Can someone help with 2 financial math problems? Please!?

Question

1) If you borrow $77,221 for 14 years and pay $810 every month, what is the equivalent annual rate (EAR) on the loan?

2) John plans to retire on a ranch worth $534,331. The plan is to save $12,120 every year starting next year. If the interest rate is 10.9%, how soon can he retire?

I can't use a financial calculator. I would be so greatful if someone could show me how to solve these using algebra. Thank you :)

Answer 1

Neither of these problems has an algebraic solution. In order to solve the first one using algebra, you would have to solve for the roots of a polynomial of degree 168.

Both are easy to do in Excel. Both are a pain to do with a calculator. Neither is easy to do by hand.

Here is a formula that may help you. An annuity is a financial instrument that pays the same amount every period for N periods. The first problem is an annuity that pays $810 per month for 168 months.

In the following formula, R is the one period interest rate, N is the number of periods and C is the amount paid -- then the value of the annuity is:

Value = C/R - C/(R*(1+R)^N)

So -- you need to solve for R when:

77,221 = 810/R - 810/(R*(1+R)^168)

R is the monthly rate. Once you find it -- multiply by 12 to get the yearly rate.

The way to do this is to start with a guess -- say 1%. If you get the right number -- stop. If your answer is too high, try a bigger rate. If you get an answer that is too low, use a lower rate.

With the second one, the thing you don't know is how long he has to invest. But you do know the rate. Guess at how many years he needs to invest & find the value now using the annuity formuls. Then multiply that by (1+R)^N to get the future value. Stop with the future value is enough.

Start with a guess at N.

Calculate present value:

PV = 12,120/0.109 - 12,120/(0.109*1.109^N)
FV = PV*1.109^N

If FV is too big, try a smaller N
If FV is too small, try a bigger N.

Answer 2

5.44% annually for question 1.