QUANT LAB
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5 questions · 1 formulas

After-tax real borrowing rate

after-tax nominal = i(1 − t); real = (1 + i(1−t)) / (1 + π) − 1

Mortgage & LeverageE-level★ High priority — official Canvas example

Nominal mortgage rate rises from 5% to 7%; tax deduction rate 30%, inflation 2%. By how much does approximate real after-tax mortgage rate rise?

→ B) 1.4 percentage points

Canvas Example MCQ — Q33

  • A) 1.0 percentage point
  • B) 1.4 percentage points
  • C) 2.0 percentage points
  • D) 40%
  • E) None of A-D is close; write the correct answer.

Initial rr = .05(.70)-.02 = .015 = 1.5%. New rr = .07(.70)-.02 = .029 = 2.9%. Change = 2.9%-1.5% = 1.4 percentage points.

Rule: A rate moving from 1.5% to 2.9% rises by 1.4 percentage points.

Mortgage & LeverageE-level★ High priority — official Canvas example

Real after-tax mortgage rate rises from 1.5% to 2.9%. Which description is correct?

→ B) It increases by 1.4 percentage points, corresponding to about 93.3%.

Canvas Example MCQ — Q34

  • A) It increases by 1.4%.
  • B) It increases by 1.4 percentage points, corresponding to about 93.3%.
  • C) It increases by 93.3 percentage points.
  • D) It increases by 2.0 percentage points, corresponding to 40%.
  • E) None of A-D is close; write the correct answer.

Percentage-point change = 2.9-1.5 = 1.4 pp. Percentage change = 1.4/1.5 = 0.9333 = 93.3%.

Rule: Never confuse percentage points with relative percent change.

Mortgage & LeverageE-level★ High priority — official Canvas example

Nominal mortgage 6%, tax deduction 30%. Inflation rises from 2% to 4%. What happens to real after-tax rate?

→ B) The real after-tax mortgage rate decreases by 2 percentage points.

Canvas Example MCQ — Q35

  • A) The real after-tax mortgage rate increases by 2 percentage points.
  • B) The real after-tax mortgage rate decreases by 2 percentage points.
  • C) The real after-tax mortgage rate decreases by 30%.
  • D) Inflation has no effect because mortgage interest is tax deductible.
  • E) None of A-D is close; write the correct answer.

Before: .06(.70)-.02 = 2.2%. After: .06(.70)-.04 = 0.2%. Change = -2.0 pp.

Rule: Higher inflation lowers the approximate real borrowing cost when nominal rate and tax treatment are fixed.

Mortgage & LeverageE-level★ High priority — official Canvas example

Initially r=5%, tax deduction=30%, inflation=2%. Later r=7%, tax deduction=30%, inflation=3%. New real after-tax mortgage rate?

→ B) 1.9%

Canvas Example MCQ — Q36

  • A) 1.5%
  • B) 1.9%
  • C) 2.9%
  • D) 4.9%
  • E) None of A-D is close; write the correct answer.

.07(.70)-.03 = .049-.03 = .019 = 1.9%.

Rule: Apply the formula to the new values, not the changes.

Mortgage & LeverageE-levelHigh priority

A mortgage has a nominal rate of 4%, a tax deduction rate of 30%, and inflation of 2%. What is the real after-tax mortgage rate, rr = r(1−t) − inflation?

→ A) 0.80%

Computer Lab 5 — Exercise 1, Task 2

  • A) 0.80%
  • B) -26.00% (4% − 30% − 2%)
  • C) 2.00%
  • D) 2.80%

Nominal after-tax rate = 4%×(1−0.30) = 2.80%. Real after-tax rate = 2.80% − 2.00% = 0.80%.

Rule: The tax deduction multiplies the interest cost by (1−t); it does not subtract t directly from the rate in percentage points.