# Swap Fixed Rate

Pretty simple question but I’m simply stuck on calculating a swap curve from spot rates. I know the answer is simple but I’m at my wits end.

I’m using CFAI L2 material at the moment, volume 5 page 28. The example is creating a swap curve from spot rates.

Given a spot rate for year 1 (5%) and year 2 (6%) the formula to solve is as follows

(SFR2/1.05) + (SFR2/(1.06)^{2}) + 1/(1.06)^{2} = 1

I get to through the 1st step and end up with (SFR2/1.05) + (SFR2/(1.06)^{2}) = 0.110004

But, it’s here that no matter how I think I should solve it, I can’t figure out what i’m doing wrong. Multiplying both sides of the equation by (1.06)^{2} and then doing the same with 1.05 ends up with values that are higher than the actual result of 5.97%

When someone has a moment, would you walk me through the algebra related to this? I have a big blind spot here.

Thanks.

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(SFR2/1.05) + (SFR2/(1.06)

^{2}) = 0.110004Multiply everything by (1.05)(1.06)

^{2}:(1.05)(1.06)

^{2}(SFR2/1.05) + (1.05)(1.06)^{2}(SFR2/(1.06)^{2}) = (1.05)(1.06)^{2}0.110004(1.06)

^{2}(SFR2) + (1.05)(SFR2) = (1.05)(1.06)^{2}0.110004 = 0.129780(1.06

^{2}+ 1.05)(SFR2) = 0.129780(2.1736)(SFR2) = 0.129780

SFR2 = 0.129780/2.1736 = 0.059707 =

5.9707%Simplify the complicated side; don't complify the simplicated side.

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By the way, it’s generally easier to do these problems by looking at the present value factors rather than the spot rates.

If the

n-period spot rate isr, then the_{n}n^{th}period PV factor is:Z

= 1 / (1 +_{n}r)_{n}^{n}In that case, the swap fixed rate is:

SFR = (1 − Z

) / ΣZ_{n}_{i}Here, for example,

_{1}= 1 / 1.05 = 0.952381_{2}= 1 / 1.06^{2}= 0.889996and,

SFR = (1 − Z

_{2}) / (Z_{1}+ Z_{2}) = (1 − 0.889996) / (0.952381 + 0.889996)= 0.110004 / 1.842377 = 0.059708 =

5.9708%Simplify the complicated side; don't complify the simplicated side.

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That’s exactly what I was looking for. Thank you.

Instead of multiplying the remaining spot rates, canceling the denominators, and leaving the

sumof the spot rates X SFR2 on the left side, I was simply working it incorrectly.The second method you say is likely easier is that covered in derivatives or is there another place which references that process so I can understand it better? I feel like I’ve seen it before but maybe I haven/t.

I haven’t seen the 2020 Level II curriculum yet, but I know that it was in the 2019 Level II curriculum (although their notation was more complicated and, frankly, confusing). I found it as equation 13 in reading 39: Pricing and Valuation of Forward Commitments.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams

http://financialexamhelp123.com/

I’m no expert, but I spent a good portion of my career working on pricing and risk applications related to interest rate derivatives, fx options, CDS, so I’d say I had an above average knowledge of derivatives going into L2 (scored 90th %tile), and I found the CFAI content to be borderline unintelligible. The notation format made absolutely no sense to me. I can’t imagine how someone going into it ‘fresh’ could find it useful.

It’s horrible.

I get what they’re trying to do with the notation, but it’s complicated beyond ridiculous.

Nor can I.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams

http://financialexamhelp123.com/