Basic Business Finance Abm

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Begin OverviewTime Value of Money

• Simple Interest• Compound Interest• Compound Interest Formula• Future Value Tables• Annuities• Present Value• Present Value of an Annuity• Intrayear Compounding• Annuities Due• Perpetuities• Using a Financial Calculator

Time Value of Money

The time value of money impacts business finance, consumer finance,and government finance. Time value of money results from the concept ofinterest.

This overview covers an introduction to simple interest and compoundinterest, illustrates the use of time value of money tables, shows a matrixapproach to solving time value of money problems, and introduces theconcepts of intrayear compounding, annuities due, and perpetuities. Asimple introduction to working time value of money problems on a financialcalculator is included as well as additional resources to help understand timevalue of money.

Simple InterestSimple interest is a topic that most people cover in elementaryschool. nterest may be thought of as rent paid on borrowedmoney. Simple interest is calculated only on the beginningprincipal.!or instance, if one were to receive "# interest on a beginningvalue of $%&&, the first year interest would be'

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$%&& ( .&") or )$" in nterest

*ontinuing to receive "# interest on the original $%&& amount,over five years the growth of the original investment would looklike'

+ear %' "# of $%&& $" - $%&& $%&"

+ear ' "# of $%&& $" - $%&" $%%&

+ear /' "# of $%&& $" - $%%& $%%"

+ear 0' "# of $%&& $" - $%%" $% &

+ear "' "# of $%&& $" - $% & $% "

Compound Interest*ompound interest is another matter. t1s good to receive compoundinterest, but not so good to pay compound interest. 2ith compound interest,interest is calculated not only on the beginning interest, but on any interestaccumulated in the meantime.!or instance, if one were to receive "# compound interest on a beginningvalue of $%&&, the first year interest would be the same as simple interest

on the $%&&, or $". The second year, though, interest would be calculated onthe beginning amount of year , which would be $%&". So the interest wouldbe'$%&" ( .&") or )$". " in nterest

This provides a balance at the end of year two of $%%&. ". f this were tocontinue for " years, the growth in the investment would look like'

+ear %' "# of $%&&.&& $".&& - $%&&.&& $%&".&&

+ear ' "# of $%&".&& $". " - $%&".&& $%%&. "

+ear /' "# of $%%&. " $"."% - $%%&. " $%%".34

+ear 0' "# of $%%".34 $".35 - $%%".34 $% %.""

+ear "' "# of $% %."" $4.&6 - $% %."" $% 3.4/

7ote that in comparing growth graphs of simple and compound interest,investments with simple interest grow in a linear fashion and compound

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interest results in geometric growth. So with compound interest, the furtherin time an investment is held the more dramatic the growth becomes.Simple Interest

Compound Interest

Compound Interest Formula

nstead of calculating interest year8by8year, it would be simple to see thefuture value of an investment using a compound interest formula. The

formula for compound interest is'9 n 9 &:% - ; n

where' 9 n

9 &


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n


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+ou will notice that this table summari=es the factors for various interestrates for various years. To use the table, simply go down the left8handcolumn to locate the appropriate number of years. Then go out along the toprow until the appropriate interest rate is located. 7ote there are three pages

containing interest rates %# through %5#.

!or instance, to find the future value of $%&& at "# compound interest, lookup five years on the table, then go out to "# interest. At the intersection ofthese two values, a factor of %. 34/ appears. @ultiplying this factor timesthe beginning value of $%&&.&& results in $% 3.4/, exactly what wascalculated using the *ompound nterest !ormula previously. 7ote, however,that there may be slight differences between using the formula and tablesdue to rounding errors.

An example shows how simple it is to use the tables to calculate futureamounts.

+ou deposit $ &&& today at 4# interest. ow much will you have in " years

The Ans er! $2000 × 1.3382 $2!"!.#0

Future Value Tables

Using Tables to Solve Future Value Problems

Practice "#cercise

The following excercise should aid in using tables to solve futurevalue problems. 9lease answer the >uestions below and proceedby clicking the button located at the bottom of the page.


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1.

You invest $5,000 today. You will earn 8% interest. How much will you have in 4years? ( ic! the closest answer."

$#,80 .50

$#,84

$ ,#&5

. You have $450,000 to invest. ' you thin! you can earn &%, how much could youaccumulate in 10 years? ( ic! the closest answer."

$ 5,415

$885, 18

$& ,#10

.' a commodity costs $500 now and in)ation is e*+ected to o u+ at the rate o10% +er year, how much will the commodity cost in 5 years?

$805. 5

$ ,05 .55

-annot tell rom this in ormation


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Annuities

Using Tables to Solve Future Value of Annuity Problems

An annuity is an e>ual, annual series of cash flows. Annuities may be e>ualannual deposits, e>ual annual withdrawals, e>ual annual payments, or e>ualannual receipts. The key is equal, annual cash flows.

2hen cash flows occur at the end of the year, this makes them an ordinaryannuity . f the cash flows were at the beginning of the year, they would bean annuity due . Annuities due will be covered a later.

Annuities work as follows'

• Annuity C>ual Annual Series of *ash !lows.• Assume annual deposits of $%&& deposited at end of year earning "#

interest for three years.

+ear %' $%&& deposited at end of year

+ear ' $%&& ( .&" $".&& - $%&& - $%&&

+ear /' $ &" ( .&" $%&. " - $ &" - $%&&

$%&&.&&

$ &".&&

$/%". "

T


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!or instance, on the three8year, "# interest annuity of $%&& per year. Eoingdown three years, out to "#, the factor of /.%" is found. @ultiply that bythe annuity of $%&& yields a future value of $/%". &.

Another example of calculating the future value of an annuity is illustrated.

+ou deposit $/&& each year for %" years at 4#. ow much will you have atthe end of that time

The Ans er! $300 × 23.2"! $! 82.80

Annuities

Using Tables to Solve Future Value of Annuity Problems

Practice "#ercise

The following excercise should aid in using tables to solve annuityproblems. 9lease answer the >uestions below and proceed byclicking the button located at the bottom of the page.

1.

You invest $ ,000 in ' / s each year or 5 years. ' interest on these ' / s is 4%,how much will you have at the end o those 5 years?


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$10,000

$10,8 .#0

$8, 0 .#0

.' you de+osit $4,500 each year into an account +ayin 8% interest, how muchwill you have at the end o years?

$1 ,500

$14,#08.80

$11,5 #. 5

Present ValueUsing Tables to Solve Present Value Problems

9resent value is simply the reciprocal of compound interest. Another way tothink of present value is to adopt a stance out on the time line in the futureand look back toward time & to see what was the beginning amount.



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Present Value

9resent


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Present Value

Using Tables to Solve Present Value Problems

Practice Excercise

2he ollowin e*cercise should aid in usin ta3les to solve +resentvalue +ro3lems. lease answer the uestions 3elow and +roceed3y clic!in the 3utton located at the 3ottom o the +a e.

%. f you want to have $%&,&&& in / years and you can earn 6#, how much wouldyou have to deposit today

$3,5/6

$ ",33%

$% ,"53

. f you think you can sell an asset for $ ",&&& in five years and you think theappropriate discount rate is "#, how much would you be will to pay for theasset today

$ ",&&&


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$%5,"66.%"

*annot Tell !rom This nformation

Present Value of an Annuity

Using Tables to Solve Present Value of an Annuity Problems

To find the present value of an annuity, use T


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The interest rate per compounding period is found by taking theannual rate and dividing it by the number of times8per8year thecash flows are compounded. The total number of compoundingperiods is found by multiplying the number of years by the

number of times8per8year cash flows are compounded.!or instance, suppose someone were to invest $",&&& at 6#interest, compounded semiannually, and hold it for five years.

The interest rate per compounding period would be 0#.

The number of compounding periods would be %&.

:6# H ;

:" ( ;

To solve, find the future value of a single sum looking up 0# and%& periods in the T


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%. +ou borrow $"&,&&& and will make monthly payments for years at % #interest. ow much will those payments be

$ ,/"0

$ 5,"60

$/,560

7one of the above

. +ou invest $6,&&& at 4# interest, which will be compounded semiannually. owmuch will you have in three years

$5," 6

$5,""

$%%,/06

7one of the above

)nnuities *ueAnnuities due are beginning-of-year annuities. To work annuitiesdue, simply set up the problem the same way as would be donewith an ordinary annuity, then multiply the resulting factor by:%- ;. This is done whether the problem is present value orfuture value.



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To illustrate'!ind the future value of a three8year 4# annuity due or $0,&&&.

so,!< a Annuity:!< ! a ;:%- ;!< a 0,&&&:/.%6/4;:%.&4;FV a $ %+,'(-./(0

+erpetuities9erpetuity is a cash flow without a fixed time hori=on.!or example, if someone were promised that they would receive acash flow of $0&& per year until they died, that would be a

perpetuity. To find the present value of a perpetuity, simply takethe annual return in dollars and divide it by the appropriatediscount rate. To illustrate this'

f someone were promised a cash flow of $0&& per year until theydied and they could earn 4# on other investments of similar>uality, in present value terms the perpetuity would be worth

$4,444.43.

or,

:$0&& H .&4 $4,444.43;

,sin( a Finan-ial Cal-ulator !inancial calculators may be used to solve time value of money problems. Touse a financial calculator, it is necessary to understand the owner1s manual.Eenerally, the following steps should be followed'

%. Iead the problem thoroughly.



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. @ake sure what is being asked in the problem.

/. *lear the calculator.

0. nput the known value.

". nput the number of compounding periods per year.4. nput the annual interest rate.

3. nput the total number of compounding periods.

6. Ie>uest the unknown. An important thing to keep in mind is that cash outflows should carry aminus sign. The examples on the following three pages illustrate the use of a financialcalculator in solving time value of money problems. There are many financialcalculators available and they all have their strengths. !or these examples,the ewlett 9ackard %&b - is used.

Using a Financial Calculator

Example 1:

ow much will $%",&&& be worth in five years if interest is 6#compounded >uarterly

Jsing an 9 10bII+



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Iemember cash outflows carry a minus sign.


2hat is the future value of the $%",&&&


C C )


1 % 0 0 0 / +V

". nput the number of compounding periods per year.

# +

4. nput the annual interest rate.

8 I


2 0 4

6. Ie>uest the unknown.

FV

Ans er! %11'1.-/1+

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Example 2:

ow much would you have in four years if you deposit $0&,&&& at thebeginning of each year for four years and interest is % compoundedannually

Jsing an 9 10bII+




2hat is the future value of a $0&,&&& four8year annuity due


C C )


# 0 0 0 0 / +MT


1 +


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1 0 I


5e( 6nd # 4


FV

Ans er! %1)('1)(/))


Example 3:

2hat would you have to deposit today to have $"&,&&& in eight years if you

can earn 4# interest, compounded semiannually

Jsing an 9 10bII+




2hat is the present value of $"&,&&&


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C C )


% 0 0 0 0 FV


2 +


! I


1 ! 4


+V

Ans er! %,+'+2./,2

C3APT"4 1/ Time Value of Money Solution 5rid

• Financial Calculator Fundamentals• The Solution 5rid• Using the Solution 5rid• Future Value• Present Value

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• Future Value of an Annuity• Payments• Interest• Intrayear Compounding• Annuity Due• Additional Problems

Time Value of Money olution !rid

al!ulating Time Value of "oney Using a Finan!ialal!ulator

Time value of money can be calculated a number of waysKusing tables,formulas, spreadsheets, and financial calculators. !inancial calculators arerelatively inexpensive, easy to use, and versatileL performing additionalfunctions besides calculating time value of money.

2hen using a financial calculator to determine time value of money, it ismost efficient to follow an orderly procedure. The Time


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Financial Calculator Fundamentals

An #mportant $ote %efore %eginning

There are many financial calculators available and they all have theirstrengths. The obGective of this overview is not to discuss how to use afinancial calculator to determine time value of money, but to adopt a

procedure to help ensure consistent results when using a financial calculatorto determine time value of money.

!or this overview, a ewlett 9ackard 10bII+ is used. Although the discussionis specific to the 9 10bII+ , concepts will apply to most popular financialcalculators. 7o matter which calculator is used, it is important to read themanual that came with it. f that manual has been misplaced, refer to themanufacturer1s website for assistance.

Cven though the primary purpose of this overview is the discussion of theTime


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!or instance, notice on the bottom row of keys, between the 0 key andthe key, there is the decimal 7 key. f you first press , then press 7 , you

will set the calculator to reflect the & 7 setting, which toggles between JS

:%,&&&.&&; and CJI :%.&&&,&&; display.

Cxample'

nput' 1 0 0 0 7 0 0

To toggle between Jnited States and nternational display, press' & 7

The display will show 1.000&00 or 1&000.00.

on!erning (e!imals

+our calculator was most likely set to decimal places. owever, you may

want to change the number of decimal places it displays. 7otice on thebottom row of keys, under , is *IS+ . 9ress *IS+ , then simply enter the

number of decimal places you wish to display.

To show four decimal places, press'

To return to two decimal places, press'

*IS+ #

*IS+ 2

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learing

t1s important to be diligent in clearing your calculator. To help understandthis process, you have to remember that your calculator is not a

sophisticated computer. t is a much more manual device and you have to doa lot of the MthinkingM for it. So when you input information, think of it as

inputting information into separate Mregisters.M +ou have to manually clearthose registers one way or another.

7otice C towards the bottom left of the keypad. C clears out your lastinput. 7ow notice under C is C ) , which stands for M*lear All.M n timevalue of money and certain other applications, C ) clears out all the

work in a given set.

To clear last input, press'

To clear all the work in a given set, press'

C

C )

T)e Time Value of "oney 'eys

!or most functions, the time value of money keys are along the top row of

the keypad.

7ote the keys'

4 , I , +V , +M T , FV and + :Shifted 9@T;

4 um3er o eriods

I /nnual 'nterest ate

+V resent 6alue

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+MT /nnuity ayment

FV 7uture 6alue

+

Periods per Year

If the calculation involves annual interest, P/YR would

be one;

semiannual interest, P/YR would be two;

quarterly interest, P/YR would be four;

monthly interest, P/YR would be 12; and so on

T"e olution !rid

$ote

There are a number of approaches to calculating time value of money, andthere is nothing wrong with any of them. The Time


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Al*ays follo* t)ese steps ea!) time you !al!ulate time value of money+

• Iead the problem thoroughly.

• @ake sure you know what you1re being asked to do.• Setup a T

Documents

Basic Business Finance Abm