Conversion Factors
How conversion factors normalize bonds for futures delivery, calculation methodology
The Problem: An Unfair Delivery
The JGB futures contract is based on a "notional" 6% coupon bond. But in the real world, the "delivery basket" contains many different bonds—some with a 0.5% coupon, some with a 2.0% coupon, and all with different maturities (e.g., 7.5 years, 10.2 years).
If you just delivered ¥100 million face value of *any* bond, it would be unfair. Delivering a 2.0% coupon bond is far more "valuable" than delivering a 0.5% coupon bond.
The Conversion Factor (CF) is the solution. It is a price adjustment factor, calculated for every bond in the delivery basket, that equalizes them all to the 6% notional standard.
The Concept: Price at 6% Yield
Conceptually, the Conversion Factor is simple:
The Conversion Factor is the price (per ¥1) that a specific bond *would have* if its yield-to-maturity was exactly 6%.
This calculation is performed by the exchange (OSE) for every eligible bond, rounding to four decimal places.
- A bond with a coupon higher than 6% will have a CF > 1.0.
- A bond with a coupon lower than 6% (i.e., every JGB issued in the last 25 years) will have a CF < 1.0.
- A bond with a coupon exactly at 6% would have a CF of 1.0.
How It's Used: The Invoice Amount
When a short-seller delivers a bond to a long-holder at settlement, the cash they receive (the "invoice amount") is not just the futures price. It is the futures price *adjusted by the Conversion Factor*.
The formula is:
\[\text{Invoice Amount} = (\text{Futures Settlement Price} \times \text{Conversion Factor}) \times \frac{\text{Face Value}}{100} + \text{Accrued Interest}\]This adjustment is what makes the delivery fair. If you deliver a "cheap" bond (low coupon, low CF), you receive less cash. If you deliver a "rich" bond (high coupon, high CF), you receive more cash. This system ensures the long-holder is economically indifferent to which bond they receive.
Calculation of the Conversion Factor
The official calculation, set by the OSE, is the present value (PV) of the bond's cash flows, discounted at the notional 6% yield (3% semi-annually), calculated to the first day of the delivery month.
While the full, precise formula accounts for non-standard first coupon periods, the standard formula for a bond with an integer number of semi-annual periods (n) until maturity is:
\[CF = \frac{1}{100} \left[ \sum_{t=1}^{n} \frac{C/2}{(1 + 0.03)^t} + \frac{100}{(1 + 0.03)^n} \right]\]Where:
- $CF$ = Conversion Factor
- $C$ = Annual coupon rate of the bond (e.g., 1.5)
- $n$ = Number of semi-annual coupon periods remaining
- $0.03$ = The notional semi-annual yield (6% / 2)
Example Calculation
Let's calculate the CF for a hypothetical JGB in the basket:
- Coupon ($C$): 1.0% (so, 0.5 per period)
- Remaining Maturity: 8 years exactly (so, n = 16)
Step 1: Present Value of Coupons $$PV(\text{Coupons}) = \sum_{t=1}^{16} \frac{0.5}{(1.03)^t}$$ $$PV(\text{Coupons}) = 0.5 \times \left[ \frac{1 - (1.03)^{-16}}{0.03} \right]$$ $$PV(\text{Coupons}) \approx 0.5 \times 12.5611 = \textbf{6.28055}$$
Step 2: Present Value of Principal $$PV(\text{Principal}) = \frac{100}{(1.03)^{16}}$$ $$PV(\text{Principal}) \approx 100 \times 0.62317 = \textbf{62.317}$$
Step 3: Total Price (per ¥100) $$\text{Price} = 6.28055 + 62.317 = 68.59755$$
Step 4: Conversion Factor (Price / 100) $$CF = 68.59755 / 100 \approx \textbf{0.6860}$$ (rounded to 4 decimal places)
This 0.6860 is the official Conversion Factor for this bond. The short-seller's goal is now to find the bond that is the *best bargain* relative to its specific CF. This is the "Cheapest-to-Deliver."
Why Conversion Factors Exist: Historical Context
Conversion factors weren't invented for JGBs—they originated in the US Treasury futures market in the 1970s when the Chicago Board of Trade (CBOT) faced a critical design challenge.
The Coupon Diversity Problem
In the 1970s-1980s, governments issued bonds with widely varying coupon rates:
- High inflation periods: New bonds issued with 8-12% coupons
- Falling rates: Older bonds still outstanding with 4-6% coupons
- Result: The delivery basket for a "10-year" futures contract contained bonds with drastically different economic values
Without conversion factors, short sellers would always deliver the lowest-coupon bond (cheapest in price), leaving long holders with systematically inferior bonds. This would destroy market confidence in futures contracts.
The 6% Notional Standard
Why 6%? In the early 1980s when JGB futures were designed:
- Japanese 10Y yields averaged 6-8%
- US Treasury futures already used 6% (established in 1977)
- 6% was a round number close to prevailing market rates
Key insight: The 6% is arbitrary. What matters is that all bonds are standardized to the same reference yield. If they had chosen 5% or 7%, the system would work identically—only the specific CF numbers would differ.
Standardization Benefits
| Benefit | Description |
|---|---|
| Fair Delivery | Long holders receive economically equivalent value regardless of which bond is delivered |
| Market Liquidity | Futures price isn't tied to one specific bond—can trade continuously even as bonds roll out of the basket |
| Hedge Efficiency | Institutions can hedge diverse bond portfolios using standardized futures contracts |
| Price Discovery | Futures reflect market consensus on the "average" 10Y JGB, not idiosyncratic bond features |
CF Calculation Deep Dive: OSE Methodology
Official OSE Calculation Rules
The Osaka Exchange publishes conversion factors for all eligible bonds approximately 2 months before each futures contract's first delivery day. The methodology follows these strict rules:
- Valuation Date: The first day of the delivery month (e.g., March 1, June 1, September 1, December 1)
- Discount Rate: Fixed at 6% annually (3% semi-annually)
- Cash Flow Calculation: Include all future coupon payments and principal redemption from valuation date to maturity
- Rounding: Final CF rounded to 4 decimal places
Handling Partial Periods
When the valuation date falls between coupon payment dates (not exactly on a coupon date), the CF formula adjusts for the fractional period:
Where:
- $v$ = Fraction of the semi-annual period from valuation date to next coupon (e.g., 0.33 = 2 months remaining in a 6-month period)
- $n$ = Total number of remaining coupon periods including the partial one
- $C$ = Annual coupon rate
Example: If the delivery month is June 1, 2025, but the bond's coupon dates are March 20 and September 20, then v = (Sept 20 - June 1) / (Sept 20 - Mar 20) ≈ 0.58 (roughly 3.5 months out of 6 months).
Rounding Conventions
OSE rounds CFs to exactly 4 decimal places using standard rounding:
- Calculated CF = 0.68597531 → Rounds to 0.6860
- Calculated CF = 0.68594999 → Rounds to 0.6859
Why this matters: For large futures positions, even a 0.0001 difference in CF translates to ¥10,000 per ¥100M contract. Professional traders verify OSE-published CFs against their own calculations to catch rare errors.
CF Behavior in Different Yield Environments
Understanding how CFs behave when market yields change is crucial for predicting CTD switches and futures pricing dynamics.
Core Principle: Duration Effects
Conversion factors are static—they're calculated once and don't change during the contract's life. But their relative attractiveness changes as market yields move.
| Market Scenario | CF Behavior | CTD Tendency |
|---|---|---|
| Yields < 6% (Japan 1995-Present) | All CFs < 1.0. Longer-duration, lower-coupon bonds have the lowest CFs | CTD = Longest maturity, lowest coupon bond in basket |
| Yields ≈ 6% | CFs cluster near 1.0. Duration differences matter less | CTD = Bond closest to 10Y par maturity |
| Yields > 6% | All CFs > 1.0 for high-coupon bonds. Shorter-duration bonds favored | CTD = Shortest maturity, highest coupon bond in basket |
Why Low-Coupon Bonds Have Lower CFs (When Yields < 6%)
Consider two bonds, both maturing in exactly 10 years:
- Bond A: 0.5% coupon → CF ≈ 0.66
- Bond B: 2.0% coupon → CF ≈ 0.81
At 6% discount rate, Bond A (paying tiny coupons) is worth only 66% of par. Bond B (paying larger coupons) is worth 81% of par. The CF reflects this economic reality.
Consequence for CTD: Since current JGB yields (~1-2%) are far below 6%, delivering Bond A costs you less (lower CF × futures price) while you paid a similar market price. Bond A becomes CTD.
Duration Matching Intuition
The conversion factor system implicitly matches duration:
- Low-coupon bonds have higher duration (cash flows weighted toward final principal payment)
- High-coupon bonds have lower duration (cash flows front-loaded with coupon payments)
When market yields are below 6%, high-duration (low-coupon) bonds are "over-penalized" by the 6% discount rate → Lower CF → Cheaper to deliver.
How Conversion Factors Drive CTD Switches
The CTD bond can "switch" from one bond to another when market conditions change. The primary driver is the relationship between CFs and actual market yields.
Scenario 1: Stable Low-Yield Environment (Typical for Japan)
Market Conditions: 10Y JGB yields at 0.75%, steady
Delivery Basket (Simplified Example):
| Bond | Coupon | Maturity (Years) | Conversion Factor | CTD Status |
|---|---|---|---|---|
| Issue #350 | 0.1% | 10.8 | 0.6245 | ⭐ CTD (lowest CF, longest duration) |
| Issue #355 | 0.5% | 10.2 | 0.6681 | — |
| Issue #360 | 1.0% | 9.5 | 0.7254 | — |
| Issue #365 | 1.5% | 8.9 | 0.7891 | — |
| Issue #370 | 2.0% | 7.6 | 0.8512 | — |
Why Issue #350 is CTD: Its 0.6245 CF is the lowest. When futures price = 143.50, invoice amount = 143.50 × 0.6245 = ¥89.62 per ¥100. If this bond trades at ~¥89.50 in the cash market, the basis is minimal → CTD.
Scenario 2: Yield Spike (Rate Hike Environment)
Market Shock: BOJ unexpectedly hikes rates → 10Y yields jump from 0.75% to 2.50%
Impact on CTD:
- As yields rise toward 6%, longer-duration bonds (high sensitivity) fall in price faster than shorter-duration bonds
- Issue #350 (longest duration) drops more in cash price than its low CF suggests
- Issue #365 or #370 (shorter duration, higher coupon) becomes relatively cheaper to deliver
- CTD switches from Issue #350 → Issue #365
Historical Example: During the 2003 VaR shock, JGB yields spiked from 0.43% to 1.60% in weeks. The CTD switched from the longest bond (11Y maturity, 0.7% coupon) to a shorter bond (8.5Y, 1.5% coupon) within days.
Monitoring CTD Switches
Professional traders track the "basis per contract" for all eligible bonds daily:
DLVY shows real-time CTD analysis for JGB futures, including basis values, conversion factors, and implied repo rates for all delivery basket bonds. Traders monitor this screen constantly during delivery months.
Worked Example: Multi-Bond CF Comparison
Let's calculate CFs for three hypothetical bonds to see how coupon and maturity affect the result.
Bond Specifications (All valued as of June 1, 2025)
| Bond | Coupon | Maturity Date | Years Remaining | Periods (n) |
|---|---|---|---|---|
| Bond X | 0.5% | Dec 20, 2034 | 9.5 | 19 |
| Bond Y | 1.0% | Dec 20, 2034 | 9.5 | 19 |
| Bond Z | 1.5% | Jun 20, 2033 | 8.0 | 16 |
Bond X: 0.5% Coupon, 9.5 Years
\[CF_X = \frac{1}{100} \left[ \sum_{t=1}^{19} \frac{0.25}{(1.03)^t} + \frac{100}{(1.03)^{19}} \right]\]\(PV(\text{Coupons}) = 0.25 \times 14.3238 = 3.5810\) \(PV(\text{Principal}) = 100 \times 0.5703 = 57.03\) \(CF_X = \frac{61.41}{100} = \textbf{0.6141}\)
Bond Y: 1.0% Coupon, 9.5 Years
\[CF_Y = \frac{1}{100} \left[ \sum_{t=1}^{19} \frac{0.50}{(1.03)^t} + \frac{100}{(1.03)^{19}} \right]\]\(PV(\text{Coupons}) = 0.50 \times 14.3238 = 7.1619\) \(PV(\text{Principal}) = 57.03\) (same as Bond X) \(CF_Y = \frac{64.19}{100} = \textbf{0.6419}\)
Bond Z: 1.5% Coupon, 8.0 Years
\[CF_Z = \frac{1}{100} \left[ \sum_{t=1}^{16} \frac{0.75}{(1.03)^t} + \frac{100}{(1.03)^{16}} \right]\]\(PV(\text{Coupons}) = 0.75 \times 12.5611 = 9.4208\) \(PV(\text{Principal}) = 100 \times 0.6232 = 62.32\) \(CF_Z = \frac{71.74}{100} = \textbf{0.7174}\)
Key Observations
- Coupon Effect: Bond X and Y have identical maturity but different coupons → CF differs by 0.0278 (2.78%)
- Maturity Effect: Bond Z has shorter maturity but higher coupon → CF is highest at 0.7174
- CTD Prediction: In current low-yield environment (yields < 6%), Bond X (lowest CF = 0.6141) would likely be CTD
Common Misconceptions About Conversion Factors
Misconception 1: "Higher CF = Better Bond"
Reality: A high CF just means the bond is worth more at 6% yield. It doesn't mean it's a better investment or cheaper to deliver. What matters for CTD is the basis (cash price - futures price × CF).
Misconception 2: "CF Adjusts for Market Yield Changes"
Reality: CFs are static once published. They don't update as yields change. This is by design—if CFs changed daily, futures pricing would be chaotic and hedging would be impossible.
Misconception 3: "All CFs Sum to 1.0"
Reality: There's no constraint on the sum of CFs. In Japan's current environment, all CFs are < 1.0 (since all coupons < 6%). The average CF for the 10Y basket is typically around 0.70-0.75.
Key Takeaways
- CFs exist to standardize: They normalize bonds with different coupons/maturities to a 6% notional standard
- CF = Price at 6% yield: Conceptually simple—what would this bond be worth if YTM = 6%?
- Low yields → Low CFs favor long bonds: In Japan (yields < 6%), longest-maturity, lowest-coupon bonds have lowest CFs → CTD
- CFs are static but relative value changes: As market yields move, the CTD can switch even though CFs don't change
- Rounding matters: OSE publishes CFs to 4 decimals; verify against own calculations for large positions
- Duration matching: CF system implicitly adjusts for duration differences between bonds
References
- Japan Exchange Group (JPX) / OSE. "Calculating Conversion Factors." Available at: https://www.jpx.co.jp/english/derivatives/products/jgb/jgb-futures/02.html.
- Japan Exchange Group (JPX). "JGB Futures Conversion Factor List (Monthly Updates)." Available at: OSE Derivatives Market.
- Burghardt, G., Belton, T., Lane, M., & Papa, J. (2005). The Treasury Bond Basis. McGraw-Hill. (Foundational reference for CF systems in government bond futures)