To solve, use a trial-and-error process

Solution = 5 years

63

Solving for n for Annuities—Financial Calculator Solution

In our example: PMT = $480, FVA = $2,816, r = 8%

N I/Y PV PMT FV


8.0 0 -480 2,816

5.00


? 8.0 -480 2,816

64

Solving for r for Uneven Cash Flows

Internal Rate of Return (IRR)—average rate of return an investment earns

Capital budgeting decisions—decisions concerning what investments a firm should purchase

65

Intra-Year Interest Compounding

Interest is compounded more than once per year—quarterly, monthly, or daily

Adjustments to computations:Use the interest rate per compounding period

and the number of interest compounding periods during the life of the investment, or

Use the effective annual rate, EAR, and the number of years to maturity

66

Intra-Year Interest Compounding—Example

How much will an amount invested today grow to in two years if interest is paid quarterly? PV = $200 and r = 8%

Quarterly interest = 8%/4 = 2% = r/mNumber of interest payments = 2 years x 4 = 8 = n

200.00x 1.02

204.00

0Quarter

Year 0

1r = 2%

2 3 4

1

8

2

…x 1.02

208.08x 1.02

212.24x 1.02

216.49

FVn = PV(1 + r)n = FVn = PV(1 + r)n = 200(1.02)8 = 200(1.17166) = 234.33

234.33

67

Intra-Year Interest Compounding

Financial calculator solution:

N I/Y PV PMT FV

8 2.0 -200 0 ?

234.33

Financial calculator solution:

68

Intra-Year Interest Compounding—Effective Annual Rate

Interest = 8%, compounded quarterlyr = 8% per year is the simple, or non-compounded

rater/m = 2% per quarter is the effective rate per

compounding period (each quarter; m=4)

Effective Annual Rate (EAR), rEAR

rEAR = (1 + r/m)m – 1.0

m = number of interest payment periods per yearrEAR = (1.02)4 – 1.0 = 0.08243 = 8.243%

69

Intra-Year Interest Compounding—Using Effective Annual Rate

In our example: PV = $200n = 2 yearsrEAR = 8.243%

FVn = PV(1 + rEAR)n

= 200(1.08243)2

= 234.33= 200(1.02)8

70

Annual Percentage Rate, APR, Versus Effective Annual Rate, EAR

EAR—the rate of return per year considering interest compounding

rEAR = (1 + r/m)m – 1.0

APR—simple rate of return; does not consider compounding

APR = r/m x m = r = simple interestrEAR = APR only if interest is paid once per year—that is, annual compoundingrEAR > APR if interest is paid more than once per year

FVA—Financial Calculator Solution

N I PV PMT FV

0.016399

In our example: n = 365, PV = -$1, PMT = 0, FV = $1.06168

365 ? -1 0 1.06168

APR = 0.016399 x 365 = 5.986Advertise: 5.986% compounded daily

72

Amortized Loans

Loan agreement requires equal periodic paymentsA portion of the payment represents interest on the debt and the remainder is applied to the repayment of the debtAmortization schedule—used to determine what portion of the total payment is interest and what portion is repayment of principalMortgage payment—only the interest portion is considered an expense for tax purposes

73

Amortization Schedule—ExampleExample: $6,655 home-equity loan; r = 6%; n = 2 years; payments are made quarterlyThe constant payments per quarter represent an annuity and the amount of the loan ($6,655) represents the present value of the loan paymentsPVA = $6,655; (n x m) = 2 x 4 = 8 payments; r/m = 6%/4 = 1.5%Financial calculator solution:

N I/Y PV PMT FV

8 1.5 6,655 ? 0

-889

74

Amortization Schedule—Example

Payment = $889 Begin. Payment Interest (I) Loan Repay.

Year Pmt # Balance (Pmt) [1.5% x Beg Bal] = Pmt – I1 1 $6,655.00 $889 $99.83 $789.17

2 5,865.83 889 87.99 801.013 5,064.82 889 75.97 813.034 4,251.79 889 63.78 825.22

2 5 3,426.57 889 51.40 837.606 2,588.97 889 38.83 850.177 1,738.80 889 26.08 862.928 875.88* 889 13.14 875.86*

* Rounding difference

75

Answers to TVM Questions

Why is it important to understand and apply time value of money concepts?To be able to compare various investments.

What is the difference between a present value amount and a future value amount?Future value adds interest (compounds);

present value subtracts interest (“de-interests”).

What is an annuity?A series of equal payments that occur at

equal time intervals.

76

What is the difference between the Annual Percentage Rate and the Effective Annual Rate?APR is a simple interest rate quoted on

loans/investments, whereas EAR is the actual interest or rate of return.

What is an amortized loan?A loan paid off in equal payments over a specified

period, which include payments of interest and principal

How is the return on an investment determined?Compute the annual rate based on the amount to which

an investment will grow in the future.

Answers to TVM Questions

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