Chapter 5 5.4

Convexity

Understanding convexity as the second-order measure of interest rate risk

Beyond Duration: The “Curve” in the Price-Yield Curve

In the last sections, we learned that Modified Duration is a linear "straight-line" estimate of price risk. However, the true relationship between a bond's price and its yield is curved (convex).

Convexity is the measure of this curvature. It is the "second derivative" of the price-yield relationship, measuring the rate of change of duration itself. For a bondholder, positive convexity is a very good thing.

Because of convexity:

  • When yields fall, the bond's price rises by *more* than duration predicts.
  • When yields rise, the bond's price falls by *less* than duration predicts.

Convexity is the "error" in the duration model, and it's an error that always works in the investor's favor (for a standard bond).

The Duration-Convexity Approximation

To get a much more accurate estimate of a price change, we combine Modified Duration (the first-order risk) with Convexity (the second-order risk).

\[\%\Delta P \approx \left[ -D_{mod} \times (\Delta y) \right] + \left[ \frac{1}{2} \times C \times (\Delta y)^2 \right]\]

Where:

  • Part 1 (Duration): $-D_{mod} \times (\Delta y)$
  • Part 2 (Convexity): $\frac{1}{2} \times C \times (\Delta y)^2$
  • $C$ = The bond's Convexity value
  • $\Delta y$ = The change in yield (e.g., +0.01 for a 100 bps rise)

Notice that the convexity term $(\Delta y)^2$ is always positive, whether the yield change is positive or negative. This is what gives the bondholder the extra gain/reduced loss.

Convexity Calculation

The formula for convexity is complex, building on the Macaulay Duration formula. For a semi-annual paying JGB, the formula for annual convexity ($C$) is:

\[C = \frac{1}{P \times (1+y/2)^2} \sum_{t=1}^{n} \frac{C_t/2}{(1+y/2)^{t}} \left( \frac{t^2 + t}{4} \right) + \frac{F}{(1+y/2)^n} \left( \frac{n^2+n}{4} \right)\]

This formula is rarely calculated by hand. It is a standard function in financial software and spreadsheets. What matters is *how to use it*.

Worked Example: Duration vs. Duration+Convexity

Let's use our 3-year JGB one last time. We'll simulate a large, 100 basis point (1.0%) yield shock.

Inputs:

  • Price: ¥1,005,903.53
  • Modified Duration ($D_{mod}$): 2.9513
  • Convexity ($C$): 9.85 (This would be calculated by the formula above)
  • Yield Change ($\Delta y$): +1.0% (or +0.01)

Scenario: What is the new price if yields rise from 0.8% to 1.8%?

Estimate 1: Duration-Only (Inaccurate)

$$\%\Delta P \approx -D_{mod} \times (\Delta y)$$ $$\%\Delta P \approx -2.9513 \times (0.01) = \textbf{-2.9513%}$$

$$\text{New Price} \approx ¥1,005,903.53 \times (1 - 0.029513) = \textbf{¥975,910}$$

Estimate 2: Duration + Convexity (More Accurate)

$$\%\Delta P \approx \left[ -2.9513 \times (0.01) \right] + \left[ \frac{1}{2} \times 9.85 \times (0.01)^2 \right]$$

$$\%\Delta P \approx \left[ -0.029513 \right] + \left[ 0.5 \times 9.85 \times 0.0001 \right]$$

$$\%\Delta P \approx -0.029513 + 0.0004925 = \textbf{-2.9021%}$$

$$\text{New Price} \approx ¥1,005,903.53 \times (1 - 0.029021) = \textbf{¥976,403}$$

Actual Price (Calculated by re-pricing at 1.8% yield)

The true price of the bond at a 1.8% YTM is ¥976,410.

Comparison:

  • Duration-Only Error: ¥976,410 - ¥975,910 = ¥500 (Overestimated the loss)
  • D+C Error: ¥976,410 - ¥976,403 = ¥7 (Almost perfect)

This example proves the value of convexity. For large yield moves, convexity is essential for accurate risk management. It is also why, given two bonds with the same duration, investors will pay a premium for the one with higher convexity.

Advanced Topic: Negative Convexity (Callable JGBs and MBS)

While standard fixed-coupon JGBs exhibit positive convexity (the desirable property we've discussed), certain securities can display negative convexity under specific conditions. Understanding this phenomenon is critical for risk management of callable bonds and mortgage-backed securities.

What is Negative Convexity?

Negative convexity means the price-yield relationship is concave rather than convex. Under negative convexity:

  • When yields fall, the bond's price rises by *less* than duration predicts
  • When yields rise, the bond's price falls by *more* than duration predicts

This is the opposite of the favorable behavior of positive convexity—it works against the investor in both directions.

Callable JGBs: Embedded Options Create Negative Convexity

While most modern JGBs are non-callable, some older issues and certain structured products contain embedded call options that give the issuer the right to redeem the bond before maturity at predetermined prices.

How Call Options Create Negative Convexity

Consider a callable JGB with the following characteristics:

  • Issue: 10Y JGB #XXX, 2.0% coupon
  • Call provision: Callable at par (¥100) after 5 years
  • Current yield: 1.5%

Scenario 1: Yields Fall to 0.5%

For a standard non-callable bond, the price would rise significantly—perhaps to ¥114 based on the new 0.5% yield. But for the callable bond:

  • The issuer will likely call the bond at ¥100 and refinance at the new lower 0.5% rate
  • Investors know this, so they won't pay much above ¥100 for the bond
  • The bond's price is capped near the call price
  • Result: Price appreciation is muted when yields fall

Scenario 2: Yields Rise to 2.5%

The issuer will not call the bond (why redeem at ¥100 when refinancing costs 2.5%?). The bond behaves like a normal bond and falls in price—perhaps to ¥96. The investor suffers the full loss.

The Convexity Profile

This creates an asymmetric payoff:

Yield Movement Non-Callable JGB Callable JGB Investor Impact
Yields fall 100bp Price rises +10% Price rises +2% (capped at call price) ❌ Limited upside
Yields rise 100bp Price falls -9% Price falls -9% (full downside) ❌ Full downside

This is negative convexity: you get the worst of both worlds.

Duration Compression

As yields approach the call strike, the bond's effective duration shrinks rapidly. This phenomenon is called duration compression:

  • At yields far above the call strike: Duration ≈ 8 years (behaves like long bond)
  • At yields near the call strike: Duration ≈ 3 years (behaves like bond maturing at call date)
  • At yields well below the call strike: Duration ≈ 0.5 years (certain to be called soon)

This makes duration-based hedging extremely difficult for callable bonds, as the hedge ratio must be constantly adjusted.

Mortgage-Backed Securities (MBS) and Prepayment Risk

While less common in Japan than in the US, mortgage-backed securities (including RMBS issued by banks or securitization vehicles) exhibit similar negative convexity due to prepayment risk:

  • When interest rates fall, homeowners refinance their mortgages
  • The MBS receives principal back early (like a bond being called)
  • Investors must reinvest at the new, lower rates
  • Result: Upside limited, just like a callable bond

Japanese regional banks and life insurers holding US MBS or domestic RMBS must model this prepayment behavior carefully, especially during periods of BOJ policy changes that affect mortgage rates.

Valuing Negative Convexity: Option-Adjusted Spread (OAS)

Because callable bonds have embedded options, their value cannot be assessed using simple yield-to-maturity. Instead, practitioners use Option-Adjusted Spread (OAS) analysis:

\[\text{Callable Bond Yield} = \text{Risk-Free Rate} + \text{Credit Spread} + \text{Option Cost}\]

The "option cost" represents the yield premium investors demand for bearing the negative convexity risk. For a callable JGB trading at 1.8% yield when a comparable non-callable trades at 1.5%:

  • Yield spread = 30bp
  • OAS analysis might show: 10bp credit spread + 20bp option cost
  • Investors earn an extra 20bp per year as compensation for giving up convexity

Risk Management Implications

For portfolios containing callable bonds or MBS:

1. Effective Duration vs. Modified Duration

Use effective duration, which accounts for changing cash flows when rates move, rather than modified duration:

\[D_{eff} = \frac{P_{-} - P_{+}}{2 \times P_0 \times \Delta y}\]

Where $P_{-}$ and $P_{+}$ are prices when yields fall/rise by $\Delta y$, calculated with a model that accounts for the call option.

2. Effective Convexity

Similarly, calculate effective convexity using option-adjusted prices:

\[C_{eff} = \frac{P_{+} + P_{-} - 2P_0}{P_0 \times (\Delta y)^2}\]

For callable bonds, $C_{eff}$ will be negative when the bond is near the call price.

3. Dynamic Hedging

Negative convexity requires more frequent rebalancing of hedges because duration changes rapidly with yield movements. A static hedge will fail.

4. Stress Testing

Standard duration/convexity approximations break down for large yield moves with callable bonds. Banks must use full revaluation under scenarios rather than Taylor series approximations.

Regulatory Considerations: IRRBB and Callable Bonds

Under Basel III's Interest Rate Risk in the Banking Book (IRRBB) framework (see Section 3.6), banks holding callable JGBs or MBS must:

  • Model prepayment/call behavior under each of the six standard shock scenarios
  • Capture the asymmetric EVE (Economic Value of Equity) impact from negative convexity
  • Explain to the FSA (Financial Services Agency) how they manage this risk

Banks with large holdings of pre-2000 callable JGBs or US MBS face particular scrutiny, as these positions amplify EVE volatility.

Historical Example: 1999 JGB Callable Issues

In the late 1990s, the Ministry of Finance issued several tranches of callable JGBs. When yields fell sharply during the 1998-2000 deflation crisis:

  • Many of these bonds were called by the MOF
  • Investors who purchased at premiums (expecting long-dated cash flows) suffered losses
  • Duration hedges based on modified duration failed spectacularly
  • Lesson learned: The market now demands much higher OAS for callable structures

When to Avoid (or Seek) Negative Convexity

Avoid callable bonds if:

  • You expect volatile interest rates (negative convexity hurts in both directions)
  • You cannot dynamically hedge (lack of systems or expertise)
  • The option cost (extra yield) is insufficient compensation

Consider callable bonds if:

  • You have a stable view that rates will stay in a narrow range
  • The extra yield meets your income requirements
  • You have sophisticated option-valuation models and can hedge effectively

Key Takeaways

  1. Negative convexity arises from embedded call options in bonds or prepayment options in MBS
  2. It creates asymmetric risk: limited upside when yields fall, full downside when yields rise
  3. Duration compression near call strikes makes hedging extremely challenging
  4. Use effective duration/convexity rather than modified duration for callable bonds
  5. OAS analysis quantifies the yield premium required to compensate for negative convexity
  6. Under IRRBB regulations, banks must model option exercise behavior in stress scenarios

Understanding negative convexity is essential for managing portfolios that include callable JGBs, structured notes, or mortgage-backed securities—particularly during periods of significant BOJ policy shifts that can trigger mass call exercises or refinancing waves.