TIME VALUE OF MONEY (TVM)

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After studying TVM , you shouldbe able to:

1. Understand what is meant by "the time value of money."2. Understand the relationship between present and future

value.3. Describe how the interest rate can be used to adjust the value

of cash flows – both forward and backward – to a single pointin time.of cash flows – both forward and backward – to a single pointin time.

4. Calculate both the future and present value of: (a) an amountinvested today; (b) a stream of equal cash flows (an annuity);and (c) a stream of mixed cash flows.

5. Use interest factor tables and understand how they provide ashortcut to calculating present and future values.

6. Use interest factor tables to find an unknown interest rate orgrowth rate when the number of time periods and future andpresent values are known.

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The Time Value of MoneyThe Time Value of Money

The Interest Rate

Simple Interest

Compound Interest

The Interest Rate

Simple Interest

Compound Interest Compound Interest

Compounding More Than Once perYear

Compound Interest

Compounding More Than Once perYear

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The Interest RateThe Interest Rate

Which would you prefer -- $10,000 today$10,000 todayor $10,000 in 5 years$10,000 in 5 years?

Obviously, $10,000 today$10,000 today.

You already recognize that there is TIME VALUETIME VALUETO MONEYTO MONEY!!

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Why TIME?Why TIME?

Why is TIMETIME such an important elementin your decision?

TIMETIME allows you the opportunity topostpone consumption and earn

INTERESTINTEREST.

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Types of InterestTypes of Interest

Simple InterestSimple Interest

• Interest paid (earned) on only the originalamount, or principal, borrowed (lent).

Compound InterestCompound Interest

Interest paid (earned) on any previous interestearned, as well as on the principal borrowed(lent).

amount, or principal, borrowed (lent).

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Simple Interest FormulaSimple Interest Formula

FormulaFormula SI = P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Period

n: Number of Time Periods

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Simple Interest ExampleSimple Interest Example

Assume that you deposit $1,000 in an accountearning 7% simple interest for 2 years. What isthe accumulated interest at the end of the 2ndyear?

SI = P0(i)(n)= $1,000(.07)(2)= $140$140

year?

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FF = P0 + SI

Simple Interest (FV)Simple Interest (FV)

What is the Future ValueFuture Value (FVFV) of the deposit?

FF = P0 + SI= $1,000 + $140= $1,140$1,140

Future ValueFuture Value is the value at some future time of apresent amount of money, or a series ofpayments, evaluated at a given interest rate.

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The Present Value is simply the

Simple Interest (PV)Simple Interest (PV)

What is the Present ValuePresent Value (PVPV) of theprevious problem?

The Present Value is simply the$1,000 you originally deposited.That is the value today!

Present ValuePresent Value is the current value of a futureamount of money, or a series of payments,evaluated at a given interest rate.

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Compound interest reflects both the remaining principal andany accumulated interest.

For $1,000 at 10%…

Period

(1)

Amount owedat beginning of

period

(2)=(1)x10%

Interestamount for

period

(3)=(1)+(2)

Amountowed at end

of period

1 $1,000 $100 $1,100

2 $1,100 $110 $1,210

3 $1,210 $121 $1,331

Compound interest is commonly used in personal andprofessional financial transactions.

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20000

Future Value of a Single $1,000 Deposit

10% Simple

Why Compound Interest?Why Compound Interest?F

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5000

10000

15000

1st Year 10th

Year

20th

Year

30th

Year

10% SimpleInterest

7% CompoundInterest

10% CompoundInterest

Fu

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Assume that you deposit $1,000$1,000 at acompound interest rate of 7% for 2 years2 years.

Future ValueSingle Deposit (Graphic)Future ValueSingle Deposit (Graphic)

0 1 22

$1,000$1,000FVFV22

7% 7%

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FF11 = PP00 (1+i)1

= $1,000$1,000 (1.07)= $1,070$1,070

Compound InterestCompound Interest

You earned $70 interest on your $1,000deposit over the first year.

This is the same amount of interest you wouldearn under simple interest.

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FF11 = PP00 (1+i)1 = $1,000$1,000 (1.07)= $1,070$1,070

FF22 = F1 (1+i)1

= PP00 (1+i)(1+i) = $1,000$1,000(1.07)(1.07)= PP00 (1+i)(1+i) = $1,000$1,000(1.07)(1.07)= PP00 (1+i)2 = $1,000$1,000(1.07)2

= $1,144.90$1,144.90

You earned an EXTRA $4.90$4.90 inYear 2 withcompound over simple interest.

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FF11 = P0(1+i)1

FF22 = P0(1+i)2

General Future ValueFuture Value Formula:

General Future Value FormulaGeneral Future Value Formula

General Future ValueFuture Value Formula:

FFnn = P0 (1+i)n

or

FFnn = P0 (FF/P, i, n/P, i, n) -- SeeTableSeeTable

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((FF/P,i,n )/P,i,n ) is found on Table at the end of thebook.

Valuation Using TableValuation Using Table

Period 6% 7% 8%Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

FF22 = $1,000 * (F/P, 7%, 2)(F/P, 7%, 2))= $1,000 (1.145)= $1,145$1,145 [Due to Rounding]

Using Future Value TablesUsing Future Value Tables

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

John wants to know how large his deposit of $10,000$10,000

today will become at a compound annual interest rateof 10% for 5 years5 years.

Story Problem ExampleStory Problem Example

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Story Problem SolutionStory Problem Solution

Calculation based on general formula:FFnn = P0 (1+i)n

FF55 = $10,000 (1+ 0.10)5

= $16,105.10$16,105.10

Calculation based on Table:FF55 = $10,000 (FF/P, 7%, 5)/P, 7%, 5)

= $10,000 (1.611)= $16,110$16,110

55

= $16,105.10$16,105.10

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Double Your Money!!!Double Your Money!!!

Quick! How long does it take to double$5,000 at a compound rate of 12% per

year (approx.)?

We will use the ““RuleRule--ofof--7272”.”.

year (approx.)?

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The “Rule-of-72”The “Rule-of-72”

Quick! How long does it take to double$5,000 at a compound rate of 12% per

year (approx.)?

Approx. Years to Double = 7272 / i%

7272 / 12 = 6 Years6 Years[Actual Time is 6.12Years]

year (approx.)?

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Assume that you need $1,000$1,000 in 2 years.2 years. Let’sexamine the process to determine how much youneed to deposit today at a discount rate of 7%compounded annually.

Present ValueSingle Deposit (Graphic)Present ValueSingle Deposit (Graphic)

compounded annually.

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PP00 = FF22 / (1+i)2 = $1,000$1,000 / (1.07)2

= FF22 / (1+i)2 = $873.44$873.44

Present ValueSingle Deposit (Formula)Present ValueSingle Deposit (Formula)

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PP00 = FF11 / (1+i)1

PP00 = FF22 / (1+i)2

General Present Value FormulaGeneral Present Value Formula

General Present ValuePresent Value Formula:

PP00= FFnn / (1+i)n

or PP00 = FVFVnn (PP/F, i, n/F, i, n) -- SeeTableSeeTable

etc.

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((PP/F, i, n )/F, i, n ) is found on Table at the end ofthe book.

Valuation Using TableValuation Using Table

Period 6% 7% 8%Period 6% 7% 8%1 .943 .935 .9262 .890 .873 .8573 .840 .816 .7944 .792 .763 .7355 .747 .713 .681

PP22 = $1,000$1,000 *(P/F, 7%, 2))= $1,000$1,000 *.(.873)= $873$873 [Due to Rounding]

Using Present Value TablesUsing Present Value Tables

Period 6% 7% 8%1 .943 .935 .9262 .890 .873 .8573 .840 .816 .7944 .792 .763 .7355 .747 .713 .681

Julie Miller wants to know how large of a depositto make so that the money will grow to $10,000$10,000in 5 years5 years at a discount rate of 10%.

Story Problem ExampleStory Problem Example

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Calculation based on general formula:PP00 = FFnn / (1+i)n

PP00 = $10,000$10,000 / (1+ 0.10)5

= $6,209.21$6,209.21

Story Problem SolutionStory Problem Solution

00

= $6,209.21$6,209.21

Calculation based on Table I:PP00 = $10,000$10,000 (PP/F, 10%, 5/F, 10%, 5)

= $10,000$10,000 (.621)= $6,210.00$6,210.00

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Types of AnnuitiesTypes of Annuities

AnAn AnnuityAnnuity represents a series of equalpayments (or receipts) occurring over aspecified number of equidistant periods.

Ordinary AnnuityOrdinary Annuity: Payments or receipts occurat the end of each period.

Annuity DueAnnuity Due: Payments or receipts occur atthe beginning of each period.

specified number of equidistant periods.

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Examples of Annuities

Student Loan Payments

Car Loan Payments

Insurance Premiums

Mortgage Payments

Retirement Savings

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Parts of an AnnuityParts of an Annuity

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Parts of an AnnuityParts of an Annuity

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Overview of an Ordinary Annuity - FVA

R R R

0 1 2 nn n+1

Cash flows occur at the end of the period

i% . . .

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FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0

FVAFVAnn

R = PeriodicCash Flow

There are interest factors for aseries of end-of-period cash flows.

How much will you have in 40 years if yousave $3,000 each year and your accountearns 8% interest each year?

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Finding the present amount from aseries of end-of-period cash flows.

How much would is needed today to providean annual amount of $50,000 each year for 20years, at 9% interest each year?

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Finding A when given F.

How much would you need to set aside eachHow much would you need to set aside eachyear for 25 years, at 10% interest, to haveaccumulated $1,000,000 at the end of the 25years?

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Finding A when given P.

If you had $500,000 today in an accountearning 10% each year, how much could youwithdraw each year for 25 years?

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Example of an Ordinary Annuity -- FVAExample of an Ordinary Annuity -- FVA

$1,000 $1,000 $1,000

0 1 2 33 47%

Cash flows occur at the end of the period

FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0

= $1,145 + $1,070 + $1,000 = $3,215$3,215

$1,000 $1,000 $1,000

$3,215 = FVA3

$1,070

$1,145

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FVAFVAnn = R (F/A, i, n)FVAFVA33 = $1,000 (F/A, 7%, 3)

= $1,000 (3.215) = $3,215$3,215

Valuation Using TableValuation Using Table

Overview of anOrdinary Annuity -- PVAOverview of anOrdinary Annuity -- PVA

R R R

0 1 2 nn n+1i% . . .

Cash flows occur at the end of the period

PVAPVAnn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n

R R R

PVAPVAnn

R = PeriodicCash Flow

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Example of anOrdinary Annuity -- PVAExample of anOrdinary Annuity -- PVA

$1,000 $1,000 $1,000

0 1 2 33 47%

$934.58

Cash flows occur at the end of the period

PVAPVA33 = $1,000/(1.07)1 +$1,000/(1.07)2 +$1,000/(1.07)3

= $934.58 + $873.44 + $816.30= $2,624.32$2,624.32

$1,000 $1,000 $1,000

$2,624.32 = PVA$2,624.32 = PVA33

$934.58$873.44$816.30

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PVAPVAnn = R (P/A, i, n)PVAPVA33 = $1,000 (P/A, 7%, 3)

= $1,000 (2.624) = $2,624$2,624

Valuation Using TableValuation Using Table

1. Read problem throughly

2. Create a time line

3. Put cash flows and arrows on time line

4. Determine if it is a PV or FV problem

Steps to Solve TimeValue of Money ProblemsSteps to Solve TimeValue of Money Problems

4. Determine if it is a PV or FV problem

5. Determine if solution involves a single CF, annuitystream(s), or mixed flow

6. Solve the problem

7. Check with financial calculator (optional)

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A cash flow diagram is an indispensabletool for clarifying and visualizing aseries of cash flows.

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Cash flow tables are essential tomodeling engineering economy problemsin a spreadsheet

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We can apply compound interestformulas to “move” cash flows alongthe cash flow diagram.

Using the standard notation, we find that apresent amount, P, can grow into a futureamount, F, in N time periods at interest rateamount, F, in N time periods at interest ratei according to the formula below.

In a similar way we can find P given F by

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It is common to use standardnotation for interest factors.

This is also known as the single paymentcompound amount factor. The term on theright is read “F given P at i% interest perright is read “F given P at i% interest perperiod for N interest periods.”

is called the single payment present worthfactor.

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Julie Miller will receive the set of cash flowsbelow. What is the Present ValuePresent Value at adiscount rate of 10%10%.

Mixed Flows ExampleMixed Flows Example

0 1 2 3 4 55

$600$600 $600 $400 $400 $100$600 $400 $400 $100

PVPV00

10%10%

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1. Solve a “piecepiece--atat--aa--timetime” by discountingeach piecepiece back to t=0.

2. Solve a “groupgroup--atat--aa--timetime” by first breaking

How to Solve?How to Solve?

2. Solve a “groupgroup--atat--aa--timetime” by first breakingproblem into groups of annuity streamsand any single cash flow groups. Thendiscount each groupgroup back to t=0.

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“Piece-At-A-Time”“Piece-At-A-Time”

0 1 2 3 4 55

$600$600 $600$600 $$400400 $400$400 $100$100

10%

$600$600 $600$600 $$400400 $400$400 $100$100$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09

$1677.15 = PV$1677.15 = PV00 of the Mixed Flowof the Mixed FlowIEG2H2-w2 51

“Group-At-A-Time” (#1)“Group-At-A-Time” (#1)

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

10%

$1,041.60$1,041.60$1,041.60$1,041.60$ 573.57$ 573.57$ 62.10$ 62.10

$1,677.27 = PV$1,677.27 = PV00 of Mixed Flowof Mixed Flow [Using Tables][Using Tables]

$600(P/A,10%,2) = $600(1.736) = $1,041.60$400(P/A,10%,2) .(P/F, 10%, 2) = $400(1.736)(0.826) = $573.57$100 (P/F, 10%,2) = $100 (0.621) = $62.10

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“Group-At-A-Time” (#2)“Group-At-A-Time” (#2)

0 1 2 3 4

$400 $400 $400 $400$400 $400 $400 $400

PVPV equals0 1 2$1,268.00$1,268.00

PVPV00 equals$1677.30.$1677.30.

0 1 2

$200 $200$200 $200

0 1 2 3 4 5

$100$100

$1,268.00$1,268.00

$347.20$347.20

$62.10$62.10

PlusPlus

PlusPlus

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General Formula:

Fn= PVPV00(1 + [i/m])mn

n : Number ofYears

m: Compounding Periods perYear

Frequency of CompoundingFrequency of Compounding

m: Compounding Periods perYear

i : Annual Interest RateFn: FV at the end ofYear n

PP00: PV of the Cash Flow today

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Julie Miller has $1,000$1,000 to invest for 2Yearsat an annual interest rate of 12%.

Annual F2 = 1,0001,000(1+ [.12/1])(1)(2)

Impact of FrequencyImpact of Frequency

Annual F2 = 1,0001,000(1+ [.12/1])(1)(2)

= 1,254.401,254.40

Semi F2 = 1,0001,000(1+ [.12/2])(2)(2)

= 1,262.481,262.48

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Qrtly F2 = 1,0001,000(1+ [.12/4])(4)(2)

= 1,266.771,266.77

Monthly F2 = 1,0001,000(1+ [.12/12])(12)(2)

= 1,269.731,269.73

Impact of FrequencyImpact of Frequency

2

= 1,269.731,269.73

Daily F2 = 1,0001,000(1+[.12/365])(365)(2)

= 1,271.201,271.20

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Effective Annual Interest Rate

The actual rate of interest earned (paid) afteradjusting the nominal rate for factors such

Effective Annual Interest RateEffective Annual Interest Rate

adjusting the nominal rate for factors suchas the number of compounding periods

per year.

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

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Welly has a $1,000 Cash Deposit at the bank. Theinterest rate is 6% compounded quarterly for 1

year. What is the Effective Annual Interest Rate(EAREAR)?

Effective Annual Interest RateEffective Annual Interest Rate

(EAREAR)?

EAREAR = ( 1 + 6% / 4 )4 - 1= 1.0614 - 1

= .0614 or 6.14%6.14%

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