Need Algebra Help: Par Rates & Forward Rates

[Rapid_777]

My algebraic skills are limited. If you could please help me with the following problems, I would greatly appreciate it. Please show all the steps so I know how you arrived at the final answer, thank you!
1)
For this question, PMT should = $5.606 (p. 432, CFAI, 2014)
100 = [ PMT / (1.05263)^1 ] + [ PMT + 100 / (1.05616)^2 ]
2)
This question relates to calculating forward rates from spot rates; 4f12 (reads: 4-year forward rate 12 years from today) should = 4.90%
I made it this far, but I have no idea how to solve this step:
(1 + 4f12)^4 = (1.046)^16 / (1.045)^12
Like I said, 4f12 should = 4.90%

 

[S2000magician]

To solve that last equation, you start by taking the fourth root if each side:
(1 + 4f12)^4 = (1.046)^16 / (1.045)^12
[(1 + 4f12)^4]^(1/4) = [(1.046)^16 / (1.045)^12]^(1/4)
1 + 4f12 = 1.210909^(1/4) = 1.049006
4f12 = 4.9006%
You can see some more examples in the articles I wrote on spot rates and forward rates:

• http://financialexamhelp123.com/calculating-forward-rates-from-spot-rates/
• http://financialexamhelp123.com/calculating-spot-rates-from-forward-rates/

 

[Rapid_777]

Thanks S2000.
Can you please offer me some help on the first question?

 

[S2000magician]

Sorry; I wasn’t thinking.
100 = [PMT / (1.05263)^1] + [(PMT + 100) / (1.05616)^2]
100 = [PMT / (1.05263)^1] + [PMT / (1.05616)^2] + [100 / (1.05616)^2]
100 = 0.95(PMT) + 0.89648(PMT) + 89.648
10.352 = 1.84648(PMT)
PMT = 5.606

 

[Rapid_777]

Thanks Bill! You’re the best!
I have a question re: Q1:
100 = [PMT / (1.05263)^1] + [(PMT + 100) / (1.05616)^2]
100 = [PMT / (1.05263)^1] + [PMT / (1.05616)^2] + [100 / (1.05616)^2]
How did you create this seperate fraction: [100 / (1.05616)^2] ?
In other words, how did you seperate the “100” value from the original: “[(PMT + 100) / (1.05616)^2]” ?
I hope I’m making sense. Really appreciate the help.
Thank you!

 

[S2000magician]

My pleasure.