Determining interest rates when comparing offers

Question

Option 1: $ 0 down, $ 424 in 1 year , $ 300 in 2 years

Option 2: $ 80 down, $ 245 in 1 year , $ 400 in 2 years

Determine the range of interest rates for which the present value of Option 2 is less than the present value of Option 1.

How would i go about solving this, I've tried using present value formulas already and solving for interest rate "i" when making the two payment methods equal but my answer is not correct.

Answer 1

Usually you would be encouraged to show your efforts and not just post a question. Especially if it looks like a homework question. So don't be surprised if you get that response now and then.

I'm gonna guess that down means right now.

We can set up some interest, say value multiplied by $x$ yearly, then the value would be:

Option one would be $0x^2 + 424*x^1 + 300$ Option two would be $80x^2 + 245*x^1 + 400$

So now the question for which values of $x$ will they be equal? This gives us the equation: $$80x^2 + 245x^1 + 400 = 0x^2 + 424x^1 + 300$$ Which we can rewrite into: $$80x^2 - 179x + 100 = 0$$

This is a second degree polynomial equation and you can solve it with completing the square or numerical program at a computer. If I don't got the numbers wrong it will have two solutions

1. $x=1.0787$ which corresponds to 7.87 percent a year.
2. $x=1.1588$ which corresponds to 15.88 percent a year.

Since the function is continous means that one is better than the other between those two percentages but that the other is better for all other percentages. Now we just need to find which of them is better in between. This should be an easy exercise to find out.