Yield-Based Bond Duration Measures and Properties

Fixed Income

Modified Duration

Learning Outcome Statement:

define, calculate, and interpret modified duration, money duration, and the price value of a basis point (PVBP)

Summary:

Modified duration measures the sensitivity of a bond's price to changes in its yield-to-maturity, calculated as the first derivative of the bond's price with respect to its yield. It can be derived from Macaulay duration and is used to estimate the percentage change in a bond's price for a given change in yield. Money duration extends modified duration by incorporating the bond's position size in currency terms. The price value of a basis point estimates the change in a bond's price for a 1 bp change in yield.

Key Concepts:

Modified Duration

Modified duration is a measure of the price sensitivity of a bond to changes in interest rates, calculated as the negative of the first derivative of the bond's price with respect to its yield, divided by the bond's price.

Money Duration

Money duration extends the concept of modified duration by incorporating the size of the bond position, calculated as the product of the bond's full price and its annualized modified duration.

Price Value of a Basis Point (PVBP)

PVBP is an estimate of the change in the full price of a bond for a 1 basis point change in the bond's yield, providing a measure of the bond's price sensitivity to small changes in yield.

Approximate Modified Duration

Approximate modified duration can be estimated by calculating the slope of the tangent to the price-yield curve of a bond, useful for bonds with unknown Macaulay duration due to features like contingency or default risk.

Formulas:

Modified Duration

\text{ModDur} = \frac{\text{MacDur}}{1 + r}

Modified duration is calculated by dividing the Macaulay duration by one plus the yield to maturity per period, reflecting the sensitivity of the bond's price to yield changes.

Variables:

$ModDur$ :: Modified Duration
$MacDur$ :: Macaulay Duration
$r$ :: Yield to maturity per period

Units: years

Percentage Change in Full Price

\%\Delta PV_{\text{Full}} \approx -\text{AnnModDur} \times \Delta \text{AnnYield}

This formula estimates the percentage change in the full price of a bond for a given change in the annualized yield, using the annualized modified duration.

Variables:

$PV_{\text{Full}}$ :: Full price of the bond
$AnnModDur$ :: Annualized Modified Duration
$\Delta \text{AnnYield}$ :: Change in annualized yield to maturity

Units: percentage

Approximate Annualized Modified Duration

\text{AnnModDur} \approx \frac{(PV_{-} - PV_{+})}{2 \times (\Delta \text{Yield}) \times (PV_{0})}

This formula approximates the annualized modified duration by using the change in bond prices resulting from small increases and decreases in yield, divided by twice the product of the change in yield and the initial full price.

Variables:

$AnnModDur$ :: Approximate Annualized Modified Duration
$PV_{-}$ :: Price of the bond when yield is decreased
$PV_{+}$ :: Price of the bond when yield is increased
$\Delta \text{Yield}$ :: Change in yield
$PV_{0}$ :: Initial full price of the bond

Units: years

Money Duration and Price Value of a Basis Point

Learning Outcome Statement:

define, calculate, and interpret modified duration, money duration, and the price value of a basis point (PVBP)

Summary:

This LOS focuses on understanding and calculating the modified duration, money duration, and price value of a basis point (PVBP) for bonds. Modified duration measures the percentage price change of a bond for a given change in yield-to-maturity. Money duration, also known as dollar duration in the US, quantifies this change in actual currency units and can be expressed per 100 of par value or for the actual position size. PVBP estimates the change in the full price of a bond for a 1 basis point change in yield-to-maturity.

Key Concepts:

Modified Duration

Modified duration is a derivative measure that indicates how much the price of a bond will change in response to a change in its yield-to-maturity. It is calculated as the Macaulay duration divided by 1 plus the yield per period.

Money Duration

Money duration, or dollar duration, measures the change in the monetary value of a bond's price due to changes in yield-to-maturity. It is calculated by multiplying the modified duration by the full price of the bond.

Price Value of a Basis Point (PVBP)

PVBP estimates the change in the full price of a bond given a 1 basis point change in its yield-to-maturity. It is calculated using the average of the price differences when the yield is increased and decreased by 1 basis point.

Formulas:

Money Duration

MoneyDur = AnnModDur \times PV_{Full}

This formula calculates the change in monetary value of a bond's price due to a change in yield, expressed either per 100 of par value or for the actual position size.

Variables:

$MoneyDur$ :: Money Duration
$AnnModDur$ :: Annualized Modified Duration
$PV_{Full}$ :: Full Price of the bond

Units: currency units

Percentage Change in Full Price

\%\Delta PV_{Full} \approx -MoneyDur \times \Delta Yield

This formula estimates the percentage change in the full price of the bond for a given change in yield-to-maturity, using the money duration.

Variables:

$\%\Delta PV_{Full}$ :: Percentage change in full price of the bond
$MoneyDur$ :: Money Duration
$\Delta Yield$ :: Change in yield-to-maturity

Units: percentage

Price Value of a Basis Point

PVBP = \frac{(PV_{-} - PV_{+})}{2}

This formula calculates the estimated change in the full price of a bond for a 1 basis point change in its yield-to-maturity.

Variables:

$PVBP$ :: Price Value of a Basis Point
$PV_{-}$ :: Full price with a decrease in yield by 1 basis point
$PV_{+}$ :: Full price with an increase in yield by 1 basis point

Units: currency units

Properties of Duration

Learning Outcome Statement:

explain how a bond’s maturity, coupon, and yield level affect its interest rate risk

Summary:

The learning outcome statement focuses on understanding how various factors such as maturity, coupon rate, and yield to maturity influence a bond's interest rate risk. It discusses the properties of duration including Macaulay duration, modified duration, money duration, and price value of a basis point, and how these metrics respond to changes in bond features.

Key Concepts:

Macaulay Duration

Macaulay duration measures the weighted average time until a bond's cash flows are received, and is influenced by the bond's yield, coupon, and time to maturity. It helps in assessing the sensitivity of a bond's price to changes in interest rates.

Modified Duration

Modified duration adjusts Macaulay duration to account for changes in yield, providing a direct measure of price sensitivity to yield changes.

Money Duration

Money duration, also known as dollar duration, multiplies the modified duration by the bond's price, providing a measure of the actual dollar change in price for a given yield change.

Price Value of a Basis Point (PVBP)

PVBP measures the change in the price of a bond for a one basis point change in yield. It is useful for assessing the impact of small yield changes on the bond's price.

Interest Rate Risk

Interest rate risk refers to the potential for investment losses due to changes in interest rates. Bonds with higher duration are more sensitive to changes in rates, implying higher interest rate risk.

Formulas:

Macaulay Duration Formula

MacDur = \frac{1 + r}{r} - \frac{1 + r + [N \times (c - r)]}{c \times [(1 + r)^N - 1 + r]} - \frac{t}{T}

This formula calculates the Macaulay duration, which is a measure of the effective maturity of a bond and is used to assess interest rate risk.

Variables:

$r$ :: yield to maturity
$c$ :: coupon rate
$N$ :: number of periods until maturity
$t$ :: time elapsed since last coupon payment
$T$ :: total time in the current coupon period

Units: years

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