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Curve-Based and Empirical Fixed-Income Risk Measures

Fixed Income

Curve-Based Interest Rate Risk Measures

Learning Outcome Statement:

Explain why effective duration and effective convexity are the most appropriate measures of interest rate risk for bonds with embedded options

Summary:

Effective duration and effective convexity are crucial for assessing interest rate risk in bonds with uncertain future cash flows due to embedded options like calls or puts. These measures account for changes in bond prices relative to shifts in the benchmark yield curve, unlike yield duration and convexity, which assume certain cash flows and may not accurately reflect the price sensitivity of bonds with embedded options.

Key Concepts:

Effective Duration

Effective duration measures the sensitivity of a bond's price to a parallel shift in the benchmark yield curve. It is particularly useful for bonds with embedded options as it considers the potential variability in cash flows.

Effective Convexity

Effective convexity measures the curvature in the relationship between bond prices and yield changes. It helps in understanding how bond prices will react to changes in yields, especially when these changes are large. For bonds with options, effective convexity can become negative, indicating limited price appreciation potential when yields fall.

Key Rate Duration

Key rate duration measures the sensitivity of a bond's price to changes in interest rates at specific points along the yield curve, rather than to parallel shifts. This allows for a more granular analysis of interest rate risk.

Analytical vs. Empirical Measures

Analytical duration and convexity use mathematical formulas and are based on assumptions of changes in yield curves, while empirical measures use historical data and consider actual market behaviors, making them potentially more accurate for bonds with credit risks.

Formulas:

Effective Duration (EffDur)

EffDur=(PV−)−(PV+)2×(ΔCurve)×(PV0)EffDur = \frac{(PV_-) - (PV_+)}{2 \times (\Delta Curve) \times (PV_0)}

This formula calculates the percentage change in the bond's price for a given parallel shift in the yield curve.

Variables:
PV0PV_0:
Present value of the bond at the current yield curve
PV+PV_+:
Price of the bond if the yield curve shifts up
PV−PV_-:
Price of the bond if the yield curve shifts down
ΔCurve\Delta Curve:
Change in the yield curve
Units: dimensionless (expressed as a factor of price change per yield change)

Effective Convexity (EffCon)

EffCon=[(PV−)+(PV+)]−[2×(PV0)](ΔCurve)2×(PV0)EffCon = \frac{[(PV_-) + (PV_+)] - [2 \times (PV_0)]}{(\Delta Curve)^2 \times (PV_0)}

This formula measures the curvature of the price-yield relationship of a bond, indicating how the price acceleration of a bond changes as yields change.

Variables:
PV0PV_0:
Present value of the bond at the current yield curve
PV+PV_+:
Price of the bond if the yield curve shifts up
PV−PV_-:
Price of the bond if the yield curve shifts down
ΔCurve\Delta Curve:
Change in the yield curve squared
Units: dimensionless (expressed as a factor of price change per square of yield change)

Bond Risk and Return Using Curve-Based Duration and Convexity

Learning Outcome Statement:

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

Summary:

This LOS focuses on using curve-based duration and convexity to calculate the percentage price change of a bond given a specified change in the benchmark yield. Effective duration and effective convexity are crucial for assessing the interest rate risk of bonds, especially those with embedded options like callable or putable bonds. These measures are derived from bond prices that are calculated using an option valuation model, considering specific changes in the underlying benchmark government yield curve.

Key Concepts:

Effective Duration (EffDur)

Effective duration measures the sensitivity of a bond's price to a parallel shift in the yield curve. It is particularly useful for bonds with uncertain future cash flows, such as callable bonds.

Effective Convexity (EffCon)

Effective convexity measures the rate of change of the bond's duration with respect to changes in yield. It helps in understanding how bond prices will react to changes in yield, especially in bonds with option-like features.

Curve-Based Interest Rate Risk Measures

These measures are used to assess the interest rate risk of bonds with embedded options and are derived from changes in bond prices due to shifts in the benchmark yield curve.

Formulas:

Effective Duration

EffDur=(PV−−PV+)2×(ΔCurve)×(PV0)EffDur = \frac{(PV- - PV+)}{2 \times (\Delta Curve) \times (PV0)}

This formula calculates the effective duration of a bond, which quantifies the bond's price sensitivity to changes in the yield curve.

Variables:
PV−PV-:
Price of the bond when the yield curve is lowered
PV+PV+:
Price of the bond when the yield curve is raised
PV0PV0:
Initial full price of the bond
ΔCurve\Delta Curve:
Change in the yield curve
Units: years

Effective Convexity

EffCon=[(PV−+PV+)−2×(PV0)](ΔCurve)2×(PV0)EffCon = \frac{[(PV- + PV+) - 2 \times (PV0)]}{(\Delta Curve)^2 \times (PV0)}

This formula calculates the effective convexity of a bond, which measures the curvature of the price-yield relationship of the bond as the yield changes.

Variables:
PV−PV-:
Price of the bond when the yield curve is lowered
PV+PV+:
Price of the bond when the yield curve is raised
PV0PV0:
Initial full price of the bond
ΔCurve\Delta Curve:
Change in the yield curve
Units: dimensionless

Percentage Price Change

%ΔPVFull≈(−EffDur×ΔCurve)+[12×EffCon×(ΔCurve)2]\%\Delta PV_{Full} \approx (-EffDur \times \Delta Curve) + \left[\frac{1}{2} \times EffCon \times (\Delta Curve)^2\right]

This formula estimates the percentage change in the full price of a bond for a given shift in the benchmark yield curve, incorporating both effective duration and effective convexity.

Variables:
EffDurEffDur:
Effective Duration of the bond
EffConEffCon:
Effective Convexity of the bond
ΔCurve\Delta Curve:
Change in the yield curve
Units: percentage

Key Rate Duration as a Measure of Yield Curve Risk

Learning Outcome Statement:

define key rate duration and describe its use to measure price sensitivity of fixed-income instruments to benchmark yield curve changes

Summary:

Key rate duration, also known as partial duration, measures the sensitivity of a bond's price to changes in specific points along the benchmark yield curve, rather than to parallel shifts. This metric is crucial for understanding how non-parallel changes in the yield curve affect the bond's price, providing insights into the bond's shaping risk, which refers to its sensitivity to changes in the shape of the yield curve.

Key Concepts:

Key Rate Duration

Key rate duration measures the sensitivity of a bond's price to a change in the interest rate at a specific maturity point on the yield curve. It helps in assessing the impact of non-parallel shifts in the yield curve on the bond's price.

Shaping Risk

Shaping risk refers to the risk associated with changes in the shape of the yield curve, such as becoming steeper, flatter, or experiencing a twist. Key rate durations help in identifying and quantifying this risk.

Effective Duration vs. Key Rate Duration

While effective duration measures sensitivity to parallel shifts in the entire yield curve, key rate duration isolates the sensitivity to changes at specific points along the yield curve, providing a more detailed analysis of interest rate risk.

Formulas:

Key Rate Duration Formula

KeyRateDurk=−1PV×ΔPVΔrkKeyRateDur_k = -\frac{1}{PV} \times \frac{\Delta PV}{\Delta r_k}

This formula calculates the key rate duration for a specific maturity point on the yield curve, indicating how much the bond's price will change in response to a 1 basis point change in the yield at that maturity.

Variables:
KeyRateDurkKeyRateDur_k:
Key rate duration for the k-th key rate
PVPV:
Present value of the bond
ΔPV\Delta PV:
Change in the present value of the bond due to the change in the k-th key rate
Δrk\Delta r_k:
Change in the k-th key rate
Units: years

Empirical Duration

Learning Outcome Statement:

describe the difference between empirical duration and analytical duration

Summary:

Empirical duration and analytical duration are two methods used to estimate the sensitivity of bond prices to changes in interest rates, but they differ in their approaches and applicability. Analytical duration uses mathematical formulas to estimate the price-yield relationship under normal market conditions, assuming independent and uncorrelated yield changes. Empirical duration, on the other hand, utilizes historical data and statistical models to account for various factors affecting bond prices, including correlations between yield spreads and benchmark yields, making it more suitable in complex scenarios such as during market crises.

Key Concepts:

Analytical Duration

Analytical duration estimates the sensitivity of a bond's price to changes in interest rates using mathematical formulas. It provides a reasonable approximation of the price-yield relationship under many circumstances, assuming that government bond yields and spreads are independent and uncorrelated.

Empirical Duration

Empirical duration uses historical data and statistical models to estimate the sensitivity of bond prices to interest rate changes. It incorporates factors such as correlations between yield spreads and benchmark yields, making it more applicable in scenarios where these factors significantly influence bond prices, such as during financial crises or in portfolios with varying credit qualities.

Key Rate Duration

Key rate duration measures the sensitivity of a bond's price to changes in the yield of a specific maturity while holding other maturities constant. It is useful for assessing the impact of non-parallel shifts in the yield curve on a bond's price.

Formulas:

Key Rate Duration Impact on Price

ΔPV/PV=−KeyRateDurk×Δrk\Delta PV / PV = -\text{KeyRateDur}_k \times \Delta r_k

This formula calculates the expected percentage change in the price of a bond given a specific change in the interest rate at maturity k, using the key rate duration for that maturity.

Variables:
ΔPV/PV\Delta PV / PV:
Percentage change in the present value of the bond
KeyRateDurk\text{KeyRateDur}_k:
Key rate duration for maturity k
Δrk\Delta r_k:
Change in the interest rate for maturity k
Units: Percentage
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