Finance loan interest rate homework

In summary, the conversation discusses finding the interest rate, compounding quarterly, at which a $30,000 loan is invested for 6 years and yields a final value of$36,295. There is disagreement about the result, with one person obtaining 6.33% and another calculating it to be 3.2%. The use of a finance solver for this problem is questioned and a different equation is proposed for a one-time repayment at the end of the loan term.
Homework Statement
A loan of $30,000 is paid back after 6 years with a final value of$36,295. At what interest rate, compounding quarterly, has this money been invested?
Relevant Equations
Finance solver
I got 6.33% but apparently it's 3.2%

And are we supposed to guess how you got that? I suppose we could just speculate about where you went wrong (or didn't) but it seems like a waste of time. How about you show your work?

Problem Statement: A loan of $30,000 is paid back after 6 years with a final value of$36,295. At what interest rate, compounding quarterly, has this money been invested?
Relevant Equations: Finance solver

I got 6.33% but apparently it's 3.2%

If an amount $A$ is invested for $N$ years at a rate $r\,\%$ and compounded quarterly, then the final value is $$F = A\left(1 + \frac{r}{400}\right)^{4N}.$$ You are given $F$, $N$ and $A$ and asked to solve for $r$.

HallsofIvy and PeroK
@paddo, what does the solver do if you enter 6 for N instead of 24? I'm thinking that N represents the number of years, not the number of payments. There is already a field for the number of payments per year (PpY). The interest rate that I get by direct calculation is a little under 3.2%.

Mark44 said:
@paddo, what does the solver do if you enter 6 for N instead of 24? I'm thinking that N represents the number of years, not the number of payments. There is already a field for the number of payments per year (PpY). The interest rate that I get by direct calculation is a little under 3.2%.

I don't think that solver is designed to deal with the situation in the OP in any event.

I think it's designed to deal with the situation where regular repayments are made throughout the term, thereby reducing the balance on which interest is charged. The OP suggests instead a single payment at the end of the term.

If an amount $P$ is lent at a rate of $r\,\%$ to repaid by $n$ equal installments of $A$ per year over $Y$ years, then the balance outstanding after $k+1$ periods is $$B_{k+1} = B_k\left(1 + \frac{r}{100n}\right) - A$$ on the assumption that the interest is calculated before the payment is deducted (reasonable, since it allows the lender to charge more interest). This recurrence relation can be solved subject to $B_0 = P$ to yield $$B_k =\left(P - \frac{100nA}{r}\right)\left(1 + \frac{r}{100n}\right)^k + \frac{100nA}{r}.$$ Now after $Yn$ periods the loan should be fully repaid, so $$B_{Yn} = \left(P - \frac{100nA}{r}\right)\left(1 + \frac{r}{100n}\right)^{Yn} + \frac{100nA}{r} = 0$$ which in terms of the total amount repaid $T = nYA$ is $$\left(P - \frac{100T}{Yr}\right)\left(1 + \frac{r}{100n}\right)^{Yn} + \frac{100T}{Yr} = 0.$$ This is much more difficult to solve for $r$ than the equation which holds where the entire amount outstanding is repaid at the end of the term:
$$T - P\left(1 + \frac{r}{100n}\right)^{Yn} = 0$$

1. What is a finance loan interest rate?

A finance loan interest rate is the percentage of the principal amount that a lender charges as interest for borrowing money. This interest rate can vary depending on the type of loan, the borrower's credit score, and market conditions.

2. How is the finance loan interest rate calculated?

The finance loan interest rate is typically calculated by taking the annual interest rate and dividing it by the number of payments in a year. This is then multiplied by the remaining balance of the loan to determine the amount of interest that will be charged for that payment period.

3. What factors can affect the finance loan interest rate?

Several factors can influence the finance loan interest rate, including the borrower's credit score, the type of loan, the length of the loan, and current market conditions. Lenders may also consider the borrower's income, debt-to-income ratio, and employment history when determining the interest rate.

4. How can I lower my finance loan interest rate?

There are a few ways to potentially lower your finance loan interest rate. These include improving your credit score, paying off any existing debt, and shopping around for different lenders to compare interest rates. You may also be able to negotiate with your lender for a lower rate.

5. What is the difference between fixed and variable finance loan interest rates?

A fixed interest rate remains the same throughout the life of the loan, while a variable interest rate can change based on market conditions. Fixed rates provide more stability and predictability, while variable rates can offer lower initial rates but may increase over time.

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