smc-2512-10-Annuity

Recursion and Finance, GEN2 2024 VCAA 7

Emi decides to invest a $300 000 inheritance into an annuity.

Let $E_n$ be the balance of Emi's annuity after $n$ months.

A recurrence relation that can model the value of this balance from month to month is

$E_0=300000, \quad E_{n+1}=1.003 E_n-2159.41$

Showing recursive calculations, determine the balance of the annuity after two months. Round your answer to the nearest cent. (1 mark)

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For how many years will Emi receive the regular payment? (1 mark)

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Calculate the annual compound interest rate for this annuity. (1 mark)

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If Emi wanted the annuity to act as a perpetuity, what monthly payment, in dollars, would she receive? (1 mark)

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Show Answers Only

a. $$297\,477.40$

b. $15\ \text{years}$

c. $3.6\%$

d. $$900$

Show Worked Solution

a.	$E_1$	$=1.003\times 300\,000-2159.41$	$=$298\,740.59$
	$E_2$	$=1.003\times 298\,740.59-2159.41$	$=$297\,477.4018$	$\approx $297\,477.40$

b. $\text{Using CAS:}$

$\text{Number of years}\ =\dfrac{180}{12}=15\ \text{years}$

c. $\text{Annual interest rate}\ =(1.003-1)\times 12\times 100\% = 3.6\%$

d. $\text{Using CAS:}$

$\text{Monthly payment}\ =$900$

$\text{Manually: Amount must be equal to the amount of monthly interest earned.}$

$\therefore\ \text{Monthly payment}\ =300\,000\times 0.003=$900$

Recursion and Finance, GEN2 2023 VCAA 6

Arthur invests $600 000 in an annuity that provides him with a monthly payment of $3973.00.

Interest is calculated monthly.

Three lines of the amortisation table for this annuity are shown below.

\begin{array} {|c|c|}
\hline
\textbf{Payment} & \textbf{Payment} & \textbf{Interest} & \textbf{Principal reduction} & \textbf{Balance} \\
\textbf{number} & \textbf{(\$) } & \textbf{(\$) } & \textbf{(\$) } & \textbf{(\$) }\\
\hline
\rule{0pt}{2.5ex} 0 \rule[-1ex]{0pt}{0pt} & 0.00 & 0.00 & 0.00 & 600\ 000.00 \\
\hline
\rule{0pt}{2.5ex} 1 \rule[-1ex]{0pt}{0pt} & 3973.00 & 2520.00 & 1453.00& 598\ 547.00\\
\hline
\rule{0pt}{2.5ex} 2 \rule[-1ex]{0pt}{0pt} & 3973.00 & 2513.90 & 1459.10 & 597\ 087.90 \\
\hline
\end{array}

The interest rate for the annuity is 0.42% per month.
Determine the interest rate per annum. (1 mark)

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Using the values in the table, complete the next line of the amortisation table.
Write your answers in the spaces provided in the table below.
Round all values to the nearest cent. (1 mark)

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Let $V_n$ be the balance of Arthur's annuity, in dollars, after $n$ months.
Write a recurrence relation in terms of $V_0, V_{n+1}$ and $V_n$ that can model the value of the annuity from month to month. (1 mark)

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The amortisation tables above show that the balance of the annuity reduces each month.
If the balance of an annuity remained constant from month to month, what name would be given to this type of annuity? (1 mark)

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Show Answers Only

a. $I\% (\text{annual}) = 12 \times 0.42 = 5.04\% $

b. $\text{Row 3 calculations are as follows:}$

$\text{Payment}\ = \$3973.00\ \text{(remains constant)}$

$\text{Interest}\ = 597\ 087.90 \times 0.0042 = \$2507.77 $

$\text{Principal reduction}\ = 3973.00-2507.77 = \$1465.23 $

$\text{Balance}\ = 597\ 087.90-1465.23 = \$595\ 622.67$

c. $V_0 = 600\ 000$

$V_{n+1} = 1.0042 \times V_n-3973$

d. $\text{Perpetuity}$

Show Worked Solution

a. $I\% (\text{annual}) = 12 \times 0.42 = 5.04\% $

b. $\text{Row 3 calculations are as follows:}$

$\text{Payment}\ = \$3973.00\ \text{(remains constant)}$

$\text{Interest}\ = 597\ 087.90 \times 0.0042 = \$2507.77 $

$\text{Principal reduction}\ = 3973.00+2507.77 = \$1465.23 $

$\text{Balance}\ = 597\ 087.90-1465.23 = \$595\ 622.67$

♦♦ Mean mark (b) 40%.

c. $V_0 = 600\ 000$

$V_{n+1} = 1.0042 \times V_n-3973$

♦ Mean mark (c) 41%.

d. $\text{Perpetuity}$

CORE, FUR2 2021 VCAA 9

Sienna invests $152 431 into an annuity from which she will receive a regular monthly payment of $900 for 25 years. The interest rate for this annuity is 5.1% per annum, compounding monthly.

Let `V` be the balance of the annuity after `n` monthly payments. A recurrence relation written in terms of `V_0 , V_{n + 1}` and `V_n` can model the value of this annuity from month to month.
Showing recursive calculations, determine the value of the annuity after two months.
Round your answer to the nearest cent. (2 marks)
After two years, the interest rate for this annuity will fail to 4.6%.
To ensure that she will still receive the same number of $900 monthly payments, Sienna will add an extra one-pff amount into the annuity at this time.
Determine the value of this extra amount that Sienna will add.
Round your answer to the nearest cent. (1 mark)

Show Answers Only

`$151 \ 925.59`
`$7039.20`

Show Worked Solution

a. `text{Payments are monthly} \ => \ r = 5.1/12 = 0.425text(%) \ text{per month}`

`R`	`= 1 + r/100 = 1.00425`
`V_1`	`= RV_0 – text{payment}`
	`= 1.00425 xx 152 \ 431 – 900`
	`= $152\ 178.83`
`:.V_2`	`= 1.00425 xx 152\ 178.83 – 900`
	`= $ 151 \ 925.59`

b. `text{Find} \ V_24 \ text{(annuity value after 2 years) by TVM Solver:}`

`N`	`= 24`
`I text{(%)}`	`= 5.1`
`PV`	`= -152 \ 431`
`PMT`	`= 900`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`=> FV = 146 \ 073.7405`

`text{Find}\ PV\ text{of annuity needed (by TVM solver):}`

`N`	`= 243 xx 12 = 276`
`Itext{(%)}`	`= 4.6`
`PV`	`= ?`
`PMT`	`= 900`
`FV`	`= 0`
`text(P/Y)`	`= text(C/Y) = 12`

`=> PV = -153\ 112.9399`

`:. \ text{Amount to add}`	`= 153 \ 112.94 – 146 \ 073.74`
	`= $ 7039.20`

CORE, FUR1 2021 VCAA 22 MC

Joanna deposited $12 000 in an investment account earning interest at the rate of 2.8% per annum, compounding monthly.

She would like this account to reach a balance of $25 000 after five years.

To achieve this balance, she will make an extra payment into the account each month, immediately after the interest is calculated.

The minimum value of this payment is closest to

$113.85
$174.11
$580.16
$603.22
$615.47

Show Answers Only

`B`

Show Worked Solution

`text{By TVM Solver:}`

`N`	`= 60`
`Itext{(%)}`	`= 2.8`
`PV`	`= -12 000`
`PMT`	`= ?`
`FV`	`= 25 000`
`text(P/Y)`	`= text(C/Y) = 12`

`:. PMT = 174.106 …`

`=> B`

CORE, FUR1-NHT 2019 VCAA 24 MC

Robyn has a current balance of $347 283.45 in her superannuation account.

Robyn’s employer deposits $350 into this account every fortnight.

This account earns interest at the rate of 2.5% per annum, compounding fortnightly.

Robyn will stop work after 15 years and will no longer receive deposits from her employer.

The balance of her superannuation account at this time will be invested in an annuity that will pay interest at the rate of 3.6% per annum, compounding monthly.

After 234 monthly payments there will be no money left in Robyn’s annuity.

The value of Robyn’s monthly payment will be closest to

$3993
$5088
$6664
$8051
$9045

Show Answers Only

`A`

Show Worked Solution

`text(Balance after 15 years (by TVM Solver):)`

`N`	`= 26 xx 15 = 390`
`Itext(%)`	`= 2.5`
`PV`	`= 347\ 283.45`
`PMT`	`= 350`
`FV`	`= ?`
`text(P/Y)`	`= 26`
`text(C/Y)`	`= 26`

`=>\ 670\ 724.87`

`text(Monthly payment (by TVM Solver)):`

`N`	`= 234`
`Itext(%)`	`= 3.6`
`PV`	`= 670\ 724.87`
`PMT`	`= ?`
`FV`	`= 0`
`text(P/Y)`	`= 12`
`text(C/Y)`	`= 12`

`=> PMT = −3993.31`

`=>\ A`

CORE, FUR2-NHT 2019 VCAA 8

A record producer gave the band $50 000 to write and record an album of songs.

This $50 000 was invested in an annuity that provides a monthly payment to the band.

The annuity pays interest at the rate of 3.12% per annum, compounding monthly.

After six months of writing and recording, the band has $32 667.68 remaining in the annuity.

What is the value, in dollars, of the monthly payment to the band? (1 mark)
After six months of writing and recording, the band decided that it needs more time to finish the album.

To extend the time that the annuity will last, the band will work for three more months without withdrawing a payment.

After this, the band will receive monthly payments of $3800 for as long as possible.

The annuity will end with one final monthly payment that will be smaller than all of the others.

Calculate the total number of months that this annuity will last. (2 marks)

Show Answers Only

`$3000`
`18`

Show Worked Solution

a. `text(By TVM Solver:)`

`N`	`= 6`
`I text(%)`	`=3.12`
`PV`	`=50\ 000`
`PMT`	`= ?`
`FV`	`=-32\ 667.68`
`text(PY)`	`= text(CY) = 12`

`=> PMT = -3000.00`

`:. \ text(Monthly payment) = $3000`

b. `text{Value after 3 more months (by TVM Solver):}`

`N`	`= 3`
`I text(%)`	`=3.12`
`PV`	`=32\ 667.68`
`PMT`	`= 0`
`FV`	`=?`
`text(PY)`	`= text(CY) = 12`

`=> FV = -32\ 923.15`

`text(Find) \ \ N \ text(when) \ FV= 0 \ text{(by TVM Solver):}`

`N`	`= ?`
`I text(%)`	`=3.12`
`PV`	`=-32\ 923.15`
`PMT`	`= 3800`
`FV`	`=0`
`text(PY)`	`= text(CY) = 12`

`=> N = 8.77`

`:. \ text(Total months of annuity)`	`= 6 + 3 + 9`
	`= 18`

CORE, FUR2-NHT 2019 VCAA 7

Tisha plays drums in the same band as Marlon.

She would like to buy a new drum kit and has saved $2500.

Tisha could invest this money in an account that pays interest compounding monthly.

The balance of this investment after `n` months, `T_n` could be determined using the recurrence relation below

`T_0 = 2500, \ \ \ \ T_(n+1) = 1.0036 xx T_n`

Calculate the total interest that would be earned by Tisha's investment in the first five months.

Round your answer to the nearest cent. (2 marks)

Tisha could invest the $2500 in a different account that pays interest at the rate of 4.08% per annum, compounding monthly. She would make a payment of $150 into this account every month.

Let `V_n` be the value of Tisha's investment after `n` months.

Write down a recurrence relation, in terms of `V_0`, `V_n` and `V_(n + 1)`, that would model the change in the value of this investment. (1 mark)
Tisha would like to have a balance of $4500, to the nearest dollar, after 12 months.

What annual interest rate would Tisha require?

Round your answer to two decimal places. (1 mark)

Show Answers Only

`$45.33`
`V_0 = 2500, \ V_(n+1) = 1.0034 xx V_n + 150`
`5.87%`

Show Worked Solution

a. `T_1 = 1.0036 xx 2500 = 2509`

`T_2 = 1.0036 xx 2509 = 2518.0324`

`vdots`

`T_5 = 2545.33`

`:. \ text(Total interest) `	`= 2545.33 – 2500`
	`= $45.33`

b. `text(Monthly interest) = (4.08)/(12) = 0.34%`

`:. \ V_0 = 2500, \ V_(n+1) = 1.0034 xx V_n + 150`

c. `text(By TVM Solver:)`

`N`	`= 12`
`I text(%)`	`=?`
`PV`	`=-2500`
`PMT`	`= -150`
`FV`	`=4500`
`text(PY)`	`= text(CY)=12`

`=> I = 5.87%`

CORE, FUR2 2019 VCAA 8

Phil invests $200 000 in an annuity from which he receives a regular monthly payment.

The balance of the annuity, in dollars, after `n` months, `A_n`, can be modelled by the recurrence relation

`A_0 = 200\ 000, qquad A_(n + 1) = 1.0035\ A_n - 3700`

What monthly payment does Phil receive? (1 mark)
Show that the annual percentage compound interest rate for this annuity is 4.2%. (1 mark)

At some point in the future, the annuity will have a balance that is lower than the monthly payment amount.

What is the balance of the annuity when it first falls below the monthly payment amount?

Round your answer to the nearest cent. (1 mark)
If the payment received each month by Phil had been a different amount, the investment would act as a simple perpetuity.

What monthly payment could Phil have received from this perpetuity? (1 mark)

Show Answers Only

`$3700`
`text(Proof)\ text{(See Worked Solutions)}`
`$92.15`
`$700`

Show Worked Solution

a. `$3700`

b.	`text(Monthly rate)`	`= 0.0035 = 0.35%`
	`text(Annual rate)`	`= 12 xx 0.35 = 4.2%`

c. `text(Find)\ N\ text(when)\ FV = 0\ \ text{(by TVM solver)}:`

`N`	`= ?`
`I(%)`	`= 4.2`
`PV`	`= 200\ 000`
`PMT`	`= 3700`
`FV`	`= 0`
`text(P/Y)`	`= 12`
`text(C/Y)`	`= 12`

`=> N = 60.024951`

`text(Find)\ \ FV\ \ text(when)\ \ N = 60.024951\ \ text{(by TVM solver):}`

`=>FV = $92.15`

d. `text(Perpetuity) => text(monthly payment) = text(monthly interest)`

`:.\ text(Perpetuity payment)`	`= 200\ 000 xx 4.2/(12 xx 100)`
	`= $700`

CORE, FUR2 2018 VCAA 6

Julie has retired from work and has received a superannuation payment of $492 800.

She has two options for investing her money.

Option 1

Julie could invest the $492 800 in a perpetuity. She would then receive $887.04 each fortnight for the rest of her life.

At what annual percentage rate is interest earned by this perpetuity? (1 mark)

Option 2

Julie could invest the $492 800 in an annuity, instead of a perpetuity.

The annuity earns interest at the rate of 4.32% per annum, compounding monthly.

The balance of Julie’s annuity at the end of the first year of investment would be $480 242.25

1. What monthly payment, in dollars, would Julie receive? (1 mark)
2. How much interest would Julie’s annuity earn in the second year of investment?
  
  Round your answer to the nearest cent. (2 marks)

Show Answers Only

`4.68 text(%)`
1. `$2800`
2. `$20\ 488.89`

Show Worked Solution

a.	`text(Annual interest)`	`= 26 xx 887.04`
		`= $23\ 063.04`

♦ Mean mark 41%.

`:.\ text(Annual percentage rate)`	`= (23\ 063.04)/(492\ 800)`
	`= 4.68 text(%)`

b.i. `text(Find the monthly payment by TVM Solver:)`

♦ Mean mark 48%.

`N`	`= 12`
`I(%)`	`= 4.32`
`PV`	`= -492\ 800`
`PMT`	`= ?`
`FV`	`= 480\ 242.25`
`text(P/Y)`	`= text(C/Y) = 12`

`=> PMT = $2800.00`

♦♦♦ Mean mark 16%.

b.ii.	`text(Year 2 start balance)`	`= $480\ 242.25`
	`text(Year 2 end balance)`	`= $467\ 131.14`
	`text(Balance reduction)`	`= 480\ 242.25 – 467\ 131.14`
		`= 13\ 111.11`

`text(Year 2 total payment) = 12 xx 2800 = 33\ 600`

`:.\ text(Interest)`	`= 33\ 600 – 13\ 111.11`
	`= $20\ 488.89`

CORE, FUR1 2018 VCAA 24 MC

Mariska plans to retire from work 10 years from now.

Her retirement goal is to have a balance of $600 000 in an annuity investment at that time.

The present value of this annuity investment is $265 298.48, on which she earns interest at the rate of 3.24% per annum, compounding monthly.

To make this investment grow faster, Mariska will add a $1000 payment at the end of every month.

Two years from now, she expects the interest rate of this investment to fall to 3.20% per annum, compounding monthly. It is expected to remain at this rate until Mariska retires.

When the interest rate drops, she must increase her monthly payment if she is to reach her retirement goal.

The value of this new monthly payment will be closest to

$1234
$1250
$1649
$1839
$1854

Show Answers Only

`E`

Show Worked Solution

`text(Find)\ FV\ text{after 2 years (by TVM solver):}`

♦ Mean mark 45%.

`N`	`= 24`
`I(%)`	`= 3.24`
`PV`	`= −265\ 298.48`
`PMT`	`= −1000`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`=>FV = $307\ 794.50\ text{(nearest cent)}`

`text(Find new monthly payment:)`

`N`	`= 8 xx 12 = 96`
`I(%)`	`= 3.20`
`PV`	`= –307\ 794.50`
`PMT`	`= ?`
`FV`	`= 600\ 000`
`text(P/Y)`	`= text(C/Y) = 12`

`=> PMT = −1854.05`

`=> E`

CORE, FUR2 2017 VCAA 7

Alex sold his mechanics’ business for $360 000 and invested this amount in a perpetuity.

The perpetuity earns interest at the rate of 5.2% per annum.

Interest is calculated and paid monthly.

What monthly payment will Alex receive from this investment? (1 mark)
Later, Alex converts the perpetuity to an annuity investment.

This annuity investment earns interest at the rate of 3.8% per annum, compounding monthly.

For the first four years Alex makes a further payment each month of $500 to his investment.

This monthly payment is made immediately after the interest is added.

After four years of these regular monthly payments, Alex increases the monthly payment.

This new monthly payment gives Alex a balance of $500 000 in his annuity after a further two years.

What is the value of Alex’s new monthly payment?

Round your answer to the nearest cent. (2 marks)

Show Answers Only

`$1560`
`$805.65 (text(nearest cent))`

Show Worked Solution

a.	`text(Monthly payment)`	`= 360\ 000 xx 0.052/12`
		`= $1560`

♦ Mean mark part (a) 50%.

b. `text(By TVM Solver,)`

`text(Find balance after 4 years:)`

`N`	`= 4 xx 12 = 48`
`I(%)`	`= 3.8`
`PV`	`= −360\ 000`
`PMT`	`= −500`
`FV`	`= ?`
`text(PY)`	`=\ text(CY) = 12`

`=> FV`	`= 444\ 872.9445`

`:.\ text(Balance is $444 872.9445)`

♦♦ Mean mark 29%.
MARKER’S COMMENT: A common error was entering the $500 payment as a positive value. Know why this is incorrect!

`text(Find)\ \ PMT\ \ text(when)\ \ FV = 500\ 000\ \ text(and)\ \ N = 24:`

`N`	`= 24`
`I(%)`	`= 3.8`
`PV`	`= −444\ 872.9445`
`PMT`	`= ?`
`FV`	`= 500\ 000`
`text(PY)`	`= text(CY) = 12`

`=> PMT`	`= −805.6505…`

`:. text(Alex’s new monthly payment = $805.65 (nearest cent))`

CORE, FUR1 2016 VCAA 24 MC

Mai invests in an annuity that earns interest at the rate of 5.2% per annum compounding monthly.

Monthly payments are received from the annuity.

The balance of the annuity will be $130 784.93 after five years.

The balance of the annuity will be $66 992.27 after 10 years.

The monthly payment that Mai receives from the annuity is closest to

$1270
$1400
$1500
$2480
$3460

Show Answers Only

`C`

Show Worked Solution

`text(By TVM solver, starting at 5 years,)`

♦♦ Mean mark 30%.
MARKER’S COMMENT: Many students incorrectly chose `E`, by allocating the same sign to both the PV and FV.

`N`	`= 5 xx 12=60`
`I(text(%))`	`= 5.2`
`PV`	`= −130\ 784.93`
`PMT`	`= ?`
`FV`	`= 66\ 992.27`
`text(P/Y)`	`= text(C/Y) = 12`

`:. PMT`	`= 1500.00…`

`=> C`

CORE, FUR1 2016 VCAA 23 MC

Sarah invests $5000 in a savings account that pays interest at the rate of 3.9% per annum compounding quarterly. At the end of each quarter, immediately after the interest has been paid, she adds $200 to her investment.

After two years, the value of her investment will be closest to

$5805
$6600
$7004
$7059
$9285

Show Answers Only

`D`

Show Worked Solution

`text(By TVM Solver, after 2 years)`

`N`	`= 2 xx 4 = 8`
`I(text(%))`	`= 3.9`
`PV`	`= −5000`
`PMT`	`= −200`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 4`

`:. FV`	`= 7059.25`

`=> D`

CORE*, FUR2 2006 VCAA 3

The company prepares for this expenditure by establishing three different investments.

$7000 is invested at a simple interest rate of 6.25% per annum for eight years.

Determine the total value of this investment at the end of eight years. (2 marks)
$10 000 is invested at an interest rate of 6% per annum compounding quarterly for eight years.

Determine the total value of this investment at the end of eight years.

Write your answer correct to the nearest dollar. (1 mark)
$500 is deposited into an account with an interest rate of 6.5% per annum compounding monthly.

Deposits of $200 are made to this account on the last day of each month after interest has been paid.

Determine the total value of this investment at the end of eight years.

Write your answer correct to the nearest dollar. (1 mark)

Show Answers Only

`$10\ 500`
`$16\ 103`
`$25\ 935`

Show Worked Solution

a.	`I`	`= (PrT)/100`
		`= (7000 xx 6.25 xx 8)/100`
		`= $3500`

`:.\ text(Total value of investment)`

`= 7000 + 3500`

`= $10\ 500`

b. `text(Compounding periods) = 8 xx 4 = 32`

`text(Interest rate)`	`= (text(6%))/4`
	`= 1.5text(% per quarter)`

`:.\ text(Total value of investment)`

`= PR^n`

`= 10\ 000(1.015)^32`

`= 16\ 103.24…`

`= $16\ 103\ \ text{(nearest $)}`

c. `text(By TVM Solver,)`

`N`	`= 8 xx 12 = 96`
`I(text(%))`	`= 6.5`
`PV`	`= 500`
`PMT`	`= 200`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`=> FV = −25\ 935.30…`

`:.\ text(Total value of investment is $25 935.)`

CORE*, FUR2 2011 VCAA 2

Tom and Patty both decided to invest some money from their savings.

Each chose a different investment strategy.

Tom's investment strategy

• Deposit $5600 into an account with an interest rate of 7.2% per annum, compounding monthly.

• Immediately after interest is paid into his investment account on the last day of each month, deposit a further $200 into the account.

Determine the total amount in Tom's investment account at the end of the first month. (1 mark)

Patty's investment strategy

• Invest $8000 at the start of the year at an interest rate of 7.2% per annum, compounding annually.

The following expression can be used to determine the value of Patty's investment at the end of the first year. Complete the expression by filling in the box. (1 mark)

At the end of twelve months, Patty has more money in her investment account than Tom.

How much more does she have?

Write your answer to the nearest cent. (2 marks)
What annual compounding rate of interest would Patty need in order to earn $1000 interest in one year on her $8000 investment?

Write your answer correct to one decimal place. (1 mark)

Show Answers Only

`$5833.60`
`text(Value of investment)\ = 8000 xx (1 + 7.2/100)`
`$78.42\ \ text{(nearest cent)}`
`text(12.5%)`

Show Worked Solution

a. `text(Tom’s investment after 1 month)`

`= 5600 xx (1 + 7.2/(12 xx 100)) + 200`

`= $5833.60`

b. `text(After 1 year,)`

`text(Value of investment) = 8000 xx (1 + 7.2/100)`

c. `text(Tom’s investment after 12 months,)`

`text(by TVM Solver,)`

`N`	`= 12`
`I(text(%))`	`= 7.2`
`PV`	`= 5600`
`PMT`	`= 200`
`text(P/Y)`	`= 12`
`text(C/Y)`	`= 12`

`=> FV = −8497.58…`

`text(Patty’s investment after 12 months)`

MARKER’S COMMENT: Many students valued Patty’s investment correctly but then deducted Tom’s after 1 month (from part a), instead of 12.

`= 8000 xx (1 + 7.2/100)`

`= $8576`

`:.\ text(Extra value of Patty’s investment)`

`= 8576 – 8497.580…`

`= 78.419…`

`= $78.42\ \ text{(nearest cent)}`

d.	`text(Using)\ \ \ I`	`= (PrT)/100`
	`1000`	`= (8000 xx r xx 1)/100`
	`r`	`= (1000 xx 100)/8000`
		`= 12.5text(%)`

CORE*, FUR2 2012 VCAA 4

Arthur invested $80 000 in a perpetuity that returns $1260 per quarter. Interest is calculated quarterly.

Calculate the annual interest rate of Arthur’s investment. (1 mark)
After Arthur has received 20 quarterly payments, how much money remains invested in the perpetuity? (1 mark)
Arthur’s wife, Martha, invested a sum of money at an interest rate of 9.4% per annum, compounding quarterly.

She will be paid $1260 per quarter from her investment.

After ten years, the balance of Martha’s investment will have reduced to $7000.

Determine the initial sum of money Martha invested.

Write your answer, correct to the nearest dollar. (1 mark)

Show Answers Only

`6.3text(%)`
`$80\ 000`
`$35\ 208`

Show Worked Solution

a. `text(Let)\ \ r =\ text(annual interest rate)`

♦♦ Mean mark of all parts (combined) was 26%.

`80\ 000 xx r/(4 xx 100)`	`= 1260`
`:. r`	`= (1260 xx 400)/(80\ 000)`
	`= 6.3text(%)`

b. `$80\ 000`

`text{(The principal invested in a perpetuity}`

`text{remains unchanged.)}`

c. `text(Find)\ PV\ text(using TVM Solver:)`

`N`	`= 4 xx 10 = 40`
`I(text(%))`	`= 9.4`
`PV`	`= ?`
`PMT`	`= 1260`
`FV`	`= 7000`
`text(P/Y)`	`= text(C/Y) = 4`

`=> PV = −35\ 208.002…`

`:.\ text{Martha initially invested $35 208 (nearest $)}`

CORE*, FUR2 2013 VCAA 2

Hugo won $5000 in a road race and invested this sum at an interest rate of 4.8% per annum compounding monthly.

What is the value of Hugo’s investment after 12 months?

Write your answer in dollars, correct to the nearest cent. (1 mark)
1. Suppose instead that at the end of each month Hugo added $200 to his initial investment of $5000.
  
  Find the value of this investment immediately after the 12th monthly payment of $200 is made.
  
  Write your answer in dollars, correct to the nearest cent. (1 mark)
2. Assume Hugo follows the investment that is described in part b.i.
  
  Determine the total interest he would earn over the 12-month period.
  
  Write your answer in dollars, correct to the nearest cent. (1 mark)

Show Answers Only

`$5245.35`
1. `$7698.86`
2. `$298.86`

Show Worked Solution

a. `text(Monthly interest rate)`

♦ Mean mark for all parts combined was 39%.

`= 4.8/12`

`= 0.4text(%)`

`:.\ text(Value after 12 months)`

`= 5000(1 + 0.4/100)^12`

`= 5000(1.004)^12`

`= 5245.351…`

`= $5245.35\ \ text{(nearest cent)}`

b.i. `text(Using TVM Solver,)`

`N`	`= 12`
`I(text(%))`	`= 4.8`
`PV`	`= −5000`
`PMT`	`= −200`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`FV = 7698.8614…`

`:. text(Value of interest) = $7698.86`

b.ii. `text(Total interest)`

`=\ text{Final value − (original + payments)}`

`= 7698.86 – (5000 + 12 xx 200)`

`= 7698.86 – 7400`

`= $298.86`

CORE*, FUR2 2014 VCAA 3

The cricket club had invested $45 550 in an account for four years.

After four years of compounding interest, the value of the investment was $60 000.

How much interest was earned during the four years of this investment? (1 mark)

Interest on the account had been calculated and paid quarterly.

What was the annual rate of interest for this investment?

Write your answer, correct to one decimal place. (1 mark)

The $60 000 was re-invested in another account for 12 months.

The new account paid interest at the rate of 7.2% per annum, compounding monthly.

At the end of each month, the cricket club added an additional $885 to the investment.

1. The equation below can be used to determine the account balance at the end of the first month, immediately after the $885 was added.
  
  Complete the equation by filling in the boxes. (1 mark)
2. What was the account balance at the end of 12 months?
  
  Write your answer, correct to the nearest dollar. (1 mark)

Show Answers Only

`14\ 450`
`text(6.9%)`
1. `text(account balance)\ = 60\ 000 xx (1 + (7.2)/(12 xx 100)) + 885`
2. `75\ 443`

Show Worked Solution

a. `text(Interest earned)`

`= text(End value − original investment)`

`= 60\ 000 – 45\ 550`

`= $74\ 450`

♦ Mean mark of all parts (combined) was 40%.

b. `text(By TVM Solver:)`

`N`	`= 4 xx 4 = 16`
`I(text(%))`	`= ?`
`PV`	`= −45\ 550`
`PMT`	`= 0`
`FV`	`= 60\ 000`
`text(P/Y)`	`= text(C/Y) = 4`

`I(text(%)) = 6.948…`

`:.\ text(Annual interest rate = 6.9%)`

c.i. `text(account balance)`

`= 60\ 000 xx (1 + 7.2/(12 xx 100)) + 885`

c.ii. `text(By TVM Solver:)`

`N`	`= 12`
`I(text(%))`	`= 7.2`
`PV`	`= −60\ 000`
`PMT`	`= −885`
`FV`	`= ?`
`text(P/Y)`	`= text(C/Y) = 12`

`FV = 75\ 443.014…`

`:. text(Balance after 12 months) = $75\ 443`

CORE*, FUR2 2015 VCAA 4

As their business grows, Jane and Michael decide to invest some of their earnings.

They each choose a different investment strategy.

Jane opens an account with Red Bank, with an initial deposit of $4000.

Interest is calculated at a rate of 3.6% per annum, compounding monthly.

Determine the amount in Jane’s account at the end of six months.

Write your answer correct to the nearest cent. (1 mark)

Michael decides to open an account with Blue Bank, with an initial deposit of $2000.

At the end of each quarter, he adds an additional $200 to his account.

Interest is compounded at the end of each quarter.

The equation below can be used to determine the balance of Michael’s account at the end of the first quarter.

account balance = 2000 × (1 + 0.008) + 200

Show that the annual compounding rate of interest is 3.2%. (1 mark)
Determine the amount in Michael’s account, after the $200 has been added, at the end of five years.

Write your answer correct to the nearest cent. (1 mark)

Show Answers Only

`$4072.54`
`3.2text(%)`
`$6664.63`

Show Worked Solution

a. `text(Amount in account after 6 months)`

`= 4000 xx (1 + 3.6/(12 xx 100))^6`

`=4072.542…`

`=$4072.54\ \ text{(nearest cent)}`

b. `text(Annual compounding rate)`

`=0.008 xx text(100%) xx 4`

`= 3.2text(%)`

c. `text(By TVM Solver, after 5 years)`

`N`	`=5 xx 4 = 20`
`I (%)`	`=3.2`
`PV`	`= – 2000`
`PMT`	`=200`
`FV`	`=?`
`text(P/Y)`	`= text(C/Y) = 4`

`FV=6664.629…`

`:.\ text(The amount in Michael’s account is $6664.63)`

CORE*, FUR1 2010 VCAA 9 MC

Rebecca invested $4000 at 5.0% per annum with interest compounding quarterly.

After interest is paid at the end of each quarter, Rebecca adds $800 to her investment.

The value of her investment at the end of the second quarter, after the $800 has been added, is closest to

A. $4101

B. $4901

C. $4911

D. $5711

E. $6060

Show Answers Only

`D`

Show Worked Solution

`P=4000,\ \ R=1 + 0.05/4`

♦ Mean mark 47%.

`text(After 1st quarter where)\ \ n=1,`

`A`	`= PR^n + 800`
	`= 4000 xx (1 + 0.05 / 4)^1 + 800`
	`= 4850`

`text(After 2nd quarter,)`

`A`	`= 4850 xx (1 + 0.05 / 4)^1 + 800`
	`= $5710.625`

`=> D`

CORE*, FUR1 2005 VCAA 7 MC

Gregor invests $10 000 and earns interest at a rate of 6% per annum compounding quarterly.

Every quarter, after interest has been added, he withdraws $500.

At the end of four years, after interest has been added and he has made the $500 withdrawal, the value of the remaining investment will be closest to

A. $3720

B. $4220

C. $5440

D. $21 660

E. $22 160

Show Answers Only

`A`

Show Worked Solution

`text(By TVM Solver:)`

`N`	`= 16`
`I text(%)`	`=6`
`PV`	`=-10\ 000`
`PMT`	`= 500`
`FV`	`=?`
`text(PY)`	`= text(C/Y) = 4`

`:. FV = 3723.67`

`=> A`

CORE*, FUR1 2014 VCAA 8 MC

Robert invested $6000 at 4.25% per annum with interest compounding quarterly.

Immediately after interest is paid at the end of each quarter, he adds $500 to his investment.

The value of Robert’s investment at the end of the third quarter, after his $500 has been added, is closest to

A. $6193

B. $7569

C. $7574

D. $7709

E. $8096

Show Answers Only

`D`

Show Worked Solution

`text(Quarterly interest rate)\ = 4.25/4 =\ text(1.0625%)`

`text(Let)\ Q_n\ text(be the value of investment)`

`text(after)\ n\ text(quarters)`

`Q_1`	`= 6000 (1.010625) + 500`
	`= 6563.75`

`Q_2`	`= 6563.75 (1.010625) + 500`
	`= 7133.49`

`Q_3`	`= 7133.49 (1.010625) + 500`
	`= 7709.28`

`=> D`

a.	\(E_1\)	\(=1.003\times 300\,000-2159.41\)	\(=$298\,740.59\)
	\(E_2\)	\(=1.003\times 298\,740.59-2159.41\)	\(=$297\,477.4018\)	\(\approx $297\,477.40\)