The hidden cost of time: Day-count conventions and effective rates in loans

Practice in the United States: Rule of 78, Actual/360 in commercial banking, and APR

The United States has a financial system where two worlds coexist: corporate banking and consumer banking. Each uses different conventions, and understanding that division is key to comparing offers.

Corporate banking: Actual/360 as the standard

In US commercial loans, the nominal rate is almost always accompanied by an Actual/360 basis. This means interest is calculated on the actual elapsed days, but divided by 360. The practice is so deeply rooted that loans tied to SOFR (the dollar benchmark rate) use Actual/360 as their conventional basis.

The consequence is direct: a commercial loan with a nominal rate of 12% under Actual/360 has an effective annual cost of approximately 12.17%, because interest accumulates over 365 actual days but is divided by 360.

Consumer banking: APR and the Truth in Lending Act

For consumer loans, the regulation changes. The Truth in Lending Act (TILA) requires lenders to disclose the APR (Annual Percentage Rate), which is a measure of the total cost of credit expressed as an annual rate. The APR is not the nominal rate: it includes fees, insurance, and other charges, and is calculated using a methodology that seeks to reflect the real cost.

However, the APR does not eliminate the ambiguity of the day-count convention. The law requires transparency, but does not impose a single convention for all products. One lender may use Actual/365 to calculate the APR on a credit card, while another uses Actual/360 for a personal loan. The difference is reflected in the APR, but the consumer must know how to read the fine print.

Rule of 78: A surviving relic

The Rule of 78 (also called the "sum of the digits method") is an interest calculation method that allocates a larger proportion of interest to the early payments of the loan. Although prohibited in many states for consumer loans, it still appears in some auto financing and personal loan contracts.

The logic is simple: if a loan has 12 months, the sum of the digits from 1 to 12 is 78. The first month pays 12/78 of the total interest, the second 11/78, and so on. This means the borrower pays more interest upfront, which increases the cost if they decide to prepay.

APR as a comparison tool

The APR is the most useful tool for comparing consumer offers because it integrates the nominal rate, fees, and insurance into a single figure. But it has a limit: it does not always reflect the underlying day-count convention. Two loans with the same APR can have different effective costs if one uses Actual/360 and the other Actual/365.

The lesson for the US borrower is clear: the APR is a starting point, but the day-count convention remains the factor that defines the real cost.

Practice in Colombia: 360 commercial, Effective Annual Rate (EA), and the Basic Legal Circular

Colombia has a distinct financial tradition shaped by the Financial Superintendency and local banking practice. The dominant convention is 30/360, but the system is more flexible than it seems.

The 30/360 convention as the traditional basis

In the Colombian banking system, the 30/360 convention (also called 360/360) is deeply rooted. A year of 360 days and months of 30 days are assumed, regardless of the actual calendar. This practice has historical roots in Latin American commercial banking and is maintained in many credit products.

However, the Financial Superintendency does not impose a single convention. Several bases coexist in the Colombian market:

• 30/360 or 360/360: the most common in traditional commercial and consumer loans.
• Actual/360: used in some products linked to international benchmark rates.
• Actual/365: appears in specific products and in transactions with foreign counterparties.

The choice depends on the product, the contract, and the negotiation between the parties.

Effective Annual Rate (EA): The comparison standard

The Effective Annual Rate (EA) is the central concept in Colombia for comparing credit costs. The Financial Superintendency requires institutions to disclose the effective rate, which reflects the real cost of the loan considering interest capitalization.

Converting a nominal rate to EA depends on the compounding frequency. If a loan has a nominal rate of 12% with monthly compounding, the EA is calculated as follows:

$$EA = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 0.1268 = 12.68\%$$

The EA is the figure the borrower should look at because it integrates the effect of compounding. But the EA does not eliminate the ambiguity of the day-count convention: a loan with an EA of 12.68% can have different costs depending on how days are counted.

Basic Legal Circular: The regulatory framework

The Financial Superintendency's Basic Legal Circular establishes the rules for disclosing interest rates and credit conditions. It requires institutions to report the nominal rate, the effective rate, and the total cost of credit, including commissions and insurance.

The circular also regulates rate advertising: a bank cannot announce a rate without disclosing the corresponding EA. This protects the consumer, but does not eliminate the need to understand the underlying day-count convention.

Colombian practice in action

For a corporate treasurer in Colombia, the lesson is twofold:

1. The dominant convention is 30/360, but it is not the only one.
2. The EA is the comparison standard, but its calculation depends on the day basis.

The combination of both variables defines the real cost of the loan. And here arises the question every treasurer must ask: how much does the difference between a 360-day and a 365-day basis cost in a real loan?

Monetary impact: The hidden surcharge of the 360 basis

The difference between 360 and 365 days seems like a technical detail. But when applied to a large corporate loan, the impact is tangible and can change the viability of a project.

The case study: 10,000 million peso loan

Suppose a Colombian company obtains a loan of 10,000 million pesos at a nominal rate of 12% per annum, with a term of one year. The bank offers two options:

• Option A: Actual/360 basis
• Option B: Actual/365 basis

Interest calculation under each convention

Option A (Actual/360):

$$Interés = 10,000,000,000 \times 0.12 \times \frac{365}{360} = 1,216,666,667$$

Option B (Actual/365):

$$Interés = 10,000,000,000 \times 0.12 \times \frac{365}{365} = 1,200,000,000$$

The difference is $16,666,667 pesos en un solo año. Eso equivale a aproximadamente $1.39 million pesos per month that the company pays extra due to the 360-day convention.

The difference in perspective

To put the impact into perspective, let's compare it with the cost of an average employee in Colombia. A minimum monthly wage in 2024 is approximately $1,300,000 pesos. La diferencia de $16.7 million equals almost 13 monthly minimum wages that the company pays extra due to the day-count convention.

The daily interest factor

The root of the problem lies in the daily interest factor. Under the 360 basis, the factor is:

$$Factor_{360} = \frac{0.12}{360} = 0.0003333$$

Under the 365 basis, the factor is:

$$Factor_{365} = \frac{0.12}{365} = 0.0003288$$

The difference is $0.0000045$ per day. It seems insignificant, but multiplied by the principal and the elapsed days, it turns into millions.

The treasurer's decision

For a corporate treasurer, choosing a convention is not academic. It is a decision that affects cash flow, project profitability, and the company's ability to meet its obligations. The right question is not "what rate are you offering me?", but "what day-count convention does that rate use?".

Amortization tables: Fixed installment (French) vs. constant principal (German)

The day-count convention does not work alone. It interacts with the amortization system, which defines how payments are distributed over time. The two predominant systems in Colombia and the United States are the fixed installment (French system) and the constant principal payment (German system).

French system: Fixed installment

In the French system, the monthly payment is constant throughout the life of the loan. Each payment consists of an interest part and a principal part, but the proportion changes: at the beginning, most is interest; at the end, most is principal.

The installment formula is:

$$Cuota = P \times \frac{i(1+i)^n}{(1+i)^n - 1}$$

Where $P$ is the principal, $i$ is the periodic rate, and $n$ is the number of periods.

Example: $100,000,000 loan at 12% per annum (1% monthly) for 12 months.

$$Cuota = 100,000,000 \times \frac{0.01(1.01)^{12}}{(1.01)^{12} - 1} = 8,884,879$$

The first month, the interest is $1,000,000$ and the principal payment is $7,884,879$. The last month, the interest is $87,969$ and the principal payment is $8,796,910$.

German system: Constant principal

In the German system, the principal payment is constant in each period, and interest is calculated on the outstanding balance. The total installment decreases over time because interest decreases as the principal is amortized.

The principal payment formula is:

$$Abono = \frac{P}{n}$$

Example: same $100,000,000 loan at 12% per annum for 12 months.

$$Abono = \frac{100,000,000}{12} = 8,333,333$$

The first month, the interest is $1,000,000$ and the total installment is $9,333,333$. The last month, the interest is $83,333$ and the total installment is $8,416,667$.

Interaction with the day-count convention

The day-count convention affects the interest calculation in each period. If the bank uses a 30/360 basis, each month is worth exactly 30 days. If it uses Actual/360, each month is worth the actual calendar days.

Comparison of total costs

| System | Initial installment | Final installment | Total interest |

|---|---|---|---|

| French | $8,884,879 | $8,884,879 | $6,618,548 |

| German | $9,333,333 | $8,416,667 | $6,500,000 |

The French system has a constant installment but higher total interest. The German system has a decreasing installment but lower total interest. The choice depends on the borrower's payment capacity and financial strategy.

Strategic decision

For a treasurer, choosing between the French and German systems is strategic. The French system offers predictability: the installment does not change, making cash flow planning easier. The German system offers a lower total cost, but requires greater payment capacity upfront.

The day-count convention adds an extra layer of complexity. A French system loan with a 30/360 basis behaves differently than the same loan with an Actual/360 basis. The treasurer must evaluate both variables together.

Total cost vs. nominal rate: fees, insurance, and internal rate of return (IRR/EACR)

The nominal rate is only the starting point. The real cost of a loan includes commissions, insurance, stamp taxes, and other charges that do not appear in the face rate. To calculate the real effective cost, the Internal Rate of Return (IRR) or Effective Annual Cost Rate (EACR) is used.

Additional costs

A typical corporate loan includes:

• Structuring fee: percentage of the loan amount, paid upfront.
• Life insurance: monthly or annual premium, mandatory in many loans.
• Stamp taxes: stamp duty, which varies by jurisdiction.
• Notary fees: in mortgage loans, the cost of the deed.

These costs do not appear in the nominal rate, but they increase the effective cost of the loan.

Calculating the IRR

The IRR is the rate that equates the present value of cash inflows (loan disbursement) with the present value of cash outflows (payments). It includes all costs, not just interest.

Example: loan of $100,000,000 a 12% nominal, con comisión de estructuración de 2% ($2,000,000) and life insurance of $500,000 upfront.

The cash flows are:

• Inflow: $100,000,000 - $2,000,000 - $500,000 = $97,500,000
• Outflows: 12 installments of $8,884,879 (French system)

The IRR is calculated by solving:

$$97,500,000 = \sum_{t=1}^{12} \frac{8,884,879}{(1 + TIR)^t}$$

The IRR turns out to be approximately 13.5%, higher than the 12% nominal rate. The 1.5 percentage point difference is the cost of the fee and insurance.

EACR as the standard

The Effective Annual Cost Rate (EACR) is the IRR expressed as an effective annual rate. It is the standard for comparing credit offers because it integrates all costs into a single figure.

In Colombia, the Financial Superintendency requires institutions to disclose the EACR in credit contracts. In the United States, the APR serves a similar function, although with slightly different methodologies.

Interaction with the day-count convention

The day-count convention affects the calculation of the EACR. If the loan uses a 30/360 basis, interest is calculated using 30-day months. If it uses Actual/360, interest varies according to actual days. The EACR reflects this difference, but the borrower must know how to read it.

The borrower's decision

For a borrower, the EACR is the figure to compare. But the EACR does not eliminate the need to understand the day-count convention: two loans with the same EACR can have different costs if one uses a 360 basis and the other a 365 basis. The EACR is a tool, not a definitive answer.

Financial decision-making and cross-border arbitrage

Comparing credit offers between Colombia and the United States is not an academic exercise. It is a real financial decision that can save or cost millions. To make the right decision, the treasurer needs an evaluation criterion that integrates all variables.

The evaluation criterion

To compare credit offers under different jurisdictions, the treasurer must:

1. Identify the day-count convention of each offer.
2. Convert the nominal rate to an effective rate considering the convention.
3. Include all additional costs (commissions, insurance, stamp taxes).
4. Calculate the EACR of each offer.
5. Compare the EACRs and choose the lowest.

Cross-border arbitrage

Cross-border arbitrage consists of taking advantage of cost differences between markets. If a loan in Colombia has an EACR of 14% and one in the US has an EACR of 12%, the company can save 2 percentage points by financing in the US market.

But arbitrage is not automatic. One must consider:

• Exchange rate risk: if the loan is in dollars, the company assumes devaluation risk.
• Transfer costs: commissions for international wire transfers.
• Regulatory requirements: compliance with local and foreign regulations.
• Terms and conditions: differences in contractual terms.

Case study: Offer comparison

Suppose a Colombian company needs 10,000 million pesos. It receives two offers:

• Colombian offer: 12% nominal, 30/360 basis, 1% fee, 0.5% insurance.
• US offer: 11.5% nominal, Actual/360 basis, 0.5% fee, 0.3% insurance.

Colombian offer:

$$TCEA_{CO} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 + 0.01 + 0.005 = 12.68\% + 1.5\% = 14.18\%$$

US offer:

$$TCEA_{US} = \left(1 + \frac{0.115}{12}\right)^{12} - 1 + 0.005 + 0.003 = 12.11\% + 0.8\% = 12.91\%$$

The US offer is cheaper in terms of APR (TCEA), but the company must consider exchange rate risk. If the peso devalues by 5% during the year, the effective cost in pesos increases.

The final decision

The final decision is not purely mathematical. It is a combination of quantitative analysis and risk assessment. The treasurer must:

• Calculate the APR (TCEA) of each offer.
• Assess exchange rate risk.
• Consider transfer costs.
• Analyze contractual terms.

Arbitrage as a strategy

For companies with operations in both markets, cross-border arbitrage can be a significant cost-saving strategy. However, it requires:

• Constant monitoring of rates in both markets.
• Risk assessment of exchange rates and regulations.
• Active negotiation with banks in both jurisdictions.

The difference between a 360-day and a 365-day basis may seem small, but in large-scale loans, it translates to millions. A treasurer who masters these conventions has a real competitive advantage.