Loan amortisation
You ask for a loan for a car, a home improvement or a business, and the bank tells you "fixed instalment of so many euros a month". That instalment does not come out of nowhere: it is the same mathematics of geometric progressions that you already know, applied to the money you owe.
The constant instalment formula
The outstanding capital after paying $k$ instalments is $C_k = C_0(1+i)^k - a\cdot\dfrac{(1+i)^k-1}{i}$. Since the loan is fully paid off, $C_n=0$: solving for $a$ from that condition gives the instalment formula.
To celebrate the end of exams, a group of friends decides to finance the purchase of a sound system for their hangout, valued at $1500 \text{ €}$. The bank offers them a…
To celebrate the end of exams, a group of friends decides to finance the purchase of a sound system for their hangout, valued at $1500 \text{ €}$. 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:
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Initial capital ($C$)
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Number of payments per year ($k$)
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Total number of installments ($n$)
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Monthly interest rate ($i$) expressed as a decimal.
A Sixth Form student wants to buy an electric mountain bike costing $2.400 \text{ €}$. The shop offers financing at a nominal annual interest rate of $4,8 \%$ spread…
A Sixth Form student wants to buy an electric mountain bike costing $2.400 \text{ €}$. The shop offers financing at a nominal annual interest rate of $4,8 \%$ spread across monthly instalments. Calculate what monthly interest rate ($i$) will be applied to each instalment, expressing it both as a percentage and as a decimal.
To set up a small recording studio and start his YouTube channel, Marcos has taken out a loan of $4\,500$ € to be repaid over 2 years in equal monthly instalments. The…
To set up a small recording studio and start his YouTube channel, Marcos 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 Marcos must pay.
Breakdown of the instalment: interest and amortisation
Each instalment $a$ is split into two parts: the interest for the period (what it costs to hold the debt) and the amortisation (what really reduces what you owe).
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…
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.
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…
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:
$a = C\cdot\dfrac{i}{1-(1+i)^{-n}}$
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.
The amortisation schedule
| Period | Instalment | Interest | Amortisation | Outstanding capital |
|---|---|---|---|---|
| 0 | — | — | — | $12\,000{,}00$ |
| 1 | $883{,}80$ | $240{,}00$ | $643{,}80$ | $11\,356{,}20$ |
| 2 | $883{,}80$ | $227{,}12$ | $656{,}68$ | $10\,699{,}52$ |
| 3 | $883{,}80$ | $213{,}99$ | $669{,}81$ | $10\,029{,}71$ |
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…
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.
A family takes out a 20-year mortgage of $160.000 \text{ €}$ at a fixed annual interest rate of $3,25 \%$ with monthly payments. After exactly 10 years of faithfully…
A family takes out a 20-year mortgage of $160.000 \text{ €}$ 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.
Early repayment
Sometimes some extra money comes your way (an inheritance, a double bonus) and you want to pay off part of the debt early. The bank usually offers two options with the outstanding capital that remains after the payment:
We compare two loan options to finance a renovation of $30.000 \text{ €}$ to be repaid over 4 years with monthly payments: Option A: Nominal annual interest of $5 \%$…
We compare two loan options to finance a renovation of $30.000 \text{ €}$ to be repaid over 4 years with monthly payments:
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Option A: Nominal annual interest of $5 \%$ with no initial fees.
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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 self-employed worker applies for a microcredit of $12.000 \text{ €}$ at $8 \%$ per year to be repaid over 3 years in semi-annual instalments. After the first year…
A self-employed worker applies for a microcredit of $12.000 \text{ €}$ 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 $3.000 \text{ €}$ 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.
Lab: building and modifying the amortisation schedule
Work with the model of 4 500 € over 24 months, annual TIN 6 %, no fees. Predict whether lowering the instalment by lengthening the term also reduces the total cost. Enter in a spreadsheet: H1 = 4500, H2 = 0,005 and H3 = 24; H4 contains =H1*H2/(1-(1+H2)^(-H3)). If H2 is zero, use =H1/H3.
In row 2, put period 0 in A2 and the initial balance =H1 in E2. Row 3 represents the first payment: A3 =A2+1, B3 =$H$4 (instalment), C3 =E2*$H$2 (interest), D3 =B3-C3 (amortisation) and E3 =E2-D3 (balance). Drag down to period 24. Format to two decimal places, keeping internal precision.
The recurrence lets you see how an instalment reduces the debt:
Modify the algorithm to include an additional repayment of 500 € just after payment 12: subtract it from the balance of that period and recalculate the instalment for the remaining 12. Compare keeping the term with keeping the instalment; in the second case, stop the process when the debt is cleared and adjust the last payment. Also try zero interest and check that the balance never goes on to become negative. Compare the recurrence and the closed formula: which helps with changes in conditions and which avoids going through all the rows. Submit a recommendation justified by both instalment and cost, not just one of them.