question about binomial interest rate tree

CFA Level II Fixed Income
#1

Exhibit 3. Binomial Interest Rate Tree Fit to the Yield Curve (Volatility = 10%)

Current Year 1 Year 2 Year 3

1.25% 1.8229% 1.828% 2.6241%

1.4925% Node 2-2 Node 3-2

1.2254% 1.759%

Node 3-4

Which of the following statements about the missing data in Exhibit 3 is correct?

  • A:Node 3–2 can be derived from Node 2–2.
  • B:Node 2–2 approximates the implied one-year forward rate one year from now.

For option B, Node 2-2 should approximate the one year forward rate two years from now because it is the rate for year 3. And one year forward rate one year from now should be the mean of 1.8229% and 1.4925%. Is that right? Thank you.

#2

You cannot derive the value for node 3-2 from the value of node 2-2

The value of node 2-2 is (often described as) the geometric mean of the values at nodes 2-1 and 2-3. More accurately, (1 + r2-2) is the geometric mean of (1 + r2-1) and (1 + r2-3):

(1 + r2-2)² = (1 + r2-1)(1 + r2-3) = (1.01828)(1.012254) = 1.030758

1 + r2-2 = √1.030758 = 1.015263

r2-2 = 0.015263 = 1.5263%

This is not the 1-year forward rate starting 2 years from now, but it’s close. It is most definitely not the 1-year forward rate starting 1 year from now; that would be approximated by (what is often described as) the geometric mean of the values at nodes 1-1 and 1-2.

So neither statement is correct.

#3

Thank you. But this question comes from practice problems in the curriculum book. The answer is the second one which is C (I wrote only two options here)

“C is correct. Because Node 2–2 is the middle node rate in Year 2, it will be close to the implied one-year forward rate one year from now (as derived from the spot curve).”

Reference: 2017 Level 2 Volume 5 Page 103.

#4

Respectfully, they’re wrong.

I’ll e-mail them and bring this up.

The one-year forward rate one year from now discounts a cash flow from two years from now back to one year from now. The rates in year 2 on the tree discount cash flows from three years from now back to two years from now.

#5

I just sent off an e-mail.

I’ll post their reply when I get it.

#6

How about the paragraph on page 80: “The middle rate will be close to the implied one year forward rate one year from now derived from the spot curve, whereas…” Is it also wrong?

Thank you.

#7

Yes.

It should also read, “two years from now”.

When someone from CFA Institute replies (I’m guessing that it’ll be Wanda Lauziere; she and I have corresponded a lot since about 2008), I’ll mention that one as well.

Sigh.

I hate this junk.

#8

_ Hot diggity! _

I just got an e-mail from my dear friend Wanda at CFA Institute. She’d e-mailed me a couple of weeks ago to say that the passage was somewhat unclear, but OK. I e-mailed back to say that I respectfully disagreed, and would count it as a personal favor if she would contact the author of that reading and present my position to him.

The upshot is that she did! She apparently has a “go-to” CFA charterholder (GTCC) whom she asks about suspected errata, and, after appropriate due diligence, her GTCC said that I was correct, and contacted the author, who also agreed that I was correct.

So, they’ll be posting an erratum for the text (changing it to the 1-year forward rate _ 2 _ years from today) and for the EOC question that you cited in your original post.

This is cool!

#9

Thank you very much!

By the way, I have posted a few questions recently. If you have time, please take a look. I would really appreciate it.

#10

Hi, have they posted the correction? You are sure I am correct, right?

#11

I am sure that you are correct.

I haven’t checked the errata yet.