Amortized money duration

What Is Amortized Money Duration?

Amortized money duration, more commonly known as money duration or dollar duration, quantifies the actual currency amount by which a bond's price is expected to change for a given shift in interest rates. It is a critical metric within fixed income analytics, providing investors with a direct measure of a bond's absolute price sensitivity to yield fluctuations in currency terms, rather than as a percentage. This concept builds upon other duration measures, such as Macaulay duration and modified duration, by incorporating the bond's full price. Amortized money duration helps investors assess interest rate risk for specific bond holdings or entire bond portfolios.

History and Origin

The foundational concept of duration, from which amortized money duration derives, was introduced by Frederick Macaulay in his seminal 1938 work, "Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856."²³ Macaulay proposed duration as a method to assess the price volatility of bonds, noting that a bond's time to maturity was an insufficient measure of its sensitivity to interest rate changes.²² Initially, few paid attention to this concept due to regulated, stable interest rates. However, with the onset of significant interest rate volatility in the 1970s, investors became increasingly interested in tools that could predict how bond prices would react to yield movements.²⁰, ²¹

This renewed interest led to the development of modified duration, which offered a more precise calculation of price changes given varying coupon rate schedules. Amortized money duration (or simply money duration) emerged as an extension of modified duration. It specifically addresses the need for a measure that translates the percentage change predicted by modified duration into an actual currency value, making it more intuitive for portfolio managers to understand the monetary impact of interest rate shifts on their holdings. The CFA Institute, among others, recognizes money duration as a key yield duration measure used to quantify interest rate risk.¹⁹

Key Takeaways

• Amortized money duration (money duration) measures the currency value change in a bond's price for a given change in its yield to maturity.
• It is calculated by multiplying the bond's modified duration by its full price.
• This metric provides a direct, dollar-denominated (or local currency) estimate of interest rate sensitivity, making it useful for risk management.
• A higher amortized money duration implies a greater monetary loss or gain for a given change in interest rates.
• It is particularly valuable for portfolio managers who need to assess the total monetary impact of interest rate movements across large bond holdings.

Formula and Calculation

The amortized money duration (MoneyDur) is calculated by multiplying the bond's annual modified duration by its full price, including accrued interest. This provides an estimate of the absolute price change in currency units.¹⁷, ¹⁸

The formula for amortized money duration is:

Where:

• (\text{Annual Modified Duration}) represents the percentage change in bond price for a 1% change in yield. It is typically derived from the Macaulay duration and the bond's yield to maturity.
• (\text{Full Price of Bond}) is the dirty price of the bond, which includes the clean price plus any accrued interest. This is the actual price an investor would pay to purchase the bond.

Another related measure, often derived from money duration, is the Price Value of a Basis Point (PVBP), also known as DV01 (Dollar Value of 01). PVBP estimates the change in a bond's full price for a minuscule 1 basis point (0.01%) change in yield-to-maturity.¹⁵, ¹⁶

This relationship highlights how amortized money duration can be scaled to understand sensitivity to very small yield changes.

Interpreting the Amortized Money Duration

Interpreting amortized money duration involves understanding the direct monetary impact of interest rate changes on a bond's value. For example, if a bond has an amortized money duration of $500,000, it suggests that for every 1% (or 100 basis point) increase in yield to maturity, the bond's value is expected to decrease by approximately $500,000. Conversely, a 1% decrease in yield would lead to an approximate $500,000 increase in value.¹³, ¹⁴

This direct monetary interpretation is highly valuable for fixed-income investors and portfolio managers. Unlike percentage-based duration measures, amortized money duration allows for immediate quantification of potential gains or losses in currency terms. This makes it easier to compare the absolute interest rate risk of different bonds or portfolios, especially when dealing with varying par values or market values. A bond with a higher amortized money duration indicates greater exposure to interest rate fluctuations in dollar terms, meaning its price will experience larger absolute swings for a given yield change.

Hypothetical Example

Consider a hypothetical bond, Bond X, with the following characteristics:

• Current Full Price: $1,020 per $1,000 par value
• Annual Modified Duration: 5.5

To calculate the amortized money duration for Bond X, we apply the formula:

This means that for every 1% (or 100 basis points) change in the bond's yield to maturity, the price of Bond X is expected to change by approximately $5,610 in the opposite direction.

For instance, if interest rates increase by 0.50% (50 basis points), the estimated decrease in Bond X's price would be:

So, the price of Bond X would be expected to decrease by approximately $28.05. This example illustrates how amortized money duration provides a clear, actionable monetary figure for assessing bond valuation sensitivity to market movements.

Practical Applications

Amortized money duration is a cornerstone metric in fixed-income investing and plays a crucial role in several practical applications:

• Risk Management: Portfolio managers use amortized money duration to directly quantify the potential monetary loss from adverse interest rate movements on their bond portfolios. This allows for precise risk budgeting and helps in setting risk limits. The Federal Reserve also considers duration in its analysis of market dynamics, noting how mutual funds manage duration risk in their Treasury holdings.¹²
• Portfolio Immunization: In asset-liability management, money duration can be used in immunization strategies to match the interest rate sensitivity of assets to liabilities, thereby protecting the net worth of an institution from interest rate changes. For instance, by aligning the duration of a bond portfolio with the duration of future liabilities, a firm can reduce the risk that changes in interest rates will negatively impact its ability to meet those obligations.¹⁰, ¹¹
• Bond Trading and Hedging: Traders utilize amortized money duration to calculate the required size of hedging instruments (e.g., interest rate futures) to offset the interest rate risk of a bond or a portfolio. If a trader has a long position in a bond with a certain amortized money duration, they can take a short position in another instrument with a comparable money duration to neutralize the interest rate exposure.
• Performance Attribution: In analyzing bond portfolio performance, money duration helps explain how much of the portfolio's return (or loss) was attributable to changes in interest rates versus other factors. This provides insight into the effectiveness of a manager's interest rate calls.

By translating interest rate sensitivity into concrete currency terms, amortized money duration offers a practical and intuitive measure for managing risk across various market conditions.

Limitations and Criticisms

While amortized money duration is a valuable tool in fixed income securities analysis, it has certain limitations, primarily stemming from the underlying assumptions of duration itself.

Firstly, like modified duration, amortized money duration assumes a linear relationship between bond prices and yields.⁹ However, the actual relationship is convex, meaning that bond prices increase at a decreasing rate when yields fall and decrease at an increasing rate when yields rise. This linearity assumption means that amortized money duration provides a good approximation for small changes in interest rates but may be less accurate for large yield swings. To address this, more advanced measures like convexity adjustments are often used in conjunction with duration to provide a more precise estimate of price changes.

Secondly, amortized money duration, like other yield-based duration measures, assumes that all cash flows from the bond are certain and that the entire yield curve shifts in a parallel fashion.⁸ This is often not the case in real markets; non-parallel shifts in the yield curve can occur, and certain bonds, such as callable bonds, have uncertain cash flows due to embedded options. For instruments with embedded options or non-fixed cash flows, effective duration (or option-adjusted duration) is typically a more appropriate measure.⁶, ⁷

Finally, a key critique relates to the practical application of immunization strategies based solely on duration matching. While a portfolio's duration can be matched to that of its liabilities to mitigate interest rate risk, this immunization is generally only perfect for infinitesimal parallel shifts in the yield curve and for a single liability. For multiple liabilities or non-parallel yield curve shifts, the strategy can become imperfect, requiring frequent rebalancing and consideration of higher-order risk measures.⁴, ⁵

Amortized Money Duration vs. Modified Duration

Amortized money duration and modified duration are both measures of a bond's interest rate sensitivity, but they express this sensitivity in different ways. The key distinction lies in their output:

While modified duration provides a standardized way to compare the interest rate sensitivity of different bonds on a percentage basis, amortized money duration translates this sensitivity into a tangible monetary value. This makes amortized money duration particularly useful for investors who need to understand the direct financial impact of interest rate changes on their specific bond holdings or overall portfolio value.

FAQs

What is the primary difference between amortized money duration and Macaulay duration?

Macaulay duration is a weighted average time (measured in years) until a bond's cash flows are received, essentially indicating the bond's effective maturity. Amortized money duration, on the other hand, is a measure of the bond's price sensitivity in actual currency units to a change in interest rates. While Macaulay duration is a precursor to modified duration, which then leads to money duration, they represent different aspects of interest rate risk.

Why is it sometimes called "dollar duration"?

"Dollar duration" is a common synonym for amortized money duration, particularly in the United States, because it expresses the price change in U.S. dollar terms. The term emphasizes that the measure provides a direct monetary value rather than a percentage.

Does a zero-coupon bond have an amortized money duration?

Yes, a zero-coupon bond does have an amortized money duration. For a zero-coupon bond, its Macaulay duration is equal to its time to maturity. This can then be converted to modified duration, and subsequently to amortized money duration, by multiplying by its present value or full price.

How does amortized money duration relate to a bond's price?

Amortized money duration indicates how much a bond's price will change in currency terms for a 1% change in its yield. Since bond prices and interest rates generally move inversely, a positive amortized money duration means that if yields rise, the bond's price will fall by that specified currency amount, and vice versa if yields fall.

Is amortized money duration the same as PV01?

PV01 (Price Value of a Basis Point) is closely related to amortized money duration. PV01 measures the change in a bond's price for a 1-basis point (0.01%) change in yield, whereas amortized money duration measures the change for a 100-basis point (1%) change. Therefore, PV01 can be calculated by dividing the amortized money duration by 100.