Question

You take out a loan to build a swimming pool in your new home's backyard. Your equal annual payments are 1/6 the amount you borrowed. If it will take you 7 years to fully repay the loan, what is the interest rate on the loan

Answer 1

To find the interest rate on the loan, we need to gather the given information and use the formula for equal annual payments (annuity).

Given:
- Equal annual payments are 1/6 of the amount borrowed.
- It will take 7 years to fully repay the loan.

Let's assume the amount borrowed is X.

The annual payment (A) = 1/6 * X.

The number of periods (n) = 7.

Using the annuity formula, we can express the amount borrowed in terms of the annual payment and interest rate (r):

A = (r * X) / (1 - (1 + r)^(-n))

Now, substituting the given values:

1/6 * X = (r * X) / (1 - (1 + r)^(-7))

To simplify, we can multiply both sides of the equation by (1 - (1 + r)^(-7)):

(1 - (1 + r)^(-7)) * (1/6 * X) = (r * X)

(1/6) - (1 + r)^(-7) * (1/6) = r ............(Eq. 1)

Now, let's solve for the interest rate (r).

At this point, finding the exact interest rate requires solving a non-linear equation, which can be a bit tedious. Alternatively, we can use estimation methods to find a reasonable approximation.

Assuming that the interest rate (r) is between 0 and 0.2, we can make a table of values and plug them into Eq. 1:

Interest Rate (r) | Left-hand side (LHS) | Right-hand side (RHS)
0.05 | 0.0188 | 0.05
0.10 | -0.0873 | 0.10
0.15 | -0.2329 | 0.15

From the table, it is clear that the interest rate must be between 0.1 and 0.15 because the left-hand side (LHS) becomes more negative as the interest rate increases.

By trying more values within that range, we can narrow down the approximation. For example:

Interest Rate (r) | LHS
0.125 | -0.0596
0.13 | -0.0567
0.135 | -0.0537
0.14 | -0.0507
0.145 | -0.0478

As the interest rate approaches 0.14, the left-hand side converges toward zero, indicating that it is nearing a solution. Based on this estimation, we can assume the interest rate is approximately 0.14 or 14%.

Therefore, the estimated interest rate on the loan is approximately 14%.