Topic 5: Loans, annuities and investments

Set up the TVM inputs with signs that follow the money: what you receive is positive and what you pay is negative, with the rate per year and the payments per year matched.

A lesson for this topic

Examination 1 style: multiple choice

Question 1Examination 1 style

A loan is modelled by the recurrence relation

V_0

= 24\,000

V_{n+1}

= 1.0055V_n - 520

, where

V_n

is the balance of the loan, in dollars, after

n

monthly repayments. Which statement describes this loan?

Question 2Examination 1 style

The balance of a loan, in dollars, after

n

monthly repayments is

V_n

, where

V_0

= 15\,000

and

V_{n+1}

= 1.004V_n - 650

. What is the balance of the loan after two repayments, correct to the nearest cent?

Question 3Examination 1 style

At the start of a month, the balance of a home loan is $212 400. Interest is charged at 5.4% per annum, compounding monthly, and a repayment of $1650 is made at the end of the month. By how much does this repayment reduce the principal?

Question 4Examination 1 style

Jack borrows $36 000 to be repaid over 5 years with equal monthly repayments made at the end of each month. Interest is charged at 6.9% per annum, compounding monthly. What is the monthly repayment, correct to the nearest cent?

Question 5Examination 1 style

A mortgage of $380 000 has interest charged at 5.85% per annum, compounding monthly. Repayments of $2400 are made at the end of each month. What is the balance of the loan after 5 years, correct to the nearest cent?

Question 6Examination 1 style

A loan of $64 000 has interest charged at 7.5% per annum, compounding monthly. Repayments of $950 are made at the end of each month, with a smaller final repayment if one is needed. How many repayments, including any smaller final repayment, are needed to repay the loan in full?

Question 7Examination 1 style

A loan of $64 000 has interest charged at 7.5% per annum, compounding monthly. Repayments of $950 are made at the end of each month until the last repayment, which is smaller and clears the loan exactly. What is the final repayment, correct to the nearest cent?

Question 8Examination 1 style

A loan of $62 000 is repaid over 7 years with monthly repayments of $932.72 made at the end of each month. Interest is charged at 6.9% per annum, compounding monthly. What is the total interest paid over the life of the loan, correct to the nearest dollar?

Question 9Examination 1 style

A $250 000 loan is to be repaid over 20 years with monthly repayments of $1791.08 made at the end of each month. Interest is charged at 6% per annum, compounding monthly. After 5 years the interest rate rises to 6.6% per annum, compounding monthly. What new monthly repayment, correct to the nearest cent, repays the loan by the end of the original 20 years?

Question 10Examination 1 style

A loan of $30 000 is repaid in full by 48 equal monthly repayments of $720, made at the end of each month. Interest is charged per annum, compounding monthly. What is the interest rate per annum, correct to two decimal places?

Question 11Examination 1 style

A $320 000 loan has interest charged at 6.3% per annum, compounding monthly. The repayment that clears it in 25 years is $2120.84 a month. The borrower instead repays $2500 at the end of each month from the start. How many monthly repayments, including a smaller final repayment, will now be needed?

Question 12Examination 1 style

A $180 000 loan has interest charged at 5.4% per annum, compounding monthly, and monthly repayments of $1461.22, made at the end of each month, that would repay it in 15 years. After 3 years the interest rate rises to 6.3% per annum, compounding monthly, and the repayments stay at $1461.22. How many repayments in total, counted from the start of the loan and including a smaller final repayment, will now be needed?

Question 13Examination 1 style

A loan of $96 000 has interest charged at 6.6% per annum, compounding monthly. Repayments of $1430.20 are made at the end of each month. How much interest is paid in the second year of the loan, correct to the nearest cent?

Question 14Examination 1 style

$640 000 is invested in a perpetuity that earns interest at 4.5% per annum, compounding quarterly. The same payment is made from it at the end of every quarter, forever. What is the quarterly payment?

Question 15Examination 1 style

A sports club wants a perpetuity that pays a scholarship of $1500 at the end of every month, forever. The perpetuity earns interest at 5% per annum, compounding monthly. How much must be invested?

Question 16Examination 1 style

Rosa invests $300 000 in an annuity that earns interest at 5.4% per annum, compounding monthly. She receives a payment of $2500 at the end of each month. What is the balance of the annuity after 2 years, correct to the nearest cent?

Question 17Examination 1 style

How much must be invested now in an annuity earning 4.8% per annum, compounding monthly, to provide a payment of $3000 at the end of each month for 20 years, after which the annuity is exhausted? Give the answer correct to the nearest cent.

Question 18Examination 1 style

Leo opens a savings account with $5000. It earns interest at 6% per annum, compounding monthly, and he adds $200 at the end of each month. What is the balance after 10 years, correct to the nearest cent?

Question 19Examination 1 style

Leo's savings account opened with $5000 and he added $200 at the end of each month for 10 years. Interest was 6% per annum, compounding monthly, and the balance after 10 years was $41 872.85. How much interest did the account earn over the 10 years?

Question 20Examination 1 style

An investment starts at $8000 and earns interest at 4.8% per annum, compounding monthly. $350 is added at the end of each month. After how many months will the balance first be more than $50 000?

Question 21Examination 1 style

Ava wants $100 000 in 8 years. Her account starts empty and earns interest at 5.4% per annum, compounding monthly. She deposits the same amount at the end of each month. What monthly deposit does she need, correct to the nearest cent?

Question 22Examination 1 style

$10 000 is invested and $400 is added at the end of every month. After 5 years the balance is exactly $40 000. Interest is paid per annum, compounding monthly. What is the interest rate per annum, correct to two decimal places?

Question 23Examination 1 style

Kai opens an account with $12 000. It earns interest at 3.6% per annum, compounding monthly, and he adds $250 at the end of each month.

V_n

is the balance, in dollars, after

n

months. Which recurrence relation models this account?

Question 24Examination 1 style

$500 000 is invested at 4.8% per annum, compounding monthly. A payment of $2600 is taken at the end of each month for 8 years. The balance then stays invested at the same rate as a perpetuity, paying the same amount at the end of each month forever. What is the monthly perpetuity payment, correct to the nearest cent?

Question 25Examination 1 style

An annuity of $400 000 earns interest at 5.1% per annum, compounding monthly. It pays $3000 at the end of each month for 5 years; from then on the payment rises to $3500 a month until the annuity runs out, with a smaller final payment. How many payments does the annuity make in total, counted from the start and including the smaller final payment?

Examination 2 style questions

A car loanExamination 2 style · 7 marks

Priya borrows $32 000 to buy a car. Interest is charged at 6.6% per annum, compounding monthly, and she repays $780 at the end of each month. The balance of the loan, in dollars, after

n

repayments is

V_n

, where

V_0

= 32\,000

and

V_{n+1}

= RV_n - 780

(a) Find the value of

R

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Complete the amortisation table for the first two repayments. Write every amount correct to the nearest cent. [2]

Repayment number	Repayment ($)	Interest ($)	Principal reduction ($)	Balance ($)
0	0.00	0.00	0.00	32000.00
1	780.00
2	780.00

Type one value in each empty cell.

0 words · a sentence or two

(d) Find the balance of the loan after 2 years, correct to the nearest cent. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Priya keeps repaying $780 a month until the loan is repaid, with a smaller final repayment. How many repayments does she make in total, counting that final one? [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

A mortgage and a rate riseExamination 2 style · 8 marks

The Nguyen family borrows $450 000 to buy a house. Interest is charged at 5.8% per annum, compounding monthly. The loan is to be repaid over 25 years with equal monthly repayments made at the end of each month.

(a) Find the monthly repayment, correct to the nearest cent. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) The family repays $2844.59 each month. Find the balance of the loan after 5 years, correct to the nearest cent. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(c) After 5 years the interest rate rises to 6.4% per annum, compounding monthly. Find the new monthly repayment, correct to the nearest cent, that repays the loan by the end of the original 25 years. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Suppose instead the family keeps repaying $2844.59 a month after the rate rise. How many repayments in total, counted from the start of the loan and including a smaller final repayment, will the loan now take? [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) In the situation of the previous part, explain why the final repayment is smaller than $2844.59. [2]

0 words · a sentence or two

Saving for retirementExamination 2 style · 7 marks

Tom invests $60 000 in an account that earns interest at 5.2% per annum, compounding monthly. He adds $1000 to the account at the end of each month for 15 years.

(a) Find the balance of the account after 15 years, correct to the nearest cent. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) How much interest does the account earn over the 15 years? Give the answer correct to the nearest cent. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(c) Tom moves the $402 467.79 into an annuity that earns 4.8% per annum, compounding monthly. It pays him the same amount at the end of each month for 20 years and then has a balance of zero. Find the monthly payment, correct to the nearest cent. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Instead, Tom could move the $402 467.79 into a perpetuity earning 4.8% per annum, compounding monthly, that pays him at the end of each month. Find the monthly payment, correct to the nearest cent. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Explain why the balance of the perpetuity never changes. [2]

0 words · a sentence or two

A savings planExamination 2 style · 8 marks

Mei opens a savings account with $2400. The account earns interest at 4.2% per annum, compounding monthly. At the end of each month, after the interest is added, she deposits $150. The balance, in dollars, after

n

months is

V_n

, where

V_0

= 2400

and

V_{n+1}

= 1.0035V_n + 150

(a) Complete the table for the first two months. Write every amount correct to the nearest cent. [2]

Month	Interest ($)	Deposit ($)	Balance ($)
0	0.00	0.00	2400.00
1		150.00
2		150.00

Type one value in each empty cell.

(b) Explain why the interest Mei earns each month keeps getting larger. [2]

0 words · a sentence or two

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Mei wants a balance of $10 000 after exactly 3 years, still starting with $2400. What monthly deposit would she need? Give the answer correct to the nearest cent. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

An annuity incomeExamination 2 style · 6 marks

Joan's annuity is modelled by the recurrence relation

A_0

= 320\,000

A_{n+1}

= 1.0045A_n - 2600

, where

A_n

is the balance of the annuity, in dollars, after

n

monthly payments to Joan.

(a) Describe what the numbers 1.0045 and 2600 represent in this recurrence relation. [2]

0 words · a sentence or two

(b) Find the interest rate per annum, compounding monthly, as a percentage. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) How much interest does the annuity earn in the first year? Give the answer correct to the nearest cent. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) How many payments, including a smaller final payment, will Joan receive before the annuity runs out? [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).