strategic finance lecture 3

8/12/2019 Strategic Finance Lecture 3

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TVM is based on the belief that peopleprefer to consume goods today rather thanwait to consume similar goods tomorrow.

The Time Value of Money

Consuming Today or Tomorrow

Money has a time value because moneytoday is worth more than money tomorrow.


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Todays money can be invested to earninterest or spent.

Value of money invested (positive interestrate) grows over time.

Rate of interest determines trade-offbetween spending today versus saving.


Consuming Today or Tomorrow


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Timeline Example

0 1 2 3 4 5

10 000 5 000 4 000 3 000 2 000 1 000

Cash Flows at the end of each Year


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Exhibit 5.1: Five-year Timeline for10 000 Investment


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The Time Value of Money Future Value versus Present Value

Future value measures what one or morecash flows are worth at the end of aspecified period.

Present value measures what one ormore cash flows that are to be receivedin the future will be worth today (at t=0).

Financial decisions are evaluated eitheron a future value basis or present valuebasis.


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Discounting is the process of convertingfuture cash flows to their present values.

Compounding is the process of earninginterest over time.

Future Value versus Present Value


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Future Value and CompoundingSingle Period Investment

We can determine the value of aninvestment at the end of one period if weknow the interest rate to be earned by theinvestment.

If you invest for one period at an interest rateof i , your investment, or principle, will grow

by (1 +i ) per unit of currency invested.

The term (1+ i ) is the future value interestfactor, often called simply the future value

factor.


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Future Value and CompoundingTwo-Period Investing

After the first period, interest accrues onoriginal investment (principle) and interestearned in preceding periods.

A two-period investment is simply twosingle-period investments back-to-back.


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The principal is the amount of money onwhich interest is paid.

Simple interest is the amount of interestpaid on the original principal amountonly.

Compounding interest consists of bothsimple interest and interest-on-interest.

Future Value and CompoundingTwo-Period Investing


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Exhibit 5.3: Future Value of 100


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Future Value and Compounding

General equation to find the future valueafter any number of periods.

The Future Value Equation

We can use financial calculators or future

value tables to find the future value factorat different interest rates and maturityperiods.

The term (1 + i )n is the future value factor.


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where:FV n = future value of investment at the end of

period n

PV = original principal (P 0) or present value i = the rate of interest per period, which is

often a year n = the number of periods

(5.1) n

i1PVFV )( n

The general equation to find the futurevalue is:

Future Value and CompoundingThe Future Value Equation


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You leave your 100 invested in the bank savingsaccount at 10 per cent interest for five years.How much would you have in the bank at the endof five years?

Future value exampleFuture Value and Compounding

05.161 6105.1100 10.1100

10.01100 5

55FV


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Exhibit 5.4: How Compound InterestGrows


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Exhibit 5.5: Future Value of 1


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Exhibit 5.6: Future Value Factors


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Compounding More Frequently Than Once aYear The more frequently the interest paymentsare compounded, the larger the future value

of 1 for a given time period.

where: m = number of compoundingperiods in a year


mnnFV =PV(1+i/m) (5.2)


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Non-annual compounding exampleFuture Value and Compounding

You invest 100 in a bank account that pays a 5 percent interest rate semi-annually for two years.How much would you have in the bank at the end

of two years?

38.110

1038.1100 025.1100

2/05.01100 4

222

FV


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When interest is compounded on acontinuous basis,we can use the equation below.

where: e = exponential function which isabout 2.71828


FV PV (5.3)i ne


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Continuous compounding exampleFuture Value and Compounding

Your grandmother wants to put 10 000 in asavings account at a bank. How much moneywould she have at the end of five years if thebank paid 5 per cent annual interestcompounded continuously?

25.84012 284025.100010 71828.200010

00010 25.0

505.05

eFV


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Exhibit 5.8: Present Value Factors


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Exhibit 5.9: Present Value of 1


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Finding the Interest Rate

A number of situations will require you todetermine the interest rate (or discountrate) for a given stream of future cashflows.

to determine the interest rate on a loan.

to determine a growth rate.

to determine the return on an investment.

For an individual investor or a firm, it may benecessary:


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The Rule of 72

Rule of 72 is used to determine the amountof time it takes to double an investment.

It says that the time to double your money(TDM) approximately equals 72/ i , where i isexpressed as a percentage.

Rule of 72 is fairly accurate for interestrates between 5% and 20%.


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Compound Growth Rates

Compound growth occurs when the initialvalue of a number increases or decreaseseach period by the factor (1 + growth rate).

(5.6)

n

g)(1PVnFV

Examples include population growth,earnings growth.


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Compound Growth Rates

Compound growth rate example

Because of an innovative advertisingcampaign, a firms sales increased from 20million in 2008 to more than 35 millionthree years later. What has been theaverage annual growth rate in sales?

3

3

13

35 = 20 (1+g)

1.75 = (1+g)

g = (1.75) 1 = 1.2051-1 = 0.2051 or 20.51%


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You have a chance to buy an annuitythat pays Rs550 at the beginning ofeach year for 3 years. You could earn5.5% on your money in otherinvestments with equal risk. What isthe most you should pay for the

annuity?

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Ali and Omar are cousins who were both born on the same day,and both turned 25 today. Their grandfather began puttingRs2,500 per year into a trust fund for Ali on his 20th birthday,

and he just made a 6th payment into the fund. The grandfather(or his estate's trustee) will make 40 more Rs2,500 paymentsuntil a 46th and final payment is made on Ali's 65th birthday.The grandfather set things up this way because he wants Ali towork, not be a "trust fund baby," but he also wants to ensurethat Ali is provided for in his old age.Until now, the grandfather has been disappointed with Omar,hence has not given him anything. However, they recentlyreconciled, and the grandfather decided to make an equivalentprovision for Omar. He will make the first payment to a trust forOmar today, and he has instructed his trustee to make 40additional equal annual payments until Omar turns 65, when the41st and final payment will be made. If both trusts earn anannual return of 8%, how much must the grandfather put intoOmars trust today and each subsequent year to enable him tohave the same retirement nest egg as Ali after the last payment

is made on their 65th birthday? 32


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Discounted Cash Flows and

Valuation


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Annuities and Perpetuities

Individual investors may make constantpayments on home or car loans, or investfixed amount year after year when savingfor retirement.

Many situations exist where businessesand individuals would face either receiving

or paying constant amount for a length ofperiod.

Level Cash Flows

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Annuity: any financial contract calling forequally spaced level cash flows over finite

number of periods.

Annuities and Perpetuities

Perpetuity: contract calling for level cashflow payments to continue forever.

Ordinary annuities: constant cash flowsoccurring at end of each period.

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Level Cash Flows


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ii)(1

11

CF

ifactor)luePresent va(1CF

annuityanforfactorluePresent vaCFPVA

n

n

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Level Cash Flows


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Level Cash Flows

A financial contract pays 2 000 at the end ofeach year for three years and the appropriatediscount rate is 8 per cent. What is the presentvalue of these cash flows?

Present Value of Annuity example

00.1545 577.20002

577.208.07938.01

Pr 1

7938.008.01

11

1Pr

3

3

factor annuity PV CF PVA

i factor valueesent

factor annuity PV

i factor valueesent n


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Finding Monthly or Yearly Payments Example

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Level Cash Flows

You have just purchased a 450 000apartment. You were able to make a 50 000down payment and obtain a 30-year fixed ratemortgage at 6.125 per cent for the balance.What are your monthly payments?

578406.1640051042.0

1599589.01

Pr 1

1599589.00051042.1

11

1Pr

%51042.012/%125.6int

360

i factor valueesent

factor annuity PV

i factor valueesent

monthsrateerest Monthly

n


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Finding Monthly or Yearly Payments Example

Level Cash Flows

45.4302 578406.164

000400 578406.164000400

CF

CF

factor annuity PV CF PVA n


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You have a chance to buy an annuitythat pays Rs550 at the beginning ofeach year for 3 years. You could earn5.5% on your money in otherinvestments with equal risk. What isthe most you should pay for the

annuity?

40


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Ali and Omar are cousins who were both born on the same day,and both turned 25 today. Their grandfather began puttingRs2,500 per year into a trust fund for Ali on his 20th birthday,

and he just made a 6th payment into the fund. The grandfather(or his estate's trustee) will make 40 more Rs2,500 paymentsuntil a 46th and final payment is made on Ali's 65th birthday.The grandfather set things up this way because he wants Ali towork, not be a "trust fund baby," but he also wants to ensurethat Ali is provided for in his old age.Until now, the grandfather has been disappointed with Omar,hence has not given him anything. However, they recentlyreconciled, and the grandfather decided to make an equivalentprovision for Omar. He will make the first payment to a trust forOmar today, and he has instructed his trustee to make 40additional equal annual payments until Omar turns 65, when the41st and final payment will be made. If both trusts earn anannual return of 8%, how much must the grandfather put intoOmars trust today and each subsequent year to enable him tohave the same retirement nest egg as Ali after the last payment

is made on their 65th birthday? 41

Exhibit 6 4: Present Value Annuity


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Exhibit 6.4: Present Value AnnuityFactors

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Preparing a Loan Amortisation Schedule

Amortisation: the way the borrowedamount (principal) is paid down over life

of loan.Monthly loan payment is structured so eachmonth portion of principal is paid off; at timeloan matures, it is entirely paid off.

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Level Cash Flows


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Amortised loan: each loan paymentcontains some payment of principal andan interest payment.


Loan amortisation schedule is a tableshowing:

loan balance at beginning and end of eachperiod.

payment made during that period.how much of payment represents interest.how much represents repayment ofprincipal.

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Level Cash Flows


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With an amortising loan, a larger proportionof each months payment goes towardsinterest in early periods.

As the loan is paid down, a greaterproportion of each payment is used to paydown principal.


Amortisation schedules are best done ona spreadsheet.

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Level Cash Flows

Exhibit 6 5: Amortisation Table for a


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Exhibit 6.5: Amortisation Table for a5-Year, 10 000 Loan

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Finding the Interest Rate

The annuity equation can also be used to findinterest rate or discount rate for an annuity.

To determine the rate of return for theannuity, we need to solve equation for theunknown value i.

Other than using a trial and error approach,it is easier to solve using a financialcalculator or spreadsheet.

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Level Cash Flows


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Finding the Interest Rate - Interpolation Example

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Level Cash Flows

Your parents are getting ready to retire anddecide to convert some of their retirementportfolio into an annuity. Their insurance agentasks for 350 000 for an annuity thatguarantees to pay them 50 000 a year for 10years. What is the return on the annuity?

i

i i

i CF PVA

n

n

n

1/1100050 000350

1/11


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Level Cash Flows

200351

024.700050 07.0

07.01/1100050

10

%7PVATry 7%

Interest rate is too lowtry a higher slightly higher interest rate.

Try 7.10%

550349 991.600050

071.0071.01/1100050

10

%10.7PVA


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Level Cash Flows

Use interpolation between the 7% and 7.10%interest rates:

%073.7%10.07273.0%0.7

%10.06501 2001

%0.7

%00.7%10.7550349 200351 000350 200351

%0.7

)2.6( low highi Highi Low

i Unknowni Low

Low Unknown i i ValueValue

ValueValue

i i


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Future Value of an Annuity

Future value annuity calculations usuallyinvolve finding what a savings orinvestment activity is worth at some futurepoint.

E.g. saving periodically for vacation,car, house or retirement.

We can derive the future value annuityequation from the present value annuityequation

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Level Cash Flows


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Future Value of an Annuity Equation

(6.2) i

1i)(1CF

i1-factor valueFuture

CF

annuityanfor factor valueFutureCFFVA

n

n

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Level Cash Flows

Exhibit 6 6: Future Value of 4-Year


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Exhibit 6.6: Future Value of 4 YearAnnuity

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Future Value of an Annuity ExampleOrbea bicycle problem in Exhibit 6.6.

Level Cash Flows

10.5064 5061.40001

5061.408.0

136049.1

136049.108.11

4

4

factor annuity FV CF FVA

i

factor valueFuturefactor annuity FV

i factor valueFuture n


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Perpetuities

A perpetuity is constant stream of cashflows that goes on for infinite period.

In stock markets, preference shares issuesare considered to be perpetuities, withissuer paying a constant dividend toholders.

Equation for present value of a perpetuitycan be derived from present value of anannuity equation with n tending to infinity.

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Level Cash Flows


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Perpetuities

(6.3) i

CF

i0)(1

CFi

i)(1

1

1CF

annuityanfor factor valuePresentCFPVA

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Level Cash Flows


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Perpetuities - Example

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Level Cash Flows

Suppose you decided to endow a chair infinance. The goal of the chair is to provide thechair holder with 100 000 of additional financial

support per year forever. If the rate of interest is8 per cent, how much money will you have togive the university foundation to provide thedesired level of support?

0002501 08.0

000100

i CF

PVA


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Important relationship between present valueof annuity and a perpetuity.

Perpetuities

Just as perpetuity equation wasderived from present value annuityequation, one can also derive presentvalue of an annuity from the equation

for a perpetuity.

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Level Cash Flows


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Annuity is called an annuity due whenthere is an annuity with payments beingincurred at beginning of each period ratherthan at end.

Annuity Due

Rent or lease payments typically made atbeginning of each period rather than atend.

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Level Cash Flows

Exhibit 6.7: Ordinary Annuity versus


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Exhibit 6.7: Ordinary Annuity versusAnnuity Due

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l h l


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Annuity Due

Annuity transformation method showsrelationship between ordinary annuity andannuity due.

Each periods cash flow thus earns extraperiod of interest compared to ordinary

annuity.Present or future value of annuity due isalways higher than that of ordinaryannuity.

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Level Cash Flows

Annuity due =Ordinary annuity value (1+i) (6.4)

l h l


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Annuity Due Example

The value of the annuity due shown in Exhibit6.7B is:

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Level Cash Flows

Annuity due = 3 312 (1.08) = 3 577

Cash Flows That Grow at a


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In addition to constant cash flow streams,one may have to deal with cash flows thatgrow at a constant rate over time.

These cash-flow streams called growingannuities or growing perpetuities.

Cash Flows That Grow at aConstant Rate

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Growing Annuity

Business may need to compute value ofmultiyear product or service contracts withcash flows that increase each year at aconstant rate.These are called growing annuities.

Example of growing annuity: valuation of

growing business whose cash flows increaseevery year at a constant rate.

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Cas ows at G ow at aConstant Rate



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Growing Annuity

Use this equation to value the presentvalue of growing annuity (equation 6.5)when the growth rate is less thandiscount rate.

(6.5

i1

g11

g-i

CFPVA

n1

n

65

Constant Rate



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Growing Annuity Example

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Constant Rate

A coffee shop will be in business for 50-years.It produced 300 000 this year and thediscount rate used by similar businesses is 15

per cent. The cash flows will grow at 2.5 percent per year. What is the estimated value ofthe coffee shop?

1284522 9968.00004602

15.1025.1

1025.015.0

500307

500307 025.01000300 50

50

1

PVA

CF



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Growing Perpetuity

When cash flow stream features constantgrowing annuity forever.

Can be derived from equation 6.5 when n tends to infinity and results in the followingequation:

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Constant Rate

1CFPVA = (6.6)i - g



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Growing Perpetuity Example

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Constant Rate

Your account reports that a firms cash flow lastyear was 450 000 and the appropriatediscount rate for the club is 18 per cent. Youexpect the firms cash flows to increase by 5percent per year and that the business willhave no fixed life. What is the value of the firm?

6156343 05.018.0

05.01000450

101 g i g CF g i CF PVA

Eff ti A l I t t R t


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Interest rates can be quoted in financialmarkets in variety of ways.

Most common quote, especially for a loan,is annual percentage rate (APR).

APR represents simple interest accrued onloan or investment in a single period;annualised over a year by multiplying it by

appropriate number of periods in a year.

Effective Annual Interest Rate

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Calculating the Effective Annual Rate (EAR) Correct way to compute annualised rate isto reflect compounding that occurs;involves calculating effective annual rate(EAR).

Effective annual interest rate (EAR) isdefined as annual growth rate that takes

compounding into account.

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Calculating the Effective Annual Rate (EAR)

EAR = (1 + Quoted rate/ m )m 1 (6.7)

m is the number of compounding periods during ayear.

EAR conversion formula accounts fornumber of compounding periods, thus

effectively adjusts annualised interest ratefor time value of money.

EAR is the true cost of borrowing andlending.

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Effecti e Ann al Interest Rate


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Effective Annual Rate (EAR) Example

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Your credit card has an APR of 12 per cent (1percent per month). What is the effectiveannual interest rate?

EAR = (1 + 0.12/12) 12 1= (1.01) 12 1= 1.1268 1= 0.1286 or 12.68%



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Consumer Protection and Interest Rate Disclosure

European Union Consumer CreditDirective ensures that APR is comparableacross countries.The Directive prescribes the pre-contractual and contract information thatmust be provided to potential borrowers forsums ranging from 200 to 75 000.

Applies for all European Union membercountries from 2008.

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Consumer Protection and Interest Rate Disclosure

Directive requires that APR be disclosedon all consumer loans and prominentlydisplayed on advertising and contractualdocuments .

APR is not the true cost of credit as thiswould require disclosure of EAR forborrowings.

Note that EAR, not APR, is theappropriate rate to use in present valueand future value calculations.


strategic finance lecture 3

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