Loan amortisation

To celebrate the end of exams, a group of friends decides to finance the purchase of a sound system for their hangout, valued at $\text{€}\ 1500$. The bank offers them a loan at a nominal annual interest rate of $6 \%$, which they must repay in monthly installments over one year. The figure summarizes the situation. Identify the elements of the loan so you can apply the constant annuity formula:

• Initial capital ($C$)

• Number of payments per year ($k$)

• Total number of installments ($n$)

• Monthly interest rate ($i$) expressed as a decimal.

To buy a moped costing $\text{€}\ 2,000$, Iratxe pays a deposit of $\text{€}\ 500$ and finances the remainder through equal monthly instalments over 2 years. If the nominal annual interest rate is $3 \%$, apply the amortisation annuity formula, clearly stating the principal $C$ that she is actually going to finance.

$$a = \frac{C \times i \times (1+i)^n}{(1+i)^n - 1}$$

Aoife is going to purchase an electric motorcycle for her urban commutes for an amount of $€4,800$. She applies for a bank loan to be repaid over 2 years (24 months) using the French amortization system at a nominal annual interest rate ($\text{TIN}$) of $6 \%$.

1. a) Before applying the financial formula, reasonably estimate a lower bound for the monthly payment (calculating how much Aoife would pay if the interest rate were $0 \%$) and an approximate upper bound (calculating the total simple interest for the 2 years on the initial principal, adding it to the principal, and dividing by 24 months).
2. b) Use the constant amortization payment formula to calculate the exact value of the monthly payment $a$ (with precision to the cent).
3. c) Check whether the exact result falls within the range estimated in part a) and assess whether it is a reasonable result. Then, calculate the total amount paid in interest over the 2 years.

To set up a small recording studio and start his YouTube channel, Cian has taken out a loan of $4\,500$ € to be repaid over 2 years in equal monthly instalments. The bank charges a nominal annual interest rate of $6\%$. Calculate the amount of the monthly instalment that Cian must pay.

We compare two loans to buy a motorbike costing $3\,500$ €:

• Option A: Monthly amortisation over 2 years at $6\%$ nominal annual interest.

• Option B: Quarterly amortisation over 2 years at $6.2\%$ nominal annual interest.

Calculate the instalment for each option and determine in which of the two options more interest is ultimately paid at the end of the two years.

A 24-year-old female entrepreneur decides to open a graphic design studio. To acquire the necessary equipment (high-end computers, software licences and furniture), she applies for a loan from a financial institution to be repaid over 5 years in equal monthly instalments of $\text{€}\ 425.50$. The institution charges a nominal annual interest rate of $5.4 \%$.

Calculate the initial principal that the bank granted her to set up her business. Round the result to the nearest euro cent.

A university student wants to buy a second-hand car costing $\text{€}\ 9,500$. He has some savings, but needs to finance the remaining $\text{€}\ 7,000$. The bank offers him a personal loan at an annual interest rate of $6.75 \%$. The student analyzes his budget and concludes that he can only allocate a maximum of $\text{€}\ 180$ per month to the instalment payment.

Using logarithms, determine the minimum number of months it will take to repay the loan so that the monthly instalment does not exceed his limit of $\text{€}\ 180$.

Aoife has taken out a loan of €3,000 to pay for her university tuition fees and course materials. The bank applies the French amortisation system (equal instalments) with a monthly interest rate of $0.5 \%$. If the monthly instalment is €133.06, calculate what amount of the first instalment corresponds exclusively to interest payment.

A young entrepreneur applies for a loan of $12\,000$ € to buy a second-hand van for her delivery business. The loan will be amortised over 4 years with quarterly payments and an annual interest rate of $8\%$. Determine the portion of the first instalment corresponding to interest payment and how much is actually used to amortise the debt.

Calculate the outstanding balance of a loan of $20\,000$ € taken out at $4\%$ per year after paying 12 monthly instalments. The loan was granted for a total of 5 years. (Hint: The outstanding balance is the present value of the remaining instalments to be paid).

A family takes out a 20-year mortgage of €160,000 at a fixed annual interest rate of $3.25 \%$ with monthly payments. After exactly 10 years of faithfully paying the monthly instalments, they receive an inheritance and consider paying off the outstanding debt.

Calculate the breakdown of the 121st instalment, specifying which part corresponds to interest payment and which part to capital amortisation.

In the French amortization system with constant payment $a$, at the beginning most of each payment is used to cover interest and very little to amortize principal, but this relationship gradually reverses as the debt decreases.

Cian and Aoife apply for a mortgage of €120,000 at an annual $\text{TIN}$ of $4.8 \%$ with monthly payments over 20 years ($n = 240$ months).

1. a) Analytically prove that the first principal amortization payment can be expressed as $A_1 = a \times (1+i)^{-n}$, starting from the relationship $I_1 = C_0 \times i$ and the constant payment formula $a$.
2. b) For the principal amortization to exceed the interest for the first time in a given month $k$, it must hold that $A_k > \frac{a}{2}$. Using the expression $A_k = A_1 \times (1+i)^{k-1}$ and the result from part a), set up the corresponding inequality and solve analytically with logarithms for the condition on $k$. Determine in which year and month of the loan this crossover occurs exactly.
3. c) Analyze the expression obtained for $k$ in part b): does the crossover month depend on the initial borrowed principal $C_0$? If they had taken out the mortgage for 30 years ($n = 360$ months) at the same interest rate, calculate in which month the crossover would occur and explain what financial consequence extending the term has on the amount of time during which interest is predominantly paid.

Given the following incomplete amortization schedule for a loan for a graphics tablet, calculate the missing values marked with letters knowing that the monthly interest rate is $1 \%$:

Month	Outstanding balance	Interest	Principal repaid	Payment (Annuity)
1	$\text{€}\ 500.00$	$\text{€}\ 5.00$	$\text{€}\ 40.00$	(A)
2	(B)	$\text{€}\ 4.60$	$\text{€}\ 40.40$	$\text{€}\ 45.00$

A group of friends decides to buy a professional mixing console that costs $1\,800$ €. They pay $300$ € in cash and finance the rest at an interest rate of $10\%$ per year (nominal, with monthly compounding period, that is, a monthly rate $i=\frac{0.10}{12}$) to pay it back over 6 months with constant monthly payments.

Remember that the constant payment $a$ that repays a principal $C$ in $n$ periods with rate $i$ per period is:

$a = C\times\dfrac{i}{1-(1+i)^{-n}}$

Create the first two rows of the amortization table, stating in each one: outstanding principal, interest for the period, principal repaid and monthly payment. Round to the nearest cent.

To finance an end-of-year trip, a student association takes out a loan of $3\,000$ € at an annual compound interest rate of $5\%$. The agreement is to repay it in a single lump sum payment after 18 months. What will the total interest amount to? If they had chosen to amortise it monthly over the same period, would they pay more or less interest in total? Reason your answer without needing to calculate the complete schedule.

We compare two loan options to finance a renovation of $\text{€}\ 30,000$ to be repaid over 4 years with monthly payments:

• Option A: Nominal annual interest of $5 \%$ with no initial fees.

• Option B: Nominal annual interest of $4.2 \%$ but with an arrangement fee of $1.5 \%$ on the capital requested, paid in cash at signing.

Calculate the total cost of each option (sum of all instalments plus fees) and determine which is more advantageous for the client.

A video rental shop wants to renew its screens and applies for a loan of $6\,000$ € at $9\%$ per year to be repaid over 3 years in equal monthly instalments. The contract includes an early redemption fee of $1\%$ of the outstanding balance. If at the end of the second year they decide to settle the remaining debt, what will be the total amount they must pay at that moment (outstanding balance plus fee)?

A logistics technology company applies for a loan of $\text{€}\ 250,000$ to renew its fleet of electric vehicles. The loan is agreed at an annual interest rate of $4.5 \%$ with quarterly payments over 6 years. The financial institution grants them a full grace period of 6 months (during the first two quarters no payment is made, but interest accumulates on the initial principal).

Calculate the amount of the quarterly instalments that they will have to pay from the third quarter until the end of the loan.

A mortgage loan of $\text{€}\ 200,000$ is taken out over a period of 25 years with monthly instalments. For the first two years, the interest rate is fixed at $2.5 \%$ per year. From the start of the third year, the interest rate becomes variable, changing to $4.75 \%$ per year.

Calculate by how much the family's monthly instalment will increase from the third year onwards due to this interest rate rise.

A self-employed worker applies for a microcredit of $\text{€}\ 12,000$ at $8 \%$ per year to be repaid over 3 years in semi-annual instalments. After the first year (after paying the second instalment), he decides to make an early repayment of $\text{€}\ 3,000$ to reduce the debt. The bank allows him to choose between reducing the loan duration or reducing the amount of the remaining instalments.

If he chooses to reduce the instalment amount while maintaining the original 3-year term, calculate what the new semi-annual instalment amount will be.