I am finding the FI of level 3 extremely challenging and I can’t digest many of the concepts they are teaching.
The butterfly spread is used to measure the curvature of the yield curve. But what’s the connection with the butterfly?
I am not sure whether this answers your question but I will try.
A butterfly spread is just a long barbell and a short bullet combined together (long/positive convexity butterfly), or a short barbell and a long bullet (short convexity butterfly). Usually on the CFA mock questions I encountered, they were talking about long convexity butterfly in fixed income questions if they just say “butterfly” for whatever that’s worth.
The idea is to generate a duration neutral position (the long and short durations cancel out) using leverage (shorting positions to help pay for your long positions)… while at the same time lasering in on which parts of the yield curve levels you think will increase (and the duration effect will cause bonds to lose value if you have positive duration exposure to them but will gain in value of you instead have a negative exposure to them), and which parts of the yield curve levels will decrease (decreasing yields resulting in higher bond values if you have positive duration exposure there, again due to duration effect). You want to have negative duration exposure to places on the yield curve where yields are expected to increase, and positive duration exposure to places on the yield curve where yields are expected to decrease, so that your bond portfolio will gain in value due to the duration effect combined with the yield changes.
Remember the [-Dur(change in yield)] effect on bond values. Also bear in mind some jargon: “bullet” refers to the middle of the yield curve, the medium duration part. “Barbell” means the short-term and long-term parts of the yield curve held together. Because the long end of the yield curve has the highest duration, you can see in the formula above that bond value changes will be greatest there for identical changes in yield because the “Dur” multiple is highest there.
So in the case of a long barbell/short bullet butterfly scenario, you are “long” the short and long duration part of the curve, and “short” the medium duration part of the curve. You are betting that the long part of the yield curve will flatten out or go down (yields decrease), because your long duration exposure to bonds there will increase your portfolio’s value. You are also betting that the medium part of the yield curve will see its yields increasing (or at least be relatively unchanged), which is why you “shorted” the medium duration part of the curve.
In the case of a short barbell/long bullet scenario, it’s the opposite. You are “long” the medium duration part of the curve, and “short” the short and long part of the curve. You are betting that the middle of the curve is where yields will decrease (so your long position in mid-duration bonds will become more valuable), and you are betting long term yields will increase (long end of the yield curve steepens) so your “short” duration exposure there benefits you too.
So having said all this, try looking at some yield curves on paper. Play around drawing different changes in curvatures. Separate the curve into three parts, the short end, the medium part, and the long end. Look at what happens to the yield curve in each part as you play with curvature changes. Are yield levels going up or are they going down? You want long (positive) portfolio exposure where yields are going down. You want short (negative) exposure where yields are going up. It’s the classic buy low sell high story. You are buying undervalued (current yields too high) bonds and also “shorting” overvalued (current yields too low) bonds… in your view based on your research and models for yield curve outlooks.
Note that in barbell strategies, and butterfly strategies, you are less concerned with what happens in the short (near term) end of the yield curve and more concerned with what is happening on the long (far term) end of the yield curve. That far end is where the biggest price movements are due to yield changes. If the short end of the yield curve is increasing and the long end is decreasing it’s still better to be in a long barbell or long butterfly than something else, all other things being equal.
So just look at a curvature change on a yield curve, and see where rates are going down and where rates are going up due to your predicted curvature change. Since the duration effect means portfolio bond values go down when yields go up, if you have positive duration exposure to them, and portfolio bond values go up when yields go up if you have negative duration exposure to them, you should start seeing in your mind’s eye why a bond investor may choose a butterfly strategy based on his or her opinion of future yield curve changes. The bond investor is trying to buy low today, and sell high in the future.
Cheers - good luck - you got this 
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Well I’ll try to explain my way… at least how I see this…
If you have yields as follows:
2s - 0.25% → short wing
10s - 1.80% → long body
30s - 2.55% → short wing
What is the yield spread? → (-0.25% + 2*(1.80%) - 2.55%) = 0.80%
What this means? That the yield curve has some curvature (literally)
Ok, what happens if 10s (mid maturity) is 1.40% now and the other (2s and 30s) stay the same? What is the yield spread now? → (-0.25% +2*(1.40%) - 2.55%) = 0%
What this means? Yield loses on curvature (it’s becoming a straight line literally - NOT FLAT!!!, as this would imply that ST rates are equal to MT and LT rates).
Try drawing yoursel a simple line chart in excel to visualize…
So what’s the purpose of butterfly?
If you have yields in the mid of the curve falling (as shown above) what would you do to profit from this situation? Obviously, long the mid of the curve; buy a bullet (the body of bfly) and sell the wings; short barbell (short and long end of the curve) → ta daaa… you onstruct a butterfly.
Please correct me if I missed something…
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