User:Winston yang/How much total interest do you pay on a loan?

How much total interest do you pay on a loan, expressed as a percentage of the loan? This percentage is called the "total-interest percentage," or TIP. It has the formula

$$\frac{in}{1 - (1 + i)^{-n}} - 1,$$

where:


 * $$n$$ is the number of periods.
 * $$i$$ is the interest rate per period.
 * You pay once per period.
 * You pay at the end of each period.
 * You pay the same amount per period.
 * You pay nothing else.

Later, we will prove the above formula.

Formula for the total-interest percentage, for a nominal annual interest rate and a number of years
Some loans have the following assumptions:


 * A period is a month.
 * The loan has a nominal annual interest rate. Call this $$r$$. Then the interest rate per period is $$i = r/12$$.
 * The loan lasts a number of years. Call this $$y$$. Then the number of periods is $$n = 12y$$.

The total-interest percentage is

$$\frac{in}{1 - (1 + i)^{-n}} - 1 = \frac{ry}{1 - \Big(1 + \frac{r}{12}\Big)^{-12y}} - 1.$$

Example: We simplify an example from the Consumer Financial Protection Bureau. A loan has a nominal annual interest rate of $$3.875%$$ and lasts $$30$$ years. What is the total-interest percentage?

Answer: The total-interest percentage is $$62.29%$$. In a previous formula, plug in $$r = 3.875%$$ and $$y = 30$$. Note: The Consumer Financial Protection Bureau calculates a total-interest percentage of $$69.45%$$, which is slightly more than our value. The reason is that their example is more complicated. It includes prepaid interest, as explained by Homes by Ardor and Jeffrey Loyd.

If you use a spreadsheet, it may have a function called CUMIPMT ("cumulative interest payment"), which you can use to calculate a total-interest percentage. Enter the following formula. The minus sign is optional.

= - CUMIPMT(3.875%/12, 30*12, 100%, 1, 30*12, 0)

We describe each argument:


 * 1) $$3.875%/12$$ is the interest rate per period.
 * 2) $$30*12$$ is the number of periods.
 * 3) $$100%$$ is the loan.
 * 4) $$1$$ is the start period.
 * 5) $$30*12$$ is the end period.
 * 6) $$0$$ means that you pay at the end of each period.

Table of total-interest percentages, for a nominal annual interest rate and a number of years
Example: If a loan has a nominal annual interest rate of $$4%$$ and lasts $$30$$ years, the total-interest percentage is $$72%$$. (This is close to the $$69.29%$$ in a previous example. In that loan, the nominal annual interest rate is close---$$3.875%$$---and lasts the same number of years.)

Table of total-interest percentages, for an interest rate per period and a number of periods
The rest of this section may be skipped. We motivate and prove a theorem about the total-interest percentage.


 * Let $$t$$ be a total-interest percentage.
 * Let $$i$$ be an interest rate per period.
 * Let $$n$$ be a number of periods.

Example: In the above table of total-interest percentages, look at cells whose value is around $$t = 100%$$. Note that $$in$$ is around $$150%$$ to $$160%$$.

Example: In the above table of total-interest percentages, look at cells whose value is around $$t = 700%$$. Note that $$in$$ is $$800%$$.

The above examples might motivate the following conjecture:

Conjecture: If $$t$$ is approximately constant, then $$in$$ is approximately constant.

We prove the above conjecture as the theorem below.

Theorem: If $$i$$ is positive, and if $$n$$ is big, then $$in \approx t + 1$$. (Consequently, if $$t$$ is constant, then $$in$$ is approximately constant.)

Proof: The total-interest percentage is

$$t = \frac{in}{1 - (1 + i)^{-n}} - 1.$$

Move the $$1$$ to the left side:

$$t + 1 = \frac{in}{1 - (1 + i)^{-n}}.$$

Look at the denominator.


 * If the denominator were $$1$$, then we would have $$t + 1 = in$$.
 * So if the denominator were close to $$1$$, then we would have $$t + 1 \approx in$$.
 * We will prove that the denominator is close to $$1$$.
 * We assumed that $$i$$ is positive. So $$1 + i$$ is greater than $$1$$.
 * So $$(1 + i)^{-1}$$ is less than $$1$$, and positive.
 * We assumed that $$n$$ is big. So $$(1 + i)^{-n}$$ is close to zero.
 * So the denominator is close to $$1$$.

Proof of a previous formula for the total-interest percentage
This section may be skipped. We use actuarial science to prove a previous formula for the total-interest percentage.


 * Let $$L$$ be the loan.
 * Let $$R$$ be the payment per period. Think of $$R$$ as standing for "repayment" or "regular payment."
 * Note that $$L = Ra(n, i)$$, where $$a(n, i)$$ is an annuity-immediate with $$n$$ periods and an interest rate of $$i$$ per period.
 * So $$R = L/a(n, i)$$.
 * Let $$v = (1 + i)^{-1}$$. Think of $$v$$ as "reverse interest." After one period, $$1$$ unit of money becomes $$1 + i$$ units of money. But if you go in reverse one period, $$1$$ unit of money becomes $$v = (1 + i)^{-1}$$ units of money.
 * Note that $$a(n, i) = (1 - v^n)/i$$.
 * The total payment is $$nR$$.
 * Subtract $$L$$ to get the total interest: $$nR - L = L\Bigg(\frac{in}{1 - v^n} - 1\Bigg)$$.
 * Divide by $$L$$ to get the total-interest percentage: $$\frac{nR - L}{L} = \frac{in}{1 - v^n} - 1$$.