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.


(SFR2/1.05) + (SFR2/(1.06)2) = 0.110004

Multiply 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)20.110004

(1.06)2(SFR2) + (1.05)(SFR2) = (1.05)(1.06)20.110004 = 0.129780

(1.062 + 1.05)(SFR2) = 0.129780

(2.1736)(SFR2) = 0.129780

SFR2 = 0.129780/2.1736 = 0.059707 = 5.9707%

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 is rn, then the nth period PV factor is:

Z_n_ = 1 / (1 + rn)n

In that case, the swap fixed rate is:

SFR = (1 − Z_n) / ΣZi_

Here, for example,

  • Z1 = 1 / 1.05 = 0.952381
  • Z2 = 1 / 1.062 = 0.889996


SFR = (1 − Z2) / (Z1 + Z2) = (1 − 0.889996) / (0.952381 + 0.889996)

= 0.110004 / 1.842377 = 0.059708 = 5.9708%

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

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

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.

Jumping back and forth between Kaplan notation and CFAI notation was confusing…because it seems like CFAI’s notation was simply overcomplicated.

The issue I had specifically with these problems is that neither the CFAI material nor the Kaplan material in their examples showed how to do the step I was simply missing

Doing them step by step by converting fractiom a to a decimal

(SFR2/1.05) + (SFR2/(1.06)2) + 1/(1.06)2 = 1

=> 0.952381*SFR2 + 0.88996*SFR2 = 1 - 0.88996 = 0.11004

=> 1.842377*SFR2 = 0.11004

=> SFR2 = 0.059707