Further'Mathematics'2018' Core:'Recursion'and'Financial'modelling' … · 2018-06-06 · Page%2%of%20 Table'of'Contents $ 9A#Combining#geometric#growth#and#decay#_#3% Worked%Example%1%_%3%

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Further'Mathematics'2018'Core:'Recursion'and'Financial'modelling'Reducing8balance'Loans'and'Annuities'Key knowledge •! Amortisation%of%a%reducing%balance%loan%or%annuity%and%amortisation%tables%•! Reducing%balance%loans,%annuities,%perpetuities%and%annuity%investments.%•! The%difference%between%nominal%and%effective%interest%rates%and%the%use%of%effective%interest%rates%

to%compare%investment%returns%and%the%cost%of%loans%when%interest%is%paid%or%charged,%for%example,%daily,%monthly,%quarterly%�%

$

Key skills •! Use%a%table%to%investigate%and%analyse%on%a%step–byAstep%basis%the%amortisation%of%a%reducing%

balance%loan%or%an%annuity,%and%interpret%amse%a%table%to%investigate%and%analyse%on%a%step–byAstep%basis%the%amortisation%of%a%reducing%balance%loan%or%an%annuity,%and%interpret%amortisation%tables%

•! Using%a%CAS%calculator,%solve%practical%problems%associated%with%compound%interest%investments%and%loans,%reducing%balance%loans,%annuities%and%perpetuities,%and%annuity%investments.%

$

Chapter$Sections Questions$to$be$completed$

9A$Combining%geometric%growth%and%decay All%9B$Analysing%reducingAbalance%loans%with%recurrence%relations All%9C$Using%financial%solver%to%analyse%reducingAbalance%loans All%9D$InterestAonly%loans% All%9E$Annuities% All%9F$Perpetuities% All%9G$Annuity%investments% All%

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Table'of'Contents$

9A#Combining#geometric#growth#and#decay#______________________________________________#3%Worked%Example%1%______________________________________________________________________________%3%Worked%Example%2%______________________________________________________________________________%3%

Using%CAS%Calculator% __________________________________________________________________________%3%

9B#Analysing#reducing:balance#loans#with#recurrence#relations# ______________________________#4%ReducingAbalance$loans$(recurrence$relation)$ __________________________________________________$4%

Worked%Example%3%______________________________________________________________________________%4%Worked%Example%4%______________________________________________________________________________%5%

Amortisation$tables$_______________________________________________________________________$6%Worked%Example%5%______________________________________________________________________________%6%

Worked%Example%5%on%CAS%calculator% _____________________________________________________________%7%

9C#Using#finance#solver#to#analyse#reducing:balance#loans#__________________________________#8%Using$Finance$Solver$for$ReducingABalance$Loans$_______________________________________________$8%

Worked%ExampleA%Monthly%repayments%and%total%interest%_______________________________________________%9%Worked%Example%on%CAS%calculator%_______________________________________________________________%9%

Worked%ExampleA%Repayment%amount%&%Principal%repaid%________________________________________________%9%Worked%ExampleA%No.%of%repayments%and%total%interest%________________________________________________%10%Worked%ExampleA%Effects%of%changing%repayment%amounts% _____________________________________________%11%Worked%ExampleA%Effects%of%changing%interest%rate%____________________________________________________%12%

9D#Interest#only#loans#_______________________________________________________________#13%Worked$Example$9$_______________________________________________________________________$13%

Worked$Example$10$______________________________________________________________________$13%

9E#Annuities#_______________________________________________________________________#14%Example%11%___________________________________________________________________________________%14%

Amortisation$tables$for$annuities$___________________________________________________________$15%

Using$Finance$Solver$for$Annuities$__________________________________________________________$15%Worked%Example%13%____________________________________________________________________________%16%

9F#Perpetuities#_____________________________________________________________________#17%Worked%Example%14%____________________________________________________________________________%17%Worked%Example%15%____________________________________________________________________________%17%Worked%Example%16%____________________________________________________________________________%18%

9G#Annuity#investments#_____________________________________________________________#19%Worked$Example$17$______________________________________________________________________$19%

Amortisation$tables$for$annuity$investments$__________________________________________________$20%Using%financial%solver%for%annuity%investments%________________________________________________________%20%Worked%Example%19%____________________________________________________________________________%20%

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9A'Combining'geometric'growth'and'decay'In%the%previous%chapter%we%looked%at%using%recurrence%relations%to%model%financial%situations%with%linear%and%geometric%growth%and%decay%separately.%They%can%also%be%used%to%model%situations%that%involve%elements%of%both%linear%and%geometric%growth%and%decay.%

The%number%of%trout%in%a%farm,%for%example,%will%grow%geometrically%over%time,%but%selling%trout%that%have%matured%will%also%impact%the%number%of%trout%present%in%the%farm%at%any%given%time.%Another%example%of%this%is%with%a%personal%loan.%The%initial%amount%is%borrowed%and%interest%is%charged%at%particular%time%periods.%This%value%needs%to%be%added%to%the%balance%of%the%loan.%But%money%is%also%paid%off%the%loan,%which%then%needs%to%be%subtracted%from%the%loan.%

A%recurrence%relation%in%the%form%

!" = $%&'%()*+,&-./, !123 = 4 × !1 ∓ 7%can%be%used%to%model%situations%that%involve%both%geometric%and%linear%growth%or%decay.%

Worked Example 1 Write%down%the%sequence%generated%by%the%recurrence%relation%!" = 3, !123 = 4!1 − 1%%%%%%%%%Worked Example 2 The%number%of%trout%in%a%fish%farm%pond%after%n%months,%Tn,%can%be%modelled%using%the%recurrence%relation%!" = 10000,!123 = 1.1!1 − 3000%

a.! Use%the%recurrence%relation%to%determine%the%number%of%trout%in%the%pond%after%2%months.%%%%%%%

b.! After%how%many%months%will%there%be%no%trout%left%in%the%pond%Using CAS Calculator%Enter%the%starting%value%and%press%enter%%Type%in%the%calculator%‘×%1.1%–%3000’%and%press%enter%%Find%the%first%year%where%the%value%is%below%zero%and%establish%how%many%months%it%has%been%%%

%%%%%

Starting%value%

1st%month%

2nd%month%

3rd%month%

4th%month%

5th%month%

%%

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There%will%be%no%trout%left%in%the%pond%after%5$months%

%

9B'Analysing'reducing8balance'loans'with'recurrence'relations'Reducing-balance loans (recurrence relation) When%money%is%borrowed%from%the%bank,%people%tend%to%pay%off%a%certain%amount%of%money%at%regular%time%periods.%The%amount%paid%off%is%calculated%to%reduce%the%balance%of%the%loan%to%zero%over%time.%When%this%occurs,%this%type%of%loan%is%called%a%reducingAbalance%loan.%It%is%effectively%a%compound%interest%loan,%with%regular%payments.%Personal%loans%and%mortgages%(home%loans)%are%examples%of%reducingAbalance%loans.%%

%%

Worked Example 3 Alyssa%borrows%$1000%at%an%interest%rate%of%15%%per%annnum,%compounding%monthly.%She%will%repay%the%loan%by%making%four%monthly%payments%of%$257.85.%Construct%a%recurrence%relation,%in%the%form%

!" = >'()?(>&-, !123 = 4!1 − 7%where%!1%is%the%balance%of%the%loan%after%n"payments.%%%%%%%%%%

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Once%we%have%the%relation%we%can%use%it%to%determine%things%such%as%the%balance%of%the%loan%after%a%given%number%of%payments.%%%%%% Worked Example 4 Alyssa’s%loan%can%be%modelled%by%the%recurrence%relation%!" = 1000, !123 = 1.0125!1 − 257.85%

a.! Use%your%calculator%to%determine%recursively%the%balance%of%the%loan%after%Alyssa%has%made%each%of%the%four%payments.%

Using CAS Calculator%Enter%the%starting%value%and%press%enter%%Type%in%the%calculator%‘×%1.0125%–%257.85’%and%press%enter%%Find%the%value%of%the%loan%after%4%iterations%%%%%The%balance%of%the%loan%after%4%payments%is%$0.04$

%%%%%

%

Starting%value%

1st%payment%

2nd%payment%

3rd%payment%

%4th%payment%

%%

%

b.! What%is%the%balance%of%the%loan%(the%amount%she%still%owes)%after%she%has%made%two%payments?%Give%your%answer%to%the%nearest%cent.%

%

Starting%value%

1st%payment%

2nd%payment%

3rd%payment%

%4th%payment%

%%

c.! Is%the%loan%fully%paid%out%after%four%payments%have%been%made?%If%not,%how%much%will%the%last%payment%have%to%be%to%ensure%that%the%loan%is%fully%repaid%after%4%payments?%

%

Starting%value%

1st%payment%

2nd%payment%

3rd%payment%

%4th%payment%

%

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% Amortisation tables When%paying%off%a%loan%it%is%often%wise%to%follow%its%progress%through%the%life%of%the%loan.%The%amortisation$of% the% loan% can% be% tracked% on% a% stepAbyAstep% basis% by% following% the% payments% made,% the% interest% and%reduction%in%the%principal.%Amortisation$is%defined%as%the%regular%decrease%in%value%(depreciation)%of%an%asset%or%the%paying%off%a%debt%over%time%through%regular%repayments.%

For%example,%Alyssa’s%loan%from%Example%4%was%a%loan%of%$1000%with%interest%rate%charged%at%1.25%%per%month.%The%loan%was%repaid%with%4%monthly%payments%of%$257.85.%Below%is%the%amortisation%table%for%Alyssa’s%loan.%

%%

Worked Example 5 A%business%borrows%$10000%at%a%rate%of%8%%per%annum.%The%loan%is%to%be%repaid%by%making%four%quarterly%payments%of%$2626.20.%Complete%the%amortisation%table%for%this%loan%(adjust%the%final%payment%so%the%loan%is%completely%paid%off%after%).%Payment$$ Principal$outstanding$($)$

(=loan%outstanding%from%previous%year)%

Interest$due$($)$(= DEFF

× GDHIJHGKL)%Payment$($)$

$Loan$outstanding$($)$

(=Principal%outstanding+interest−repayment)%

0$ % 0% 0% 10000%

1$ 10000% M÷O3""

× 10000 =+200% 2626.20% 10000%+%200%A%2626.20=%7573.80%

2$ 7573.80% % % %

3$ % % % %

4$ % % % %

%%%%%%

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%%%%%%

Worked Example 5 on CAS calculator •! Enter%the%labels%“n+1”,%“Vn”,%“Pmt”%(for%Payment),%“Vn+1”%Note:"You"can’t"use"+"on"the"CAS"so"spell"it"out"%•! Next%enter%1%to%5%in%column%A,%and%the%starting%values%for%

Vn=b2=10,000,%Pmt=c2=2626.20%in%cells%b2%and%c2%respectively.%

•! In%cell%d1%insert%the%equation%%

= P2Q1 +8 ÷ 4100

S − ?2%

Note:"where"0.625"is"r"the"interest"rate"per"period"(7.5/12)""%

%•! In%cell%b3%enter%

=d2%%%

This%is%just%using%the%previous%answer%as%the%starting%value%of%the%next.%

%Now%fill%down%the%equations%of%cells%b3,%c2%and%d2,%downward%for%each%of%columns%b,%c%and%d.%%

%%%%%%%%%%%%%%% %

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9C'Using'finance'solver'to'analyse'reducing8balance'loans'As%seen%in%the%previous%exercise,%reducingAbalance%loans%are%loans%where%interest%is%added%during%each%time%period,%but%payments%are%also%taken%from%the%loan,%hence%reducing%the%balance%each%time%period.%%%One%of%the%most%common%types%of%reducing%balance%loans%are%home%loans.%Typical%home%loans%are%large%sums%of%money%borrowed%and%then%repaid%with%regular%payments%over%a%number%of%years.%Generally%the%payments%are%made%fortnightly%for%a%period%of%25A30%years.%Using%Finance$Solver$in%the%CAS%calculator%makes%analysing%these%loans%much%easier.%%%

%%%Using Finance Solver for Reducing-Balance Loans $Paying$off$a$loanA$REDUCING%BALANCE%LOANS%%Entry$ Sign$ Description$

PV% +ve% The%amount%of%money%that%is%given%to%the%person%at%the%start%of%the%loan%

Pmt% Ave% The%amount%of%each%payment%made%to%the%bank%to%pay%off%the%loan%

FV% Ave% The%amount%of%money%stull%owing%to%the%bank%after%N%payments%have%been%made%

0% Nothing%is%owing%to%the%bank%(the%loan%is%fully%paid%off)%Note:%If%FV%is%+ve,%this%represents%the%amount%of%money%overpaid%which%is%to%be%refunded.%%

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Worked Example- Monthly repayments and total interest Rob%wants%to%borrow%$2800%for%a%new%sound%system%at%7.5%%p.a.,%interest%adjusted%monthly.%a)! What%would%be%Rob’s%monthly%repayment%if%the%loan%is%fully%repaid%in%1%½%years?%b)%%%What%would%be%the%total%interest%charged?%

Worked Example on CAS calculator Using%the%Financial%Solver%%Enter%the%following:%

n"(N:)%=%_______%r"(I(%):)%=%_______�%V0"(PV:)%=%_______�"Pmt:%=%_______%%Vn"(FV:)%=%_______�%PpY:%=%_______�%CpY:%=%_______%

Place%the%cursor%on%Pmt:.%%Press%ENTER%o%to%solve.% %Total%Interest%paid%=%total%repayments%–%amount%borrowed%%

Total%interest%=%164.95%x%18%–%2800%%%%%%%%%%%%%%%%%%%%%%%%%%=%2969.10%–%2800%%%%%%%%%%%%%%%%%%%%%%%%%%=%$169.10%

Worked Example- Repayment amount & Principal repaid Josh%borrows%$12%000%for%some%home%office%equipment.%He%agrees%to%repay%the%loan%over%4%years%with%monthly%instalments%at%7.8%%(adjusted%monthly).%Find:%a)%the%instalment%value.%%% Calculate%the%value%of%n:%n%=%4%x%12%=%48%Using%the%Financial%Solver%%Enter%the%following:%

n"(N:)%=%�_______%r"(I(%:)%=%�_______%P"(PV:)%=%�_______"Pmt:%=%�_______"FV:%=%�_______%PpY:%=%�_______%CpY:%=%�_______%

Place%the%cursor%on%Pmt:%Press%ENTER%to%solve.%�% %The%monthly%repayment%over%the%4%year%period%is%$291.83%

b)%the%principal%repaid%and%interest%paid%during%the:%%i)%10th%repayment%

To%calculate%this%we%need%to%find%the%difference%between%the%9th%and%10th%repayments.%Using%the%CAS%financial%solver,%this%means%we%need%to%find%the%amount%owed%(FV)%after%the%9th%and%10th%payments.%Using%the%Financial%Solver%%Enter%the%following:%r"(I(%))%=%7.8�%P"(PV):%=%12%000�"Pmt:%=%�A291.82"(FV):%=%unknown%�%PpY:%=%12�%CpY:%=%12��%

%With"n"(N:)%=%9%Place%the%cursor%on%FV:%Press%ENTER%o%to%solve.%�%


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Principal%owing%after%9th%repayment%is%$10024.73,%Principal%owing%after%10th%repayment%is%$9798.06.%So,%the%principal%repaid%during%the%10th%repayment%is%$10024.73%A%$9798.06%=%$226.67%If%$291.83%is%the%monthly%repayment%and%$226.67%is%the%principal%repaid,%then%the%interest%paid%is:%

$291.83%A%$226.67%=%$65.16%Answer%in%words,%In%the%10th%repayment,%$226.67%of%the%Principal%is%repaid%and%$65.16%interest%is%paid.%

ii)%40th%repayment%

%Using%the%Financial%Solver%%Enter%the%following:%r"(I(%)):%=%_______�%P"(PV):%=%_______�"Pmt:%=%_______"(FV):%=%_______�%PpY:%=%_______%CpY:%=%_______�%



Principal%owing%after%39th%repayment%is%$2543.10,%Principal%owing%after%40th%repayment%is%$2267.80.%So,%the%principal%repaid%during%the%40th%repayment%is%$2543.10%A%$2267.80%=%$275.30%So,%if%$291.83%is%the%monthly%repayment%and%$275.30%is%the%principal%repaid%then%%

$291.83%A%$275.30%=%$16.53%In%words,%In%the%40th%repayment,%$275.30%of%the%Principal%is%repaid%and%$16.53%interest%is%paid.%This%makes%sense!!%At%the%40th%repayment%there%is%less%money%owed%so%therefore%there%is%less%interest%to%pay.%%

%Worked Example- No. of repayments and total interest A%reducing%balance%loan%of%$60%000%is%to%be%repaid%with%monthly%instalments%of%$483.36%at%an%interest%rate%of%7.5%%p.a.%(debited%monthly).%Find:%%a)%the%number%of%monthly%repayments%(and,%hence,%the%term%of%the%loan%in%more%meaningful%units)%needed%to%repay%the%loan%in%full%Using%the%Financial%Solver%%Enter%the%following:%

n"(N:)%=%unknown%�%r"(I(%):)%=%_______��%P"(PV:)%=%�_______�"Pmt:%=%�_______�"FV:%=%�_______�%PpY:%=%�_______�%CpY:%=%�_______�%

Place%the%cursor%on%N,%Press%ENTER%to%solve.%� %Answer:%n%=%240%months%is%240/12%=%20%years,%%

Hence,%the%term%of%the%loan%needs%to%be%20%years%

%% %

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b)%the%total%interest%charged%TUVWX+YZV[\[]V+ = +TUVWX+^[_W`a[ZV]+– +c\dZed_WX+^[_Wdf$

%%%%%%%

Worked Example- Effects of changing repayment amounts Brad% borrowed% $22% 000% to% start% a% business% and% agreed% to% repay% the% loan% over% 10% years%with% quarterly%instalments%of%$783.22%and%interest%debited%at%7.4%%p.a.%However,%after%6%years%of%the%loan%Brad%decided%to%increase%the%repayment%value%to%$879.59.%Find:%

a)%the%actual%term%of%the%loan%

% Calculate%the%value%of%n:% n%=%6%x%4%=%24%

Using%the%Financial%Solver*%%Enter%the%following:%

n"(N:)%=%�_______�%r"(I(%):%=�_______�%P"(PV:)%=%�_______�"Pmt:%=%�_______�"FV:%=%�_______�%PpY:%=%�_______�%CpY:%=%�_______�%

Place%the%cursor%on%FV,%Press%ENTER%to%solve.%��%

Now%we%need%to%find%the%n%value%to%repay%the%loan%in%full,%in%other%words%reduce%$10761.83%to%$0.%%Enter%the%following:%


Place%the%cursor%on%N,%Press%ENTER%to%solve.%�% %

b)%the%total%interest%paid%%%%%%b)! the%interest%saving%achieved%by%increasing%the%repayment%value.%%%%%%

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Worked Example- Effects of changing interest rate Natsuko%and%Hymie%took%out%a%loan%for%home%renovations.%The%loan%of%$42%000%was%due%to%run%for%10%years%and%attract%interest%at%7%%p.a.,%debited%quarterly%on%the%outstanding%balance.%Repayments%of%$1468.83%were%made%each%quarter.%After%4%years%the%rate%changed%to%8%%p.a.%(debited%quarterly).%The%repayment%value%did%not%change.%

a)! Find%the%amount%outstanding%when%the%rate%changed.%%

Using$the$Financial$Solver$$Enter%the%following:%n"(N:)%=%�_______�%r"(I(%):%=�_______�%P"(PV:)%=%�_______�"Pmt:%=%�_______�"FV:%=%�_______�%PpY:%=%�_______�%CpY:%=%�_______��%

b)! Find%the%actual%term%of%the%loan.%%

Using$the$Financial$Solver$$Enter%the%following:%n"(N:)%=%�_______�%r"(I(%):%=�_______�%P"(PV:)%=%�_______�"Pmt:%=%�_______�"FV:%=%�_______�%PpY:%=%�_______�%CpY:%=%�_______�%

%c)%compare%the%total%interest%paid%to%what%it%would%have%been%if%the%rate%had%remained%at%7%%p.a.%for%the%10%years.%%%$ $

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9D'Interest'only'loans'Interest%only%loans%are%where%the%borrower%only%pays%back%the%interest%that%is%charged.%This%means%that%the%amount%paid%is%exactly%the%same%as%the%amount%charged%(interest=repayment),%and%the%balance%remains%unchanged.%

%%

Worked Example 9 Jane%borrows%$50%000%to%buy%some%shares.%Jane%negotiates%and%interestAonly%loan%for%this%amount,%at%an%interest%rate%of%9%%per%annum,%compounding%monthly.%What%is%the%monthly%amount%Jane%will%be%required%to%pay?%%%%%%%%%Worked Example 10 Stuart%borrows%$180%000%to%buy%a%house.%He%negotiates%an%interestAonly%loan%for%this%amount,%at%an%interest%rate%of%7.6%%per%annum,%compounding%fortnightly.%What%is%the%fortnightly%payment,%correct%to%the%nearest%cent?%Now%we%need%to%find%the%Pmt%value%when%the%PV%and%FV%are%the%same%%Enter%the%following:%


Place%the%cursor%on%Pmt,%Press%ENTER%to%solve.%�%%

%%%%%%

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9E'Annuities'Annuities%are%investments%that%provide%regular%payments%over%a%fixed%period%of%time%(e.g.%superannuation).%The%calculations%used%to%model%them%are%identical%to%those%used%to%calculate%reducing%balance%loans,%except%the%future%value%represents%how%much%money%is%left%in%the%investment%instead%of%how%much%is%owing.%%It%can%be%calculated%recursively%by%using%the%recurrence%relation%shown%below.%%

%Example 11 Reza%plans%to%travel%overseas%for%6%months.%He%invests%$12%000%in%annuity%that%earns%interest%at%the%rate%of%6%%per%annum,%providing%him%with%a%monthly%income%of%$2035%per%month%for%6%months.%

a)! model%this%annuity%using%a%recurrence%relation%in%the%form%V0%=%the%principal,%Vn+1%=%RVn"–%D,%where%Vn"is%the%value%of%the%annuity%after%n"payments%have%been%received.%

%%%%%%

b)! Use%your%calculator%to%determine%recursively%the%value%of%the%annuity%after%Reza%has%received%three%payments%from%the%annuity.%

c)! Is%the%annuity%fully%paid%out%after%six%monthly%payments%have%been%made?%If%not,%how%much%will%the%last%payment%have%to%be%to%ensure%that%the%annuity%terminates%after%6%months?%

Enter%the%first%term%of%12000%%Type%×1.005−2035%and%continue%to%press%enter%until%the%6th%payments%%%%%%

%Starting%value%

1st%month%

2nd%month%

3rd%month%

4th%month%

5th%month%

%6th%month%

%

%b.%%

c.$$

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Amortisation tables for annuities An%amortisation%table%can%be%used%to%summarise%the%key%properties%of%an%annuity.%It%shows%the%payment$number,$payment$received,$interest$earned,$principal$reduction$and%balance$of$the$annuity.%The%amortisation%table%for%Reza’s%annuity%(the%above%example)%can%be%seen%below.%

%%

%%%

%Note:%the%last%payment%was%increased%by%88%cents%so%the%balance%was%exactly%$0.00%after%6%payments%had%been%made.%

Total$return$from$the$annuity$=$the$sum$of$payments$made$

Total$interest$earned$=$total$payments$received$–$principal$

$

$

$

Using Finance Solver for Annuities Receiving$paymentsA$ANNUITY%(PAYMENTS)%Entry$ Sign$ Description$

PV% Ave% The%amount%of%money%that%has%been%put%away%in%the%annuity%account%at%the%start%of%the%payments%

Pmt% +ve% The%amount%of%each%payment%made%to%the%person%from%the%annuity%account%FV% +ve% The%amount%of%money%that%is%still%available%in%the%annuity%account%after%N%payments%

have%been%made%0% Nothing%is%left%in%the%annuity%account%(all%funds%have%been%used)%

Note:%If%FV%is%–ve,%this%is%the%amount%of%money%overdrawn%which%is%to%be%repaid.%%%%%%

Interest earned =interest rate/compounding period × balance

Principal reduction=payment made – interest earned

Balance of loan=previous balance – principal reduction

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Worked Example 13 Joe%invests%$200%000%into%an%annuity,%paying%5%%compound%interest%per%annum,%compounding%monthly.%

a)! If%he%wishes%to%be%paid%monthly%payments%for%10%years,%how%much%will%he%receive%each%month?%%Using%the%Financial%Solver*%%Enter%the%following:%


Place%the%cursor%on%Pmt,%Press%ENTER%to%solve.%��

%%%%

b)! If%he%receives%a%regular%monthly%payment%of%$3000,%how%long%will%the%annuity%last?%Give%your%answer%correct%to%the%nearest%month.%



Place%the%cursor%on%N,%Press%ENTER%to%solve.%��

%%%%

c)! What%interest%rate,%correct%to%one%decimal%place,%would%allow%Joe%to%withdraw%$2500%each%month%for%10%years.%



Place%the%cursor%on%I(%),%Press%ENTER%to%solve.%��

%%

%%%%%

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9F'Perpetuities'A%perpetuity%is%a%type%of%annuity%which%continues%for%an%infinite%period%of%time.%The%way%this%happens%is%the%amount%of%the%payment%that%is%received%from%the%perpetuity%is%equal%to%the%amount%of%interest%that%is%earned%on%it.%This%means%that%when%the%interest%is%earned,%that%exact%amount%is%the%payment%amount,%and%the%balance%of%the%perpetuity%remains%unchanged.%The%calculations%of%a%perpetuity%account%are%exactly%the%same%as%interestAonly%loans.%

%Worked Example 14 Elizabeth%invests%in%her%superannuation%payout%of%$500%000%into%a%perpetuity%that%will%provide%a%monthly%income%without%using%any%of%the%initial%investment.%If%the%interest%rate%of%the%perpetuity%is%6%%per%annum,%what%monthly%payment%will%Elizabeth%receive?%%%%%%%%%%%%Worked Example 15 How%much%money%will%need%to%be%invested%in%a%perpetuity%account,%earning%interest%of%4.2%%per%annum%compounding%monthly,%if%$200%will%be%withdrawn%every%month?%Write%your%answer%to%the%nearest%dollar.%%%%%%%%%%%%%%

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Worked Example 16 A%university%mathematics%faculty%has%$30%000%to%invest.%It%intends%to%award%an%annual%mathematics%prize%of%$1500%with%the%interest%earned%from%investing%this%money%in%a%perpetuity.%What%is%the%minimum%interest%rate%that%will%allow%this%prize%to%be%awarded%indefinitely?%Using%the%Financial%Solver*%%Enter%the%following:%


Place%the%cursor%on%I(%),%Press%ENTER%to%solve.%�� %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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9G'Annuity'investments'When%having%an%investment,%in%order%to%increase%the%rate%at%which%it%grows,%additional%payments%can%be%made%at%a%regular%basis.%

%Worked Example 17 Nancy%plans%to%travel%overseas%when%she%finishes%her%VCE.%She%has%already%saved%$1200%and%thinks%that%she%can%save%an%additional%$50%each%month%that%she%plans%to%add%to%her%savings%account.%The%account%pays%interest%at%a%rate%of%3%%per%annum,%compounding%monthly.%

a)! model%this%investment%using%a%recurrence%relation%of%the%form%V0%=%the%principal,%Vn+1%=%RVn"+%D%where%Vn%is%the%value%of%the%investment%after%n%payments%(additions%to%the%principal)%have%been%made.%

%%%%%

b)! Use%your%calculator%to%determine%recursively%the%value%of%the%investment%after%Nancy%has%made%three%additional%payments%to%her%investment.%

c)! What%will%be%the%value%of%her%investment%after%1%year?%Enter%the%first%term%of%1200%%Type%×1.0025+50%and%continue%to%press%enter%until%the%12th%month%%%%%%

%Starting%value%

1st%month%

2nd%month%

3rd%month%

4th%month%

…%%

9th%month%%

10th%month%

%11th%month%

%

12th%month%

%%

%b.%%

c.$$

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Amortisation tables for annuity investments %%

%%

Total interest earned = balance of loan – (principal + additional payments)

Using financial solver for annuity investments Making$an$investmentA$ANNUITY%INVESTMENTS%Entry$ Sign$ Description$

PV% Ave% The%amount%of%money%that%is%given%to%the%financial%institution%at%the%start%of%the%investment%

Pmt% Ave% The%amount%of%each%payment%made%to%the%financial%institution%to%increase%the%value%of%the%investment%

FV% +ve% The%amount%of%money%accumulated%in%the%account%returning%to%the%person%after%N%payments%have%been%made%

Worked Example 19 Lars%invests%$500%000%at%5.5%%per%annum,%compounding%monthly.%He%makes%a%regular%deposit%of%$500%per%month%into%the%account.%What%is%the%value%of%his%investment%after%5%years?%Using%the%Financial%Solver*%%Enter%the%following:%


Place%the%cursor%on%FV,%Press%ENTER%to%solve.%��%

%

Interest earned =interest rate/compounding period × previous balance

Principal increase =payment made + interest earned

Balance of investment=previous balance + interest + payment made

Documents

Further'Mathematics'2018' Core:'Recursion'and'Financial'modelling' … · 2018-06-06 · Page%2%of%20 Table'of'Contents $ 9A#Combining#geometric#growth#and#decay#_____#3% Worked%Example%1%_____%3%

Further'Mathematics'2018' Core:'Recursion'and'Financial'modelling' … · 2018-06-06 · Page%2%of%20 Table'of'Contents $ 9A#Combining#geometric#growth#and#decay#_#3% Worked%Example%1%_%3%