Need Algebra Help: Par Rates & Forward Rates

CFA Level I Fixed Income
#1

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%

#2

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:

#3

Thanks S2000.

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

#4

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

#5

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!

#6

My pleasure.

#7

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!

#8

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!

#9

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!!!

#10

My pleasure.