# What is effective interest rate?

I'm reading the book The Big Short. In the first chapter, it says that around 2002, Household Finance Corporation was offering home equity loans that were lying to the consumers. They were actually 15 year fixed rate loans but were disguised as a 30 year loan. The book says,

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?

Then a line later it says,

The borrower was told he had an "effective interest rate of 7 percent" when he was in fact paying something like 12.5 percent.

How do I get the numbers that he has calculated here? I'm not getting the numbers right when I try to use the popular expression for the effective rate:

EIR = (1+AIR/n)^n - 1

P.S. I'm not from finance background so that might be the issue.

• Possible duplicate of Effective Interest Rate Interpretation Sep 17, 2019 at 7:55
• I am not able to get the numbers using the answer. Maybe this is something different. Sep 17, 2019 at 8:47
• If you were paying the same dollar payments over 30 years, you'd be paying twice as much as if you only paid for 15 years (\$x per year for 15 years totals \$15x. \$x per year for 30 years totals \$30x). Do you mean that a bigger 30-year loan was compared with a smaller 15-year loan, or that each of the 15-year-loan payments was a lot bigger than each of the 30-year-loan payments? Sep 17, 2019 at 9:08

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
``````

$s=\sum_{k=1}^{n}\frac{d}{(1+r)^k}=\frac{d-d(r+1)^{-n}}{r}$

$\therefore d=rs(\frac{1}{(1+r)^n-1}+1)$

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
``````

and solving `s = (d - d (1 + r)^-n)/r` for `r`

``````r = 13.7986 %
``````

Plot of `s` and `(d - d (1 + r)^-n)/r` over a range of `r`

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.

To recap

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.