MODULE 1: THE EXECUTIVE SUMMARY (The "Elevator Pitch")

Definition

Duration and Convexity are the quantitative twin pillars of bond risk measurement. Duration (the first derivative) measures a bond's price sensitivity to interest rate changes—expressed as the percentage price change per 100 basis points of yield movement. Convexity (the second derivative) captures the nonlinear, or curved, relationship between bond prices and yields—it measures how duration itself changes as rates move. Together, they decompose a bond's total price risk into its linear (duration) and nonlinear (convexity) components.

For credit investors, this framework is essential, but incomplete: you must separate interest rate duration (sensitivity to benchmark rate moves) from spread duration (sensitivity to credit spread widening/tightening). Most institutional credit losses don't come from parallel rate moves; they come from the combination of simultaneous rate and spread movements, often with amplification from negative convexity embedded in callable bonds.

The Intuition

At its core, duration and convexity capture a market reality: bond prices do not move in a straight line with rates; they move along a curve. A Treasury bond with 10 years to maturity won't lose exactly 10% if rates rise 1%—the actual loss depends on the curvature of the price-yield relationship. Why? Because as rates fall, each additional unit of fall produces progressively larger price gains (positive curvature for most bonds). Conversely, for callable bonds, as rates fall and the call moves into-the-money, the issuer refinances and you lose the upside—the curve bends the wrong way.

The deeper intuition: Duration is the market's linear approximation of a nonlinear reality. For a risk manager or trader, this means:

  • Small rate moves? Duration alone suffices.
  • Large rate moves or credit stress? Convexity becomes a first-order risk factor.
  • Callable bonds in a falling-rate environment? Negative convexity can wipe out your predicted gains.

For credit models, the revelation is that credit risk and interest rate risk are not independent. When a borrower deteriorates, spreads typically widen at the same time rates are moving (often declining as central banks ease into recessions). A bond with long spread duration combined with negative convexity can suffer catastrophic losses—not from either risk alone, but from their combination.

Core Discipline

Stochastic Calculus & Numerical Optimization (for modeling), combined with Applied Statistics (for empirical calibration) and Market Microstructure (for understanding why spreads move the way they do). The practical discipline is derivatives pricing (treating bonds as complex options on rates and spreads) and portfolio risk attribution (decomposing returns into duration, spread, convexity, and credit event components).

MODULE 2: THE MECHANICS (How It Works)

Key Inputs/Variables

For Interest Rate Duration:

  • Benchmark yield curve (Treasuries, swaps)
  • Bond's coupon, maturity, and cash flows
  • Expected call/prepayment schedule (for embedded options)
  • Volatility of interest rates (for option valuation)

For Spread Duration:

  • Credit spread (yield-to-maturity minus benchmark yield)
  • Haircut/recovery assumptions (for default scenarios)
  • Macro variables affecting credit: unemployment, GDP growth, sector health
  • Issuer-specific metrics: leverage, interest coverage, liquidity

For Convexity Analysis:

  • Bond price sensitivity to large rate moves (calibrated via numerical differentiation)
  • Embedded option parameters: call price, first call date, coupon, refunding likelihood
  • Credit spread volatility (often underestimated, especially pre-crisis)
  • Correlation between rate moves and spread moves (typically negative: rates down, spreads widen in flight-to-quality)

The Mechanism: Mathematical Foundation

1. Modified Duration (Linear Approximation)

The percentage change in a bond's price for a 1% change in yield is approximated as:

%ΔP ≈ −ModDur × ΔY

Where Modified Duration is:

ModDur = Macaulay Duration / (1 + y)

Sensitivity Interpretation: If a bond has ModDur = 5 and yields rise 100 bps, the bond price falls approximately 5%. The negative sign reflects the inverse relationship: higher yields → lower prices.

Critical nuance for credit: The modified duration you calculate from a bond's YTM is actually a blend of two distinct components:

ModDurTotal ≈ ModDurRate + ModDurSpread

  • Rate Duration: Sensitivity to parallel shifts in the benchmark curve (Treasury rates).
  • Spread Duration: Sensitivity to credit spread changes (the yield premium above Treasuries).

Why it matters: In a recession, rates often fall (duration works for you), but credit spreads widen dramatically (spread duration works against you). The net effect depends on the relative magnitudes and the correlation between the two.

2. Convexity (The Curvature Adjustment)

The second-order price approximation is:

%ΔP ≈ (−ModDur × ΔY) + [1/2 × Convexity × (ΔY)²]

The convexity adjustment becomes significant only when yield changes are large (100+ basis points). For smaller moves, duration dominates.

Numerical Example (from institutional practice):

  • Bond price: 100
  • Modified Duration: 9 years
  • Convexity: 100
  • Scenario: Yields rise 100 bps

Duration effect: −9% × 1% = −9%

Convexity adjustment: 0.5 × 100 × (0.01)² = +0.005 = +0.5%

Net price change: −8.5% (convexity partially offsets the loss)

Compare to a 200 bps move:

  • Duration effect: −18%
  • Convexity adjustment: 0.5 × 100 × (0.02)² = +2%
  • Net: −16% (convexity benefit now substantial)

3. Positive vs. Negative Convexity

Positive Convexity (straight bonds, most Treasuries, non-callable corporates):

  • As yields fall, bond prices rise more than predicted by duration (the curve is steeper).
  • As yields rise, bond prices fall less than predicted by duration (the curve flattens).
  • Asymmetric payoff: gains exceed losses.

Negative Convexity (callable bonds, most corporate and municipal bonds, MBS):

  • Embedded option: Issuer owns a call option (right to redeem bond at par/call price).
  • Conceptually: Long straight bond − Short call option = Callable bond.
  • As yields fall:
    • Straight bond would gain (say, +15% for a 200 bps move).
    • Call moves in-the-money; issuer refinances.
    • Your bond price is capped at the call price (usually par or slightly higher).
    • You keep only +2% gain (the call option value you sold is now realized by the issuer).
  • As yields rise:
    • Call moves out-of-the-money.
    • Bond behaves like a straight bond, but with duration that has extended (the call is now worthless, so you're holding a longer bond).
    • Price falls more than expected: −12% (instead of predicted −9%).

Key metric for credit: Negative convexity is most pronounced when the bond is near-par (at-the-money). When deep in-the-money or deep out-of-the-money, convexity is muted because the probability of the option being exercised is already extreme (close to certain or impossible).

4. Spread Duration: The Credit Dimension

Spread duration measures the bond's sensitivity to credit spread changes:

%ΔPspread ≈ −SpreadDur × Δs

Where Δs is the change in credit spread (not yields).

Empirical observation: Credit spreads tend to move relatively (not absolutely). A bond with a 500 bps spread might widen to 550 bps (a 10% relative move), while a bond with a 100 bps spread might widen to 110 bps (also 10%). This is because the underlying credit fundamentals (macro conditions, liquidity, leverage) affect all credits proportionally.

Duration Times Spread (DTS), the industry standard for credit risk:

DTS = SpreadDur × Spread

This metric answers the question: Which bond will have higher volatility—a 1-year bond with a 500 bps spread, or a 5-year bond with a 100 bps spread?

  • Bond A: DTS = 1 × 500 = 500
  • Bond B: DTS = 5 × 100 = 500

Answer: Both have roughly the same expected volatility, despite very different maturity profiles. This is because the shorter-duration bond has much worse credit quality.

5. Effective Duration: Decomposing Rate vs. Spread

For bonds with embedded options or uncertain cash flows, calculate effective duration by shocking the benchmark curve, not the bond's own yield:

EffDur = (P− − P+) / (2 × P0 × ΔCurve)

Where:

  • P− = Bond price if benchmark yields fall by Δ (say, 25 bps)
  • P+ = Bond price if benchmark yields rise by Δ
  • P0 = Current bond price
  • ΔCurve = Change in benchmark curve

Contrast: Modified duration assumes the spread stays constant, so it mixes rate and credit sensitivity. Effective duration isolates just the rate sensitivity by varying the benchmark curve while holding the spread constant (implicitly).

The difference between effective duration and modified duration reveals the extent to which a bond's cash flows depend on embedded options or credit deterioration.

Sensitivity Illustration: How Key Variables Drive Duration

Variable Effect on Duration Why
Coupon increases Duration decreases Higher coupons return principal faster (weighted avg maturity shortens)
Maturity increases Duration increases More cash flows in the future → longer wait for principal return
Yield decreases Duration increases Lower discount rate → more value placed on distant cash flows
Credit quality deteriorates Spread duration increases Higher spread means more sensitivity to spread changes
Call option probability increases Negative convexity increases Duration becomes more volatile as call likelihood shifts
Rate volatility increases Effective duration changes Option value (and thus effective duration) depends on path-dependent scenarios

MODULE 3: APPLICATION IN TRADING & INVESTMENT (The "Alpha")

For Long-Term Investors & Asset Allocators

Strategic Application: Duration as a Tactical Lever

Institutional investors (pension funds, endowments, insurance companies) face a fundamental liability mismatch problem: they have long-term obligations (pensions, claims) that are often fixed-rate or inflation-adjusted, while they can dynamically manage assets.

1. Liability-Driven Investment (LDI) & Duration Targeting

If a pension fund has 10 billion in liabilities with an effective duration of 15 years, they want to match this duration on the asset side. They construct a portfolio:

  • 40% long-duration Treasuries (duration ~12 years)
  • 35% investment-grade corporates (duration ~5 years, but higher yield)
  • 25% high-yield bonds (duration ~3 years, even higher yield, but more credit risk)

The weighted average duration is roughly 15 years. Why this matters: If rates fall 200 bps uniformly:

  • Liabilities gain ~30% in value (funded ratio improves).
  • Assets also gain ~30%, creating a hedged position.
  • Net result: funded ratio is stable, reducing the need for additional contributions.

Convexity consideration: If all positions have positive convexity (no callable bonds), the asset gains from a large rate fall exceed the liability gains (which are linear). This is the "convexity crush" or "convexity gift" that LDI managers exploit: they deliberately take positive convexity positions because they expect to benefit from volatility.

2. Credit Allocation: Spread Duration as a Macro Bet

Portfolio managers use credit spreads as a forward-looking indicator of economic health. The Signal:

  • Spreads Narrowing → Market expects: low default rates, stable growth → Allocate to high-yield, extend duration.
  • Spreads Stable but High → Market prices in: elevated defaults, but they're priced in → Selective bottom-fishing in specific issuers.
  • Spreads Widening Rapidly → Market fears: systemic stress, credit crunch → De-risk, reduce spread duration, shorten maturity.

Institutional Playbook:

In 2023-2024, investment-grade spreads were extraordinarily tight (OAS ~80-100 bps). A systematic LDI manager would:

  1. Recognize that spreads are pricing in near-zero default probability (inconsistent with macro risk).
  2. Reduce allocation to IG corporates (less compensation for duration risk).
  3. Rotate into agency MBS (positive convexity, more protection if spreads widen).
  4. Overweight Treasuries for the carry + convexity benefit.

The timing: This signals a manager is moving before spreads widen, locking in gains when credit is still "expensive" and switching into cheaper long-duration assets.

3. Sector & Security Selection: OAS-Based Relative Value

Option-Adjusted Spread (OAS) is the fair-value spread that accounts for embedded options. A straightforward application:

Bond X: OAS = 150 bps (vs. comparable non-callable bond)

Bond Y: OAS = 140 bps (same issuer, similar maturity, but callable)

Conclusion: Bond X is cheap relative to Bond Y. If credit conditions remain stable, Bond X will outperform as its OAS compresses toward Bond Y's level.

For a quant team, this becomes a multi-factor signal:

Fair-Value Spread = f(Leverage, Interest Coverage, Liquidity, Sector Health, …)

Overvalued bonds (trading tighter than model-implied spread) are shorts (or underweight).

Undervalued bonds (trading wider than fair-value) are longs (or overweight).

For Short-Term Traders & Relative Value Desks

1. Identifying Dislocations: Spread vs. Credit Model

A credit model (whether a simple regression on leverage and coverage ratios, or a sophisticated machine-learning classifier) outputs a fair-value spread for each bond.

Setup for the Trade:

  • Bond trading at 250 bps
  • Credit model fair value: 220 bps
  • Interpretation: Market is pricing 30 bps of extra risk

Buy Signal:

  • If you believe the model's estimate is correct, the bond is wide (cheap).
  • Execute long; target: spread tightens to 220 bps.
  • P&L driver: credit spread compression (positive carry if you hold to maturity or until spread re-rates).

Catalyst Trade: Earnings are announced tomorrow and you expect positive revisions.

  • If accurate, the credit model's estimated PD will fall.
  • Fair-value spread should tighten (fewer defaults baked in).
  • Trade: Long the bond now; sell after earnings surprise.

2. Duration Hedging & Relative Value Positioning

A trader on a relative-value desk identifies:

  • Corporate Bond XYZ: Effective duration 6.0 years, spread duration 4.5 years
  • Treasury 5-year: Duration 4.8 years

Thesis: The credit spread is unjustifiably wide; XYZ will outperform.

Execution:

  • Long XYZ (long both rate and spread duration).
  • Short Treasury 5-year (short rate duration).
  • Net position: Long spread duration only; rate-hedged.

Why this matters: The trader isolates the idiosyncratic credit bet from macro rate risk. If rates fall uniformly, the long Treasury short offsets the rate gain on XYZ. Only the spread bet remains, allowing precise alpha capture.

3. Convexity Positioning: Anticipating Volatility

During low-volatility environments (e.g., post-2009 to 2019), credit spreads are tight, rates are low, and embedded option values are depressed. This creates a convexity imbalance:

  • Callable corporates are priced with low embedded option values (vol is low).
  • Treasuries trade with positive convexity premium (investors pay up for asymmetry).

Trade:

  • Long callable corporates (you're short the embedded call, but the call is underpriced given historical vol is low).
  • Short Treasuries (positive convexity is expensive).

Execution Timing: When volatility spikes (or is expected to spike):

  • Embedded option values soar → Callable bond prices compress more than expected.
  • Treasury prices rise more (positive convexity amplifies gains).
  • Your long corporates underperform → Negative convexity bite.

Conversely, if volatility declines (mean reversion after a spike):

  • Embedded options lose value → Callable bonds rally (negative convexity reduces drag).
  • You profit if you positioned for falling vol.

4. The "Negative Convexity + Widening Spreads" Trade (The Black Swan)

This is the most dangerous combination and where alpha is made or lost:

Scenario: A sector (e.g., commercial real estate, retail) shows early stress signals:

  • Delinquency rates ticking up
  • Refinancing risks rising
  • Spreads still tight (market hasn't repriced yet)

Trade Signal:

  • Long duration exposure in CRE bonds (negative convexity due to call features)
  • Spread duration: ~5 years
  • Expected spread move: widening 200 bps over 12 months

Predicted P&L (naïve, duration-only):

  • Spread widening 200 bps on 5-year spread duration = −10% loss
  • Rate duration 6 years: if rates fall 50 bps, +3% gain
  • Net: −7% expected return

Actual P&L (with negative convexity):

  • Spreads widen 200 bps + rates rise 50 bps (flight-to-quality fails; fiscal stress instead)
  • Duration loss: −6% (rates up 50 bps)
  • Spread duration loss: −10% (spreads widen 200 bps)
  • Negative convexity: As rates rise, call option value drops; bond duration extends, amplifying rate losses to −6.5% instead of −6%.
  • Total: −16.5% instead of −7% (your prediction was off by nearly 10 percentage points!)

The lesson: Negative convexity turns a bad outcome into a catastrophic one precisely when you're already taking losses.

Signals: Buy, Sell, and Warning

Buy Signal (Relative Value)

  1. Credit model estimates PD at 1.2% annually; bond pricing in 2.1% (based on spread).
    • Spread is 90 bps wide to fair value.
    • Catalyst: Earnings beat or rating upgrade narrower spreads.
    • Position: Long bond, hedge rate duration with Treasury short.
  2. Effective duration = 5.0; spread duration = 4.5; bond trading at 200 bps.
    • Comparable corporate with 180 bps OAS is tighter.
    • Why the 20 bps gap? Near-term refinancing risk is priced in, but long-term credit is solid.
    • Catalyst: Refinancing closed successfully, spread re-rates tighter.
  3. Convexity-rich position: Positive convexity bond trading at yield parity to negative convexity bond.
    • Positive convexity underpriced relative to volatility regime.
    • Expected vol to increase → positive convexity gains value.

Sell/Warning Signal

  1. Spread duration indicator flashing red: High-yield spreads narrowing to historical tights while delinquency rates rise.
    • Delinquency rates (credit card, auto) at multi-year highs, but spreads at multi-year tights.
    • Historical pattern: Spreads widen after delinquency spikes, not before.
    • Model prediction: Spreads likely to widen 200-300 bps in 12-18 months.
    • Action: Reduce high-yield exposure; rotate into investment-grade or Treasuries.
  2. Negative convexity concentration: Portfolio heavily weighted to callable corporates trading near par.
    • Callable bonds near par have maximum negative convexity.
    • If rates fall from here, upside is capped (calls exercised).
    • If rates rise sharply, losses are amplified (negative convexity bite).
    • No asymmetric payoff available; the risk-reward is balanced or worse.
    • Action: Swap into non-callable bonds or reduce position size.
  3. Illiquidity + negative convexity: Private credit funds showing liquidity stress, callable structures.
    • Private credit spreads may not reflect true default risk if they're illiquid (no mark-to-market).
    • Negative convexity + illiquidity = trapped position (can't exit at fair value).
    • Action: Avoid new allocations; evaluate exit options for existing holdings (secondary sales, fund redemptions).

MODULE 4: THE "INVERSION" (Risks & Limitations)

Apply Munger's Principle: How Smart Investors Get Burned

Failure State 1: The Regime Change

Assumption embedded in duration/convexity models: Historical correlations and volatility regimes persist.

What changes: Interest rate volatility regime, credit correlation structure, or central bank policy stance.

Example: The 2022 Shock

Pre-2022 regime (2010-2021):

  • Low rates, falling volatility.
  • Credit and rate spreads negatively correlated: When rates rose, credit spreads tightened (equities rallied, credit fundamentals improved).
  • Models were calibrated to this regime.
  • Callable bonds had low embedded option value (low vol → cheap options).

Post-2022 reality (2022-2024):

  • Aggressive Fed rate hikes (425 bps in 12 months).
  • Credit and rate spreads positively correlated: As rates rose, spreads widened (flight-to-quality reversed; credit fundamental concerns emerged).
  • Duration models failed because the historical relationship inverted.
  • Callable bonds with negative convexity suffered twice as much: +200 bps in rates + +200 bps in spreads = massive losses amplified by extending duration.

Red flag: If your model assumes a correlation of −0.5 between rates and spreads, but it flips to +0.3, your duration hedge becomes a risk concentrator.

Failure State 2: Black Swan / Systemic Crises

The Problem: Duration and convexity are local measures—they assume a smooth, continuous price function. They break down in discontinuous markets.

2008 Financial Crisis Example:

  • Lehman Brothers bonds were trading at 60 cents on the dollar on Friday.
  • On Monday (post-bankruptcy), they traded at 10 cents.
  • Duration models estimated a −75% move if spreads widened 500 bps → implied 500 bps / (6 years × 5% loss per 100 bps) ≈ −375% loss, which is nonsensical.
  • Why: Duration assumes you can continuously sell at market price. In a liquidity crisis, there's no bid—duration is infinite (the price falls to zero or near-zero before you can exit).

Liquidity evaporation amplifies convexity failures:

  • Callable bonds trade at par or slightly above; in a crisis, the call option becomes worthless (issuer goes bankrupt, refinancing impossible).
  • Your negative convexity "protection" (lower prices protect you from further losses) disappears; instead, the price falls to zero, wiping out your principal.

Failure State 3: Pro-Cyclical Selling & Margin Calls

Dynamic hedging problem:

Suppose a large hedge fund is long high-yield bonds with spread duration 4 and has hedged interest rate risk with short Treasury positions. This hedge is delta-neutral in a normal market.

In a stress event (e.g., European sovereign crisis, 2011):

  • High-yield spreads widen (bonds down −8%).
  • Investor marks-to-market, faces a loss.
  • To meet margin calls, the hedge fund must sell assets, not cover shorts.
  • They sell corporates (adding to selling pressure) and cover short Treasuries (buying Treasuries back).
  • This pro-cyclical selling widens spreads further and tightens Treasury yields more than model-predicted.
  • Duration model breaks because it assumed spreads move independently; in a deleveraging spiral, all credits sell together.

Failure State 4: Private Credit Mispricing (The "Blind Spot")

In private credit markets (2+ trillion AUM), bonds are illiquid and not marked-to-market daily. This creates a hidden convexity risk:

Pre-2023 Narrative: Private credit spreads are "stable" (200-300 bps above cost of capital). Models suggest low default risk, attractive risk-adjusted returns.

2023-2024 Reality:

  • Interest rates rose from 0% to 5%+ (central banks tightened).
  • Private credit fund valuations were slow to adjust (quarterly or annually).
  • Underlying borrowers faced refinancing risk: what was a 5% floating-rate loan is now a 10%+ refinancing option.
  • Funds began paying "interest-in-kind" (PIK) toggles and amend-and-extend maneuvers (hiding defaults in extended terms).
  • Spread duration calculations (based on quarterly NAVs) were wildly optimistic; true spread duration was much higher, and volatility was not captured.

The blind spot: Duration/convexity models require accurate market prices. When prices are stale (weeks or months old), the model thinks risk is lower than it actually is. Negative convexity (embedded in the call feature of leveraged loans and preferred structures) becomes a hidden risk that explodes when a mark-to-market is finally forced.

Blind Spots: What This Framework Ignores

1. Credit Events & Recovery Rates

Duration/convexity models assume the bond remains outstanding. They fail to price:

  • Default probability (credit models do this; duration doesn't).
  • Loss given default (LGD): If the issuer defaults, you recover 30 cents on the dollar, not 100 cents. Duration assumes you get paid in full; it doesn't account for tail risk.
  • Credit event optionality: Distressed-debt investors can extract value through exchange offers, restructurings, etc. Duration doesn't capture this.

Blind spot in practice: A bond with 5 years to maturity and high probability of default (say, 8% annually) might have a negative expected return even if the spread looks attractive. Duration says it's "worth" the yield; credit analysis says it's a value trap.

2. Correlation Breakdown

Duration assumes:

  • Spreads move gradually, proportional to fundamental changes.
  • Rate and spread moves are independent (or have a stable correlation).
  • Liquidity is always available (implied by the ability to continuously trade at model prices).

Blind spot: During crises, correlation spikes to 1.0. All spreads widen together. The diversification benefit (long some credits, short others) evaporates. Duration hedges (short Treasuries to offset credit exposure) become liability, not protection.

3. Embedded Option Bifurcation

Models treat callable bonds as a single security. In reality:

  • The issuer's call option is a separate instrument with its own Greeks (delta, gamma, vega, rho).
  • As the option moves from out-of-the-money to at-the-money to in-the-money, its sensitivity changes.
  • A bond that is deep out-of-the-money has low negative convexity; one at-the-money has high negative convexity.
  • Duration models often fail to capture this non-linearity in the option's own delta.

Blind spot: A callable bond that has just moved to be at-the-money (struck by a move in rates) may have dramatically higher negative convexity than it did yesterday, even though its duration hasn't changed much. The nonlinearity in the option's convexity is ignored.

4. Path Dependency

Convexity calculation assumes a one-step shock to rates. In reality, rates follow paths:

Scenario A: Rates fall 200 bps in one day.

  • Callable bond loses upside immediately (call is in-the-money).

Scenario B: Rates fall 50 bps, then recover to +100 bps above original, then fall again 200 bps total.

  • Callable bond's refinancing incentive changes over time (path-dependent).
  • The option's value depends on the route rates take, not just the endpoint.

Duration/convexity are static measures; they don't capture path dependency. An options-based pricing model using Monte Carlo simulation would, but it's far more complex.

Contrarian View: How Smart Investors Get Burned by Over-Reliance

The SVB Collapse: A Convexity Risk Case Study (2023)

The Setup:

  • Silicon Valley Bank held 91 billion in Treasury and mortgage-backed securities.
  • Purchase history: Heavily during 2010-2021 (low rates, low yields).
  • Financing: Funded with deposits (low rates, stable).
  • Model assumption: Bond valuations are stable; duration hedging via derivatives is unnecessary.

The Blind Spot:

  • Bank managers knew their duration (estimated ~6-7 years).
  • They didn't fully account for the convexity of deposit liabilities.
  • Deposit convexity: Deposits have negative convexity (hidden option).
    • When rates are low, deposits stay (rate-locked, no reason to move).
    • When rates are high, deposits flee (to higher-yielding money market funds).
    • In rising-rate environment, deposit duration collapses (opposite of a callable bond, but also negative convexity).

The Failure:

  • Rates rose 425 bps (2022-2023).
  • Bond portfolio marked down ~15 billion (duration: 6 × 4.25% = ~25% unrealized loss, worse due to negative convexity + MBS repayment risks).
  • Depositors fled (negative convexity on liability side): 40 billion in deposits withdrawn in a few days.
  • Bank needed to sell bonds at fire-sale prices to meet withdrawals.
  • Cascade: losses are realized, capital is wiped out, bank fails.

The Lesson: Smart risk managers calculated duration and it seemed reasonable. But they missed the dynamic interaction between the duration of assets and the negative convexity of liabilities. In stress, the two move together, not independently.

MODULE 5: REAL-WORLD CASE STUDY

Scenario: The 2022-2024 Credit Cycle Whiplash

The Setup: 2021-Mid 2022

Market Context:

  • Federal Funds Rate: 0-0.25% (kept low post-COVID).
  • 10-year Treasury yield: ~1.5-2.0%.
  • Credit spreads: Compressed to historic tights.
    • IG corporate spreads: ~80 bps (vs. 15-year average of 120 bps).
    • HY spreads: ~350 bps (vs. 15-year average of 550 bps).
  • Credit volumes: Record issuance; LBOs and leveraged buyouts booming.
  • Narrative: "Everything is fine, growth is stable, spreads are pricing in low default rates."

The Model-Informed Analysis (June 2022)

A sophisticated credit investor constructs a forward-looking credit model that incorporates:

  • Input 1: Current leverage ratios for major HY issuers (median net debt/EBITDA ~4.2x, at high end of normal).
  • Input 2: Refinancing risk: 400 billion HY debt maturing 2023-2024, most issued at 3-4% coupons; refinancing rates now 6-7%.
  • Input 3: Macro stress scenario: Fed funds peak at 4.25%, causing:
    • GDP growth to slow to 0.5% annually.
    • Unemployment to rise from 3.5% to 5.5%.
    • Credit card delinquencies to rise from 1.5% to 3.5% (doubled).
    • HY default rate to rise from 2% historical average to 5.5%.
  • Input 4: Historical relationship between default rates and credit spreads:
    • Every 1% increase in realized defaults corresponds to ~150 bps of spread widening.
    • Current spreads assume ~2% default rate; 5.5% default rate implies spreads of ~350 + (5.5 − 2.0) × 150 = 875 bps.

Model Output:

  • Current HY spreads: 350 bps.
  • Fair-value spreads (given macro stress scenario): 875 bps.
  • Implied P&L if scenario unfolds: −525 bps of spread widening on 3-year average duration HY bond = −15.75% loss.

Consensus View (Traditional, Discretionary):

  • "Spreads can't widen that much; equities have already priced in rate increases."
  • "Refinancing risk is manageable; this cohort has strong coverage ratios."
  • "Credit spreads are reasonable; the market is fairly priced."
  • Recommendation: Hold/overweight credit; "recession is priced in."

The Divergence

Model-Informed Investor Actions (Mid-2022):

  1. Reduce HY allocation: From 20% to 10% of bond portfolio.
  2. Shorten duration of remaining credit: Swap long-dated HY for shorter-dated IG; extend Treasuries for carry.
  3. Implement downside hedge: Buy index CDS (credit default swap indices) to protect portfolio; cost: ~100 bps annually.
  4. Position in "safe" credits: Rotate into mega-cap IG corporates (Apple, Microsoft, JPMorgan), which have short spread duration and high recovery rates if defaults do occur.
  5. Lock in Treasury yields: Allocate to 10-year Treasuries at 3.5% yield (attractive given forward rate expectations).

Consensus Actions:

  1. Maintain HY allocation; view volatility as buying opportunity.
  2. Extend duration to capture yield.
  3. Avoid "expensive" CDS hedges; deploy cash in higher-yielding HY names.
  4. Lean into private credit; comfortable with illiquidity for illiquidity premium.

2024-2025 Reality

Macro unfolded roughly as model predicted:

  • Fed hiked rates to 4.25-4.50%, held steady through 2023-2024.
  • 10-year Treasury yield peaked near 5.0% (mid-2023), then normalized to 4.0-4.2% (2024).
  • Refinancing wave hit in 2023-2024.
  • Credit card delinquency rates rose to 3.2% (close to model scenario).
  • HY default rate rose to 4.8% (close to model's 5.5% scenario).
  • HY spreads widened from 350 bps to peak of 850-900 bps (mid-2023), then partially recovered to 700 bps by end-2024.

Model-Informed Investor Outcome:

  • HY portfolio losses were limited (reduced allocation + hedge).
  • CDS hedge paid off: ~300 bps of spread widening × 100 bps notional on hedge = +3% offset to portfolio losses.
  • Treasury extension provided +5-7% total return (rates fell from peak).
  • Net portfolio return: −1 to +2% (weathered the storm).
  • Dry powder: Maintained reduced allocation, deployed it when spreads widened to 800+ bps, buying distressed credits at attractive prices.
  • By 2025: Repositioned into credit as spreads stabilized; benefited from credit recovery rally.

Consensus Investor Outcome:

  • Heavy HY allocation + negative convexity risk (many holdings were callable): −20 to −25% losses on credit holdings through peak stress.
  • CDS hedges spurned (expensive); no protection.
  • Treasury extension helped (+5% return), offsetting credit losses partially.
  • Net portfolio return: −15 to −20%.
  • Trapped capital: Low ammunition for opportunities at trough spreads; missed the bounce-back rally.

Key Differences: Model vs. Consensus

Dimension Model-Informed Consensus (Discretionary)
Default rate forecast 5.5% (stress scenario) 2% (long-term average)
Spread sensitivity 150 bps per 1% default rate No formal model
Refinancing risk Quantified; incorporated into model Qualitatively assessed as manageable
Downside hedging CDS, rotate to IG Avoid hedging; deploy in dislocations
Duration positioning Shortened credit; extended Treasuries Extended both (higher yield)
Private credit Avoided (illiquidity + negative convexity) Overweighted (illiquidity premium attractive)
Outcome 2022-2024 −1 to +2% −15 to −20%

MODULE 6: IMPLEMENTATION CHECKLIST

Critical Questions for Traders & Portfolio Managers Before Executing Any Trade Where Duration/Convexity is Primary

1. Decompose Your Duration Exposure

Questions to Ask:

  • Is my duration exposure primarily interest-rate driven or spread-driven?
    • Calculate effective duration (rates shock only) vs. spread duration (spread shock only).
    • Sum should equal modified duration.
  • What's my rate duration hedge ratio? If I'm long credit (spread duration 4), do I have short Treasuries equal to 4 DV01 (dollar-value-of-1-bps)?
  • Have I accounted for the correlation between rates and spreads in my hedging? Historically, rates down → spreads tighten (good for credit). But in crises, rates down → spreads widen (bad for credit). Which regime am I hedging for?

Why it matters: You might think you're long credit, but if your hedge is sloppy, you're actually long duration in a crisis (rates fall, spreads widen simultaneously), amplifying losses.

2. Stress Your Convexity Assumptions

Questions to Ask:

  • Does my bond have embedded options (callable, putable, convertible, MBS-like prepayment)?
    • If yes, calculate effective duration (not modified duration), which accounts for option risk.
    • Is the bond near-par (maximum negative convexity)?
    • What's the probability that the option moves in-the-money in my stress scenario?

Example: You own a 5-year callable corporate, currently trading at 102 (above par). Coupon 5%, call price 100, first call date 1 year out.

  • If rates fall 200 bps, is the call exercised? (Yes, issuer refinances.)
  • Your effective duration is much shorter than modified duration implies; upside is capped at ~102.
  • If rates rise 200 bps, the call expires; bond acts like a 6-year bond now (extended duration).
  • Test: Calculate price at −200 bps (capped around 102-103) vs. +200 bps (should be lower than duration alone predicts due to extending duration).

3. Check for Duration × Spread Regime Correlation

Questions to Ask:

  • What's the implied correlation between 10Y rate moves and HY spread moves in my model?
    • Historical (2010-2019): ~−0.5 (rates up, spreads tighten; rates down, spreads widen).
    • Recent (2022-2024): ~+0.3 (rates up, spreads widen; rates down, spreads tighten).
    • Implication: Your hedge assumptions might be inverted.
  • What happens in your portfolio if rates and spreads both widen 200 bps simultaneously? (This is the "double duration pain" scenario.)
    • Rate duration loss: −6% (for 6-year duration).
    • Spread duration loss: −8% (for 4-year spread duration on 200 bps move).
    • Total: −14% (not −6% or −8%, but the sum, approximately).
    • Negative convexity amplification: Add another −1% to −2%.

4. Validate Data Freshness & Source Integrity

Questions to Ask:

  • Are my input data (leverage ratios, interest coverage, financial metrics) current?
    • Many banks and data providers publish quarterly data; some update only annually.
    • If I'm trading on an old balance sheet, my credit model is backwards-looking, not forward-looking.
    • Red flag: You're in June 2024, your leverage data is from Q4 2023 (8 months old). If the business is deteriorating, your model doesn't know it yet.
  • Are the spreads I'm quoting realistic?
    • Bid-ask spreads in bonds can be 10-20 bps (especially in tighter credits).
    • Are my fair-value calculations based on executable prices or stale EOD quotes?
    • Red flag: You calculate a fair-value spread of 150 bps, but the current bid is 145 bps, ask is 165 bps. Your trade is in the wide side of the market; execution risk is high.
  • For private credit or illiquid positions, what's the mark-to-model date?
    • If the valuation is 30 days old, duration/spread calculations are unreliable.
    • Your true P&L exposure might be 2-3x higher than your model implies (negative convexity is hidden).

5. Liquidity Test: What's Your Exit Price & Speed?

Questions to Ask:

  • If I'm wrong and this position moves 100 bps against me, can I exit at model prices?
    • Liquid bonds (mega-cap IG corporates, on-the-run Treasuries): Exit within minutes, minimal slippage.
    • Less liquid bonds (smaller IG, regional HY): Exit within hours/days, slippage 5-10 bps.
    • Illiquid (private credit, bonds trading infrequently): May not be able to exit at any price in a stress scenario.
  • For hedges (CDS, futures), is liquidity adequate?
    • 5-year HY CDS index: Highly liquid, tight bid-ask, can trade 100 million+.
    • Single-name CDS on smaller issuer: Wide bid-ask, harder to trade size.
    • Treasury futures: Ultra-liquid, but basis risk to your specific bond position.
  • Hard stop-loss question: At what loss threshold do I exit, regardless of thesis?
    • If you can't define a quantitative exit (e.g., "if spread widens to 400 bps or P&L hits −5%, I'm out"), you're taking unlimited downside risk.

6. Scenario Analysis: The Stress Test

Questions to Ask:

  • Run a minimum of three scenarios through your model:
    • Scenario A (Bull Case): Rates fall 100 bps, spreads tighten 50 bps.
      • Expected return: ?% (your target).
    • Scenario B (Base Case): Rates stable, spreads move 0 bps.
      • Expected return: Carry (coupon − cost of funding).
    • Scenario C (Bear Case): Rates rise 150 bps, spreads widen 250 bps.
      • Expected return: ?% (your downside).
      • Does this scenario make you uncomfortable? If yes, position sizing is too large or hedge is insufficient.
    • Scenario D (Tail Risk): Rates rise 200 bps, spreads widen 400 bps, negative convexity amplifies by 3%.
      • Expected return: ?% (catastrophic loss scenario).
      • Is this tail risk acceptable given your risk budget? If no, reduce position or change strategy.

CONCLUSION: From Theory to Alpha

Duration and convexity are not just risk metrics; they are alpha generators for investors who understand their mechanics, limitations, and interactions with credit spreads.

The Strategic Takeaway:

  1. Duration is your baseline: All bond investing starts with understanding how interest rates affect valuations. But duration is incomplete for credit portfolios.
  2. Spread duration is your credit edge: The bond's sensitivity to credit spread changes is where you monetize credit views and differentiate from consensus.
  3. Convexity is your tail risk: Positive convexity provides asymmetry (upside > downside); negative convexity does the opposite. In a crisis, negative convexity becomes your enemy. Understand where it's hidden (callable corporates, MBS, leveraged loans).
  4. Regime awareness is critical: The relationship between rates, spreads, and default rates is not stationary. Each economic regime has different correlation structures. Model flexibility matters.
  5. Data freshness and liquidity are underrated: You can have the best model on Earth, but if your inputs are stale or your position is illiquid, you're blind in a crisis—the exact moment when alpha is made or lost.

For Junior Analysts & Traders: Master the decomposition of return (rate effect + spread effect + convexity adjustment + credit event). Learn to hedge precisely: isolate the bet you want to make, eliminate the bets you don't want. This discipline is what separates sustainable alpha from luck.

For Portfolio Managers: Use credit models not as gospel, but as frameworks for structured thinking. Combine quantitative signals with macro judgment. When the model screams "sell" because spreads are tight and default indicators are deteriorating, listen. That's not theory; that's pattern recognition on steroids.

Key Metrics for Implementation (Reference Table)

Metric Formula Interpretation When to Use
Modified Duration (Macaulay Duration) / (1 + y) % price change per 1% yield move Quick approximation for straight bonds
Effective Duration (PV− − PV+) / (2 × PV0 × Δr) % price change accounting for embedded options Bonds with calls, puts, prepayment options
Spread Duration % price change per 1% spread move Credit-specific sensitivity Credit portfolio management
DTS (Duration Times Spread) Spread Duration × Spread Relative volatility across credits Comparing bonds with different spreads/maturities
Convexity (PV− + PV+ − 2 × PV0) / (Δr)² Curvature of price-yield relationship Large rate moves, assessing asymmetry
Effective Convexity Same formula but using benchmark curve shocks Curvature for embedded option bonds Callable/putable bonds, MBS
OAS (Option-Adjusted Spread) Z-spread − Option cost Fair-value spread accounting for optionality Relative value, fair-value identification