1.7 Amortisation and annuities questions for IB Maths AI

Find the amount of each monthly payment. Give your answer correct to two decimal places.

Find the total amount Diego will repay.

Find the total interest Diego will pay.

Find the value of the investment after 5 years if Amara chooses Account A.

Find the value of the investment after 5 years if Amara chooses Account B.

State which account Amara should choose. Justify your answer.

After 5 years, Amara uses the full value of her chosen account as a deposit on a house costing 340 000 EUR. She borrows the rest of the price.

Find the amount she borrows.

The loan is repaid over 25 years with equal payments at the end of each month, at a nominal annual interest rate of 4.4%, compounded monthly. Find the monthly payment.

Find the total interest Amara will pay on the loan.

Find the monthly payment for Option A.

Find the total amount of interest Ines pays with Option A.

Find the amount Ines would still owe after 2 years with Option A.

Option B is advertised as a "flat-rate loan at only 4.2 % per year". For each year of the loan, interest equal to 4.2 % of the original amount borrowed is added to the debt, so the total debt after $n$ years, $u_n$, forms an arithmetic sequence. The total debt after 5 years is then repaid in 60 equal monthly payments.

Show that the total debt after 5 years is EUR 29 040.

Hence write down the monthly payment for Option B.

Find the nominal annual interest rate, compounded monthly, that Ines is actually being charged with Option B.

Ines says that Option B is cheaper than Option A because 4.2 % is less than 6.0 %. Explain why she is wrong.

Option C is a "growing payment" loan with the same interest rate as Option A, 6.0 % compounded monthly, which is 0.5 % per month. The first payment is EUR $P$, and each later payment is 1 % larger than the previous one, so that the payments form a geometric sequence.

Write down an expression, in terms of $P$ and $k$, for the $k$th payment.

The present value of a payment of $X$ made at the end of month $k$ is $\frac{X}{{1.005}^k}$.

Show that the sum of the present values of all 60 payments is

The loan is fully repaid when the sum of the present values of the payments equals the amount borrowed.

Hence find the value of $P$.

Find

(i) the size of the final payment;

(ii) the total amount Ines repays with Option C.

Option C costs more in total than Option A. Suggest one reason why Ines might still choose Option C.

Ines later estimates the value of the bakery as the sum of the present values of all its future monthly profits, again discounting at 0.5 % per month. She models the profit at the end of the first month as EUR 1500, with each month's profit 0.3 % greater than the previous month's, and assumes this continues indefinitely.

Find the value of the bakery according to this model.

Explain why this method of valuation could not be used if Ines assumed that the monthly profit grew by 0.6 % each month.

Find the monthly payment. Give your answer correct to two decimal places.

Find the total interest they will pay over the 25 years.

Find the value of the fund when Leo is 18 years old, if no further money is added.

Use an annual real interest rate of $5.2\mathrm{\%}-2.7\mathrm{\%}$, compounded monthly, to find the real value of the fund when Leo is 18 years old.

Explain what the answer to part (b) represents in this context.

Mia and Jack estimate that university will cost 120 000 AUD in today's money. They decide to add a fixed amount to the fund at the end of every month, starting one month after Leo is born, so that the real value of the fund when Leo is 18 is 120 000 AUD.

Using the same real interest rate, find the amount they must add each month.

Find the total amount of money Mia and Jack pay into the fund, including the initial deposit.

State one reason why the real value of the fund after 18 years might be different from 120 000 AUD.

Find the number of payments Rowan makes.

Find the amount of the final payment. Give your answer correct to two decimal places.