Time Value of Money

Section 1 Basic Ideas of Time Value of

Money Concept

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The Core Question of Finance

Congratulations!!! You have won a cash prize! There are two optional payment schedules: A - receive $100,000 now B - receive $100,000 in five years. Which option would you choose?

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Time Value of Money Concept

In simple termsthe concept implies that money today is always better than money tomorrow.

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Why Time Value of Money Exists?

Risk and Uncertainty-future always involves some risk, especially in respect to cash inflows of company as they are highly uncontrollable;

Inflation-in an inflationary economy a dollar today has always more purchasing power in compared to a dollar some point in future;

Consumption Preference- individuals generally prefer current consumption to a future one;

Investment Opportunities-an investor can profitably use money received today by investing it immediately;

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Allows investors to adjust cash flows for the passage of time;

It’s an integral part of Capital Budgeting Processes;

Applied in present and future value calculations;

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Section 2Interest Rates

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FormulaFormula SI = P0(i)(n)

SI: Simple InterestP0: Deposit today (t=0)

i: Interest Rate per Periodn: Number of Time Periods

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

Simple Interest Example

Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

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Compound Interest Yields higher return forinvestors or deposit holders; Cumbersome for borrowers; Makes borrowers to be more adhere to their payment schedule,for example.credit cards;

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Assume that you deposit $1,000$1,000 at a compound interest rate of 7% for 2 2

yearsyears.

Compound Interest Example

0 1 22

$1,000$1,000FVFV22

7%

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At the end of first yearPP00 (1+i)1 = $1,000x$1,000x (1.07)

= $1,070$1,070Compound Interest

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

This is the same amount of interest you would earn under simple interest.

Compound Interest Example

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At the end of first yearAt the end of first year = $1,000$1,000 (1.07) = $1,070$1,070

At the end of second yearAt the end of second year = 1070 (1+i)1 1070x(1.07)

$1,144.90$1,144.90You earned an EXTRA $4.90$4.90 in Year 2 with

compound over simple interest.

Compound Interest Example(cont.)

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Section 3Present Value vs Future

Value

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Valuation Concepts

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Future Value

The value at some future time of a present amount of money, or a series of payments evaluated at a given interest rate;

The interest earned on the initial principal amount becomes a part of the principal at the end of the compounding period;

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Future Value Example Problem

Suppose you invest $1000 for three years in a saving account that pays 10 % interest per year. If you let your interest income be reinvested, your investment will grow as follows:First year : Principal at the beginning $1000 Interest for the year ($1,000 × 0.10) $100 Principal at the end $1,100 Second year : Principal at the beginning $1,100 Interest for the year ($1,100 × 0.10) $110 Principal at the end $1,210Third year : Principal at the beginning $1,210 Interest for the year ($1210 × 0.10) $121 Principal at the end $1,331

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FormulaFormula FV = P0(1+i)n

FV: Future ValueP0: Deposit today (t=0)

i: Interest Rate per Periodn: Number of Time Periods

In the previous example FV=1000(1+0.1)3=1,331

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Double Your Money!We will use

the““Rule-of-72Rule-of-72””

Quick! How long does it take to

double $5,000 at a compound rate of

12% per year (approx.)?

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Approx. Years to Double = 7272 / i%

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

The “Rule-of-72”

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

12% per year (approx.)?

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Present ValueWhich one would you prefer assuming that the rate is 8%?a)$1000 today or,b)$2000 10 years later?

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To answer this question we have to express $2000 in today’s money. PV=FV/(1+i)n

$926=2000/(1+0.8)10

Types of Annuities

• Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period(coupon);

• Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period(rent);

An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

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Parts of an Annuity

0 1 2 3

$100 $100 $100

(Ordinary Annuity)EndEnd ofPeriod 1

EndEnd ofPeriod 2

Today EqualEqual Cash Flows Each 1 Period Apart

EndEnd ofPeriod 3

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Parts of an Annuity

0 1 2 3

$100 $100 $100

(Annuity Due)BeginningBeginning ofPeriod 1

BeginningBeginning ofPeriod 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3

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

= $1,145 + $1,070 + $1,000 = $3,215 or R(FVIFA$3,215 or R(FVIFAi,ni,n))

Example of anOrdinary Annuity -- FVA

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

0 1 2 3 3 4

$3,215 = FVA$3,215 = FVA33

7%

$1,070

$1,145

Cash flows occur at the end of the period

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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 or R(PVIFA$2,624.32 or R(PVIFAi,ni,n))

Example of anOrdinary Annuity -- PVA

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

0 1 2 3 3 4

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

7%

$934.58$873.44 $816.30

Cash flows occur at the end of the period

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FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1

= $1,225 + $1,145 + $1,070 = $3,440 or R(FVIFA$3,440 or R(FVIFA i,ni,n)(1+i))(1+i)

Example of anAnnuity Due -- FVAD

$1,000 $1,000 $1,000 $1,070

0 1 2 3 3 4

$3,440 = FVAD$3,440 = FVAD33

7%

$1,225$1,145

Cash flows occur at the beginning of the period

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PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02 $2,808.02

or or R(PVIFAR(PVIFA’,n-1’,n-1+1)+1)

Example of anAnnuity Due -- PVAD

$1,000.00 $1,000 $1,000

0 1 2 3 3 4

$2,808.02 $2,808.02 = PVADPVADnn

7%

$ 934.58$ 873.44

Cash flows occur at the beginning of the period

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Julie Miller will receive the set of cash flows below. What is the Present Value Present Value at a discount rate of 10%10%.

Mixed Flows Example

0 1 2 3 4 55

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

PVPV00

10%10%

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How To Solve 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $10010%

$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09

$1677.15 $1677.15 = = PVPV00 of the Mixed Flowof the Mixed Flow31

The actual rate of interest earned (paid) after adjusting the nominal rate for

factors such as the number of compounding periods per year.

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

Effective Annual Interest Rate

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Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6%

compounded quarterly for 1 year. What is the Effective Annual Interest Rate

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

= 1.0614 - 1 = .0614 or 6.14%!6.14%!

BW’s Effective Annual Interest Rate

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Julie Miller is borrowing $10,000 $10,000 at a compound annual interest rate of 12%.

Amortize the loan if annual payments are made for 5 years.

Step 1: Payment PVPV00 = R (PVIFA i%,n)

$10,000 $10,000 = R (PVIFA 12%,5)

$10,000$10,000 = R (3.605)RR = $10,000$10,000 / 3.605 = $2,774$2,774

Amortizing a Loan Example

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Amortizing a Loan ExampleEnd ofYear

Payment Interest Principal EndingBalance

0 --- --- --- $10,0001 $2,774 $1,200 $1,574 8,4262 2,774 1,011 1,763 6,6633 2,774 800 1,974 4,6894 2,774 563 2,211 2,4785 2,775 297 2,478 0

$13,871 $3,871 $10,000

Thank You