Butterfly Spread Yield Curve Impact
Please could someone help me understand the impact of yield curve changes on a long butterfly spread?
Question: If you are long butterfly spread with calls in fixed income securities (assume long 2yr + long 30yr  short 2 x 10yr), what is the impact of:

The butterfly spread increasing?

The curvature increasing?

The curvature decreasing?

Yields in the 2yr + 30yr increasing?

Yields in the 10yr increasing?

Price of 10yr increasing?
Waffle
I understand the mechanics of a butterfly spread, the concept is simple to understand with stock prices, the long position is designed to have a high probability of earning a limited profit when the future volatility of the underlying asset is expected to be low.
What I have difficulty understanding is the butterfly spreads in fixed income.
From the CFA reading below:
A butterfly long in the wings and short in the body is long (has positive) convexity and benefits from volatile interest rates. This butterfly also benefits from a yield curve flattening (unrelated to its convexity).
So a long butterfly spread (2yr + 30yr  2 x 10yr) profits from a flatter yield curve. How does this relate to the actual butterfly spread? Correct my if I’m wrong  as I understand it if a curve flattens, yields in the 2yr + 30yr increase (or 10yr yields decrease) then their respective call prices will be reduced (10yr price increased) which in turn will reduce the butterfly spread  so a decrease in the butterfly spread is equivalent to a decrease in curvature i.e. spread = curvature?
So a long butterfly spread profits from the spread decreasing? Intuitively that sounds wrong  if I’m long the spread I would expect to profit from the spread going up.
What confuses the matter for me is that the scenario I’ve described above is a “positive butterfly”.
Also how does convexity feature in this?
Very confused.
I’ve used the following references:
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Studying With
2 things here:
1st is the portfolio position is a butterfly (a long barbell short mid term bullet)
2nd is the yield curve change can be a positive butterfly (short rates and long rates go up, mid go down) or a negative butterfly (short rates and long rates go down, mid go up).
in your example, you stated flattening yield curve. i think you misunderstood flattening to be decrease in curvature. flattening is simply long rates go gown, short rates go up, which a butterfly position (long barbell) can benefit from.
Thanks for the response.
You’re right, I mistook flattening for a decrease in curvature.
So for a positive butterfly (short rates and long rates go up, mid go down) then I would expect the spread (2yr + 30yr  2 x 10yr) to decrease as the price of the short and long bonds will go down and the price of the mid bonds will go up. In which instance the butterfly is unprofitable?
Again I feel as though there’s something missing in my understanding as a “positive butterfly” is a decrease in curvature, which leads to a decrease in the spread, which is a loss for the butterfly trade. Is that right?
Studying With
Isn’t a flat yield curve also a decrease in curvature? If the curve is flat, then short, intermediate and long rates will be the same resulting in essentially no curvature.
Studying With
Better to think of a butterfly position as a long barbell/short mid bullet; though I agree it is super confusing. But assuming you are holding that position, a positive butterfly yield movement will not be to your advantage.
Just a quick Q, I thought spread is just long rate minus short rate? Hence, a butterfly shift assuming same magnitude in the short rate and long rate should not change the spread right?