I've looked for a question that hits on something similar without success. The problem is this: at time 10, an asset pays 1000. Then the teacher gave us a bunch of random interest rates for times 0 to 2, 2 to 5, 5 to 8, and 8 and beyond. I'm positive I converted them all properly to an annual effective rate (used a financial calculator and Wolfram Alpha both and my answers each way matched). We're supposed to find the PV of this asset. I tried discounting from time 10 back to 8 with the discount rate based on the interest rate for 8 and beyond, etc., then the discount rate from 8 to 5, from 5 to 2, and then 2 back to time 0. I also tried starting at time 0 and advancing by the interest rate all the way to 10 and then discounting back to 2 and so on. I get the same answer both ways, but it's NOT one of the multiple choice answers.
If anyone knows of a YouTube channel that uses really complex financial math problems, I'd die grateful to you. He's the worst teacher I've ever had -- teaches concepts and then gives us very complex problems, and when I ask for an example problem of similar complexity, he gets mad and implies I'm stupid. Which I probably am, but I don't think needing an example problem is why. Blech.
Edit: for times 0 to 2, annual interest rate of 8%. For 2 to 5, force of interest of 0.015t. For 5 to 8, d of 6%. For 8 and beyond, i^(4) of 10%. I got annual interest rates of .08, .0151130646, .0638298, and .10381289 respectively, and calculated the discount factor by using 1/1+i for each. PV=1000*(1/1.08)^2 * (1/1.0151130646)^3 * (1/1.0638298)^3 * (1/1.10381289)^2. I get $558.73 on both a calculator and wolfram alpha. Not one of the choices available.
I agree with your present value at time $0$ of $558.73$ to $2$ decimal places. In detail, my calculations are as follows:
$PV(10) = 1000 \\ PV(8) = \frac{PV(10)}{1.025^8} = 820.746571 \\ PV(5) = PV(8) \times 0.94^3 = 681.698970 \\ PV(2) = PV(5) \times e^{-0.015 \times 3} = 651.702498 \\ PV(0) = \frac{PV(2)}{1.08^2} = 558.729851$
where $PV(n)$ is the present value at time $n$.
So either we have both misunderstood something about the conditions of the problem, or the "official" answer is incorrect.