Financial mathematics

Apply geometric series to calculate future and present values of annuities; analyse reducing-balance loans using recurrence relations; compare investment and loan scenarios.

Algebra / Functions · MAV-12-08

Sub-topics

Annuities and superannuation

Calculate future and present values of annuities using geometric series; model superannuation accumulation; solve annuity problems using recurrence relations.

Loans and investments

Analyse reducing-balance loans using recurrence relations and geometric series; calculate loan repayments and outstanding balances; compare investment and loan scenarios.

Open Math

Financial mathematics

Algebra / Functions · MAV-12-08

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Date:

Annuities and superannuation

Q1Straightforward

An annuity account satisfies the recurrence relation

A_{n+1} = 1.05 A_n + 2000

, with

A_0 = 0

. Find

A_3

, the value of the account after 3 contributions.

Q2Straightforward

Use the future-value formula

FV = M \dfrac{(1+r)^n - 1}{r}

to find the future value of an annuity with annual contribution

M = \$1000

, interest rate

r = 5\%

per annum, over

n = 4

years. Give your answer in dollars, rounded to the nearest cent.

Q3Straightforward

Alicia deposits $500 at the end of each year for 3 years into an account earning 4\% per annum. How much is in the account at the end of 3 years? Give your answer in dollars, rounded to the nearest cent.

Q4Straightforward

What is the present value of an annuity that pays $1500 per year for 3 years, if the interest rate is 4\% per annum? Use

PV = M \dfrac{1-(1+r)^{-n}}{r}

. Round to the nearest cent.

Q5Moderate

Raj contributes $3000 at the end of each year into a superannuation fund earning 6\% per annum. Find the total value of the fund after 10 years. Round to the nearest cent.

Q6Moderate

Deposits of $1000 are made at the end of each of 4 years into an account paying 5\% per annum. Explain why the accumulated value is a geometric series, and find the total. Round to the nearest cent.

Q7Moderate

How much must be deposited at the end of each year for 5 years, into an account earning 4\% per annum, to accumulate $20{,}000? Round your answer to the nearest cent.

Q8Moderate

How much must be in a superannuation fund today to allow annual withdrawals of $20{,}000 for 10 years, starting in one year, if the fund earns 5\% per annum? Round to the nearest cent.

Q9Challenging

Mei has $5000 in a savings account and also deposits an additional $1000 at the end of each year. The account earns 5\% per annum. How much will she have after 5 years? Round to the nearest cent.

Q10Challenging

Priya deposits $4000 at the end of each year into an account earning 6\% per annum. After how many full years will the fund first exceed $50{,}000?

Q11Challenging

Leo invests $2000 per year for 5 years into an account earning 4\% per annum. After 5 years he stops making deposits, but the account continues to earn interest at 6\% per annum for a further 5 years. Find the total value of the account after 10 years. Round to the nearest cent.

Q12Challenging

Sam contributes $6000 at the end of each year into a superannuation fund earning 7\% per annum for 30 years. On retirement, the entire fund is transferred to an account earning 4\% per annum. Sam then makes equal annual withdrawals for 20 years before the fund is exhausted. Find the annual withdrawal Sam can afford. Round to the nearest cent.

Loans and investments

Q13Straightforward

A loan of $20{,}000 charges interest at 0.5\% per month. At the end of each month a repayment of $400 is made. The balance satisfies

A_{n+1} = 1.005\,A_n - 400

, with

A_0 = 20{,}000

. Find

A_1

, the outstanding balance after the first repayment.

Q14Straightforward

A loan of $5000 charges 1\% interest per month. Monthly repayments are $200. Starting from

A_0 = 5000

, apply the recurrence

A_{n+1} = 1.01\,A_n - 200

to find

A_3

, the balance after 3 months. Round to the nearest cent.

Q15Straightforward

Find the monthly repayment required to repay a loan of $12{,}000 over 12 months at an interest rate of 1\% per month. Use

M = P\dfrac{r(1+r)^n}{(1+r)^n - 1}

. Round to the nearest cent.

Q16Moderate

The outstanding balance after

n

repayments on a reducing-balance loan is given by

A_n = P(1+r)^n - M\frac{(1+r)^n - 1}{r}

A loan of $5000 at 2\% per month has monthly repayments of $300. Find the outstanding balance after 3 months. Round to the nearest cent.

Q17Moderate

A loan of $10{,}000 at 1\% per month is repaid over 12 months at $888.49 per month. Calculate the total interest paid over the life of the loan. Round to the nearest cent.

Q18Moderate

Find the monthly repayment to fully repay a $30{,}000 loan over 5 years (60 months) at an interest rate of 6\% per annum, compounded monthly. Round to the nearest cent.

Q19Moderate

Loan A: $20{,}000 at 6\% per annum (monthly compounding) repaid over 3 years.
Loan B: $20{,}000 at 5\% per annum (monthly compounding) repaid over 4 years.

Calculate the total interest paid on Loan A. Round to the nearest cent.

Q20Moderate

For the loans from Q7:
- Loan A: $20{,}000 at 6\% p.a. (monthly) over 3 years — total interest $1903.79
- Loan B: $20{,}000 at 5\% p.a. (monthly) over 4 years

Calculate the total interest paid on Loan B. Round to the nearest cent.

Q21Challenging

A reducing-balance loan satisfies the recurrence

A_{n+1} = (1+r)A_n - M

, with

A_0 = P

. By recognising the geometric series that arises when the recurrence is expanded, show that

A_n = P(1+r)^n - M \cdot \frac{(1+r)^n - 1}{r}

Then use this formula to find the outstanding balance after 24 months on a $30{,}000 loan at 0.5\% per month with monthly repayment $333.06. Round to the nearest cent.

Q22Challenging

A loan of $8000 at 1\% monthly interest is repaid in monthly instalments of $500. After how many full months will the loan first be fully repaid?

Q23Challenging

A borrower takes out a $200{,}000 home loan at 6\% per annum compounded monthly over 25 years. The monthly repayment is $1288.60. After 10 years (120 months) of repayments, the borrower makes a one-off extra payment of $20{,}000 off the principal.

How many months does this shorten the loan by?

Q24Challenging

An investor deposits $500 per month into a managed fund returning 5\% per annum (compounded monthly) for 10 years.

A friend takes out a $50{,}000 personal loan at 8\% per annum compounded monthly and repays it over 10 years.

Find the difference between the investor's fund value and the total interest paid by the friend. Round to the nearest dollar.

Worked solutions and answers at openmath.au/year-12/advanced/financial-mathematics