# Return Impact

The return impact of a 25 bp widening in yield spread for an option free bond is greatest if it has a duration of:

A) 4 and convexity of 24

B) 5 and convexity of 32

C) 6 and convexity of 90

Answer: C

I got this correct by merely thinking the highest duration has the highest impact on return. However, can someone take me through calculations. It seems to be [Duration x change in i] + [1/2 x convexity x change in i^2].

This seems to be very similar to formula for price change…but you multiply convexity by (1/2)?.. can someone expand on this. Thank you.

The return impact of spread changes is made up of 3 things: 1) the spread change (Δ spread) 2) the bond’s modified duration 3) convexity return impact = - modified duration x Δspread+ (0.5)(convexity)(Δspread)² As spreads get wider, bond prices decrease and the impact on return is negative (inverse relationship between price and rates). Longer maturity bonds have higher duration and consequently higher spread sensitivity; their prices and returns are more sensitive to changes in spread. So in this case you could simply answer this question without having to make any calculations but I’d be careful as these answer choices get more narrow… My answers for the following (you can plug them into the formula) A) -0.009925 B) -0.0124 C) -0.0140

As spreads widened. The biggest impact on return was to bond C.

This is obviously from the section on credit analysis. In that section only, you need to include the ½ in the convexity term; everywhere else, you don’t include it (it’s already incorporated into the convexity number. It’s stupid that they give you different formulae for the same calculation.

There is disagreement in the fixed income community about whether the convexity number should include the ½ or not, but CFA Institute should pick one formulation and stick to it.

That’s ridiculous, indeed! I just finished a mock exam. Question 94 was answered using the whole convexity number. And question 103 used 0.5 of it! And no clue under the question if it was a credit analysis or plain fixed income calculation (they were both under “Asset Valuation” section).

I guess we should simply try to work with a number around 100, since question 94 presented a convexity of 0.88 (which I turned to 88) and question 103 presented a convexity of 190 (which they divided by 2 to answer).

Any thoughts?

Oops, just realised that in Znieh’s example the convexity was already 90 and they divided as well… gosh. Similarities in both Znieh’s example and question 103 was a change in the spread. So, I am going to try to find any mention to a spread widening/tightening in the question.