smc-2512-20-Perpetuity

Financial Maths, 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

$E_0=300\,000, \quad E_{n+1}=1.003 E_n-2159.41$

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. $E_0=300\,000$

$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 $297\,477.40$

♦ Mean mark (a) 49%.

b. $\text{Using CAS:}$

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

♦ Mean mark (b) 42%.

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

d. $\text{Method 1: Using CAS}$

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

$\text{Method 2:}$

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

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

♦ Mean mark (d) 47%.

Financial Maths, 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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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 6

Sienna invests $420 000 in a perpetuity from which she will receive a regular monthly payment of $1890.

The perpetuity earns interest at the rate of 5.4% per annum.

Determine the total amount, in dollars, that Sienna will receive after one year of monthly payments. (1 mark)

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Write down the value of the perpetuity after Sienna has received one year of monthly payments. (1 mark)

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Let `S_n` be the value of Sienna's perpetuity after `n` months.
Complete the recurrence relation, in terms of `S_0`, `S_{n + 1}` and `S_n`, that would model the value of this perpetuity over time. Write your answers in the boxes provided. (1 mark)

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`S_n =`

, `S_{n+1} =`

`xx S_n-1890`

Show Answers Only

`$ 22 \ 680`
`$ 420 \ 000`
`S_n = 420 \ 000, \ \ S_{n+1} = 1.0045 xx S_n-1890`

Show Worked Solution

a.	`text{Total amount}`	`= 12 xx 1890`
		`= $ 22 \ 680`

b. `text{Value} = $ 420 \ 00 \ text{(balance remains the same after each payment).}`

c. `S_0 = 420 \ 000 \ => \ S_n = 420 \ 000`

`S_{n+1} = RS_n-1890 \ \ text{where}\ \ R = 1 + r/100`

`r = 5.4/12 = 0.45`

`:. \ S_{n+1} = 1.0045 xx S_n-1890`

Financial Maths, GEN2 2019 NHT 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)

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Show that the annual percentage compound interest rate for this annuity is 4.2%. (1 mark)

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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)

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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)

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`$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)

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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)
  
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2. How much interest would Julie’s annuity earn in the second year of investment?
3. Round your answer to the nearest cent. (1 mark)
  
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`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, 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)

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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)

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`$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 21 MC

Juanita invests $80 000 in a perpetuity that will provide $4000 per year to fund a scholarship at a university.

The graph that shows the value of this perpetuity over a period of five years is

Show Answers Only

`B`

Show Worked Solution

`text(A perpetuity lasts indefinitely by only)`

`text(paying out interest.)`

`:.\ text(The graph should show a value that)`

`text(does not change over the years.)`

`=> B`

CORE*, FUR2 2010 VCAA 2

$360 000 is invested in a perpetuity at an interest rate of 5.2% per annum.

Find the monthly payment that the perpetuity provides. (1 mark)

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After six years of monthly payments, how much money remains invested in the perpetuity? (1 mark)

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`$1560`
`$360\ 000`

Show Worked Solution

a. `text(Monthly repayment)`

`= 1/12 xx (360\ 000 xx 5.2/100)`

`= 1/12 xx 18\ 720`

`= $1560`

♦♦ Part (b) was poorly answered.
MARKER’S COMMENT: Many students did not understand the concept of a perpetuity.

b. `$360\ 000`

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)

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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)

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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)

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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)

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`$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)

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After Arthur has received 20 quarterly payments, how much money remains invested in the perpetuity? (1 mark)

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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)

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`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 2014 VCAA 2

A sponsor of the cricket club has invested $20 000 in a perpetuity.

The annual interest from this perpetuity is $750.

The interest from the perpetuity is given to the best player in the club every year, for a period of 10 years.

What is the annual rate of interest for this perpetuity investment? (1 mark)

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After 10 years, how much money is still invested in the perpetuity? (1 mark)

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The average rate of inflation over the next 10 years is expected to be 3% per annum.
1. Michael was the best player in 2014 and he considered purchasing cricket equipment that was valued at $750.
2. What is the expected price of this cricket equipment in 2015? (1 mark)
  
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3. What is the 2014 value of cricket equipment that could be bought for $750 in 2024? Write your answer, correct to the nearest dollar. (1 mark)
  
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Show Answers Only

`3.75`
`$20\ 000`
1. `$772.50`
2. `$558\ \ text{(nearest dollar)}`

Show Worked Solution

a.	`20\ 000 xx r`	`= 750`
	`:. r`	`= 750/(20\ 000)`
		`= 0.0375`

`:.\ text(Annual interest rate = 3.75%)`

b. `$20\ 000`

`text{(A perpetuity’s balance remains constant.)}`

c.i. `text(Expected price in 2015)`

`= 750 xx (1 + 3/100)`

`= 750 xx 1.03`

`= $772.50`

c.ii. `text(Value in 2014) xx (1.03)^10 = 750`

`:.\ text(Value in 2014)\ `	`= 750/((1.03)^10)`
	`= 558.07…`
	`= $558\ \ text{(nearest dollar)}`

CORE*, FUR2 2015 VCAA 3

Jane and Michael decide to set up an annual music scholarship.

To fund the scholarship, they invest in a perpetuity that pays interest at a rate of 3.68% per annum.

The interest from this perpetuity is used to provide an annual $460 scholarship.

Determine the minimum amount they must invest in the perpetuity to fund the scholarship. (1 mark)

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For how many years will they be able to provide the scholarship? (1 mark)

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`12\ 500`
`text(It will last forever.)`

Show Worked Solution

a. `text(Let)\ \ $A\ =\ text(Perpetuity amount)`

`3.68text(%) xx A`	`= 460`
`:.A`	`=460/0.0368`
	`= $12\ 500`

b. `text(It will last forever.)`

CORE*, FUR1 2008 VCAA 2 MC

Pia invests $800 000 in an ordinary perpetuity to provide an ongoing fortnightly pension for her retirement.

The interest rate for this investment is 5.8% per annum.

Assuming there are 26 fortnights per year, the amount she will receive at the end of each fortnight is closest to

A. $464

B. $892

C. $1422

D. $1785

E. $3867

Show Answers Only

`D`

Show Worked Solution

`I`	`= (PrT)/100`
`I`	`= (800\ 000 xx 5.8 xx 1)/100`
	`= 46\ 400`

`text(Fortnightly payment)`	`= (46\ 400)/26`
	`= $1784.62…`

`=> D`

CORE*, FUR1 2006 VCAA 3 MC

Grandpa invested in an ordinary perpetuity from which he receives a monthly pension of $584.

The interest rate for the investment is 6.2% per annum.

The amount Grandpa has invested in the perpetuity is closest to

A. $3600

B. $9420

C. $94 200

D. $43 400

E. $113 000

Show Answers Only

`E`

Show Worked Solution

`I = 584, \ r = 6.2, \ T = 1/12`

♦♦ Mean mark 23%.
MARKERS’ COMMENT: Nearly half of students read the monthly pension amount as a yearly amount. Be careful!

`I`	`= (PrT)/100`
`584`	`=(P xx 6.2 xx 1/12)/100`
`:. P`	`=(584 xx 100)/(6.2 xx 1/12)`
	`=$113\ 032.25…`

`=> E`

CORE*, FUR1 2011 VCAA 5 MC

Jane invests in an ordinary perpetuity to provide her with a weekly payment of $500.

The interest rate for the investment is 5.9% per annum.

Assuming there are 52 weeks per year, the amount that Jane needs to invest in the perpetuity is closest to

A. $26 000

B. $102 000

C. $154 000

D. $221 000

E. $441 000

Show Answers Only

`E`

Show Worked Solution

♦ Mean mark 44%.

`text{Total payment ($) per year}`	`= 500 xx 52`
	`= 26\ 000`

`text(Perpetuity)\ xx text(5.9%)`	`= 26\ 000`
`:.\ text(Perpetuity)`	`= (26\ 000) / text(5.9%)`
	`= 440\ 677.96…`

`=> E`

CORE*, FUR1 2013 VCAA 5 MC

$100 000 is invested in a perpetuity at an interest rate of 6% per annum.

After 10 quarterly payments have been made, the amount of money that remains invested in the perpetuity is

A. $15 000

B. $40 000

C. $85 000

D. $94 000

E. $100 000

Show Answers Only

`E`

Show Worked Solution

`text(A perpetuity, by definition, is designed to last)`

♦ Mean mark 38%.

`text(indefinitely which is done by only paying out)`

`text(the interest it receives.)`

`=> E`