Introduction

The mortgage interest deduction allows taxpayers to deduct interest paid on qualified residence loans. Under the Tax Cuts and Jobs Act (TCJA), the maximum loan amount eligible for deduction is $750,000 for mortgages incurred after December 15, 2017.

This post provides a mathematically accurate method for computing qualified mortgage interest directly, as well as a refined approach to defining the average mortgage balance. Before introducing the new method, however, we will first review the methods currently provided by the IRS.

IRS provided methods

Under Treasury Regulation §1.163-10T(h), several acceptable methods are available for determining the average balance of debt secured by a qualified residence:

  • Daily basis method (Reg. §1.163-10T(h)(3)): This method calculates the average mortgage balance by weighting the actual principal outstanding each day over the course of the year.

    This method can favor taxpayers by averaging down the balance when you do not have a loan during some months but while owning the home as a residence. For example, John Smith starts 2024 with a $1.5 million mortgage and pays it off on June 30. Under this method, his average balance is about $750,000, allowing full deduction of the interest paid.

  • Interest Rate Method (Reg. §1.163-10T(h)(4)): This method calculates the average balance based on the ratio between the interest paid and the stated interest rate of the debt.

  • Beginning and Ending Balance Method (Reg. §1.163-10T(h)(5)): This approach allows the taxpayer to determine the average balance by taking the simple average of the beginning and ending loan balances for the year. However, this method may only be used when no new borrowing occurred during the tax year.

  • Highest Balance Method (Reg. §1.163-10T(h)(6)): Alternatively, the average balance can be approximated by taking the highest principal balance outstanding at any time during the year.

    This method is generally the most disadvantageous for taxpayers because it often lowers the deductible mortgage interest by locking in the highest debt level, even if the balance dropped later.

    For this reason, I specifically ask for the year-end mortgage balance in my checklist, so that I can apply a more favorable calculation method - which may lower the average balance and thus maximize the mortgage interest deduction.

Furthermore, Reg. §1.163-10T(h)(8) establishes an anti-abuse provision. Under this rule, if the IRS determines that a taxpayer’s chosen method materially overstates the interest deduction, the IRS has the authority to adjust the calculation using a more appropriate method to correctly reflect the average balance of the debt.

Direct Method - Concept

Although the IRS provides several methods based on average mortgage balance for simple cases, they do not address more complex real-world situations.

To handle such cases, we rely on both the law and mathematics. Instead of calculating a single average mortgage balance for the entire year, we compute qualified mortgage interest period-by-period throughout the year.

Within each period, if multiple loans exist, the method applies either a “Simplified Method” or an “Exact Method” similar to the principles outlined in Treasury Regulation §1.163-10T(d) and §1.163-10T(e).

Although not explicitly listed among the IRS’s provided methods, this calculation approach is fully defensible because it is based on statutory law and sound mathematical principles.

IRC 163(h)(3)(B)(ii) establishes the basis for calculating qualified mortgage interest:

The aggregate amount treated as acquisition indebtedness for any period shall not exceed $1,000,000 ($500,000 in the case of a married individual filing a separate return).

This limit was subsequently modified by IRC 163(h)(3)(F)(i):

Limitation on acquisition indebtedness. Subparagraph (B)(ii) shall be applied by substituting “$750,000 ($375,000” for “$1,000,000 ($500,000”.

From a purely mathematical perspective, independent of tax law, the definition of interest amount for loan \(i\) is the integral of the interest per dollar per day (\(\alpha_i\)) multiplied by the loan amount (\(L_i\)) over time. The total interest paid across all loans can be expressed as:

\[\text{Total Interest} = \sum_{i=1}^{n} \int_0^{T} \alpha_i \times L_i \, dt \tag{1}\]

Apart from the factor of \(\alpha_i\), the total interest corresponds to the total area under the curve (the light purple and dark purple areas combined) shown in the digram.

The law simply imposes a restriction on the loan amounts eligible for deduction:

\[\sum_{i=1}^{n} L_i \leq 750,000 \tag{2}\]

Thus, the Qualified Interest corresponds only to the dark purple area.

Please note that the law restricts only the total loan amount eligible for the deduction; it does not dictate exactly which loans must be included. Therefore, we have flexibility: we can allocate proportionally across loans or allocate optimally by prioritizing loans with the higher interest rate \(\alpha\).

Direct Method - Implementation

To implement the direct method, we follow these steps:

  • Step 1: For each loan \(i\), compute the interest rate \(\alpha_i\) using the total interest paid, beginning and ending principal balances, and the loan period. This assumes the principal balance declines linearly over time.
  • Step 2: Divide the year into several periods such that the total principal balance behaves like a continuous piecewise-linear function - for example, when loans are added, paid down, or change use (e.g., converted to rental).
  • Step 3: For each period, estimate the interest for loan \(i\) using the interest rate \(\alpha_i\) and principal balance approximation from Equation (1) in the previous section.
  • Step 4: Apply the loan cap condition in Equation (2), ensuring that the total principal subject to deduction does not exceed $750,000 at any given time. Use this to determine the qualified mortgage interest.

Example Scenario:

First loan:

  • 01/01/2024: Balance of $500,000
  • 09/01/2024: Converted to rental property.
  • 12/31/2024: Year-end balance of $400,000, interest paid: $25,000

Second loan:

  • 04/30/2024: Borrowed $900,000 to buy a new home.
  • 12/31/2024: Year-end balance of $800,000, interest paid: $20,000

We divided the year into three periods:

  • 01/01/2024 - 04/30/2024: Only the first loan is present
  • 04/30/2024 - 09/01/2024: Both loans are active.
  • 09/01/2024 - 12/31/2024: Only the second loan is active (the first loan has been converted to rental use).

During each period, we can calculate interest rate and interest of each loan based on the principal amount and numver of days in the period.

For periods 1 and 3, since only one loan qualifies in each, the deductible portion is calculated directly:

\[\text{Qualified Interest} = \text{Total Interest} \times \min\left(1, \frac{750,000}{\text{Average Balance}}\right) \tag{3}\]

For period 2, where both loans are active, we have two options:

  • Optimal method: Prioritize including the loan with the higher interest rate first, up to the $750,000 limit.
  • Average method: Combine the balances and interests of both loans and apply the same formula to the combined amounts.

After calculating the qualified interest for each period, we sum the results to arrive at the annual deductible amount.

Automation:

Since spreadsheets are surprisingly powerful (and even Turing complete), we developed a Google Sheets tool to automate the entire process.

  • You enter the inputs (highlighted in purple) such as loan dates, balances, and use type.
  • The qualified interest results (highlighted in blue) are calculated automatically.
  • A sample spreadsheet screenshot demonstrates the structure and formulas used.

Average Loan Balance Defined

Since we have already calculated the yearly qualified mortgage interest directly, we no longer need to compute the yearly average mortgage balance to derive it.

However, we can define the average mortgage balance as the hypothetical loan amount that, if kept constant throughout the year, would generate the same qualified interest we calculated.

In other words, it’s the balance that would make the following equation true:

\[\frac{\text{750,000}}{\text{Average Mortgage Balance}} \times \text{Total Interest Paid} = \text{Qualified Interest} \tag{4}\]

Solving for the average mortgage balance:

\[\text{Average Mortgage Balance} = 750,000 \times \frac{\text{Total Interest Paid}}{\text{Qualified Interest}} \tag{5}\]

You might wonder why we still need to calculate the average mortgage balance if the qualified interest is already known. There are two important reasons:

  • First, the IRS requires the reporting of average mortgage balances in Table 1 of Publication 936 to calculate qualified mortgage interest using the total interest paid and the average balance.
  • Second, we must provide the total interest information for state tax returns. While federal tax law caps the deductible amount at $750,000, some states apply different limits - or no limits at all - allowing full mortgage interest deductions without federal restrictions.

Thus, by calculating the Average Principal, we retain flexibility to apply either the $750,000 federal limit or more generous state-specific rules as needed.