The effective interest rate (EIR) is the annual equivalent rate. So, for example, 7% EIR would imply a quarterly rate
qr = (1 + 0.07)^(1/4) - 1 = 1.70585 % and an AIR nominal rate compounded quarterly of
4 qr = 4*0.0170585 = 6.82341%. Hence
EIR = (1 + AIR/n)^n - 1 = (1 + 0.0682341/4)^4 - 1 = 0.07
However, converting between AIR and EIR does not appear to account for the discrepancy in your question, which seems to be more concerned with spreading a loan over 30 years.
Using the loan equation where
s is the principal
d is the annual payment
r is the effective annual rate
n is the number of years
and with some example figures, e.g.
d = 10000
r = 7 % = 0.07
n = 30
the loan principal is
s = (d - d (1 + r)^-n)/r = 124090.41
Doubling the payments and halving the number of years
d = 20000
n = 15
s = (d - d (1 + r)^-n)/r for
r = 13.7986 %
(d - d (1 + r)^-n)/r over a range of
Paying for a $124,090 loan over 15 years with an annual payment of $20,000 implies an effective annual interest rate of 13.8%.
Conversely, a $124,090 loan over 30 years with an annual payment of $10,000 implies an effective annual interest rate of 7%.
The total payments for both loans are the same, at $300,000.
It took the stream of payments the homeowner would make to Household over 15 years, spread it hypothetically over 30 years and asked: if you were making the same dollar payments over 30 years that you are in fact making over 15, what would your "effective rate" of interest be?
So $300,000 in repayments spread over 30 years is 7%. If you are repaying $300,000 over 15 years the interest rate is 13.8%.
The borrower was told he had an "effective interest rate of 7 percent" when he was in fact paying something like 12.5 percent.
If the borrower is repaying a $124,090 loan over 15 years at a true 7% the repayments should be $13,624 not $20,000.
s = 124090.41
r = 0.07
n = 15
d = r s (1/((1 + r)^n - 1) + 1) = 13624.46
Total repayments being
n d = 15*13624.46 = $204367, not $300,000.