Cheapest-to-Deliver Analysis
Understanding CTD bond selection, conversion factors, and delivery basket dynamics
Cash Settlement vs Physical Delivery
Before understanding the Cheapest-to-Deliver (CTD) concept, it's critical to understand how futures contracts settle. There are two fundamental settlement methods used in derivatives markets:
Futures Settlement Methods
Two fundamental ways to close out a futures position at expiration
- Simple and efficient
- No logistics/storage concerns
- Lower transaction costs
Short delivers: Actual bonds from delivery basket
- Requires delivery basket specification
- Conversion factor calculations
- CTD bond selection strategy
Why JGB Futures Use Physical Delivery:
- Price Convergence: Physical delivery forces the futures price to converge to the cash market price at expiration. If there's a deviation, arbitrageurs can exploit it by buying cash/selling futures (or vice versa) and delivering into the contract.
- Market Integrity: Ensures futures track real market conditions rather than becoming detached from underlying JGB values.
- Hedging Effectiveness: Institutional investors (banks, insurers) actually hold JGBs. Physical delivery allows them to truly hedge by offsetting actual bond positions.
- Historical Precedent: Bond futures globally (US Treasuries, German Bunds) use physical delivery—it's the market standard for government debt.
The Short-Seller's Choice
Because JGB futures use physical settlement, anyone still holding a short position at expiration must deliver actual JGBs to a long-holder.
However, the short-seller doesn't have to deliver the "notional" 6% bond (which doesn't exist). Instead, the exchange (OSE) allows them to deliver any bond from the official "delivery basket" (e.g., for the 10-year contract, any JGB with 7-11 years to maturity).
Why Use a Delivery Basket? The Design Rationale
The delivery basket is a carefully designed feature that balances market liquidity, hedging effectiveness, and fairness. Understanding why exchanges use baskets (instead of specifying a single bond) reveals fundamental insights about how futures markets work.
Why Delivery Baskets Exist
Why Specifically 7-11 Years for the 10Y Contract?
The maturity range is not arbitrary—it reflects a balance between contract precision and market liquidity:
| Design Choice | Rationale | Trade-off |
|---|---|---|
| Lower Bound: 7 Years | Ensures delivered bonds still have substantial duration (interest rate sensitivity) similar to the notional 10Y bond. Bonds shorter than 7 years behave more like 5Y bonds. | If the floor were too low (e.g., 5 years), the contract would lose its "10-year" character and fail to hedge 10Y exposures effectively. |
| Upper Bound: 11 Years | Prevents very long-duration bonds (12+ years) from being delivered, which would have significantly different price sensitivity than a true 10Y bond. | If the ceiling were too high (e.g., 15 years), the CTD could shift to super-long bonds, making the futures price unreliable for hedging 10Y risk. |
| 4-Year Window (7-11) | Provides ~10-15 eligible bonds at any time, ensuring deep liquidity. The window is wide enough to prevent squeezes but narrow enough to maintain contract homogeneity. | A wider window (e.g., 5-15 years) would dilute the contract's hedging precision. A narrower window (e.g., 9-11 years) would risk liquidity shortages. |
Comparative Example: Other Bond Futures
- US 10Y Treasury Futures: Delivery basket includes bonds with 6.5-10 years remaining maturity (slightly tighter than JGBs)
- German Bund Futures: 8.5-10.5 years (much narrower, reflecting deeper Bund market liquidity)
- JGB 20Y Futures: 15-21 years (wider window due to less frequent super-long issuance)
The OSE's choice of 7-11 years for the 10Y contract is calibrated to Japan's issuance patterns, market depth, and institutional hedging needs. It's reviewed periodically but has remained stable because it effectively balances all competing demands.
This creates a valuable strategic decision for the short-seller: which of the 10-15 eligible bonds should I deliver to minimize my cost?
The answer is always the same: they will deliver the bond that is the Cheapest-to-Deliver (CTD). This is the bond that minimizes their cost.
Finding the CTD: The Math
A bond trader's goal is to find the bond that is the biggest "bargain" relative to the futures price. The answer lies in a simple formula called the Basis.
The Basis Formula
Formula Name: Net Basis (Simplified)
Where:
- $P_{\text{cash}}$ = Dirty price of the bond in the cash market (clean price + accrued interest)
- $P_{\text{fut}}$ = Futures price (quoted clean price)
- $\text{CF}$ = Conversion Factor for that specific bond (explained in Section 2.7)
The CTD Rule: The bond with the lowest (most negative) Basis is the Cheapest-to-Deliver.
Why This Works: The basis measures the "cost" of buying a bond in the cash market versus what you receive for delivering it into the futures contract. A more negative basis means you're getting a better deal—you pay less in the cash market relative to what the futures contract values it at.
Step-by-Step: Identifying the CTD
Step 1: Gather the Data
For every bond in the delivery basket, collect:
| Data Point | Description | Source |
|---|---|---|
| Clean Price | Current quoted price in the secondary market | Bloomberg, bond dealers, MOF reference rates |
| Accrued Interest | Interest earned since last coupon payment | Calculate using Actual/365 convention (Section 1.9) |
| Conversion Factor | Official factor published by OSE for each bond | JPX Conversion Factor List |
| Futures Price | Current trading price of the futures contract | OSE market data, Bloomberg ticker: JGB<Maturity> |
Step 2: Calculate Dirty Price
\(P_{\text{cash}} = \text{Clean Price} + \text{Accrued Interest}\)
Step 3: Calculate Basis for Each Bond
\(\text{Basis}_i = P_{\text{cash},i} - (P_{\text{fut}} \times \text{CF}_i)\)
Step 4: Identify the Minimum
\(\text{CTD Bond} = \arg\min_i(\text{Basis}_i)\)
The bond with the lowest (most negative) basis is your CTD.
What Makes a Bond the CTD?
The CTD bond is not static; it can "switch" from one bond to another as market conditions change. The key driver is the relationship between the bond's coupon and the current market yield (relative to the 6% notional yield).
A simple (but not perfect) rule of thumb for a low-yield environment (like Japan's):
- If Yields are < 6% (and the curve is normal): The CTD bond will generally be the one with the longest maturity in the basket.
- Why: The conversion factor formula "punishes" long-duration, low-coupon bonds the most (gives them a very low CF). This makes them the cheapest to deliver.
- If Yields are > 6% (and the curve is inverted): The CTD bond will generally be the one with the shortest maturity and highest coupon in the basket.
Because JGB yields have been below 6% for decades, the CTD for the 10-year future is almost always the eligible bond with the longest duration (i.e., the longest maturity, lowest coupon).
This is critical: the 10-year JGB futures contract (JGB L) does not track the 10-year benchmark bond. It tracks the theoretical CTD, which is often a bond with 10.5 or 11 years to maturity. This is a key concept for hedging.
Worked Example: CTD Calculation for 10Y JGB Futures
Let's determine the CTD bond for the December 2024 10-year JGB futures contract using actual market data.
Market Setup (November 2024)
Futures Contract: 10Y JGB Futures (JGB L), December 2024 settlement
Futures Price: 144.50 (quoted price, clean)
Settlement Date: December 20, 2024
Current Date: November 15, 2024 (35 days to settlement)
General Collateral (GC) Repo Rate: 0.10% (overnight, compounded)
Delivery Basket (Simplified - 3 bonds)
| Bond | Coupon | Maturity | Remaining Maturity | Clean Price (¥) | Accrued Interest (¥) | Conversion Factor (CF) |
|---|---|---|---|---|---|---|
| Bond A (JGB #370) | 0.40% | Mar 2035 | 10.3 years | 98.25 | 0.18 | 0.9245 |
| Bond B (JGB #371) | 0.50% | Jun 2034 | 9.6 years | 99.10 | 0.21 | 0.9356 |
| Bond C (JGB #372) | 0.60% | Sep 2033 | 8.8 years | 99.85 | 0.15 | 0.9471 |
Step 1: Calculate Dirty Price for Each Bond
\[\text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest}\]- Bond A: $98.25 + 0.18 = 98.43$
- Bond B: $99.10 + 0.21 = 99.31$
- Bond C: $99.85 + 0.15 = 100.00$
Step 2: Calculate Futures Invoice Price
If you deliver a bond, you receive:
\[\text{Invoice Price} = (\text{Futures Price} \times \text{CF}) + \text{Accrued Interest at Settlement}\]Assuming accrued interest remains approximately the same at settlement (simplified):
- Bond A: $(144.50 \times 0.9245) + 0.18 = 133.59 + 0.18 = 133.77$
- Bond B: $(144.50 \times 0.9356) + 0.21 = 135.19 + 0.21 = 135.40$
- Bond C: $(144.50 \times 0.9471) + 0.15 = 136.86 + 0.15 = 137.01$
Step 3: Calculate Gross Basis
\[\text{Gross Basis} = \text{Dirty Price (Cash)} - \text{Invoice Price}\]- Bond A: $98.43 - 133.77 = \mathbf{-35.34}$
- Bond B: $99.31 - 135.40 = \mathbf{-36.09}$
- Bond C: $100.00 - 137.01 = \mathbf{-37.01}$
Note: Negative basis means futures are trading at a premium to cash (common in low-yield environments).
Step 4: Adjust for Carry (Net Basis)
If you buy the bond today and hold it for 35 days until delivery, you earn:
\[\text{Carry} = \text{Coupon Income} - \text{Repo Financing Cost}\]For 35 days (~0.096 years):
- Bond A: Coupon = $0.40\% \times 0.096 = 0.038$; Repo = $98.43 \times 0.10\% \times 0.096 = 0.009$; Net Carry = $0.029$
- Bond B: Coupon = $0.50\% \times 0.096 = 0.048$; Repo = $99.31 \times 0.10\% \times 0.096 = 0.010$; Net Carry = $0.038$
- Bond C: Coupon = $0.60\% \times 0.096 = 0.058$; Repo = $100.00 \times 0.10\% \times 0.096 = 0.010$; Net Carry = $0.048$
- Bond A: $-35.34 + 0.029 = \mathbf{-35.31}$
- Bond B: $-36.09 + 0.038 = \mathbf{-36.05}$
- Bond C: $-37.01 + 0.048 = \mathbf{-36.96}$
Step 5: Identify the CTD
The CTD is the bond with the least negative (highest) net basis:
Bond A (JGB #370): Net Basis = -35.31 → CTD ✓
Why Bond A? It has the longest maturity (10.3 years) and lowest coupon (0.40%), making its conversion factor the smallest. In a low-yield environment (current 10Y yield ~0.7%), long-duration bonds are cheapest to deliver.
Implied Repo Rate Verification
The Implied Repo Rate (IRR) for Bond A:
\[\text{IRR} = \frac{(\text{Invoice Price} - \text{Dirty Price}) + \text{Coupon}}{\text{Dirty Price}} \times \frac{360}{\text{Days to Settlement}}\] \[\text{IRR} = \frac{(133.77 - 98.43) + 0.038}{98.43} \times \frac{360}{35} = \frac{35.378}{98.43} \times 10.29 = \mathbf{3.70\%}\]Compare to market repo rate of 0.10%: The high IRR confirms this is the most attractive bond to deliver (though in practice, arbitrage would compress these spreads).
Key Takeaway: Understanding CTD dynamics is essential for futures hedging. Since the 10Y futures tracks the CTD (not the benchmark 10Y bond), duration hedges must account for the actual delivered bond's characteristics.
References
- Japan Exchange Group (JPX) / OSE. "Eligible Issues/Deliverable Grades." Available at: https://www.jpx.co.jp/english/derivatives/products/jgb/jgb-futures/02.html.