time value of money (mba)

8/3/2019 Time Value of Money (MBA)

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2011 Financial Management

Fundamentals of Financial Management,Prepared by: Amyn Wahid


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

You already recognize that there isTIME VALUE TO MONEYTIME VALUE TO MONEY!!

Time Value of MoneyTime Value of MoneyTime Value of MoneyTime Value of MoneyWhich would you prefer -- $10,000$10,000

todaytoday or$10,000 in 5 years$10,000 in 5 years?


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We know that receiving $10,000 today is worth

more than $10,000 after 5 years. This is due to

OPPORTUNITY COSTS.

The opportunity cost of receiving $10,000 in thefuture is the interest we could have earned if

we had received the $10,000 sooner.

Today Future

Time Value of MoneyTime Value of MoneyTime Value of MoneyTime Value of Money


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Time Value of MoneyTime Value of Money

How about a choice between $1,000today or $1,060 one year from today?

If the money is not needed immediately and saving accountsinterest rate is 6%, then one would be

indifferent about the two alternatives


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TIMETIME allows you the opportunityto

postpone consumption and earnINTERESTINTEREST.

Why TIME?Why TIME?Why TIME?Why TIME?

Why is TIMETIME such an important

element in your decision?


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Time Value TerminologyTime Value Terminology

Future value (FV) is the amount aninvestment is worth after one or moreperiods. (for future value you always

compound) Present value (PV) is the current value

of future cash flows of an investment.(for present value you always discount)

0 1 2 3 4

PV FV


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Future & Present ValueFuture & Present Value

Translate $1 today into its equivalent in thefuture (COMPOUNDING).

Translate $1 in the future into its equivalenttoday (DISCOUNTING).

?

?

Today Future

Today Future


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Time Value TerminologyTime Value Terminology

The number of time periods betweenthe present value and the future

value is represented by t orn.

The rate of interest for discounting or

compounding is called r ori.

All time value questions involve fourvalues: PV, FV, n and i. Given threeof them, it is always possible tocalculate the fourth.


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TimelinesTimelines

0 1 2 3 4 5

PV FV

Today

XA timeline is a graphical device used to clarify thetiming of the cash flows for an investment

XEach tick represents one time period

TimelinesTimelines


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Types of InterestTypes of InterestTypes of InterestTypes of Interest

Compound InterestCompound Interest

Interest paid (earned) on any previousinterest earned, as well as on theprincipal borrowed (lent).

Simple InterestSimple Interest

Interest paid (earned) on only the

original amount, or principal borrowed(lent).


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S

imple Interest FormulaS



imple Interest Formula

FormulaFormula SI= P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Period

n: Number ofTime Periods


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SI = P0(i)(n)

= $1,000(.07)(2)= $140$140

S

imple Interest ExampleS



imple Interest ExampleAssume that you deposit $1,000 in an

account earning 7% simple interest for

2 years. What is the accumulatedinterestat the end of the 2nd year?


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

Future ValueFuture Value is the value at some futuretime of a present amount of money, or aseries of payments, evaluated at a giveninterest rate.

S

imple Interest (FV)S



imple Interest (FV)What is the Future ValueFuture Value (FVFV) of the

deposit?


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FVFV = P0+ SI= P0+ P0(i)(n)

= P0[1 + (i)(n)]

= 1000[1 + (0.07)(2)]

= 1,000[1.14]= $1,140

S




imple Interest (FV) To find the Future ValueFuture Value (FVFV) of the deposit

directly we can use the following formula?


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

Present ValuePresent Value is the current value of afuture amount of money, or a series ofpayments, evaluated at a given interest

rate.

S

imple Interest (PV)S



imple Interest (PV)What is the Present ValuePresent Value (PVPV)?


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PVPV0= P0 = FVFVn / [1 + (i)(n)]

= 1140 / [1 + (0.07)(2)]

= $1000

Simple Interest (PV)Simple Interest (PV)Simple Interest (PV)Simple Interest (PV)

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


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0

5000

10000

15000

20000

1st Year 10th

Year

20th

Year

30th

Year

Future Value of a Single $1,000 Deposit

10%SimpleInterest

7% CompoundInterest

10% CompoundInterest

Why Compound Interest?Why Compound Interest?Why Compound Interest?Why Compound Interest?

FutureValue(U.S.

Dollars)


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Another ExampleAnother Example

You invest $100 in a savings account that earns

10% interest per annum (compounded) for threeyears.

After one year: $100 v (1+0.10) = $110

After two years: $110 v (1+0.10) = $121

After three years: $110 v (1+0.10) = $133.10


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The accumulated value of this investment at theend of three years can be split into twocomponents:

original principal $100

interest earned $33.10

Using simple interest, the total interest earnedwould only have been $30. The other $3.10 isfrom compounding.

Another ExampleAnother Example


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Assume that you deposit $1,000$1,000 ata compound interest rate of7% for

2 years2 years.

Future ValueFuture ValueS

ingle Deposit (Graphic)S

ingle Deposit (Graphic)




0 1 22

$1,000$1,000

FVFV22

7%


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FVFV11 = PP00 (1+i)1 = $$11,,000000 (1.07)= $$11,,070070

Compound Interest

You earned $70 interest on your $1,000

deposit over the first year.This is the same amount of interest youwould earn under simple interest.


ingle Deposit (Formula)S

ingle Deposit (Formula)





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FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07)

= $1,070$1,070

FVFV22 = FV1 (1+i)1

= 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$4.90in Year 2 with

compound over simple interest.








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FVFV11 = P0(1+i)1

FVFV22 = P0(1+i)2

General Future ValueFuture Value Formula:

FVFVnn = P0 (1+i)n

or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I

General FutureGeneral Future

Value FormulaValue Formula

General FutureGeneral Future


etc.


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FVIFFVIFi,n is found on Table I at the endof the book

Valuation Using Table IValuation Using Table IValuation Using Table IValuation Using Table I

Period 6% 7% 8%1 1.060 1.070 1.080

2 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.360

5 1.338

1.4

03 1.469


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FVFV22 = $1,000 (FVIFFVIF7%,2)= $1,000 (1.145)

= $1,145$1,145 [Due to Rounding]

Using Future Value TablesUsing Future Value TablesUsing Future Value TablesUsing Future Value Tables

Period 6% 7% 8%

1 1.060 1.070 1.080

2 1.124 1.145 1.1663 1.191 1.225 1.260

4 1.262 1.311 1.360

5 1.338 1.403 1.469


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ExampleExample

What will $1 000 amount to in 5 years time if interest is 12% per annum, compounded annually?

From the example, now assume interest is 12% perannum, compounded monthly.

Always remember that n is the number of compounding periods, not the number of years.

5

FV $1000 1 0.12

$1000 1.76234

$1 762.34

!

! v

!

$1816.70

1.8167$1000

0.011$1000FV60

!

v!

!


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Julie Miller wants to know how large her deposit

of$10,000$10,000 today will become at a compound

annual interest rate of10% for5 years5 years.

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000

FVFV55

10%


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Calculation based on Table I:

FVFV55 = $10,000 (FVIFFVIF10%, 5)= $10,000 (1.611)= $16,110$16,110 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem SolutionStory Problem Solution

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

n

FVFV55 = $10,000 (1+ 0.10)5= $16,105.10$16,105.10


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Assume that you need $1,000$1,000 in 2 years.2 years.Lets examine the process to determinehow much you need to deposit today at a

discount rate of7% compounded annually.

0 1 22

$1,000$1,000

7%

PV1PVPV00

Present ValuePresent ValueS







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PVPV00 = FVFV22/ (1+i)2 = $1,000$1,000/ (1.07)2

= FVFV22/ (1+i)2

= $873.44$873.44







0 1 22

$1,000$1,000

7%

PVPV00


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PVPV00 = FVFV11/ (1+i)1

PVPV00 = FVFV22/ (1+i)2

General Present ValuePresent Value Formula:

PVPV00 = FVFVnn/ (1+i)n

or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II

General PresentGeneral Present


General PresentGeneral Present


etc.


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PVPV22 = $1,000$1,000 (PVIF7%,2)= $1,000$1,000 (.873)

= $873$873 [Due to Rounding]

Using Present Value TablesUsing Present Value TablesUsing Present Value TablesUsing Present Value Tables

Period 6% 7% 8%

1 .943 .935 .926

2 .890 .873 .8573 .840 .816 .794

4 .792 .763 .735

5 .747 .713 .681


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ExampleExample

Your rich grandmother promises to giveyou $10000 in 10 years time. If interest

rates are 12% per annum, how much isthat gift worth today?

219.73$3

0.321973000$10

0.121000$10PV10

!

v!

v!


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Julie Miller wants to know how large of adeposit to make so that the money will

grow to $10,000$10,000 in 5 years5 years at a discountrate of10%.

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000

PVPV00

10%


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Calculation based on general formula:PVPV00 = FVFVnn/ (1+i)

n

PVPV00 = $10,000$10,000/ (1+ 0.10)5= $6,209.21$6,209.21

Calculation based on Table I:

PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)= $10,000$10,000 (.621)= $6,210.00$6,210.00 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem SolutionStory Problem Solution


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Solving for the Discount RateSolving for the Discount Rate (i)(i)

You currently have $100 available for investment for a21 year period. At what interest rate must you investthis amount in order for it to be worth $500 at

maturity? Given any three factors in the present value or future

value equation, the fourth factor can be solved.

ican be solved by one of the 2 ways:

take the nth root of both sides of the equation; or

use the future value tables to find a correspondingvalue. In this example, the FVIF after 21 years is 5.

r = 7.97%


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Solving for the Time PeriodsSolving for the Time Periods (n)(n)

How long does it take to double $5,000 at acompound rate of 12% per year (approx.)?

n can be solved by one of the 2 ways:

take logs on both sides of the equation; or

use the future value tables to find a

corresponding value.

n= ln2 / ln1.12n= ln2 / ln1.12

== 6.116 yrs6.116 yrs


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The Rule of 72The Rule of 72

The Rule of 72 is a handy rule of thumbthat states:

If you earn r % per year, your money will double in about 72 / r%years.

For example, if you invest at 6%, your money will double in about 12 years.

This rule is only an approximate rule.


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I love this

stuff!Can we do

some more?


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Practice QuestionsPractice Questions

6. Your company proposes to buy an asset for $335. Thisinvestment is very safe. You will sell off the asset in 3 years for$400. You know that you could invest the $335 elsewhere at 10%with very little risk. Should you go for the investment?

7. You are considering a one year investment. If you put up $1,250,

you will get back $1350. What rate is the investment paying?8. You estimate that you will need about $80,000 to send your child

to college in 8 years. You have about $35,000 now. If you earn12% per year, will you make it?At what rate, will you just reachyour goal?

9. Suppose we are interested in purchasing an asset that cost$50,000. We currently have $25,000. If we can earn 12% on this$25,000, how long until we have the $50,000

10. You have been saving funds to buy the Giant Company. The total costwill be $10 million. You currently have about $2.3 million. IF you can earn5% on your money, how long will you have to wait?


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AnswersAnswers

1. $201.1357

2. $561.9712

3. $373.83

4. $735.03

5. You are still about$2,375 short

6. No. You can make $445.89in the other alternative

7. 8%8. $86658.71( YES), 10.89%

9. 6.1163 years

10. 30.13 years


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AnnuitiesAnnuities

AnAn AnnuityAnnuityrepresents a series ofequal payments (orreceipts) occurring over a specified number of

equidistant periods. Payments or receipts normallyoccur at the end of each period.

Examples include consumer loans, car loanpayments, student loan payments, insurance

premiums and home mortgages. A perpetuityperpetuity is an annuity in which the cash flows

continue forever.


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Parts of an AnnuityParts of an AnnuityParts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

EndEnd of

Period 1

EndEnd of

Period 2

Today EqualEqual Cash FlowsEach 1 Period Apart

EndEnd of

Period 3


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

... + R(1+i)1 + R(1+i)0

Overview of anOverview of an

Ordinary AnnuityOrdinary Annuity ---- FVAFVA



R R R

0 1 2 nn n+1

FVAFVAnn

R = PeriodicCash Flow

Cash flows occur at the end of the period

i% . . .


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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$3,215

Example of anExample of an




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

0 1 2 33 4

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

7%

$1,070

$1,145



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FVAFVAnn = R (FVIFAi%,n)FVAFVA33 = $1,000 (FVIFA7%,3)

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

Valuation Using Table IIIValuation Using Table IIIValuation Using Table IIIValuation Using Table III

Period 6% 7% 8%

1 1.000 1.000 1.000

2 2.060 2.070 2.080

3 3.184 3.215 3.246

4 4.375 4.440 4.506

5 5.637 5.751 5.867


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Future Value of an AnnuityFuture Value of an Annuity

FVA =FVA =R[(1+i)n1]i

1000[(1+0.07)3

1] =0.07

$3214.9


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ExampleExample

What is the future value $200 deposited at the

end of every year for 10 years if the interest rate

is 6% per annum?

10

1.06 - 1FV $200

0.06$200 13.1808

$2 636.16

-

! v

! v

!


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

+ ... + R/(1+i)n


Ordinary AnnuityOrdinary Annuity ---- PVAPVA



R R R

0 1 2 nn n+1

PVAPVAnn

R = PeriodicCash Flow

i% . . .



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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$2,624.32





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

0 1 2 33 4

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

7%

$ 934.58$ 873.44$ 816.30



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PVAPVAnn = R (PVIFAi%,n)PVAPVA33 = $1,000 (PVIFA7%,3)

= $1,000 (2.624) = $2,624 (D to R)$2,624 (D to R)

Valuation Using Table IVValuation Using Table IVValuation Using Table IVValuation Using Table IV

Period 6% 7% 8%

1 0.943 0.935 0.926

2 1.833 1.808 1.783

3 2.673 2.624 2.577

4 3.465 3.387 3.312

5 4.212 4.100 3.993


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Present Value of anPresent Value of an

AnnuityAnnuity PVA =PVA =R 1 1

(1+i)n

i

10001 1 (1 + 0.07)3 =0.07

$2624.3


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Example 1You will receive $500 at the end of each of the next 5 years. Thecurrent interest rate is 9% per annum. What is the present value ofthis series of cash flows?

Example 2You borrow $7 500 to buy a car and agree to repay the loan by wayof equal monthly repayments over 5 years. The current interest rateis 12% per annum, compounded monthly. What is the amount ofeach monthly repayment?

_ a51 - 1/ 1.09PV $500

0.09

$500 3.8897

$1 944.83

! v -

! v

!

_ a601 - 1/ 1.01$7 500 R

0.01

R $7 500 44.955

$166.83

! v -

! z

!

ExamplesExamples


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MOREEXAMPLESMOREEXAMPLES


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Solving for theSolving for the

AnnuityPaymentAnnuityPayment(R)(R)Suppose we want to know the

amount that we have to deposit in

order to accumulate a given sumafter a number of years

e.g $10,000 down payment required

after 5 years How much you need tosave every year at 4 % interest rate?


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ComputationsComputations UsingUsing

Table IIITable IIIFVAFVAnn = R (FVIFAi%,n) see slide 34

$10,000 =R

(FVIFA4%,5)

$10,000 = R (5.416)

R = $1846.38


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ComputationsComputations

Using FormulaUsing Formula

n

5

1 + i - 1FV R

i

1.04 - 110000 R

0.04

R 5.416

10000R

5.416

$1846.27

- ! v

- ! v

! v

!

!


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Solving for the numberofSolving for the numberof

periods in an annuityperiods in an annuity(n)(n)

Suppose we want to know thenumber of years it would take a

certain amount to accumulate agiven sum

E.g the same question as before butnow we are given the annualpayment of $1846.27 and we have tofind the number of years


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Table IIITable III FVAFVAnn = R (FVIFAi%,n)

$10,000 = 1846.27 (FVIFA4%,n)

(FVIFA4%,n) = (5.416)

n = 5 periods

i,e 5 years



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n

n

n

1 + i - 1FV R

i

1.04 - 110000 1846.27

0.04

0.2167 1.04

Apply logs or ln on both sides

ln 1.2167n =

ln 1.04

n = 5 years

- ! v

- ! v

!


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Solving for the interestrateSolving for the interestrate

in an annuityin an annuity(i)(i)Suppose we want to know

interest rate now and the otherthings are known to us.

E.g using the same example we

would find the interest rate andhence verify that it is 4%.


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Table IIITable III FVAFVAnn = R (FVIFAi%,n)

$10,000 = 1846.27 (FVIFAi%,5)

(FVIFAi%,5) = (5.416)

i = 4%

C t tiC t ti


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The equation becomes really complex and canThe equation becomes really complex and canonly be solved by trial and error approach,only be solved by trial and error approach,NewtonNewton Raphson or bisection methodsRaphson or bisection methods

n

5

5

1 + i - 1

FV Ri

1 + i - 1

10000 1846.27i

5.416i = 1 + i - 1

- ! v

- ! v


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Julie Miller will receive the set ofcashflows below. What is the Present ValuePresent Value

at a discount rate of10%10%?

Mixed Flows ExampleMixed Flows ExampleMixed Flows ExampleMixed 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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1. Solve a piecepiece--atat--aa--timetimeby

discounting eachpiecepiece back to t=0.2. Solve a groupgroup--atat--aa--timetimeby first

breaking problem into groups of

annuity streams and any singlecash flow group. Then discounteach groupgroup back to t=0.

How to Solve?How to Solve?How to Solve?How to Solve?


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PiecePiece--AtAt--AA--TimeTimePiecePiece--AtAt--AA--TimeTime

0 1 2 3 4 55

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

10%

$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 Flow


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GroupGroup--AtAt--AA--Time (#1)Time (#1)GroupGroup--AtAt--AA--Time (#1)Time (#1)

0 1 2 3 4 55

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

10%

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

$1,677.30$1,677.30 == PVPV00 of Mixed Flowof Mixed Flow [Using Tables][Using Tables]

$600(PVIFA10%,2) = $600(1.736) = $1,041.60$400(PVIFA10%,4) $400(PVIFA10%,2)

=$400(3.170) $400(1.736) = $573.60

$100 (PVIF10%,5) = $100 (0.621) = $62.10


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GroupGroup--AtAt--AA--Time (#2)Time (#2)GroupGroup--AtAt--AA--Time (#2)Time (#2)

0 1 2 3 4

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

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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Present Value ofPresent Value ofMultiple Cash Flows ExampleMultiple Cash Flows Example

You deposit $1,500 in one year, $2000 in twoyears and $2,500 in three years in an accountpaying 10% interest per annum. What is thepresent value of these cash flows?

$2 500 v (1.10)-3 = $1 878

$2 000 v (1.10)-2 = $1 653

$1 500 v (1.10)-1

= $1 364Total = $4 895


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You deposit $1,000 now, $1,500 in one year,$2,000 in two years and $2,500 in three years inan account paying 10% interest per annum. Howmuch do you have in the account at the end of the third year?

$1 000 v (1.10)3 = $1 331

$1 500 v (1.10)2 = $1 815

$2 000 v (1.10)1 = $2 200$2 500 v 1.00 = $2 500

Total = $7 846

Future Value ofFuture Value ofMultiple Cash Flows ExampleMultiple Cash Flows Example


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Uneven Series ofPayment DateUneven Series ofPayment Date

(A

nEx

ample)(A

nEx

ample)

Year 1 2 3 4 5 6

Payment $500 $500 $700 $700 $700 $1,000

Karee Brow will receive the set of cash flows below. Whatis the Present ValuePresent Value at a discount rate of 10%10%? If Karee

Brow was depositing the cash flows instead determine theFuture ValueFuture Value at the same discount rate

Karee Brow will receive the set of cash flows below. Whatis the Present ValuePresent Value at a discount rate of 10%10%? If Karee

Brow was depositing the cash flows instead determine theFuture ValueFuture Value at the same discount rate

PV = $2870.92

FV = $5086.01

PV = $2870.92

FV = $5086.01


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BW Eff tiBW Eff tiBW Eff tiBW Eff ti


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

is 6% compounded quarterly for 1year. What is the Effective Annual

Interest Rate (EAREAR)?

EAREAR = ( 1 + 6%/ 4 )4 - 1= 1.0614 - 1 = .0614 or6.14%!6.14%!

BWs EffectiveBWs EffectiveAnnual Interest RateAnnual Interest Rate

BWs EffectiveBWs EffectiveAnnual Interest RateAnnual Interest Rate


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General Formula:

FVn = PVPV00(1 + [i/m])

mn

n: Number of Yearsm: Compounding Periods per Yeari: Annual Interest RateFVn,m: FV at the end of Year n

PVPV00: PV of the Cash Flow today

Frequency of CompoundingFrequency of CompoundingFrequency of CompoundingFrequency of Compounding


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Julie Miller has $1,000$1,000 to invest for2years at an annual interest rate of

12%.

Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)

= 1,254.401,254.40

Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)

= 1,262.481,262.48

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency


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Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)

= 1,266.771,266.77

Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)

= 1,269.731,269.73

Daily FV2 = 1,0001,000(1+[.12/365])(365)(2)= 1,271.201,271.20

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency

C i diff tC i diff t


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Comparison differentComparison different

effective rates of return?effective rates of return? An investment with monthly payments is different

from one with quarterly payments. Must put eachreturn on an EFF% basis to compare rates of

return. Must use EFF% for comparisons.S

eefollowing values of EFF% rates at various

compounding levels.

EARANNUAL 10.00%

EARQUARTERLY 10.38%EARMONTHLY 10.47%

EARDAILY (365) 10.52%

Can the EAR ever beCan the EAR ever be


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Can the EAR ever beCan the EAR ever beequal to the nominal rate?equal to the nominal rate?

Yes, but only if annual

compounding is used, i.e., if m= 1.

If m > 1, EFF% will always be

greater than the nominal rate.


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Amortizing a loanAmortizing a loan

Installment type loan that is repaidin equal periodic payments that

include both interest and principal.These payments can be madeannually, semi annually, monthly

etc

Steps to Amortizing a LoanSteps to Amortizing a LoanSteps to Amortizing a LoanSteps to Amortizing a Loan


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1. Calculate the payment per period.

2. Determine the interest in Period t.

(Loan balance at t-1) x (i% / m)3. Compute principal paymentprincipal payment in Period t.

(Payment-interestfrom Step 2)

4. Determine ending balance in Period t.(Balance -principal paymentprincipal paymentfrom Step 3)

5. Start again at Step 2 and repeat.

Steps to Amortizing a LoanSteps to Amortizing a Loan( An Overview)( An Overview)

Steps to Amortizing a LoanSteps to Amortizing a Loan( An Overview)( An Overview)


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Julie Miller is borrowing $10,000$10,000 at acompound annual interest rate of12%.

Amortize the loan ifannual payments are

made for5 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 ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan Example


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Amortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan Example

End ofYear

Payment Interest Principal EndingBalance

0 --- --- --- $10,000

1 $2,774 $1,200 $1,574 8,4262 2,774 1,011 1,763 6,663

3 2,774 800 1,974 4,689

4 2,774 563 2,211 2,478

5 2,775 297 2,478 0

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

[Last Payment Slightly Higher Due to Rounding]


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Illustration with aIllustration with a

simple examplesimple exampleSuppose you borrow $22,000 at 12%

compound annual interest to berepaid over the next 6 years.

The first step is to calculate R ( annualpayment)

PVPV00

= R (PVIFAi%,n

)

$22,000$22,000 = R (PVIFA 12%,6)

$22,000$22,000 = R (4.111)

RR = $22,000$22,000/ 4.111 = $5350.97$5350.97


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End of

Chapte

rEnd of

Chapte

r

time value of money (mba)

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