# Need Algebra Help: Par Rates & Forward Rates

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!

For this question, PMT should = \$5.606 (p. 432, CFAI, 2014)

100 = [PMT / (1.05263)^1] + [PMT + 100 / (1.05616)^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%

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:

Thanks S2000.

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

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

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!

My pleasure.

Dear S2000magician,

You are so genius!

Appreciate you could teach me how to calculate the bold wording part as highlighted below:

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

I tried to press “y^x” in the calculator to calculate 1.05263^1, it equals to 1.05263 instead of 0.95; and 1.05616^2 = 1.115474 instead of 0.894648.

Thanks very much!

You are very kind.

Be careful reading what I wrote.

**PMT / (1.05263)**¹ is PMT divided by 1.05263

0.95(PMT) is PMT multiplied by 0.95.

1 / 1.05264 = 0.9500

**PMT / (1.05616)**² is PMT divided by 1.115474

0.89648(PMT) is PMT multiplied by 0.89648.

I hope that that helps.

You’re quite welcome.

Merry Christmas!

Dear S2000magician,

You are so wonderful and have solved my puzzle and helped me to solve my problem that had bothered me for so many days. Now I feel so happy and relieve!

Once again thanks for your patience and kindness.

Merry Christmas! Ho Ho Ho!!!

My pleasure.