# Conversion factor for bonds

Hull defines the conversion factor for a bond as the "quoted price the bond would have per dollar of principal on the first day of the delivery month on the assumption that the interest rate for all maturities equals 6% per annum."

My understanding is that the hypothetical bond underlying the futures contract has a 6% coupon, and that the (quoted) price of this bond varies with the zero curve. If this zero curve is flat and equal to 6% for all maturities, then the quoted price of the hypothetical bond should be 100 and the product of this hypothetical price and the conversion factor for a particular bond is exactly the quoted price of the bond in question. However, if the zero curve is flat and not equal to 6%, or more generally is not flat at all, then why should this conversion factor be considered a good approximation of the quoted price of the actual bond?

Suppose for the moment that the zero curve is flat. Let P(N,y,c) be the present value of a bond with semiannual coupon c and time to maturity N when the zero rate is y (for all maturities). The present value of the hypothetical bond is P(N,y,0.06). What naturally seems to be the correct conversion factor to get the quoted price of a bond with coupon c and time to maturity M is

CF = P(M,y,c)/P(N,y,0.06).

When y=0.06, this conversion factor is the same one defined by Hull, but otherwise they need not be the same. Is there some reason why it is assumed that the conversion factor is constant? y need not be close to 0.06, and M can be different from N, so it doesn't seem clear that the (constant) conversion factor gives anything useful.

• Spent hours looking for the third doc you mentioned, but in vain so far...any chance you/someone can kindly email that doc to me please? Looking for: Koenigsberg, Mark. "Deciding Between Futures and Cash: A Closer Look at the Basis." Salomon Brothers Bond Portfolio Analysis Group (1991).
– V K
Jan 9 at 16:11

First, the exact computation of conversion factor is actually quite tricky. The "6% yield" rule is really an approximation (although a very good one). CME provides a spreadsheet that you can use to compute the exact conversion factor for each bond and each contract (http://www.cmegroup.com/trading/interest-rates/us-treasury-futures-conversion-factor-calculator.html). The main reason for the discrepancy comes from rounding of term to maturity. For example, if we are looking at the classic bond (US) contract and a deliverable has 15 years 2 months to maturity on the first day of the delivery month, then you need to round it down to 15 years.

Second, it's not a good idea to think of the underlying as a hypothetical 6% coupon bond. What makes bond futures so complex and so interesting is that a full basket of deliverables act as the underlying. For example, for TYZ4 (10-year Treasury note expiring in December of this year), the deliverable basket includes 21 Treasury notes (5 of which haven't even been auctioned!) Right now, TYZ4 behaves very similarly to 2.25% 31-July-2021 (the cheapest-to-deliver into TYZ4). You can see that 1) it behaves nothing like a 10-year bond -- in fact, over past few years, the 10-year Treasury contract has mostly behaved like a 7-year bond; and 2) it behaves nothing like a bond with a 6% coupon.

The definitive guide to bond futures is The Treasury Bond Basis: An in-Depth Analysis for Hedgers, Speculators, and Arbitrageurs. I read it cover to cover a few times and learned new things every time I read it... In addition, there are a few Salomon Brothers research notes that I found tremendously helpful:

• Koenigsberg, Mark. "Understanding Treasury Bond Futures: Questions and Answers." Salomon Brothers Bond Portfolio Analysis Group (1990).

• Koenigsberg, Mark. "The Salomon Brothers Delivery Option Model: Understanding Treasury Bond Futures." Salomon Brothers Bond Portfolio Analysis Group (1991).

• Koenigsberg, Mark. "Deciding Between Futures and Cash: A Closer Look at the Basis." Salomon Brothers Bond Portfolio Analysis Group (1991).

• Thanks for responding. Do you happen to know of a reference that rigorously treats these issues? Hull's Options, Futures, and Other Derivatives only gives a cursory treatment and I am having trouble finding a suitable alternative. Aug 29, 2014 at 19:55
• @S.Dupree I updated my answer with a few suggested readings. Enjoy! Aug 29, 2014 at 20:53
• Where can I get access to these Salomon reports? Dec 2, 2018 at 14:57
• You are unlikely to find 30-year-old Citi/Salomon research notes anywhere. However doi://10.3905/jfi.1992.408047 Mark Koenigsberg, Yiorgos Bqurtzos. Basis and the Choice Between Futures and Cash. The Journal of Fixed Income, Fall 1992, 2 (2) 45-56 probably covers all the same material. Jan 9 at 17:02