Interest Calculation Design

Research the design of daily interest calculation for go-luca, covering AER, day-count conventions, rounding strategies, and a discrete-integer formula that is uniform, fair, and correct for small balances over long periods.

1. AER and Gross Rates

AER (Annual Equivalent Rate) is the effective annual return including the effect of compounding. It is the standard comparison rate quoted to consumers (FCA BCOBS). A gross rate is the simple annual rate before tax, which may or may not equal AER depending on compounding frequency.

The relationship:

AER = (1 + gross_rate / n)^n - 1

where n is the number of compounding periods per year. When interest is compounded daily (n = 365 or 366):

AER = (1 + gross_daily_rate)^365 - 1

Conversely, to derive the gross daily rate from an AER:

gross_daily_rate = (1 + AER)^(1/365) - 1

For small rates this is very close to AER / 365, but the difference matters over long periods and for regulatory accuracy.

Should systems quote gross daily rates? Generally no. Products are quoted as AER or gross annual rate. The daily rate is an internal calculation detail. However the precision of the internally derived daily rate is critical (see Section 4).

2. Day-Count Conventions

Day-count conventions determine how many days of interest accrue over a period. The two most relevant for UK banking:

Actual/365 (fixed)

Numerator: actual number of calendar days in the period.
Denominator: always 365, regardless of leap years.
Used by most UK banks for savings and loan interest.
In a leap year, a full year accrues 366/365 of the annual rate — the customer earns slightly more.
Simple and predictable. No special leap-year logic needed in the daily engine.

Actual/Actual (ISDA)

Numerator: actual days.
Denominator: 365 in a non-leap year, 366 in a leap year (or pro-rated if the period spans both).
Used in some bond markets and interbank lending (ISDA conventions).
A full year always accrues exactly the annual rate, regardless of leap years.
More complex: when a period spans a year boundary (e.g. 28 Dec – 3 Jan across a leap/non-leap transition), days must be split and weighted.

Leap year implications

Convention	Daily rate (non-leap)	Daily rate (leap)	Full year interest
Actual/365	rate / 365	rate / 365	366/365 * rate ≈ 1.00274 * rate
Actual/Actual	rate / 365	rate / 366	exactly rate

For go-luca, Actual/365 (fixed) is the right default — it matches UK retail banking practice, requires no leap-year branching in the daily engine, and is the convention behind the discrete formula in Section 4.

3. Daily Interest and AER Comparison

When interest is calculated daily and added to the balance (compounded daily), the effective rate over a year exceeds the gross rate due to compounding.

Example: 4.00% gross, Actual/365, non-leap year.

Daily rate     = 0.04 / 365 = 0.00010958904...
Daily interest = balance * daily_rate

After 365 days of compounding on £10,000.00:

Final balance = 10000 * (1 + 0.04/365)^365 = 10,408.08
Effective AER = 4.0808%

The gap between gross rate and AER grows with the rate:

Gross rate	AER (daily compound)	Difference
0.01%	0.01%	< 0.001%
1.00%	1.005%	0.005%
4.00%	4.081%	0.081%
10.00%	10.516%	0.516%

Rounding limits

The core problem: at 2-decimal-place precision (pence), daily interest on small balances rounds to zero.

£1.00 at 4.00% gross: daily interest = £0.000109589... → rounds to £0.00.
The balance never grows. Interest is silently lost.

This is the "rounding trap" — it is not a numerical curiosity but a fairness issue. A customer with £1 in a savings account earning 4% should eventually see interest credited.

Threshold for 1p daily interest at 2dp:

balance * rate / 365 >= 0.005  (rounds up to 0.01)
balance >= 0.005 * 365 / rate

At 4% gross: balance >= £45.63. Below this, daily interest rounds to zero every day forever.

The 10,000-year test

At 0.01% gross on 1p (£0.01), using exact arithmetic:

0.01 * 0.0001 / 365 = 0.00000000027397...

At 2dp, this rounds to £0.00 — no interest ever accrues. Even at 4dp (£0.0001 precision), daily interest = £0.0000000027... which still rounds to zero.

To earn 1p of interest on 1p at 0.01% with no rounding loss requires continuous/exact accumulation over ~10,000 years:

0.01 / (0.01 * 0.0001 / 365) ≈ 3,650,000 days ≈ 10,000 years

This motivates the need for an accumulation method that preserves fractional interest across days.

4. Discrete Interest Formula

The problem with naive daily rounding

If we round interest to the account's exponent (e.g. 2dp) each day, small balances lose interest permanently. If we use arbitrary-precision decimals, we lose the uniformity and auditability of integer arithmetic.

The intermediate-precision approach

Use 4-digit precision (exponent -4, i.e. hundredths of a penny) as the interest accumulator, while keeping account balances at their native exponent (e.g. -2 for GBP).

The formula for daily interest at 4-digit precision, using only integer arithmetic:

interest_4dp = (daily_amount * num_days * gross_rate_scaled) / divisor

where:

gross_rate_scaled = gross_annual_rate * 365 * 100000    (integer)
divisor           = 365 * 10000                         (integer, = 3650000)

Worked example: £100.00 at 4.00% gross, 1 day.

daily_amount      = 10000        (£100.00 in pence, at exponent -2)
                  = 1000000      (at exponent -4: 10000 * 100)
num_days          = 1
gross_rate_scaled = 0.04 * 365 * 100000 = 1460000
divisor           = 365 * 10000 = 3650000

interest_4dp = (1000000 * 1 * 1460000) / 3650000
             = 1460000000000 / 3650000
             = 400000
             → £0.0400 per day (i.e. 400000 at exponent -6? No...)

Let's restate more carefully. Express amounts in the 4dp unit (1 unit = £0.0001):

balance_4dp       = 1000000      (£100.00 = 1,000,000 units of £0.0001)
gross_rate_int    = 4000         (4.00% expressed as rate * 100000)
daily_interest_4dp = balance_4dp * gross_rate_int / (365 * 100000)
                   = 1000000 * 4000 / 36500000
                   = 4000000000 / 36500000
                   = 109 (truncated)
                   → £0.0109 per day

Check: £100.00 * 0.04 / 365 = £0.01095890... → at 4dp this is £0.0109 (truncated) or £0.0110 (rounded). The integer formula gives 109 units of £0.0001 = £0.0109. Correct.

The accumulator pattern

Each account maintains an interest accumulator at exponent -4 (or whatever extended precision is chosen). Each day:

Compute daily_interest_4dp using integer arithmetic.
Add to the accumulator: accumulator += daily_interest_4dp.
When accumulator >= rounding_threshold (e.g. 50 units of £0.0001 = £0.005, the half-penny that rounds up), extract whole pence and post as a movement.

Alternatively, accumulate for a fixed period (monthly, quarterly) and round once at posting.

Small balance behaviour

£0.01 at 0.01% gross, per day:

balance_4dp    = 100          (£0.01 = 100 units of £0.0001)
gross_rate_int = 10           (0.01% * 100000)
daily_4dp      = 100 * 10 / 36500000 = 0 (truncated)

Even at 4dp, £0.01 at 0.01% produces zero daily interest. The accumulator stays at zero. This is mathematically correct — the amount is so small and the rate so low that even 4dp cannot represent it.

To handle this extreme case, either:

Accept it (£0.01 at 0.01% = £0.000001/year — below any reasonable materiality threshold).
Use 6dp or 8dp accumulators for extreme precision (but adds complexity for negligible real-world benefit).

For practical rates (>= 0.1%) and practical balances (>= £1.00):

balance_4dp    = 10000        (£1.00)
gross_rate_int = 100          (0.1%)
daily_4dp      = 10000 * 100 / 36500000 = 0 (still zero)

balance_4dp    = 100000       (£10.00)
gross_rate_int = 100          (0.1%)
daily_4dp      = 100000 * 100 / 36500000 = 0 (still zero)

balance_4dp    = 100000       (£10.00)
gross_rate_int = 4000         (4.00%)
daily_4dp      = 100000 * 4000 / 36500000 = 10
               → £0.0010/day. Accumulates to £0.01 in 10 days. ✓

At 4dp, the minimum balance to accrue non-zero daily interest at 4.00% is:

balance_4dp * 4000 / 36500000 >= 1
balance_4dp >= 36500000 / 4000 = 9125
→ £0.9125, i.e. about 91p

This is a substantial improvement over the 2dp threshold of £45.63.

Properties of this approach

Uniform: same formula for every account, every day. No special cases.
Fair: fractional interest accumulates rather than being silently discarded.
Integer-only: no floating point in the hot path. Deterministic, auditable, reproducible.
Works with small values: the 4dp accumulator captures interest that 2dp would lose.
Carries forward: the accumulator persists across days, so sub-unit interest is never lost (only deferred until it's large enough to round).
Discrete: at any point, the accumulator contains an exact integer. No representation error.

Bank practice: adjusting the gross rate

Banks sometimes achieve the same effect differently: they adjust the quoted gross rate slightly so that daily interest at 2dp precision matches the target AER over a year. This avoids accumulators but means:

The effective rate varies by balance band (since rounding effects differ).
Very small balances still earn nothing.
The adjusted rate must be recalculated whenever the base rate changes.

The accumulator approach is more principled and treats all customers uniformly.

5. Methods of Calculating Interest

Method 1: Simple daily (current go-luca)

interest = balance * annual_rate / 365

Round to account precision each day. Post as movement. Subject to the rounding trap for small balances.

Method 2: Daily with extended-precision accumulator (recommended)

daily_interest_4dp = balance_4dp * rate_int / (365 * 100000)
accumulator += daily_interest_4dp

Post to account when accumulator exceeds rounding threshold. Eliminates the rounding trap for all practical balances and rates.

Method 3: Product-of-sums (batch)

For Actual/365:

interest = SUM(daily_balance * num_days_at_that_balance) * annual_rate / 365

Compute the weighted sum over a period (e.g. a month), then round once. Minimises rounding events. Commonly used in bank batch processing. Compatible with the 4dp accumulator approach.

Method 4: Continuous compounding

interest = balance * (e^(rate * days/365) - 1)

Theoretical maximum. Not used in retail banking. Included for completeness.

6. Recommendations for go-luca

Adopt the 4dp accumulator as the core interest engine. Store the accumulator as a field on the account (or a separate interest_accrual table) at exponent -4.
Keep Actual/365 (fixed) as the default day-count convention. Support Actual/Actual as an option for bond/interbank use cases.
Derive daily rate from gross annual rate using integer arithmetic as shown in Section 4. Do not store or quote daily rates.
Post accumulated interest on a configurable schedule (daily/monthly/quarterly/annually). The posting rounds the accumulator to account precision and records a movement.
The interest rate field (annual_interest_rate on Account) should remain the gross annual rate. AER is a derived/display value computed from the gross rate and compounding frequency.
Validate with the benchmark: run 10,000 years at 0.01% on 1p — the accumulator should eventually reach 1p without loss (at sufficient accumulator precision), confirming no systematic rounding drain.