Math Lessons

8/18/2019 Math Lessons

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Chapter 1. Simple Interest and Simple Discount1.1 Simple Interest

− It is the interest on the amount invested or borrowed at a given rate and for a giventime.

−It is usually associated with loan or investments which are short term in nature.−It is computed entirely on the original principal by simply multiplying together the

principal, rate and, time.I=Prt

Where I – interestP−principal (in pesos)r− interest rate per period of time, epressed as percent or a fractiont−time (in years) between the date the loan is made and the date itmatures or becomes repayable to the lender

Maturity value (F) – the total amount the borrower would need to pay bac!a. F=P+Ib. F=P + Prt leading to F=P (1+rt

Illustrative example: ". #ind the interest on a loan of $",%%% for one year if the interestrate is "&'.Solution:

P$",%%% r"&' ."&t" yearIPrt ($",%%%) (."&) (" year) $"&%

If the term is & years, then the interest isP$",%%% r"&' ."&t& yearsIPrt ($",%%%) (."&) (&) $&*%

!"ercises 1.1". + student loan $"%% at -' was repaid at the end of - years. ow much interest

did he pay/Solution:

I prt I "%%(.%-) (-)

I $"-.%%&. +n interest of $0&% was paid on a $-%%% simple1interest loan at the end of &

years. What was the rate of interest charged/Solution:

Iprt0&%-%%%(r) (&)2"&'

-. $0%%% is deposited to an account paying 3' simple interest. ow much is in theaccount after years/Solution:

Iprt I0%%%(.%3) () I&"%% +fter years i4p &"%%40%%%$5"%%.%%

*. + company has issued a 1year loan of $0%%%% to a new vice president to

6nance a home improvement pro7ect. 8he terms of the loan are to be paid bac!


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in full at the end of years, with simple interest at 3'. 9etermine the interest

which must be paid.Solution:

Iprt I0%%%%() (.%3) I$&&%%%.%%

. + student has received a $"%%%% loan from a wealthy aunt in the :nited ;tatesin order to 6nance his *1year college course. 8he term is that the student is to

repay his aunt in full at the end of "% years with a simple interest of &' per year.

ow much should he repay his aunt after "% years/Solution:

I"%%%%("%) (.%&) I-%%%% +fter "% years -%%%%4"%%%%$" set from 2ollie ?ompany, which is o@ered for $3%%

cash or $30%% if purchased at the end of - months. If money is worth ' simple

interest per month, how much would he save by borrowing the money and

paying cash/Solution:

Iprt I3%%(.%) (-A"&)


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- months

"-.#ind the maturity value on a $%%%% loan at "*1

4 ' for 0 months.

Solution:Iprt

I%%%%(."*&) (0A"&) I*"3.&

=aturity value p 4 i %%%%4*"3.& $*"3.&

"*.#ind the interest on a $3%%% loan at "0' for * months.Solution:

Iprt I3%%%(."0) (*A"&)I$-*%.%%

".=ary ;antos bought a new car. 8o pay for the car, which was priced at $


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b. Cact InterestP$",%%% r"&' ."&93% days

Solution:

Ie Pr ( D365 )

($",%%%) (."&) ( 60365 ) $&5.


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$&&, %%%(".%-*0&&&&&) $&&,03-. I 4 P

$-,0


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.&*. Bn ;eptember ;an Duan went to +H? ban! to borrow $&-%,%%% at 5' interest. Dodyrepaid the loan on Danuary &0, &%%-. +ssuming the loan is on eact time, ordinary interest,how much did Dody repay on the maturity date/Solution:

Iprt => I 4 P

$&-%,%%% (.%5) (137

360 ¿ $&-%,%%% 4$0,


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I 4 P

$&", *(."%%) (89

360 ¿ $3.5* 4 $&", *

$3.5* $&&,%"".5*

!"ercise 1.'!"act and ppro"imate )ime ,eteen )o Dates

Date o* investment Date o* Maturity !"act o. o* Daysppro"imate o. Days

". Duly ", &%%" =arch "%, &%%& &-< &-

&. #ebruary &, &%%- +ugust"&, &%%- "5" "5%-. +pril &%, &%%- Govember 3, &%%- &%% "53*. Dune , &%%- Bctober "&, &%%- "&5 "&0. =arch &


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Pro,lem Solving on $rdinary and !"act Interest/ !"act and ppro"imate)ime

". #ind the ordinary interest and the amount on $3.&%% at 03

4 'for 3 days.

Solution:

I Prt

$3,&%% (.%00) (65

360 ¿

$


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Solution:I Prt

$*,&%% (.%%


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d= D

Ft , t =

D

Fd and F =

D

dt ,

Example: ". #ind the present value of $&, %%% which is due at the end of 5% days at ' simplediscount.

Given: # $&,%%% t 14 d .%

Solution:9 #dt P # − 9

$&,%%% (.%) (1

4¿ $&,%%% − $&

$& $".50 Alternative solution:

9 # ("−dt)

$&,%%%

[1

−(.05

)(1

4

)] $",50&. +ccumulate $ *,&%% for & years and - months at 3

1

2 ' simple discount.

Given: P $*,&%% t &1

4 years d .%3

Solution:

# P

1−dt

₱ 4,200

1−( .65 )(2 14 ) $*, 5"5.*0

!"ercise 1.0". #ind the present value of $,%%% due at the end of 3 months if the discount

rate is 3

4 '.

P #98

, %%%(6

12 )(.%0)

$ "*-.0P $,%%% – "*-.0


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&. 9iscount $3,%% for "&% days at *1


P #dt P

$,%%% ( 6

12

)(.0575 )

$"*-.0

-. 9iscount $*,&% for " year and & months at 01


*. +ccumulate $-,&%% for & years and 3 months at 1


. +ccumulate $-,%%% for " year and 5 months at 31

4 simple discount.

3. If $&,%% is due on ;eptember &%, &%%-, 6nd its value on Dune &&, &%%- if thediscount rate is 3 '.0. #ind the discount rate if $0,%%% is the present value of $0,%% which is due at the

end of 3 months.


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I Pr ( days360 ) or I Pr tn# P 4 I=d 9n 4 tn

8d =d – 9d

9 #d ( days360 )∨ D= Fd td

Proceeds F [1−d ( days360 )] orProceeds # – 9 or f ("−d td)

Where r, rate of interest

d, rate of simple discounttn, term of the note in yearstd, term of discount in years I, amount of simple interest9, amount of simple discountP, Principal or face value#, 6nal amount or maturity of the

note

9d, date of discounttn, term of the note9n, date of the note or initial date=d, maturity date or 6nal datePr, amount of proceeds, amountpayable at the discount date

Illustrative Example:=r. =ananNuil has a 6ve month note of $",5%% dated #ebruary 3, &%%& bearing

interest at <3

5 '. If she sells the note on +pril "%, &%%&at a ban! discount rate at


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₱ 16,469.75[1−.08 ( 87360 )] $"3, "".--

!"ercise 1. Pro,lem Solving Involving Promissory ote

". + "%−day note for $"&,%% dated Dune -, &%%& with interest at ""' is discounted on

;eptember -, &%%& at <1

2 ' simple discount. #ind the

a. maturity dateSolution

Dune -, &%%& "*150

304 days

b. maturity valueSolution

# P 4 I

180/360

c. term of the discountSolution

I $"&,%% F ."" F150

360

$"0&.5& 4 $"&,%%d. discountSolution

9 # F d Ftd

360

$"-%0&.5& F .%


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Chapter #4 Compound Interest#.1 Finding the Compound mount

J ?ompound Interest is the sum by which the original principal has been increasedby the end of the contract. 8he original principal plus the compound interest, is called the?ompound +mount.

+nnually m";emi1annually m&Ouarterly m*=onthly m"&

!"ample 1

#ind the compound amount of $&%,%%% compounded semi1annually for & years at"&'.

m& i12

2 3'

t& years nm t & & *Solution

$&%,%%%.%% original principal 4 ",&%%.%% interest for si months (&%,%%% .%3)$&",&%%.%% new principal at the end of 3 months 4 ",&0&.%% interest for " year (&",&%% .%3)

$&&,*0&.%% new principal at the end of " year 4 ",-*


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#.# Finding the Compound mount%hen n is ot an Integer

!"ample4

+ccumulate $3,%% for * years and months at 12 ' compounded semi1

annually. (using scienti6c calculator)

Let P $3,%% n mt & (2 512 ) & ( 5312 ) 10612


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Siven 7"&' 7 "%' 7


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P $3,%%5.


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",%%% ( ".%&"%&5)<

$ ", "


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+nother eNuivalent relation states that the nominal rate 7 c, compounded

continuously is eNuivalent to the e@ective rate w, if and only if the accumulated value of

any amount P at the e@ective rate w is eNual to the accumulated value of the same

amount P at the force of interest 7c for any time t, that is shown in the wor!ing eNuation.

ence

Br

Cample

#ind the accumulated value of $%,%%% if it is invested for years at "%' interest

compounded continuously.

;olution

Let P %,%%% t years 7c "%' %." and # /

8hen # Pe 7c(t)

%,%%%e(%.")()

%,%%%(&.0"


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ence, the formula for i is given by

i antilog ( log F −log Pn ) – "

and 6nally, from i

j

m , the un!nown periodic rate is 7 (m)(i)

,y !"ponential Method

+gain, to 6nd the periodic rate of interest 7, begin from the formula

# P (" 4 i )n

F

P ( " 4 i )n

" 4 i ( F P )

8he formula for 7 is given by

D m ( F

P ) 1

n

¿ 1 "U

!"ample4

+t what rate compounded Nuarterly will $0,%%% become $"


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#.; Finding the )ime or )erm o* Investment

It is very important for an investor to !now how long it will ta!e his money to

accumulate into his desired amount. 8he term of investment or the length of time, t, for a

given amount P to accumulate into an amount, #, at a given interest rate (7, m), may be

obtained through a Logarithmic =ethod.

8o derive the formula for time, t, begin from the formula

# P(" 4 i)n

log ( F P )=¿ n log (" 4 i)

ence, n log( F P )log (1+i)

or n log F −log Plog(1+i)

+nd for n mt, the un!nown time in years is therefore,

t n

m

!"ample4

Bn Danuary &,&%%- =rs ;ally LorenMo had $*%%,%%% on a trust fund which earns

"*' converted Nuarterly. ;he plans to put up a ca!e and pastry business as soon as the

fund contains an amount of $%%,%%%. Bn what date will that amount be available/

;olution

Let P *%%,%%% # %%,%%% 7 %."* m * and

t /

8hen from the formula

# P ( j

m

)*t

%%,%%% *%%,%%% (1+ 0.144 ) *t".& (".%-)*t

log ".& *t log ".%-

*t log1.25

log1.035


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0.09691001

0.01494035

t ".3&"3 years or

t " year, 0 months and "* days

#.1< 6aluation o* Contracts and their Comparison!"ample4

$",%%% is due in 0 years. If money is worth 5.' compounded semi1annually, 6nd

a. Its value on the present date

b. Its value if payment will be given - years before its due

c. Its value if payment will be given & years after its due date

Solution4

a. Let # ",%%% i 0.0952 =0.0475 n &(0 years) "* and /

ILL:;82+8IBG

8hen

P ",%%%(".%*0)1"*

",%%% (%.&&&%


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# ",%%% (".%*0)*

",%%% (".&%-50"&


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;tep &. ?onstruct the 8ime 9iagram

a. Locate the due amounts of the set obligations on the upper part of the diagram,

b. Locate the due amounts of the set of new obligations on the lower part of the

diagram

;tep -. ;elect the most convenient ?omparison or #ocal 9ate, #9. 9raw arcs from

each due amount of the old and new obligations, all arcs pointing towards the #ocal

9ate.

;tep *. Prepare an eNuation of values by ad7usting the eponent of each

accumulation factor, (" 4 i)n, according to the chosen #ocal 9ate.

Cample

+ debtor owes $&%,%%% at the end of & years and $%,%%% at the end of < years. If

interest is at 0' e@ective rate, what single payment at the end of 3 years would

liNuidate both debts/

;olution

Let be the single payment. 9raw a time diagram and locate the two old

obligations and the new single payment.

9iagram

8he focal date may be chosen arbitrarily but for convenience, choose end of

3 years as focal date. 8hus $&%,%%% must be accumulated for * years at 0'

e@ective rate and $%,%%% must be discounted for & years at 0' e@ective rate. 8he

single payment which lies at the focal date is neither accumulated nor discounted.

8he eNuation of values may now be epressed as

Gew payment ;um of Bld Payments (at focal date –end of 3 years)

&%,%%% (".%0)* 4 %,%%% (".%0)1&

&%,%%% (".-"%05%3%") 4 %,%%% (%.


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'.1 2asic Concepts and )erminologies Annuity− a seNuence of payments, usually, eNual, made at regular intervals. Annuity certain – it is certain when the term of annuity is 6edContingent annuity or annuity uncertain – is one for which the 6rst and last payments orboth cannot be foretold accuratelyPayment – 8he time between successive payments whereas interval or conversion period

Periodic payment− the siMe of each paymenterm o! annuity− the time from the start of the 6rst payment to the last.Simple annuity− when the payments interval coincides with the interest conversionperiod.Ordinary annuity− is one for which the periodic payment are made at the end of eachperiod

Annuity due – when the payments are made at the beginning of each periodDeferred annuity − one whose 6rst payment is due at some later time

8he following symbols will be used in dealing with ordinary annuity formulas.2 periodic payment of the annuityn total number of payments i interest per conversion period

; amount of annuity+ present value of an annuity

'.# mount and Present 6alue o* an $rdinary nnuity 8he amount or 6nal value, denoted by ;, of an ordinary annuity is the sum of all

accumulated value of the sets of payments due at the end of the term, while the presentvalue of an annuity, denoted by +, is the sum of all discounted value of several paymentsdue at the beginning of the term.

+ and ; are related by the eNuations+ ; ("4i)1n and ; + ("4i) n

Where, + is the present value of ; due in the periods; is the amount of + for n periods

Illustrative example?onsider an ordinary annuity of $"%,%%% per year payable for - years with money

worth "%'. 8o 6nd the amount ; of the annuity, add the accumulated payments of eachperiod at the end of - years.


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$"%,%%% P"%,%%% $"%,%%%(" 4."%)" "",%%%$"%,%%%(" 4."%)& "&,"%%

; P--,%%%

8erm

% " & -

$"%,%%%(" 4 ."%)1" $5,%5%.5" $"%,%%%(" 4."%)1&


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n1" n (periods)22(" 4 i)"...

2(" 4 I)n1-2(" 4 I)n1&2(" 4 I)n1"

2 2 2 2 2

8erm

% " & -

ence, from the relation+ ;(" 4 i)1n and ; +(" 4 i)n

+ --,"%%(" 4 ."%)1- and ; &*,


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if we let

then the formula of + can be written as

where + is the present value of the annuity an I, is read as Ya angle n at iY

Illustrative Example "#ind the amount and present value of an annuity of $",%% payable for

& years if money is worth "% compounded semi1annually.Siven 2 P",%%

7 "%' 1 ."%

i & yearsm &

8o 6nd ;; 2sn i; ",%%svn .%; ",%%(*.-"%"&); P3,*3."5

or ; 2 [(1+.05)4−1

i ]s ",%% [(1+.05)

4−1.05 ]

s ",%%(*.-"%"&)s $3,*3.*

to 6nd ++ 2an i

+ ",%%* .%+ ",%%(-.*5)+ P,-"


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2 ,%%% m *

Solution: 8he down payment is not part of the annuity. 8he cash value of the

house is $*%%,%%% plus the present value of the ordinary annuity of *%

monthly payments of P,%%% each. 8he cash value is?ash >alue down payment 4 2a n I

$*%%,%%% 4 ,%%% *% .%-0 *%%,%%% 4 ,%%%(&%.%5555) *%%,%%% 4 "%&,0*.5

?ash >alue $%&,0*.5*

'.0 Periodic Payment o* an $rdinary nnuitya.) Periodic payments of ;

b.) Periodic payment of +

Illustrative Example "In order to have $-%%,%%% at the end of " years, how much must be

deposited in a fund every - months if money Is worth


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=r. 8uy bought a refrigerator that cost $"5,%%. e paid $3,%%% as downpayment and the balance will ne paid in -3 eNual monthly payments. #ind themonthly payment if money is "' compounded monthly.

Solution:

+ $"5,%% – $3,%%% $"-,%%n -3

i .15

12 .%"&

Find $:

2 A

an¬i

13,500

a.36¬.0125

2

13,500

1−(1+.0125 )−36

.0125

2 13,500

28.84726737

Exercise 3.1" +nalyn bought a dining set. ;he paid $&,%% as down payment and promised

to pay $


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3 What is the cash value of a set of ;cience Cncyclopedia that can bepurchased for $"&,%% as down payment and $",*% at the end of eachmonth for "< months at "' compounded monthly/

0 In a series of monthly payments of $",


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a (n& 1 ")i& – 3(n 1 ")i 4 "& (1−nRS ) %

b and (n& 1 ")i& 4 3(n 4 ")i 4 "& (1−nRS ) %

Example "+ refrigerator can be purchased for $5,%%% cash down and P3-% a month for

"< months. #ind the interest rate charged if it is compounded monthly.

Solution:+ 5,%% – 5%% –


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; $,"%%2 $0%n mt &(-) 3

Formula: ;ince ; is !nown

(n& 1 ")i& – 3(n 1 ")i 4 "&

(1−

nR

S

) %

(3& 1 ")i& – 3(3 1 ")i 4 "& (1−6(750)5,100 ) %-i& – -%i 4 ".*""03*0 %

8o 6nd i, we use the Nuadratic formula. We have

i 30± √ 30

2−4 (35)(1.4117647)2(35)

here we set two values of ii .


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Brdinary annuity of &* months

%5%5%5% 5%5% . . ." & - &- &*


%5%5%5% 5%5%

. . ." & - &- &*

>ictoria borrows $",%%% with interest at "' compounded Nuarterly. ;hewill discharge the debt by paying P5% Nuarterly.

a ow many payments of $5% are reNuired/b ow much would the 6nal payment be if it is made the day after the last

P5% payment/

c ow much would the 6nal payment be if it is made - months after the last P5%payment/

Solution:Given: + $",%%% 2 5% i .%-0a ;ubstitute the given values into the formula

n log [1− Ac R ]log (1+ i)

n

log

[1−

15,000(.0375)

950

]log (1+.0375)n

log.407894737

log (1.0375)

n .389451898

.015988105

n &*.-


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% " & - && &- &* &. . .5% 5% 5% 5% 5% 5% y

ordinary annuity of &* payments

:sing the present as the focal date

y(".%-0)1& 4 5% an ¬ .0375 ",%%%

y(.-5


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% " & - n1" n. . .)erm

2 2 2 2 . . . 2

% " & - n1" n. . .2 2 2 2 . . . 2

2 periodic payment of the annuity duen total number of paymentsi interest per conversion period

S amount of an annuity due

+ present value of an annuity due

'.9.1 Present 6alue o* an nnuity Due

8he present value of an annuity due, which is denoted by +, is the sum of thepresent values of the payments.

8he formula in 6nding the present value of an annuity due is+ ("st payment) 4 (present value of remaining payments)+

Example "=arlon purchased a car. e paid $"%,%%% down payment and $"%,%%%

payable at the beginning of each month for years. If money is worth "&'compounded monthly, what is the eNuivalent cash price of the car/

Given:2 $"%,%%% 7 "&'

9own payment $"%,%%% i 12

12 – "'

m "& + / ?ash CNuivalent 1 /t yearsn 3%

+ 2 4 Ran−1¬i

+ $"%,%% 4 ₱ 10,50060−1¬1

+ $"%,%% 4 $"%,%% a59¬1

+ $"%,%% 4 $"%,%% [1−(1.01)−59

.01 ]+ $"%,%% 4 $*33,&*


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8he formula in 6nding the amount of an annuity due is; Z>alue of an annuity of (n 4") payments on the last payment date[ – 2;

Example "If $%% is deposited in a ban! at the beginning of each - months for "%

years and money is worth


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; ₱ 5,000 [(1.045)13−1

.045 ] 1$,%%%; $&&,


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& +n investment of $,%%% is made at the beginning of each month for < yearsand 0 months. If interest is "&' compounded monthly, how much will theinvestment be worth at the end of the term/

- +t retirement, =r. Golram Bcampo 6nds his share f a pension fund is

$&,%%,%%%. What payment will this provide at the beginning of each monthfor & years, for him or his estate, if the fund invested at "


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% " & - d d" d& d4n1& d4n1" d4n. . . deferment period n periods

. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d

% " & - * 3 0 < 5 "%& yrs 0

"%

"" &5222222222222 \ \ \ \ \ \ \ \ \

22222 \ \ \

*%

'.: De*erred nnuity

Is an annuity whose term does not begin until the epiration of aspeci6ed time

8he formula in 6nding the present value Ad of the deferred annuity is

Ad Zpresent value of annuity with term (d 4 n) periods[ – Zpresentvalue of annuity with term d periods[

Where Ad present value of deferred annuity

d=¿ value of deferment

n=¿ total number of payments

R=¿ periodic payment of the deferred annuity

i=¿ interest per conversion period

Example "+ seNuence of Nuarterly payments of $-,%% each, with the 6rst one due

at the end of - years and the last of the end of "% years. #ind the presentvalue of the deferred annuity, if money is worth "3' compounded Nuarterly.Given:

2 $-,%% 7 "3' m *

Solution:

i

16

4 *'

d - * "&Q "& – " ""n 0 * &


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% " & - * 3 0 < 5 "% "" "& "-& yrs

&%2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 \ \ \ \ \ \ \ \ \ \

&

+d

Ad Ran+d ¬ i− Rad ¬ i

Ad ₱ 3,500a29+11¬4 − ₱ 3,500a11¬4

Ad ₱ 3,500a40¬ 4− ₱ 3,500a11¬4

Ad $-,%% [1−(1.04)

−40

.04 ] 1 $-,%% [1−(1.04)

−11

.04 ] Ad $35,&0*.0"1$-%,33".30

Ad $-


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Example "+le borrows $"%%,%%% with interest at the rate "&' compounded semi1

annually. he agrees to discharge his obligation by paying a seNuence of "%

eNual semi1annual payments, the 6rst being due at the end of 1

2 years. #ind

the semi1annual payment.

Given: Ad $"%%,%%

D "&'m &n "%d "%

i 12

2 3'

Solution:

2 Ad

ad+n¬i−ad¬ i

2 P100,000

a20+6 ¬ i−a10¬6

2

P100,00

[1−(1.06)

−20

.06 ]−[1−10.06 ]

2 P100,000

11,46992122−7.360087051

2 P100,000

4.109834169

2 $&*,--".


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the end of - years, if money is worth "*' compounded semi1annually, 6ndthe cash value of the house and lot.

- #ind the present value of a series of Nuantity payments of $&,%% each, the6rst payment is due at the end of * years and 3 months, and the last at the

end of "% years and - months, if money is worth "3' compounded Nuarterly.

* #ind the cash eNuivalent of a computer set that sells for $"%,%%% cash and aseNuence of "% semi1annual payments of $-,%% each, the 6rst is due at the

end of - years and 3 months, if money is worth "%1

2 ' compounded semi1

annually.

Dason borrows $%,%%% and agrees to pay his obligation by ma!ing " eNualannual payments, the 6rst is due at the end of - years . #ind the annualpayment, if money is worth "&'.

3 2yan obtains a loan of $&%,%%% with interest t "*' compounded Nuarterly.e will discharge his debt by a seNuence of eNual Nuarterly payments, the6rst is due at the end of years and the last at the end of "% years 6nd theperiodic payment.

0 #ind the present value of the pension of a man, now 3% years old, and whowill receive a pension of $"%,%%% per month for " years, with the 6rstpayment to occur one month after his 3th birthday, if money is worth "


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% " & - * 3 0 < 5 "% "" "& "- "* " yr. & yrs. - yrs. * yrs. yrs. 3 yrs. 0 yrs.

2 2 2 2 2

+ seNuence of semi1annual payments of $&,%% each will start with a

payment at the end of &1

2 years. If money is worth "&' compounded semi1

annually, 6nd the sum of the values of these payment at the end of a)& years,

b)*

1

2 years, c) 3 years, d)the actual present value of the payments.

SivenQ2 $&, %%

7 "&'m &

i 12

2 3'

n

8ime 9iagram

Solution%

a) +t the end of & years b) +t the end of 4 1

2 years

+] $ &, %% a5¬6 ; $&,%% a5¬6

+& $ &, %%

1−(1.06 )⁻ ⁵.06 ; $&,%%

[ (1.06 )−5−1

.06

] +] $ "%, -%.5" ; $"*, %5&.0-

c) 8o 6nd the value of the annuity at the end of 3 yrs, accumulate a) for * years, or

the result of b) for "1

2 years.

>alue at the end of 3 years is $&,%%( a5¬ 6 )(".%3)<

$"3, 0alue at the end of 3 years is $&,%%( a5¬6 )(".%3)-

$"3,05*.30d) 8o 6nd the actual present value, discount the result of a) for & years or the

result of b) for *1

2 years.

+ctual Present >alue $&,%%

a

¿¿¿

)(".%3)1*

$


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+ctual Present >alue $&,%%( a5¬6 )(".%3)-

$ation and Sin3ing Fund

+mortiMation+mortiMation is a means of repaying a debt by a series of eNual time interval.

the periodic payments form an annuity in which the present value is the principalof an interest1bearing debt. ence we use the following annuity formulasQ

+ 2 [ 1−(1+i) ⁻ ⁿi ]


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% " & - *

-,""5.%5-,""5.%5-,""5.%5

2 2 2 2 2 2 2 2yrs

remaining liability after the th payments is the present value of the remaining periodic payments.

2 [ Ai1−(1+i)⁻ⁿ ]&'ere: + principal

2 periodic payment i interest per period n total number of payment periods

Examples: ". +n obligation of $&",%%% with interest of


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*. "-,


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0. +ster borrows $-3%, %%%. ;he plans to amortiMe her debt with eNual Nuarterlypayments for & years. Interest is allowed at ""' converted Nuarterly.a) #ind the Nuarterly cost of his debt.b) Hy how much is the debt reduced by the -rd payment/c) ?onstruct the amortiMation schedule.


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;-,%%%(3.""-3-"


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3 *,


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2 $3,-*".%-

8he interest payable Nuarterly to the creditor is $*%,%%%(.%&&) $5%%. 8herefore, the Nuarterly epense of the debt is $3,-*".%- 4 $5%% $0

Chapter Depreciation- Capitali>ation and Perpetuities

Bb7ectives +t the end of the lesson students should be able to a. calculate the rate of depreciation, periodic payments using di@erent methods

b. participate actively in class discussion.1 )ypes o* depreciation". Gormal 9epreciation

a. P'ysical Depreciation− is due to the lessening of the ability of a property toproduce a result

b. Functional Depreciation−is due to the lessening in the demand for the function,which the property was designed to render.

&. 9epreciation 9ue to ?hanges in Price Levels-. 9epletion−refers to the decrease in the value of a property due to the gradualetraction of its contents.Formula: WP−L Q H P− C where Wwearing value H boo! value

Pprice C accrued value Lsalvage value

.# )hree Most Common )ypes o* Depreciation1. Straight?line Method

9enote P original cost L scrap valueAresaleAsalvage value G number of yearAs (estimated life)

9 annual depreciation d "An Caccrued value (t9) H boo! value Formulas:

9 P – L

n or 9 d (W)

C t9Q HP1C or HP1 t9In ;traight line method, the total depreciation over the given life of a property is evenlydistributed, and this yearly depreciation is congruent to each other during the propertyKsestimated life or period in n years.

8hus, d d"d&d-RRdn9An 8he total depreciation 9 after a certain period of time or at the end of n years can beobtained by the given !ormula: 9nn (d) Where 9 total depreciation of the property over a period of time` n number of years (estimated)

d annual depreciation+ boo! value after a particular yearFormula: H P1n (d)


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Where H boo! value P original cost 9 annual depreciation G number of yearExample: + motorcycle cost is $


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5−115

.&3333339d& (W)

.&333333($",%%)

$*%%

d-n−215

5−215

.&9 d- (W) .& ($",%%) $-%%

d* n−315

5−315

."------9 d* (W)

."------($",%%)

$&%%

d n−415

(5−4 )15

.%------9 d (W)

.%------($",%%) $"%%

'. Sin3ing Fund Method". W r where 2 – replacement deposit

; n idepreciation charge for any year increase in the depreciation fund at the end of theyear.

the amount of fund after the (!1")th is&. 2s!1" i the depreciation charge at the end of the !th year.-. 9! 2s! i 2s!1" iExample:

+ machine which costs $&%,%%% will have a scrap value of $&,%%% when worn out atthe end of years. :nder the sin!ing fund method at ', 6nd the annual charges fordepreciation. Prepare a schedule of depreciation.Given P $&%,%%%L $&,%%%Solution: a. W P−L

$&%,%%% −$&,%%% $"


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(9) (C)

% $% $% $% $% $&%,%%%.%% " $%.%% $-,&0. $-,&0. $-,&0. $"3,0*&.* & $"3&.


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.' Depreciation ,y a Constant Percentage o* Declining 2oo3 value 9epreciation charge for any year (boo! valueA the beg of the year) F d

d 1

n Q (natural rate of depreciation)

d 2n (#ederal revenue act of "5*)

Example:". Hy the method using a 6ed percentage of declining boo! value with rate eNual or

twice the natural rate, 6nd the depreciation charges and boo! value during the lifeof $*,%%% with life of - years.

Solution:

8he natural rate of depreciation is1

n which is 1

3

ence d

2

n , d

2

3 .33333 or 30 '

9 .30 ($*, %%%) – the depreciation charge at the end of 6rst year.9 $&,3


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L$3%% $"%,%%%−$3%% ₱ 9,400

12

n"& years $5,*%% $0


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+ R

i , " j

m (Present >alue of ;imple Brdinary Perpetuity)

Example "If money is worth 3' compounded Nuarterly, 6nd the present valuea. Bf a perpetuity of $"%% payable Nuarterly

b. Bf an annuity of $"%% payable Nuarterly for &% yearsSolution:Given:

D 3' m * 2 $"%% t &% years

a. + R

i ₱ 100

.015

$3,333.30b. 8o solve the annuity use 2an i

Where an i " − (" 4 I )1n

i $"%% a


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12monts

6moonts &

i .10

2 .%

+ $ "%,%%%. ".% s &.%

+ $&%%,%%% (.*


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Example:". It is estimated that the maintenance of a certain section of a L28 railroad will

reNuired $&,%%% per !ilometer at the end of every * years. If money is worth"%', 6nd the capitaliMed cost of the maintenance per !ilometer.

Solution:2eplacement cost per period

2 W $ &,%%% s ! I s * "%'

₱ 2,000

4.641

$*-%.5*"3%0*?apitaliMed cost per !ilometer

? R

I ₱ 430.9416074

0.1

? $ *,-%5.*&? W $ &,%%%

is ! i %."(*.3*")? $*,-%5.*&

Formula:+nnual investment cost

= Pi 4W , W P − L;n i

P Briginal costW Wearing valueL ;crap value

?apitaliMed costs Hased +nnual Investment ?ost of the +sset Example:

&. #ind annual investment cost (=) and also the capitaliMed cost () of the machineby 6rst 6nding the value of and then using = i. =oney worth *' machinewill cost $",%%% life < years#inal salvage value $-,%%%

Solution:=ethod "

W P – L P $",%%% L $-,%%%W $",%%% − $-,%%%W $"&,%%%

P 4 W is ! i

$",%%% 4 $ "&,%%% (%.%*)(s


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= $",5%&.--=ethod &

= Pi 4 W , W P − Lsni

= ($",%%%)(%.%*) 4 $ "&,%%% s


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$"%,%%% P"%,%%% $"%,%%%(" 4."%)" "",%%%$"%,%%%(" 4."%)& "&,"%%

; P--,%%%

8erm

% " & -

$"%,%%%(" 4 ."%)1" $5,%5%.5" $"%,%%%(" 4."%)1&


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; sum of the accumulated values of 2 at the end of the term(") ; 2 4 2(" 4 i) 4 . . . 2(" 4 i)n1- 4 2(" 4 i)n1& 4 2(" 4 i)n1"

multiplying (") by ("4i)(&) (" 4 i); 2(" 4 i) 4 2(" 4 i)& . . . 2(" 4 i)n1- 4 2(" 4 i)n1" 4 2(" 4 i)n

subtracting (") from (&) we get

(" 4 i); – ; 2(" 4 i)n –2i; 2V(" 4 i)n –"U

; R (1+ i)n−1

1

or

if we let

the formula for ; can be written as

where ; id the amount of an ordinary annuity2 is the periodic payment

I j

m interest rate per period

G mt number of payments

8he symbol, ;n I, is read as X; angle n at IY

8o derive the present value of an ordinary annuity we use the relation

+ ;(" 4 i)1n

+ 2;n i(" 4 i)1n

if we let

then the formula of + can be written as

where + is the present value of the annuity an I, is read as Ya angle n at iY

Illustrative Example "#ind the amount and present value of an annuity of $",%% payable for

& years if money is worth "% compounded semi1annually.Siven 2 P",%%

; 2

;n i

;n i

;n i

+ 2

n

+ 2

−n

n i −

+

an i −

i j

m .10

2 .%

n mt &(&) *


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7 "%' 1 ."%i & yearsm &

8o 6nd ;

; 2sn i; ",%%svn .%; ",%%(*.-"%"&); P3,*3."5

or ; 2 [(1+.05)4−1

i ]s ",%% [(1+.05)

4−1.05 ]

s ",%%(*.-"%"&)

s $3,*3.*

to 6nd ++ 2an i+ ",%%* .%+ ",%%(-.*5)+ P,-"alue $%&,0*.5*

or by using the relation+ ;(" 4 i)1n

+ 3,*3."5(" 4 .%)1*

+ P,-"


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'.0 Periodic Payment o* an $rdinary nnuitya.) Periodic payments of ;

b.) Periodic payment of +

Illustrative Example "In order to have $-%%,%%% at the end of " years, how much must be

deposited in a fund every - months if money Is worth


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2 A

an¬i

13,500

a.36¬.0125

2

13,500

1−(1+.0125 )−36

.0125

2 13,500

28.84726737

Exercise 3.1

". +nalyn bought a dining set. ;he paid $&,%% as down payment and promisedto pay $


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". ow much should one deposit monthly in a fund that earns "&1

4

compounded monthly in order to have $-%,%%% in - years and 3 months/

'. Finding the Interest 5ate o* an $rdinary nnuity 8hese formulas are

a) (n& 1 ")i& – 3(n 1 ")i 4 "& (1−nRS ) %

b) and (n& 1 ")i& 4 3(n 4 ")i 4 "& (1−nRS ) %

Example "+ refrigerator can be purchased for $5,%%% cash down and P3-% a month for"< months. #ind the interest rate charged if it is compounded monthly.

Solution:+ 5,%% – 5%% –


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ere we get two values of i, we disregard the negative value of i. 8herefore,the periodic interest rate is

i -.


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-. + television set costs P"%,**%. It is purchased by a down payment of $*,**%and monthly payments of P -


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%5%5%5% 5%5%

. . ." & - &- &*


%5%5%5% 5%5%

. . ." & - &- &*

% " & - && &- &* &. . .5% 5% 5% 5% 5% 5% y

ordinary annuity of &* payments

Let be the irregular payment to be paid on the &*th payment. =a!e atime diagram using the present as the focal date.

#ocal date

(".%-0)1&* 4 5% a24¬ .0375 ",%%%

(.*"--"5%5) 4 "*,


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% " & - n1" n. . .2 2 2 2 . . . 2

Given:2 $"%,%%% 7 "&'

9own payment $"%,%%% i 12

12 – "'

m "& + / ?ash CNuivalent 1 /

t yearsn 3%

+ 2 4 Ran−1¬i

+ $"%,%% 4 ₱ 10,50060−1¬1

+ $"%,%% 4 $"%,%% a59¬1

+ $"%,%% 4 $"%,%% [1−(1.01)−59

.01 ]+ $"%,%% 4 $*33,&*


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; ₱ 500 s40+1¬ 2− ₱ 500

; $%% [(1.02)41−1

.02 ] 1$%%; $-",-%.%" – $%%

; -%,


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Example "What eNual deposits should be placed in a fund at the beginning of each

year for " years in order to have $",%%,%%% in the fund at the end of "years, if money accumulates "&'/

Given:; $",%%,%%% 7 "&'m " i "&'t " 2 /n "

2 A

Sn+1¬i−1

2 P1,500,000

S15+1¬12−1

2 P1,500,000S16¬12 −1

2 P1,500,000

41.75328042

2 $-,5&.-&

Exercise 3.4#or each problem draw a diagram illustrating the data and the solution

". If money is worth "3' compounded Nuarterly, 6nd the present value and theamount of annuity due of $",%% payable Nuarterly for "% years.

&. +n investment of $,%%% is made at the beginning of each month for < yearsand 0 months. If interest is "&' compounded monthly, how much will theinvestment be worth at the end of the term/

-. +t retirement, =r. Golram Bcampo 6nds his share f a pension fund is$&,%%,%%%. What payment will this provide at the beginning of each monthfor & years, for him or his estate, if the fund invested at "


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. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d

. + house and lot is bought for $",%%%,%%% down payment and $,%%% payableat the beginning of each month for years. What is the eNuivalent cash priceof the house and lot, if interest rate is "*' compounded monthly/

3. Bn Danuary "5, &%%-, =r. Herlindo ;antos opened a saving account for hiswife with an initial deposit of $"%,%%% in a ban! paying 3' compoundedNuarterly. If =r. ;antos continues to ma!e Nuarterly deposits of the sameamount until Duly "5, &%%


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. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d


. . .

(d)Go payments

Payments are made

+d;d

% " & - * 3 0 < 5 "%& yrs 0

"%

"" &5222222222222 \ \ \ \ \ \ \ \ \

22222 \ \ \

*%

8he formula in 6nding the present value Ad of the deferred annuity is

Ad Zpresent value of annuity with term (d 4 n) periods[ – Zpresentvalue of annuity with term d periods[

Where Ad present value of deferred annuity

d=¿ value of deferment

n=¿ total number of payments

R=¿ periodic payment of the deferred annuity

i

=¿ interest per conversion period

Example "+ seNuence of Nuarterly payments of $-,%% each, with the 6rst one due

at the end of - years and the last of the end of "% years. #ind the presentvalue of the deferred annuity, if money is worth "3' compounded Nuarterly.Given:

2 $-,%% 7 "3' m *

Solution:

i

16

4 *'

d - * "&Q "& – " ""n 0 * &


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% " & - * 3 0 < 5 "% "" "& "-& yrs

&%2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 \ \ \ \ \ \ \ \ \ \

&+d

Ad $35,&0*.0"1$-%,33".30

Ad $-


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eNual semi1annual payments, the 6rst being due at the end of 1

2 years. #ind

the semi1annual payment.

Given: Ad $"%%,%%

D "&'m &n "%d "%

i 12

2 3'

Solution:

2

Ad

ad+n¬i−ad¬ i

2 P100,000

a20+6 ¬ i−a10¬ 6

2

P100,00

[ 1−(1.06)−20

.06 ]−[ 1−10.06 ]2

P100,000

11,46992122−7.360087051

2 P100,000

4.109834169

2 $&*,--".


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-. #ind the present value of a series of Nuantity payments of $&,%% each, the6rst payment is due at the end of * years and 3 months, and the last at theend of "% years and - months, if money is worth "3' compounded Nuarterly.

*. #ind the cash eNuivalent of a computer set that sells for $"%,%%% cash and aseNuence of "% semi1annual payments of $-,%% each, the 6rst is due at the

end of - years and 3 months, if money is worth "%1

2 ' compounded semi1

annually.

. Dason borrows $%,%%% and agrees to pay his obligation by ma!ing " eNualannual payments, the 6rst is due at the end of - years . #ind the annualpayment, if money is worth "&'.

3. 2yan obtains a loan of $&%,%%% with interest t "*' compounded Nuarterly.e will discharge his debt by a seNuence of eNual Nuarterly payments, the6rst is due at the end of years and the last at the end of "% years 6nd theperiodic payment.

0. #ind the present value of the pension of a man, now 3% years old, and whowill receive a pension of $"%,%%% per month for " years, with the 6rstpayment to occur one month after his 3th birthday, if money is worth "


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% " & - * 3 0 < 5 "% "" "& "- "* " yr. & yrs. - yrs. * yrs. yrs. 3 yrs. 0 yrs.

2 2 2 2 2

'.:.# 6alue o* an nnuity in an r,itrary DateExample "

+ seNuence of semi1annual payments of $&,%% each will start with a

payment at the end of &1

2 years. If money is worth "&' compounded semi1

annually, 6nd the sum of the values of these payment at the end of a)& years,

b)*1

2 years, c) 3 years, d)the actual present value of the payments.

SivenQ2 $&, %%

7 "&'m &

i 12

2 3'

n

8ime 9iagram

Solution%

a) +t the end of & years b) +t the end of 4 1

2 years

+] $ &, %% a5¬ 6

; $&,%% a5¬6

+& $ &, %% [ 1−(1.06 )⁻ ⁵.06 ] ; $&,%% [ (1.06 )−5−1

.06 ] +] $ "%, -%.5" ; $"*, %5&.0-

c) 8o 6nd the value of the annuity at the end of 3 yrs, accumulate a) for * years, or

the result of b) for "1

2 years.

>alue at the end of 3 years is $&,%%( a5¬6 )(".%3)<

$"3, 0alue at the end of 3 years is $&,%%( a5¬6 )(".%3)-

$"3,05*.30d) 8o 6nd the actual present value, discount the result of a) for & years or the

result of b) for *1

2 years.


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+ctual Present >alue $&,%%

a

¿¿¿

)(".%3)1*

$ation and Sin3ing Fund

+mortiMation


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% " & - *

-,""5.%5-,""5.%5-,""5.%5

2 2 2 2 2 2 2 2yrs

remaining liability after the th payments is the present value of the remaining periodic payments.

+mortiMation is a means of repaying a debt by a series of eNual time interval.the periodic payments form an annuity in which the present value is the principalof an interest1bearing debt. ence we use the following annuity formulasQ

+ 2

[1−(1+i)⁻ ⁿ

i

]2 [ Ai1−(1+i)⁻ ⁿ ]

&'ere: + principal 2 periodic payment i interest per period n total number of payment periods

Examples: ". +n obligation of $&",%%% with interest of


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+ -,""5.%< ⁻1−(1.04)−3

0.04 ⁻

$


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. +n item worth $--, 0%% is purchased for a down payment of $, %%%. Interestis computed at 3' converted monthly and the balance to be settled withpayment of $", %% at the end of each month.a) 9etermine the number of $", %%1payment needed.b) #ind the siMe of 6nal payment.

c) ?onstruct the amortiMation schedule.

3. + loan of $"*, %%% bearing a "' interest converted annually is to beamortiMed with eNual yearly payments for "% years.a) #ind the siMe of each payment.b) What should the outstanding liability be 7ust after the *th payment/c) ?onstruct the amortiMation schedule.

0. +ster borrows $-3%, %%%. ;he plans to amortiMe her debt with eNual Nuarterlypayments for & years. Interest is allowed at ""' converted Nuarterly.a) #ind the Nuarterly cost of his debt.

b) Hy how much is the debt reduced by the -rd payment/c) ?onstruct the amortiMation schedule.


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2 S (i)

1−(1+i)⁻ⁿ

Illustrative Examples". + fund is created by ma!ing eNual monthly deposits of $-,%%% at 5'converted monthly.

a. 9etermine the sum after half year.b. What is the amount in the fund after the *th deposit/c. ?onstruct the sin!ing fund schedule for a 31month period.

Siven 2 $-,%%% 7 5' or .%5 m "&

Solution:

a) ;-,%%%[ (1.0075 )6−1]

.0075

;-,%%%(3.""-3-"


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2 30,000(.05)

[(1.015)6−1 ]

2 450

.093443263

2 $*,


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Illustrative Example". 8he principal of a loan $*%,%%% will be at the end of " years by the

accumulation of a sin!ing fund by Nuarterly deposits, and interest will bepayable on the debt Nuarterly at the rate of 5'.

a. #ind the Nuarterly epense of the loan to debtor if his sin!ing fund isinvested at

Math Lessons

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