Finance loan interest rate homework

[paddo]

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%

 

[phinds]

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?

 

[paddo]

 

[paddo]

 

[pasmith]

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 <br /> F = A\left(1 + \frac{r}{400}\right)^{4N}. You are given F, N and A and asked to solve for r.

 

[Mark44]

@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%.

 

[pasmith]

@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 <br /> 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 <br /> 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} = <br /> \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:
<br /> T - P\left(1 + \frac{r}{100n}\right)^{Yn} = 0