Duration and convexity are key concepts in fixed-income analysis. They measure a bond's price sensitivity to interest rate changes, helping investors assess risk and make informed decisions. Duration quantifies the linear relationship between price and yield, while convexity captures the curvature.

These tools are crucial for portfolio management and risk assessment. They enable strategies like immunization and hedging, and help optimize portfolios. Understanding duration and convexity allows investors to better navigate the complex world of fixed-income securities and manage interest rate risk effectively.

Definition of duration

• Measures the sensitivity of a fixed-income security's price to changes in interest rates
• Represents the weighted average time until all cash flows from a bond are received
• Serves as a fundamental tool in financial mathematics for assessing bond price volatility

Types of duration

• Macaulay duration calculates the weighted average time to receive cash flows
• Modified duration estimates the price change for a given change in yield
• Effective duration accounts for embedded options in bonds
• Key rate duration measures sensitivity to shifts in specific points on the yield curve

Macaulay duration

• Developed by Frederick Macaulay in 1938
• Expresses the weighted average time to receive all cash flows from a bond
• Calculated using the present value of each cash flow and the bond's yield to maturity
• Formula: $Macaulay Duration = \frac{\sum_{t=1}^{n} t \times PV(CF_t)}{\text{Bond Price}}$
  • Where t represents the time period, PV(CF_t) is the present value of cash flow at time t

Modified duration

• Adjusts Macaulay duration to estimate price sensitivity to yield changes
• Represents the approximate percentage change in bond price for a 1% change in yield
• Calculated by dividing Macaulay duration by (1 + yield/n)
  • Where n is the number of coupon payments per year
• Formula: $Modified Duration = \frac{Macaulay Duration}{1 + \frac{YTM}{n}}$

Calculating duration

• Involves determining the present value of all future cash flows
• Requires knowledge of bond characteristics (coupon rate, face value, maturity)
• Utilizes time value of money concepts and discounting techniques
• Applies to various fixed-income securities and portfolios

Formula for duration

• General formula for Macaulay duration: $Duration = \frac{\sum_{t=1}^{n} t \times C_t \times (1+r)^{-t}}{\sum_{t=1}^{n} C_t \times (1+r)^{-t}}$
  • Where C_t represents cash flow at time t, r is the yield to maturity
• Modified duration formula: $Modified Duration = -\frac{1}{P} \times \frac{dP}{dr}$
  • Where P is the bond price, dP/dr is the derivative of price with respect to yield

Duration of bonds

• Zero-coupon bonds have duration equal to their time to maturity
• Coupon bonds have duration less than their maturity due to interim cash flows
• Factors affecting bond duration:
  • Maturity: longer maturity generally increases duration
  • Coupon rate: higher coupons decrease duration
  • Yield to maturity: higher yields typically decrease duration

Duration of portfolios

• Calculated as the weighted average of individual security durations
• Formula: $Portfolio Duration = \sum_{i=1}^{n} w_i \times D_i$
  • Where w_i is the weight of security i, D_i is the duration of security i
• Allows for assessment of interest rate risk at the portfolio level
• Used in portfolio immunization strategies to match assets and liabilities

Interpreting duration

• Provides a measure of interest rate risk for fixed-income securities
• Helps investors and portfolio managers assess potential price volatility
• Facilitates comparison of interest rate sensitivity across different bonds

Interest rate sensitivity

• Duration quantifies the approximate percentage change in bond price for a 1% change in yield
• Negative relationship between interest rates and bond prices
• Higher duration indicates greater sensitivity to interest rate changes
• Used to estimate potential gains or losses from interest rate movements

Price volatility

• Duration serves as a proxy for bond price volatility
• Longer duration bonds experience larger price fluctuations for given yield changes
• Price change approximation: $\Delta Price \approx -Duration \times \Delta Yield \times Price$
• Useful for comparing relative riskiness of different fixed-income securities

Duration as risk measure

• Provides a single number to assess interest rate risk exposure
• Allows for quick comparison of risk across different bonds or portfolios
• Used in various risk management applications (Value at Risk, stress testing)
• Helps in setting risk limits and designing hedging strategies

Limitations of duration

• Assumes a linear relationship between price and yield changes
• Accuracy decreases for large interest rate movements
• Does not account for changes in the shape of the yield curve
• May not fully capture risks associated with complex bond structures

Non-parallel yield curve shifts

• Duration assumes parallel shifts in the yield curve
• Real-world yield curve changes often involve twists or changes in steepness
• Key rate duration or partial durations address this limitation by measuring sensitivity to specific points on the yield curve

Large interest rate changes

• Duration provides a good approximation for small yield changes (typically <0.5%)
• For larger yield changes, duration can significantly underestimate price changes
• Convexity becomes more important for accurately estimating price changes in these scenarios

Embedded options

• Traditional duration measures do not account for embedded options (call, put features)
• Options can significantly alter a bond's interest rate sensitivity
• Effective duration or option-adjusted duration should be used for bonds with embedded options

Introduction to convexity

• Measures the rate of change of duration with respect to yield changes
• Provides a more accurate estimate of price changes for larger yield movements
• Complements duration in fixed-income analysis and risk management

Definition of convexity

• Second derivative of a bond's price with respect to yield
• Measures the curvature of the price-yield relationship
• Formula: $Convexity = \frac{1}{P} \times \frac{d^2P}{dr^2}$
  • Where P is the bond price, d^2P/dr^2 is the second derivative of price with respect to yield
• Expressed as a percentage change in price per 1% change in yield squared

Positive vs negative convexity

• Positive convexity: price increases more when yields fall than it decreases when yields rise
  • Typical for standard bonds without embedded options
  • Beneficial for bondholders as it provides price protection
• Negative convexity: price decreases more when yields rise than it increases when yields fall
  • Often associated with mortgage-backed securities or callable bonds
  • Less desirable for bondholders due to asymmetric price behavior

Calculating convexity

• Involves complex mathematical calculations
• Requires knowledge of bond cash flows and yield to maturity
• Can be approximated using finite difference methods or calculated analytically

Formula for convexity

• General formula for convexity: $Convexity = \frac{\sum_{t=1}^{n} \frac{t(t+1)C_t}{(1+r)^{t+2}}}{P(1+r)^2}$
  • Where C_t represents cash flow at time t, r is the yield to maturity, P is the bond price
• Approximation formula using finite differences: $Convexity \approx \frac{P_+ + P_- - 2P_0}{2P_0(\Delta y)^2}$
  • Where P_+, P_-, and P_0 are bond prices at yields y+Δy, y-Δy, and y respectively

Convexity of bonds

• Zero-coupon bonds have the highest convexity for a given maturity
• Factors affecting bond convexity:
  • Maturity: longer maturity generally increases convexity
  • Coupon rate: lower coupons increase convexity
  • Yield level: lower yields typically increase convexity
• Callable bonds may exhibit negative convexity at low yield levels

Convexity of portfolios

• Calculated as the weighted average of individual security convexities
• Formula: $Portfolio Convexity = \sum_{i=1}^{n} w_i \times C_i$
  • Where w_i is the weight of security i, C_i is the convexity of security i
• Used in conjunction with portfolio duration for more accurate risk assessment
• Helps in designing convexity-matching strategies for asset-liability management

Convexity adjustment

• Improves the accuracy of price change estimates based on duration alone
• Particularly important for large yield changes or high convexity securities
• Allows for better risk management and valuation of fixed-income instruments

Price-yield relationship

• Convexity captures the non-linear relationship between bond prices and yields
• Explains why actual price changes differ from duration-based estimates
• Demonstrates asymmetric price behavior for yield increases vs decreases
• Illustrated by the convex shape of the price-yield curve

Improving duration estimates

• Convexity adjustment formula: $\Delta Price \approx -Duration \times \Delta Yield \times Price + \frac{1}{2} \times Convexity \times (\Delta Yield)^2 \times Price$
• Provides more accurate price change estimates, especially for larger yield movements
• Reduces estimation errors in portfolio valuation and risk assessment

Convexity vs duration

• Duration measures the first-order effect of yield changes on price
• Convexity captures the second-order effect, improving accuracy
• Higher convexity generally considered favorable for bondholders
• Trade-off between duration and convexity in portfolio construction

Applications in risk management

• Duration and convexity serve as key tools in fixed-income risk management
• Enable more precise measurement and control of interest rate risk
• Facilitate development of sophisticated hedging and investment strategies

Immunization strategies

• Aim to protect portfolio value against interest rate changes
• Duration matching: aligns portfolio duration with investment horizon
• Convexity matching: further refines immunization by accounting for non-linear price changes
• Cash flow matching: precise matching of asset and liability cash flows

Portfolio optimization

• Incorporates duration and convexity targets in portfolio construction
• Balances risk and return objectives using duration and convexity constraints
• Allows for customized risk profiles based on investor preferences
• Facilitates efficient allocation of interest rate risk across portfolio components

Hedging interest rate risk

• Utilizes duration and convexity to design effective hedging strategies
• Duration hedging: offsets first-order interest rate risk
• Convexity hedging: addresses second-order effects for more comprehensive protection
• Employs derivatives (futures, swaps, options) to achieve desired risk exposure

Duration and convexity in practice

• Essential concepts for fixed-income portfolio managers and traders
• Applied in various areas of financial markets and risk management
• Require ongoing monitoring and adjustment due to dynamic market conditions

Trading strategies

• Relative value trades based on duration and convexity differentials
• Yield curve positioning using duration-targeted portfolios
• Convexity trading to capitalize on expected yield volatility
• Basis trading between cash bonds and derivatives using duration-matched positions

Asset-liability management

• Aligns duration and convexity of assets with liabilities
• Crucial for insurance companies and pension funds
• Involves dynamic rebalancing to maintain desired risk profile
• Incorporates scenario analysis to assess impact of various interest rate environments

Performance attribution

• Decomposes portfolio returns into duration and convexity components
• Assesses manager skill in positioning portfolios for interest rate movements
• Identifies sources of outperformance or underperformance
• Helps refine investment strategies and risk management techniques

Advanced concepts

• Extend basic duration and convexity measures to address complex scenarios
• Provide more nuanced analysis of interest rate risk and fixed-income securities
• Require sophisticated mathematical and financial modeling techniques

Key rate duration

• Measures price sensitivity to changes in specific points on the yield curve
• Allows for analysis of non-parallel yield curve shifts
• Calculated for various maturities along the yield curve
• Enables more precise hedging and risk management strategies

Effective duration

• Accounts for embedded options in bonds (callable, putable)
• Calculated using small positive and negative yield shocks
• Formula: $Effective Duration = \frac{P_- - P_+}{2 \times P_0 \times \Delta y}$
  • Where P_-, P_+, and P_0 are bond prices at yields y-Δy, y+Δy, and y respectively
• More accurate than modified duration for bonds with embedded options

Spread duration

• Measures sensitivity of bond price to changes in credit spread
• Particularly relevant for corporate and emerging market bonds
• Calculated similarly to effective duration, but with respect to spread changes
• Helps assess and manage credit risk in fixed-income portfolios