Yield-Based Bond Convexity and Portfolio Properties - Fixed Income | CFA® Level 1 Guide

Bond Convexity and Convexity Adjustment

Learning Outcome Statement:

calculate and interpret convexity and describe the convexity adjustment

Summary:

Bond convexity is a measure of the curvature in the relationship between bond prices and bond yields, reflecting the sensitivity of the bond price to changes in yield. Convexity adjustment refers to the addition of the convexity measure to the duration effect to get a more accurate estimate of bond price changes due to yield changes. This adjustment is crucial for assessing the impact of yield changes on bond prices, especially for large changes in yield.

Key Concepts:

Bond Convexity

Bond convexity is a second-order measure of the bond's price sensitivity to interest rate changes and provides a way to account for changes in bond duration as yields change. It is always positive for option-free bonds and increases the accuracy of price change estimates from duration alone.

Convexity Adjustment

The convexity adjustment is added to the duration effect to provide a more accurate estimate of the new bond price after a change in yield. This adjustment accounts for the curvature of the price-yield relationship and is particularly important for large yield changes.

Modified Duration

Modified duration measures the first-order sensitivity of a bond's price to yield changes, estimating the percentage price change per unit change in yield. It is a linear approximation.

Annualized Convexity

Annualized convexity is calculated by summing the product of squared time to cash flow receipt, the cash flow's present value, and a discount factor, then dividing by the square of the number of periods per year.

Formulas:

Percentage Change in Full Price

\%\Delta P_{V_{Full}} \approx (-\text{AnnModDur} \times \Delta \text{Yield}) + \left[\frac{1}{2} \times \text{AnnConvexity} \times (\Delta \text{Yield})^2\right]

This formula calculates the estimated percentage change in the full price of a bond due to changes in yield, incorporating both the linear effect through modified duration and the non-linear effect through convexity.

Variables:

$AnnModDur$ :: Annualized Modified Duration
$AnnConvexity$ :: Annualized Convexity
$DeltaYield$ :: Change in Yield to Maturity

Units: percentage

Approximate Convexity

\text{ApproxCon} = \frac{(P_{V_{+}} + P_{V_{-}} - [2 \times P_{V_{0}}])}{(\Delta \text{Yield})^2 \times P_{V_{0}}}

This formula provides an approximation of the bond's convexity using prices calculated at slightly higher and lower yields, useful for bonds with uncertain cash flows or additional features.

Variables:

$PV_+$ :: Bond price when yield is increased
$PV_-$ :: Bond price when yield is decreased
$PV_0$ :: Original bond price
$DeltaYield$ :: Change in Yield

Units: dimensionless

Bond Risk and Return Using Duration and Convexity

Learning Outcome Statement:

calculate the percentage price change of a bond for a specified change in yield, given the bond’s duration and convexity

Summary:

This LOS focuses on estimating the percentage price change of a bond when there is a specified change in yield, considering both the bond's duration and convexity. It emphasizes the importance of incorporating convexity along with duration for a more accurate prediction of price changes, especially under significant yield fluctuations. The content covers the calculation of modified duration and convexity using price values at different yields and demonstrates the application through various bond examples.

Key Concepts:

Bond Convexity and Convexity Adjustment

Bond convexity measures the curvature in the relationship between bond prices and yields, providing a necessary adjustment to the duration measure, especially when there is a large change in yield. Convexity helps in predicting the bond price more accurately by considering the non-linear price-yield relationship.

Approximate Modified Duration (ApproxModDur)

Approximate Modified Duration is calculated to estimate the sensitivity of a bond's price to changes in yield. It is derived from the price changes at slightly higher and lower yields, providing a linear estimate of price volatility.

Approximate Convexity (ApproxCon)

Approximate Convexity is calculated to measure the rate of change of the bond's duration with respect to yield. It provides a second-order measure to account for the curvature in the price-yield relationship of the bond.

Money Duration (MoneyDur) and Money Convexity (MoneyCon)

Money Duration and Money Convexity are used to estimate the actual dollar change in a bond's price due to yield changes. Money Duration represents the first-order effect, while Money Convexity captures the second-order effect, enhancing the accuracy of the price change estimation.

Formulas:

Approximate Modified Duration

\text{ApproxModDur} = \frac{PV^- - PV^+}{2 \times \Delta y \times PV_0}

This formula calculates the approximate modified duration, indicating how much the bond price will change per unit change in yield.

Variables:

$PV^-$ :: Bond price if yield decreases
$PV^+$ :: Bond price if yield increases
$PV_0$ :: Initial bond price
$\Delta y$ :: Change in yield (decimal form)

Units: years

Approximate Convexity

\text{ApproxCon} = \frac{PV^- + PV^+ - 2PV_0}{(\Delta y)^2 \times PV_0}

This formula calculates the approximate convexity, measuring the curvature of the price-yield relationship and providing a second-order adjustment to the price change estimate.

Variables:

$PV^-$ :: Bond price if yield decreases
$PV^+$ :: Bond price if yield increases
$PV_0$ :: Initial bond price
$\Delta y$ :: Change in yield (decimal form)

Units: years^2

Full Price Change Estimate

\Delta PV_{\text{Full}} \approx -(\text{MoneyDur} \times \Delta \text{Yield}) + \left(\frac{1}{2} \times \text{MoneyCon} \times (\Delta \text{Yield})^2\right)

This formula combines Money Duration and Money Convexity to estimate the full price change of a bond due to a specified change in yield, providing a more accurate and less risky prediction.

Variables:

$MoneyDur$ :: Money Duration
$MoneyCon$ :: Money Convexity
$\Delta \text{Yield}$ :: Change in yield (decimal form)

Units: currency units

Portfolio Duration and Convexity

Learning Outcome Statement:

calculate portfolio duration and convexity and explain the limitations of these measures

Summary:

Portfolio duration and convexity are measures used to assess the interest rate risk of a bond portfolio. These metrics can be calculated using the weighted averages of the durations and convexities of the individual bonds in the portfolio. While this method is practical and commonly used, it assumes a parallel shift in the yield curve, which is a limitation since yield curves often experience non-parallel shifts such as steepening, flattening, or twisting.

Key Concepts:

Portfolio Duration

Portfolio duration is calculated as the weighted average of the durations of the individual bonds in the portfolio, using the market values of the bonds as weights. It measures the sensitivity of the portfolio's value to changes in interest rates.

Portfolio Convexity

Portfolio convexity is calculated similarly to duration, using the weighted average of the convexities of the individual bonds. It measures the rate of change of duration and provides a more accurate estimate of price changes for large shifts in interest rates.

Limitations of Duration and Convexity

The main limitation of using duration and convexity for portfolio management is the assumption of parallel shifts in the yield curve. In reality, yield curves can change in shape, affecting the accuracy of these measures.

Formulas:

Weighted-average Modified Duration

Weighted\text{-}average\ Modified\ Duration = (Duration_1 \times Weight_1) + (Duration_2 \times Weight_2) + \ldots + (Duration_n \times Weight_n)

This formula calculates the overall duration of a bond portfolio by taking the sum of the products of each bond's duration and its respective weight in the portfolio.

Variables:

$Duration_i$ :: Duration of the ith bond
$Weight_i$ :: Portfolio weight of the ith bond based on market value

Units: years

Weighted-average Convexity

Weighted\text{-}average\ Convexity = (Convexity_1 \times Weight_1) + (Convexity_2 \times Weight_2) + \ldots + (Convexity_n \times Weight_n)

This formula calculates the overall convexity of a bond portfolio by taking the sum of the products of each bond's convexity and its respective weight in the portfolio.

Variables:

$Convexity_i$ :: Convexity of the ith bond
$Weight_i$ :: Portfolio weight of the ith bond based on market value

Units: dimensionless

Percentage Change in Portfolio Value

\%\Delta PV_{Full} \approx (-Weighted\text{-}average\ Modified\ Duration \times \Delta Yield) + \left[\frac{1}{2} \times Weighted\text{-}average\ Convexity \times (\Delta Yield)^2\right]

This formula estimates the percentage change in the market value of a bond portfolio due to changes in yield, incorporating both duration and convexity effects.

Variables:

$Weighted\text{-}average\ Modified\ Duration$ :: Calculated weighted-average modified duration of the portfolio
$Weighted\text{-}average\ Convexity$ :: Calculated weighted-average convexity of the portfolio
$\Delta Yield$ :: Change in yield

Units: percentage