# Conversion factor and CTD Bond

I'm reading the book 'Options, Futures and Other derivatives' an having a hard time to understand Conversion factor and CTD bond.

1. Conversion factor

I understand this as a factor to adjust the price of the bond delivered to the hypothetical bond(with yield = 6% in my case)

ex. maturity = 20 years. Coupon rate = 10% with semiannual payment. Face value = $100 then in the book, after discounting future cash flow with annual rate = 6%$\sum_{i=1}^{40} \dfrac5{1.03^i} +\dfrac{100}{1.03^{40}} = $146.23 author says conversion factor is$\dfrac{$146.23}{$100}\$ = 1.4623

(If I'm wrong, please tell me. )

I wonder why we deal with only yield. I think the calculation must reflect the difference in maturities of each bonds.

1. CTD Bond.

In the book, author said 'when bond yields are in excess of 6%, the conversion factor system tends to favor the delivery of low-coupon, long-maturity bonds.'

I think, this means that under that circumstance, conversion factor increases as coupon rate decreases and maturity increases. However, I can't show that mathematically. I guess I misunderstood or there must be more variables that needs to be considered from the words 'tends to'. How can I understand that description both intuitively and mathematically.

• maybe you want to quote the formula for the conversion factor in your case (Bund?)
– Ric
Feb 26 '16 at 8:07

First of all, the idea that bond futures track a hypothetical 6% coupon bond with 20 years to maturity is a false one (although it frequently appears in textbooks). In reality, the "classic" bond futures contract tracks a basket of bonds between 15 and 25 years to maturity. Currently, the classic bond futures contract does behave similarly to a 20-year bond (because its CTD has 19.97 years to maturity), but until recently, it behaved more like a 15-year bond over the past few years.

Back to your question, conversion factor is (approximately) the price (divided by 100) of a bond assuming its yield to maturity as of the first delivery date is 6%. So it does depend on the maturity of the bond in question. For example, the 4.75s of 02-15-2041 are deliverable into USH6 (the March 2016 expiry bond futures contract). To calculate its conversion factor involves these steps:

1. calculate its time to maturity as of first delivery date; since the first delivery date for USH6 is 3/1/2016, the time to maturity is 24 years, 11 months, 14 days;

2. round time to maturity to the nearest whole quarter, which is 24 years and 9 months;

3. price a 4.75% coupon bond with 24 years and 9 months to maturity at 6% yield, which get you 83.9809. Hence the conversion factor is 0.8398.

With regard to CTD and yield curve environment, here are some intuitions. First of all, remember that the purpose of conversion factor is to make bonds approximately equally deliverable when the yield curve is flat at 6%. Now think about what happens when yields increase – high duration bonds, with higher sensitivity to yields, will see their prices decline faster, making them more likely to be cheaper to deliver. Conversely, when yields decline, low duration bonds will see their prices increase less, making them more likely to be cheaper.