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Yield-Based Bond Convexity and Portfolio Properties

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Bond Convexity and Price Estimation5 min

Convexity measures the curvature of the bond price-yield relationship. It improves upon the linear estimate provided by duration. For a standard option-free bond, convexity is always positive, meaning the price-yield curve opens upward. This characteristic ensures that for a given magnitude of yield change, the capital gain from a yield decrease exceeds the capital loss from a yield increase. The formula for estimating the percentage price change combines modified duration and convexity: %ΔPrice ≈ -Duration(ΔYield) + 0.5 × Convexity × (ΔYield)². The convexity adjustment is always positive, reflecting the benefit of curvature to the investor.

Key Points

Convexity accounts for the curvature in the price-yield relationship.
Positive convexity means prices rise more for yield drops than they fall for yield hikes.
The price change formula includes a convexity adjustment: 0.5 * Convexity * (Δy)².
Convexity bias explains why duration alone underestimates price increases and overestimates price decreases.

Money Convexity and Volatility Structure5 min

Money Convexity translates the convexity statistic into currency units, providing an estimate of the absolute price change in dollars (or other currency) rather than a percentage. It is calculated as Annual Convexity multiplied by the Full Price of the bond. The formula for the estimated change in full price is: ΔPVFull ≈ -(MoneyDur × ΔYield) + [0.5 × MoneyCon × (ΔYield)²]. The section also covers the term structure of yield volatility, observing that short-term yields tend to exhibit higher volatility compared to long-term yields, which has implications for risk management across different maturities.

Key Points

Money Convexity = Annual Convexity × Full Price.
Money Convexity helps estimate the absolute change in bond value.
Short-term yields are generally more volatile than long-term yields.
Yield volatility structure influences the risk assessment of bonds with different maturities.

Portfolio Duration and Horizon Analysis7 min

Portfolio duration can be calculated either by aggregating cash flows (calculating the IRR of the portfolio) or by taking the weighted average of individual bond durations. The weighted average method is standard in practice because it accommodates bonds with embedded options, though it relies on the assumption of a parallel shift in the yield curve. The chapter concludes with Horizon Analysis, comparing the Investment Horizon (HP) to the Macaulay Duration (MD). The 'Duration Gap' (MD - HP) indicates the dominant risk. If HP < MD (Positive Gap), price risk dominates; if HP > MD (Negative Gap), reinvestment risk dominates. Immunization occurs when HP = MD, offsetting price and reinvestment risks.

Key Points

Weighted average portfolio duration assumes parallel yield curve shifts.
Duration Gap = Macaulay Duration - Investment Horizon.
HP < MD: Price risk dominates; rising rates lower realized yield.
HP > MD: Reinvestment risk dominates; rising rates increase realized yield.
HP = MD: Immunized; realized yield approximates initial YTM.

Questions

Question 1

What does bond convexity primarily measure in the context of the price-yield relationship?

Yield-Based Bond Convexity and Portfolio Properties

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