8/10/2019 KWK 4th Append D(1)

1/6

D-l6 aBpendix

D Time Value of

Money

Lfsing

Financsn'E *ai*uEat*=r=

Use

a financial calculator

to solve

time

value of

money

problems.

lllustration D-25

Financial

calculator

keys

lllustration

D-26

Calculator solution

for

present

value of a single

sum

Business

professionals,

once

they have mastered

the

underlying concepts

in

sections

1

and 2,

often

use

a

financial

calculator

to

solve

time

value

of

money

problems.

In

many

cases,

they must

use calculators

if interest rates or time

periods

do not

correspond with

the information

provided

in

the

compound

interest tables.

To

use

financial

calculators,

you

enter the

time

value

of

money variables

into

the calculator. Illustration

D-25 shows

the five

most common

keys used to solve

time value of money

problems.l

where:

N

:

number

of

periods

I

:

interest rate

per period

(some

calculators

use VYR or

i)

PV

:

present

value

(occurs

at the

beginning

of the first

period)

PMT

:

payment

(all

payments

are

equal,

and

none

are skipped)

FV

:

future value

(occurs

at the

end

of

the

last

period)

In

solving time

value of money

problems

in

this appendix,

you

will

gener-

ally

be

given three

of four

variables

and

will

have

to

solve

for

the

remaining vari-

able. The

fifth

key

(the

key not

used)

is

given

a

value of zero

to

ensure

that this

variable is not used in the computation.

Present

Value

of a Single

Sum

To

illustrate how

to solve a

present

value

problem

using a

financial

calculatoL

assume

that

you

want to know the

present value of

$84,253

to be

received

in

five

years,

diicounted at llo/o

compounded

annually.

Illustration

D-26

depicts

this

problem.

2On

many calculators,

these keys

are

actual

buttons

on the

face

of the

calculator;

on

others,

they

appear

on the display after the user

accesses

a

present

value menu.

lnputs: 5 11

?

0

84,253

ffiffiffiffiry

nswer:

*50,000

ryryryffiltre

8/10/2019 KWK 4th Append D(1)

2/6

Present

Value

of

an

AnnuitY

D'17

Illustration

D-26

shows

you

the

information

(inputs)

to enter

into

the

calculator:

N

:

5,

I

:

il,

pMT

:

b,

and

FV

:

84,253.

You

then

press PV

for

the answer:

-5so,ooo.

As

indicated,

the

PMT

key

was

given

a

value

of

zero

because

a

series

of

pal,.nnents

did

not

occur

in

this

probiem'

pg-tJs

A*\$s

nfififr*6"is

The

use

of

plus

and

minus

signs

in

time

value

of

money

problems

with

a

finan-

cial

calculaior

can

be

confusirg.

Most

financial

calculators

are

programmed

so

that

the

positive

and

negative

cash

flows

in

any

problem

offset

each

other'

In

tfr"

pr"r"rrt

value

probl"ir

ubou",

we

identified

the

$84,253

future

value

initial

irr,restment

as

a

positive

(inflow); the

answer

-$50,000

was

shown

as

a negative

amount,

reflecting

a cash

outflow.

If the

84,253

were entered

as

a negative,

then

the

final

answer

would

have

been

reported

as

a

positive

50,000.

Hopefull5,,thesignconventionwillnotcauseconfusion.Ifyouunderstand

what

is

required

in

a-problem,

you should

be

able

to

interpret

a

positive

or

neg-

ative

amount

in

determining

the

solution

to

a

problem'

ffisMp##ruffiBruffi

pffiffism#s

In

the

problem

above,

we assumed

that

compounding

occurs

once

a

year'

Some

financial

calculators

iru.r"

u

default

setting,

which

assumes

that

compounding

occurs

12

times

ayear.

You

must

determine

what

default

period

has

been

pro-

grammed

into

your calculator

and

change

it

as

necessary

to

arrive

at

the

proper

compounding

Period.

RSq.iNWEruffi

Most

financial

calculators

store

and

calculate

using

12

decimal

places'

As

a

result,

because

.o*forrnd

interest

tables

generally

have

factors

only

up

to

five

a".i*.f

places,

a

,tig1tt

difference

in

the

final

answer

can

result'

In

most

time

value

of

money

proil.-r,

the

final

answer

will

not

include

more

than

two

decimal

places.

Fresent

Value

of

an

Annuity

To

illustrate

how

to

solve

a

present

value

of

an

annuify

problem using

a finan-

cial

calculator,

assume

that

ytu are

asked

to determine

the

present

value

of

rental

receipts

of

$6,000

each

to

be

received

at

the

end

of

each

of

the

next

five

vears,

*h"n

dir"ounted

at

1'2o/o,

as

pictured

in

IllustrationD-27'

lllustration

D-27

Calculator

solution

for

present value

of

an

annuiiY

In

this

case,

You

enter

N

:

5,

I

:

12,

PMT

:

to

arrive

at the

answer

of

-$21,628'66'

6,000,

FV

:

0,

and

then

Press

PV

8/10/2019 KWK 4th Append D(1)

3/6

D-l8

;ippsr'ldix

#

Tirne

Value

of

Money

lllustration

D.28

Calculator solution

for

auto

loan

payments

lllustration

D.29

Calculator

solution

for

mortgage

amount

Useful

Applications

cf

the

Financial

Calculat

8/10/2019 KWK 4th Append D(1)

4/6

D-20 mppenc:x

*

Tirne

Value of

Money

Bnief

Fxercises

(Use

tables to solve

exercises BED-I to

BED-25.)

Compute

the

future

value

BED-I

Randy

Owen

invested

$8,000

at

5o/o

annual

interest,

and left the money

invested

of a single amounr.

without

withdrawing any of the interest

for 12

years. At the

end of the

12

years,

Randy

($0

2),

AP

withdrew

the accumulated amount of

money.

(a)

What amount

did

Randy

withdraw,

assuming

the investment earns simple

interest?

(b)

What amount

did

Randy

withdraw

assuming

the investment

earns

interest compounded

annuaily?

Use

future

value tables.

BED-?

For each of the following

cases, indicate

(a)

to

what

interest rate

columns and

iSil

:,3), {}

(b)

to

what number of

periods you

would

refer

in

looking

up the

future value

factor.

(1)

In

Table

1

(future

value

of 1):

Annual

Number

of

Rate

Years Invested

Compounded

Case

A 6a/o

Case

B 8o/o

(2)

In

Table

2

(future

vaiue

of

an annuity

of 1):

Annual

Number

of

Rate

Years Invested

Compounded

Case

A 5o/o

Case

B 60/o

3

Annually

4

Semiannually

8

Annually

6

Semiannuallv

Computethefuturevalue

BED-3

,,Joyce

Company

signed a

iease

for an

office

building

for a

period

of

12

years.

of a single

amount.

.. U-n-derthe

lease agreement, a

security

deposit of

$9,200

is made.

The deposit

will be re-

(S02),SP

'-'--turned

at

the expiration of the

lease with interest

compounded

at

4o/o

per

year. What

amount

will

Joyce receive at the

time

the lease expires?

Compute

the

future

value

BED-4., Bates

Company

issued

$1,000,000,

l0-year bonds

and agreed

to

make

annual

sink-

of an

annuity.

-.

ingfund

deposits of

$78,000.

The deposits are made

at

the

end

of each

year

into

an account

{S0

3), 4p

payrng

5o/o

annual

interest. What amount

will

be

in the sinking

fund

at the end

of 10

years?

Compute

the

future

talue

BED-5

,.

Frank and Maureen Fantazzi invested

$6,000

in

a

savings

account paying

4o/o

of asingleqmountandof

annual interest

when

their

daughte4

Angela,

was born.

They also

deposited

$1,000

on

an

annuity.

each

of

her

birthdays

until she

was 18

(including

her 18th

birthday).

How

much

was in

is{j

l.:),

AP

the savings

account on her 18th

birthday

(after

the

last

deposit)?

Cornpute

the

future

value

tsED-6

Hugh Curtin borrowed

$34,000

on July

1,

2012. This

amount

plus

accrued

in-

of a single

amount.

.

terest

at 9o/o compounded

annually is to

be repaid on

July

l,2AI7.

How

much

will

Hugh

iSO

Z}, ilF

have to repay on July

1,2017?

(Jse

present

value

tables.

BED'? For each of

the

following

cases, indicate

(a)

to

what

interest rate

columns

and

isg

5

:l) if

(b)

to

what number of

periods you

would refer

in looking

up the

discount

rate.

(1)

in

Table 3

(present

value of 1):

Annual

Number

of

Discounts

Rate

Years

Involved

Per Year

CaseA

1.2o/o

6

AnnuallY

Case

B 10a/o

11

AnnuallY

CaseC 60/o

9

SemiannuallY

(2)

In

Table

4

(present

value of an annuity

of

1):

Annual

Nurnber of

Number

of

Frequency

of

Rate Years

Involved

Pa)rrnents

Involved

Payments

Case

A L2o/o

20

20

AnnuallY

CaseB 10o/o

5

5

AnnuallY

CaseC 8a/o

4

8

Semiannually

8/10/2019 KWK 4th Append D(1)

5/6

Brief

Exercises

D-21

BED-8(a)Whatisthepresentvalueof$23,000duegperiodsfromnowdiscountedatDeterminepresentvalue

sB

5,6),

AF

'

.-

l0o/o?

(b)

What

is

the

present

ytl":-:i

$28'000

to

be

received

at

the

end

of

each

of

BED-e

J::::rl";::::::*T:y.:"investment

that

w*i

return

a

lump

sum

of

compute

the

present

tatue

of

five

years

f.;;;*.

What

amo'r,f;fua

Chaffee

C"'t;;;;;v

fo'

this

invest-

a

singte

amount

investment'

i$0

5i

&r

ffi;:"ffi;'::ffi

earns roo/o::

u,

investment

that

will

return g480,000

computethepresentuarueof

eight

years

t

orrr

rro*-w#t

is

the

^*o*rrr,

rifi

should

i.rrr".r

rro*io

".r-t-,rtir

rate

of

a

singreamount

investmen,

{S"l

5;.

&F

return?

BED_I1

Arthur

company

is

consjdering

investing

in

an

annuity

contract

that-will,re-

compute

the

present

value

of

turn

945,000

"rr"iroliii'eend

of

"""h

v;;;;;;'i"t;;;t''

wh""";;;;

sno"ld

Arthur

an

annuitv

iwestmen'

company

pay

for;i:

ffiJi;;;

";;;

5o/o

reiurn?

_,

o*o

ge0,000

at

th

(5i

6;

AF'

l

BED-12

'Kaehler

Enterprises

":y,

q7,,:'

an

investment

that

pays

back

$90,000

at

the

Compute

the

present

tlalue

,

dnrl_of

each

of

the

next

6

years.

what

is

,h"

;;;*

Kaehrer

u*"tinr"r"i"*li"a

,o

"u*

of

an

annuitv

i'Nestment'

is{l

-il,

sr

H|T:|]ffioad

co.

i1

about

to

issue

g300,000 of

l0-vear

bonds

pavin

s

?

s%

compute

the

present

ratue

interest

rate,

with

i*"r".t

payable

,"-itrr,t"^*y'

The

discount

tli"

i"t

t""h^securities

is

of

bonds'

8o/o.

Howmuch

can

Hanna

expect

,"

."""*ll"i

i-t'"-'^f"

of

these

bonds?

f't

I' i

l;'

&r'

BED-14

Assume

the

same

information

as

BED-13

except

that

the discount

rate.was

computethepresentvalueof

l0olo

instead

of

ga/o.In

this

case,

how

*rr"h

"r'

Hanna

expect

t"

t"t"l*

from

the

sale

of

'u,Jfi

,,

no

ffi:"T;mas

raco

companv

o':i"":

i,J^'--in'i,::::,xil;J:,5;:fi;:l1:Ti$:?

"iT'l:::":-presentvatue

(paid annually)

lrofr-;

"rr.a-o'*.i

^,

a

time

*f'"t

'f'"

discount

tut.-

it-Sq"'

frhat

ls

the

pres-

of

a

note'

ent

value

of

the

n;;;"tei"ed

bY

Tomas?

.sfb_re

Gleason

Enterprises

issued

9,o/o,

g-year,$2,600,000

par

value

bonds

that

pay

in-

compute

the

present

value

of

idierse*lu"""utiv

o"

o"tober

1

and

o#ii

i;"-;;i;

"'"'d";"Jap-r

t'2012'

""q:'"

bonds'

i,.'"a

on

that

d#

;i";;;;

;ryi'J*#tfii#:f*#"t

i'

zot'z'

is

roo/o

{sG

}'

E

?)'

AF

,H::;-;;::::"T,:r;.*"d

is

contempratingpurchasing

a

tire

rerreadin

E

comnltl

t\orelfltvatue

of

m'aihine

for

$18,000.

After

estimat,*

.".,,

"'J,".,"''",,

M"'ffi;j:"?'.;;;;;'h

fffi

a machine

for

purposes

of

from

rhe

."tr"uai.,g

machine

of

$3,30i ;;;;;li;

io'

a

v'*''

ruruti

iJp"'

to

earn

i

::y;

makins

a

purchase

decision'

of

t0

percent.";H';;;;il""ir.

wr'.it;;;;;;;#"alue

of

th'

retreading

operation?

ist

l'

7'

A'r

Should

fUutt

p"tt:ft*e

the

retreading

machine?

BED_'g

Fraziercompany

issues

an.golo,

5-year

mortga,ge

note

on

January

r,

-2orz,

to

compute

the

present

value

of

obtain

financing

for

new

equipm".rr.

r-una-iJ;;;J"t

"9rlilta

for

the

note'

The

terms

anote'

nrovide

for

semiannual

installmen,

o"#l*

l"rtfou,,to'

wr'^t

*"t"

the

cash

proceeds

l$3

8)

eF

i;;;;^ft""'

the

issuance

of

the

note?

:

BED-19

Leffler

company

_is

considering

purchasing

equ]prnent.

The

equipment

will

compute

the

maximum

price

,

pioduce

the

following

"..h

no*r,

""u"'i:-$#;'*ff;;

fi9'ffi;

-JY;-*:'

$s0'000'

to

pav

for

a

machine'

Leffler

requires

u

*iiri*rr*

rate

of

t;'-;;1-0

il"'

wt'''utit

Jrt"

*t-i"tum

price

Leffler

($$

[,

ii

AF

J"Ja

pty

fot

this

equipment?

-:-.^

o 1^ Aoo

qr

rhe compute

the

interest

rate

on

BgnliO If

Colleen

Mooney

invest-s.$4'172'65

low'and

she-will

receive $10'000

at

the

computetheinter

end*of-l5

y".rr,'*fruiunnual

rate

of

ini"t"'t

will

Colleen

";;;

tttt

lt"""tt*.'l? (Hint:

a

single

amoun,

is8

5)

&fi

use

Table

3')

r

'1-^

^--^*"'itw

of

investing $25'490

now'

The

compute

the

number

of

.l?3,j"".Y,f:#f,ffi.:E:ffi*:l::x5Ti#l'.sjffiii;1ffi;i?1";J;'Jd;:

}inJil."i"

'',-ngle

amount.

How

many

y"^.r;Jw;yre

wait

r"

*-."i""

Isot,ool?

(Hint: use

Table

3')

{si:5;'

AF"

B]ED-22JoanneQuickmadeaninve.stmentof$g,128.55.Fromthisinvestment,shewillComputetheinterestrateon

receive

g1,000

.rrrarv

for

the

.r"*t

i0

y"..s

starting

on"

d;;il*

"o*'

what

rate

of

anannuity'

interest

*il

ioi""Jt

ii*t*t"t

b"

";"i;;;"t

i"tiiai"'

use

Table

4')

(s

Ei'

Art

BED-23

?attv

Schleis

invests

F:iii,i'3"H:

:::'ilTi"i,"1,

T;ttl1"il$

'-Ti:tr*H

""#:f'"";n"ffif;'

inning

one

Year

from

now'

Pltl{

Yl

How

manv

t""'-tur

pavrnents

of

$1'00il;ift;6;;

"*tt

tiii''

use

Table

4')

igt

5;

&tt.

8/10/2019 KWK 4th Append D(1)

6/6

D'22 appendix

D

Time

Value

of MoneY

Compute

the

present

value

of

a machine

for

purposes of

making

a

purchase decision.

isc

t], AP

Compute

the

maximum

price

to

pay

for a

machine.

lsc

8),

$il

Detennine

interest

rate.

rsO

g).

3p

Deteftnine

interest

rate.

{sfi

s},

AP

Deteftnine

interest

rate

iso

3),

nF

Various

time

value

of money

situations.

it0

),

iiP

Vaious

time

value of

money

situations.

13il

i.

ilF

BED-24 Barney Googal

owns

a

Earage

and is contemplating

purchasing

a

tire

retread-

"in

aaCfrine

for

$14,280.

After estimating

costs

and revenues,

Barney

projects a

net

cash

flow

from

the retreading machine

of

$2,900

annually

for

8

years.

Barney

hopes

to

earn

a

return of

l|o/o on such

investments.

What

is the

present

value of

the retreading

opera-

tion?

Should

Barney Googal

purchase the

retreading

machine?

BED-25

Ramos Company

is considering

purchasing equipment.

The equipment

will

produce the

following

cash

flows: Year

1,

$30,000; Year

2,

$40,000;

Year

3,

$50,000.

Ramos

requiresaminimumrateof

return of

l2o/o.

WhatisthemaximumpriceRamosshouid

pay

for this

equipment?

BED-25

Carly

Simon

wishes to invest

$13,000

on July

I,2012,

and

have

it

accumulate

to

$50,000

by

July 1,

2022.

Instructions

Use

a financial

calculator

to determine

at

what exact

annual

rate

of

interest

Carly

must

invest

the

$18,000.

BED-27

On July

17

,2012,

James

Taylor borrowed

$60,000

from his

grandfather

to open

a clothing

store. Starting

July

17, 2018, James

has to

make

10 equal

annual

pa1'rnents

of

$8,860

each

to

repay the loan.

Instructions

Use

a

firrancial calculator

to

determine what interest rate

James

is

paying.

BED-28 As

the

purchaser of a new

house, Carrie

Underwood

has signed

a

mortgage

=-note'to

pay

the Nashville

National

Bank and

Trust Co.

$8,400

every 6

months

for

20

years,

at

the

"nd

of

which time she

will

own

the

house. At

the date

the mortgage

is

signed,

the

purchase

price was

$198,000

and Underwood

made

a

down

payment of

$20,000.

The

first

payment

will be made

6 months after

the date

the

mortgage

is

signed.

Instructions

Using

a

financial

calculator, compute

the

exact

rate of

interest

earned

on

the

mortgage

by the

bank.

BED-29

Using

a financial

calculato4

solve

for the

unknowns

in

each

of

the

following

"situaticins.

(a)

On June

1,

2012,

Holly

Golightly

purchases lakefront

properry from

her

neighbo4

George Peppard, and

agrees

to

pay the purchase

price

in

seven

payments

of

$16,000

each, the

first

payment

to

be

payable

June

l,2Ol3.

(Assume

that

interest

compounded

at an annual

rate of 6.9a/o

is implicit

in the

payments.)

What

is the

purchase

price

of

the

property?

(b)

Onl.6rry

1,201.2, SammisCorporationpurchased200of

the91,000facevalue,

7

o/o

cottpon,

I

0-year

bonds

of Malone

Inc. The

bonds

mature

on

January

I

,

2020

,

and

pay interest annually

beginning

January

1, 2013.

Sammis

purchased the

bonds

to

yteld

8.650/o.

How

much

did

Sammis

pay

for

the

bonds?

BED-30

Using

a financial

calculatot

provide a solution

to

each

of

the

following

'Situatibns.

(a)

Lynn

Anglin

owes a debt

of

g42,000

from

the

purchase of

her new

sport

utility

vehi-

cie.

The debt

bears annual

interest

of

7.8a/o compounded

monthly.

Lynn

wishes

to

pay

the

debt and

interest

in equal

monthly

pa)ments

over 8

years, beginning

one

month

hence.

What

equal

monthly

pa).rnents

will

pay

off

the debt

and

interest?

(b)

On January

l,2Ol2,

Roger

Molony offers

to buy

Dave

Feeney's

used

snowmobile

for

$8,000,

payable

in

five equal

annual

installments,

which

are

to

include

7.25o/o

intet-

est on

the unpaid

balance

and a

portion of

the

principal.

If

the

first

pal'ment is

to

be made

on December

31,2012,

how

much

will

each

payment be?