Loan Amortization
You take out a loan for a computer, a delivery truck, or a business, and the bank tells you "fixed monthly payment of such-and-such." That payment doesn't come out of nowhere: it is the same math as geometric sequences and compound interest, applied to debt. How is that fixed payment calculated, and how much of it actually reduces what you owe?
The constant monthly payment formula
The outstanding principal after $k$ payments is $C_k = C_0(1+i)^k - a\cdot\dfrac{(1+i)^k-1}{i}$. Since the loan ends up fully paid off, $C_n=0$: solving that condition for $a$ gives the payment 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…
To celebrate the end of exams, a group of friends decides to finance the purchase of a sound system for their hangout, valued at $\$1500$. 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 $\$48\,000$. The shop offers financing at a nominal annual interest rate of $4.8 \%…
A Sixth Form student wants to buy an electric mountain bike costing $\$48\,000$. 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 buy a moped costing $\$2,000$, Iratxe pays a deposit of $\$500$ and finances the remainder through equal monthly instalments over 2 years. If the nominal annual…
To buy a moped costing $\$2,000$, Iratxe pays a deposit of $\$500$ 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.
$$a = \frac{C \times i \times (1+i)^n}{(1+i)^n - 1}$$To set up a small recording studio and start his YouTube channel, Marcos has taken out a loan of $4\,500$ pesos to be repaid over 2 years in equal monthly instalments.…
To set up a small recording studio and start his YouTube channel, Marcos has taken out a loan of $4\,500$ pesos 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 payment: interest and amortization
Each payment $a$ is split into two parts: the interest for the period (what it costs to carry the debt) and the amortization (what truly reduces what you owe).
Marta has taken out a loan of $\$3,000…
Marta 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 $12\,000$ pesos 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 $12\,000$ pesos 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 24-year-old female entrepreneur decides to open a graphic design studio. To acquire the necessary equipment (high-end computers, software licenses and furniture), she…
A 24-year-old female entrepreneur decides to open a graphic design studio. To acquire the necessary equipment (high-end computers, software licenses and furniture), she applies for a loan from a financial institution to be repaid over 5 years in equal monthly installments of $\$8\,510$. 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 pesos cent.
The amortization table
| Period | Payment | Interest | Amortization | Outstanding balance |
|---|---|---|---|---|
| 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 | $883.80$ | $213.99$ | $669.81$ | $10{,}029.71$ |
Each row is filled in the same way: interest = previous balance $\times\, i$; amortization = payment − interest; new balance = previous balance − amortization.
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…
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 | $\$500.00$ | $\$5.00$ | $\$40.00$ | (A) |
| 2 | (B) | $\$4.60$ | $\$40.40$ | $\$45.00$ |
Given the following row from the amortization table of a loan at $12\,\%$ per year with monthly payments (the monthly interest is calculated by dividing the annual rate…
Given the following row from the amortization table of a loan at $12\,\%$ per year with monthly payments (the monthly interest is calculated by dividing the annual rate by $12$, that is, $1\,\%$ per month):
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Month: $10$
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Initial debt for the month: $14\,250,80$ pesos
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Monthly payment: $450$ pesos
Calculate:
- a) The interest for month $10$ (round to the nearest cent).
- b) The principal repaid in that month.
- c) The outstanding debt at the start of month $11$.
A group of friends decides to buy a professional mixing console that costs $1\,800$ pesos. They pay $300$ pesos in cash and finance the rest at an interest rate of…
A group of friends decides to buy a professional mixing console that costs $1\,800$ pesos. They pay $300$ pesos 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.
Single payment at the end versus monthly payments
Not all loans are repaid in monthly installments. Sometimes, as with a promissory note, you pay nothing during the term and settle everything at once at the end. Then the debt grows just as in the compound interest you already know.
When the term is not a whole number of periods, the exponent can be fractional: 18 months with an annual compound rate is $n=1.5$ years.
The difference can be explained without building the whole table. With a single payment, the full $\$5{,}000$ generates interest during all 18 months. With monthly payments, each payment reduces the balance, so the average balance is much lower.
To finance an end-of-year trip, a student association takes out a loan of $3\,000$ pesos 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 $3\,000$ pesos 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.
Outstanding balance and term
To find out how much you owe after $k$ payments you can follow the table, or use the fact that the balance is the present value of the remaining payments:
Calculate the outstanding balance of a loan of $20\,000$ pesos taken out at $4\%$ per year after paying 12 monthly instalments. The loan was granted for a total of 5…
Calculate the outstanding balance of a loan of $20\,000$ pesos 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 university student wants to buy a second-hand car costing $\$190\,000$. He has some savings, but needs to finance the remaining $\$140\,000$. The bank offers him a…
A university student wants to buy a second-hand car costing $\$190\,000$. He has some savings, but needs to finance the remaining $\$140\,000$. 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 $\$3\,600$ 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 $\$3\,600$.
Early repayment
Sometimes extra money comes your way (a year-end bonus, an inheritance) and you want to pay down part of the debt early. The bank usually offers two options with the balance left after the extra payment:
A video rental store wants to renew its screens and applies for a loan of $120\,000$ pesos at $9\%$ per year to be repaid over 3 years in equal monthly installments. The…
A video rental store wants to renew its screens and applies for a loan of $120\,000$ pesos at $9\%$ per year to be repaid over 3 years in equal monthly installments. 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 self-employed worker applies for a microcredit of $\$240\,000$ at $8 \%…
A self-employed worker applies for a microcredit of $\$240\,000$ 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 $\$60\,000$ 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.
Comparing loans and special cases
To decide between two loans, looking at the monthly payment is not enough: compare the total cost (sum of all payments plus fees).
Other cases you will see in the exercises:
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Grace period: if interest is added to the principal during $m$ periods, the new principal is $C_0(1+i)^m$ and from there you calculate the payment for the remaining periods.
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Rate change: calculate the balance at the moment of the change and recalculate the payment with the new rate and the periods that remain.
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Prepayment penalty: the amount to pay is the balance plus the fee percentage on that balance.
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0% rate: the payment is $a=\dfrac{C_0}{n}$ and the balance always drops by the same amount. This follows from seeing that, if $i=0$, every payment is worth the same as its present value, so $n\cdot a=C_0$.
Lucas finances his new next-generation games console, costing $\$12\,000…
Lucas finances his new next-generation games console, costing $\$12\,000$, in 12 equal monthly instalments. At the end of the year, he realizes that he has paid a total of $\$12\,720$. Calculate:
- a) The amount of the monthly instalment.
- b) The total cost of the interest paid over the whole loan.
We compare two loans to buy a motorbike costing $3\,500$ pesos: Option A: Monthly amortisation over 2 years at $6\%$ nominal annual interest. Option B: Quarterly…
We compare two loans to buy a motorbike costing $3\,500$ pesos:
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Option A: Monthly amortisation over 2 years at $6\%$ nominal annual interest.
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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.
We compare two loan options to finance a renovation of $\$600\,000$ 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 $\$600\,000$ 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.
Back to the start: the bank's fixed monthly payment comes from $a=\dfrac{C_0\, i}{1-(1+i)^{-n}}$; at the beginning almost the whole payment is interest, and little by little most of it goes toward reducing the debt.