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If the bond is trading at 5% and it becomes special, how do you quantify the impact this has on the bond? Or am I misunderstanding the concept of specialness?

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I'll use a real life example back from the early 2000s, since the specialness effect was much more pronounced.

Back on Feb 14, 2001, the 10-year on-the-run note (5s of Feb 15, 2011) traded at a yield of 5.121% / price of 99.062. We assume that the bond will lose its specialness in three months, so we reference the 3-month forward market for clues.

On this day, this issue's 3-month repo rate was 3.14%, so the 3-month forward price is simply: $$ 99.062 \times \left(1 + 3.14\% \times \frac{89}{360}\right) - 1.230 = 98.602, $$ where 1.230 was the accrued interest as of the forward settlement date.

The general collateral repo rate on the same day was over 200 bp higher at 5.19%. If the bond weren't trading special and had to be financed at GC, the forward price calculated above would imply a spot price as follows: $$ P \times \left(1 + 5.19\% \times \frac{89}{360}\right) - 1.230 = 98.602. $$ So without repo advantage, the bond's price should be $P = 98.567$, or a yield of 5.18%. This is 6 bp higher than the quoted yield of 5.12%, indicating that specialness richened the bond by 6 bp.

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