Duration & Convexity (Gentle)
Duration and Convexity Cheat Sheet
Duration and Convexity Cheat Sheet
This chapter has covered duration from first principles through real-world complexity. Here is the essential knowledge distilled into one page for rapid reference.
Key takeaways
- Duration is the weighted-average time to receive a bond's cash flows, measured in years; it predicts percentage price changes from yield moves.
- Convexity is the curvature of the price-yield relationship; positive convexity favors you, negative convexity works against you.
- Three types of duration exist: Macaulay (time in years), modified (percentage price change per 1% yield move), and effective (for bonds with options).
- The core formula for price changes is: ΔP/P ≈ −D × Δy + ½ × C × (Δy)² (duration effect + convexity effect).
- Use duration to assess interest rate risk; use convexity to understand asymmetry; use both together for accurate predictions.
Duration quick reference
| Concept | Definition | Use |
|---|---|---|
| Macaulay Duration | Weighted-average time to receive cash flows (in years) | Historical/theoretical; not directly used for price prediction |
| Modified Duration | % price change per 1% yield change | Main measure for Treasuries and non-option bonds; simpler, more intuitive |
| Effective Duration | % price change accounting for embedded options | For callable bonds, MBS, convertibles; more accurate but complex |
| Duration in months | Macaulay duration × (coupon rate / yield to maturity) | Adjusting for reinvestment; rarely used in practice |
Duration interpretation
- 2-year duration bond: 1% yield rise → 2% price fall. 1% yield fall → 2% price rise.
- 5-year duration bond: 1% yield rise → 5% price fall. 1% yield fall → 5% price rise.
- 10-year duration bond: 1% yield rise → 10% price fall. 1% yield fall → 10% price rise.
For a £100 bond:
- 2-year duration: 1% move = £2 price change.
- 5-year duration: 1% move = £5 price change.
- 10-year duration: 1% move = £10 price change.
Typical durations
| Bond Type | Maturity | Duration (approx.) |
|---|---|---|
| Treasure bill | 6 months | 0.5 years |
| Money market fund | 30–90 days | 0.05 years |
| Short-duration corporate bond | 2 years | 1.8 years |
| Intermediate Treasury | 5 years | 4.5 years |
| Long-duration Treasury | 10 years | 9 years |
| Very long Treasury | 30 years | 18–20 years |
| Total bond market fund (BND, AGG) | Mixed | 5.5 years |
| High-yield bond fund | Mixed | 3.5 years |
| Mortgage-backed security (agency MBS) | 30 years stated, 6–10 effective | 6–10 years effective |
Convexity quick reference
| Type | Characteristic | Example | Impact |
|---|---|---|---|
| Positive Convexity | Price-yield curve bends upward | Plain-vanilla bonds, Treasuries | Gains from falling rates are larger than losses from rising rates (good for bondholder) |
| Negative Convexity | Price-yield curve bends downward | Callable corporate bonds, MBS | Gains from falling rates are capped; losses from rising rates are amplified (bad for bondholder) |
The price-change formula
For small moves (0–0.5% yield change): Use duration alone: ΔPrice/Price ≈ −Duration × ΔYield
Example: 5-year duration, yield rises 0.4%: ΔPrice/Price ≈ −5 × 0.004 = −0.02 = −2%
For large moves (1%+ yield change): Include convexity: ΔPrice/Price ≈ −Duration × ΔYield + ½ × Convexity × (ΔYield)²
Example: 5-year duration, positive convexity 50, yield falls 2%: ΔPrice/Price ≈ −5 × (−0.02) + ½ × 50 × (0.02)² = +0.10 + 0.01 = +0.11 = +11%
Duration alone would predict +10%; convexity adds 1%.
When to use each duration metric
Is the bond traded without options?
├─ Yes → Use modified duration
└─ No → Is it a callable bond, MBS, or convertible?
├─ Yes → Use effective duration
└─ For educational purposes: understand Macaulay duration (theoretical foundation)
Strategic vs. tactical duration decisions
| Scenario | Action | Rationale |
|---|---|---|
| Setting a long-term portfolio | Choose strategic duration = time horizon (5–8 years for retirees; 6–10 for pre-retirees) | Matches risk to life stage |
| Rate forecast: rates will fall | Increase duration (buy longer bonds; shift to TLT or long Treasuries) | Longer duration = larger gains when rates fall |
| Rate forecast: rates will rise | Decrease duration (buy shorter bonds; shift to SHY or money market) | Shorter duration = smaller losses when rates rise |
| Uncertain about rates | Maintain strategic duration; rebalance annually | Avoid costs of frequent trading; benefit from both directions |
Portfolio duration calculation
Weighted-average duration: Portfolio Duration = Σ (Weight in each bond × Duration of that bond)
Example:
- 60% in BND (duration 5.5) + 40% in TLT (duration 18) = 0.6 × 5.5 + 0.4 × 18 = 3.3 + 7.2 = 10.5 years
Red flags: when duration misleads
| Risk | Example | Mitigation |
|---|---|---|
| Yield curve shift (non-parallel) | Short rates rise 2%, long rates fall 0.5% (curve steepens) | Use multiple maturity points; monitor curve shape |
| Credit spread widening | Corporate yields rise 150 bp (100 bp from Treasury rise, 50 bp from spread widening) | Monitor credit conditions; diversify across credit qualities |
| Embedded options | Callable bond effective duration shrinks as rates fall (call becomes likely) | Use effective duration for bonds with options |
| Currency risk | Foreign bond falls 5%, currency depreciates 3% (net loss 8%) | Monitor currency and interest rates separately |
| Liquidity evaporation | Bond's bid-ask spread widens 0.5% during a crisis | Hold highly liquid bonds (Treasuries, agency MBS) for emergency reserves |
Quick decision tree: choosing a bond
Do you want to:
├─ Preserve capital, minimize volatility? → Buy short-duration bonds (1–2 years)
├─ Balance growth and income? → Buy intermediate-duration bonds (4–6 years)
├─ Maximize current income, accept volatility? → Buy long-duration bonds (8–15 years)
└─ Hedge against rate cuts/declining growth? → Buy very long-duration bonds (15–30 years) or add positive convexity
Practical portfolio examples
Example 1: Conservative retiree
- Goal: Stability, modest income
- Portfolio: 100% in BND (duration 5.5) + some cash
- Why: Medium duration cushions against moderate rate moves; still provides yield
Example 2: Moderate pre-retiree (age 50)
- Goal: Balanced growth and income
- Portfolio: 50% equities, 50% bonds (mix of BND 5.5 and TLT 18 for average 11 years)
- Why: Equity growth potential, bond stability; adequate duration to capture falling rates if recession occurs
Example 3: Aggressive saver (age 30)
- Goal: Maximum long-term growth
- Portfolio: 90% equities, 10% bonds (BND for ballast)
- Why: Bonds are minimal; equities drive growth; bond allocation is for downside cushion
Example 4: Tactical trader (professional/experienced)
- Goal: Outperform by adjusting duration based on rate forecasts
- Portfolio: Varies; might be 80% TLT if expecting rates to fall, 80% SHY if expecting rates to rise
- Why: Duration swaps capture directional rate bets efficiently; requires market timing skill
Books and further reading
- Frederick Macaulay: "Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the United States Since 1856" (1938)—the original duration paper, dense but foundational.
- Frank Fabozzi: "Bond Markets, Analysis, and Strategies" (10th ed.)—comprehensive bond textbook, the standard reference.
- Andrew Ang: "Asset Management" (2014)—modern treatment of bonds in portfolio context.
Summary: the one-sentence version
Duration measures how much a bond's price will change when yields move; convexity measures whether that change is symmetrical or asymmetrical; use both together to predict returns accurately.
Flowchart: Risk assessment at a glance
Next
You have now completed the chapter on duration and convexity. You understand:
- How duration is calculated and interpreted
- Why duration is useful but imperfect
- What convexity is and how it affects returns
- How to manage duration in a portfolio
- When duration and convexity mislead
This knowledge is the foundation for being a thoughtful bond investor. In practice, you will not need to calculate duration yourself (fund managers do that), but understanding these concepts lets you:
- Choose appropriate bond funds and holdings
- Assess risk accurately
- Avoid surprises when rates move
- Make decisions aligned with your time horizon and risk tolerance
As you continue with bonds or other assets, return to this chapter whenever you encounter a new bond concept. Duration and convexity are the bedrock.