Duration

Duration has become one of the most fundamental analytical tools in fixed income investment management, appearing throughout risk measurement, portfolio construction, hedging strategy, and the management of interest rate exposure across the full spectrum of fixed income instruments from short-term Treasury bills to long-dated corporate bonds. It is directly and heavily tested on securities industry examinations and is central to the practice of any investment professional who manages or advises on fixed income portfolios.

Macaulay Duration: The Weighted Average Time to Cash Flow Receipt

Macaulay duration is the original and most conceptually fundamental form of duration, measuring the weighted average time to receipt of all cash flows from a bond, where the weights are the present values of each cash flow as a proportion of the bond's total price.

The Macaulay duration formula weights each cash flow by the time period in which it is received and by its present value as a proportion of the bond's total price, then sums these weighted time periods to produce a single number expressed in years. For a coupon-paying bond, the Macaulay duration will always be less than the bond's maturity because the coupon payments received before maturity reduce the weighted average time to receipt of all cash flows below the time of the final principal payment.

A practical example clarifies the calculation. Consider a two-year bond with a face value of one thousand dollars, an annual coupon rate of eight percent paying annually, and a yield to maturity of ten percent. The bond generates two cash flows: eighty dollars at the end of year one and one thousand and eighty dollars at the end of year two.

The present value of the year one cash flow is eighty dollars divided by one point one, equalling seventy-two dollars and seventy-three cents. The present value of the year two cash flow is one thousand and eighty dollars divided by one point one squared, equalling eight hundred and ninety-two dollars and fifty-six cents. The total price of the bond is seventy-two dollars and seventy-three cents plus eight hundred and ninety-two dollars and fifty-six cents, equalling nine hundred and sixty-five dollars and twenty-nine cents.

The weight of the year one cash flow is seventy-two dollars and seventy-three cents divided by nine hundred and sixty-five dollars and twenty-nine cents, equalling approximately zero point zero seven five three. The weight of the year two cash flow is eight hundred and ninety-two dollars and fifty-six cents divided by nine hundred and sixty-five dollars and twenty-nine cents, equalling approximately zero point nine two four seven. The Macaulay duration is zero point zero seven five three multiplied by one year plus zero point nine two four seven multiplied by two years, equalling approximately one point nine two years.

This result means that despite having a two-year maturity, this bond has a Macaulay duration of approximately one point nine two years, meaning the weighted average time to receipt of all its cash flows is one point nine two years rather than two years. The difference arises because the year one coupon payment, which is received one year before maturity, pulls the weighted average receipt time below the maturity date.

Modified Duration: The Price Sensitivity Measure

While Macaulay duration has intuitive conceptual appeal as a measure of when investors receive their money, the more practically useful form of duration for risk management is modified duration, which directly measures the percentage change in a bond's price for a given change in yield.

Modified duration is calculated by dividing the Macaulay duration by one plus the yield to maturity divided by the number of coupon payments per year. For the bond in the example above with a Macaulay duration of one point nine two years and a yield of ten percent paid annually, the modified duration is one point nine two divided by one point one, equalling approximately one point seven five.

The modified duration can be interpreted directly as a price sensitivity measure: a bond with a modified duration of five point zero will change in price by approximately five percent for a one percentage point change in yield. If yields rise by one percentage point, the bond's price falls by approximately five percent. If yields fall by one percentage point, the bond's price rises by approximately five percent. This linear approximation of price sensitivity to yield changes is the primary practical application of duration in portfolio management and risk assessment.

The modified duration formula for the percentage price change is: percentage price change equals negative modified duration multiplied by the change in yield. The negative sign reflects the inverse relationship between bond prices and yields. A positive change in yield produces a negative change in price, and the modified duration determines the magnitude of that price change.

Dollar Duration and DV01

Duration is also expressed in dollar terms rather than percentage terms, providing a direct measure of the dollar change in portfolio value for a given change in yields.

Dollar duration is calculated by multiplying the modified duration by the price of the bond or portfolio. For a bond with a modified duration of five point zero and a price of nine hundred and fifty dollars per one thousand dollar face value, the dollar duration is five point zero multiplied by nine hundred and fifty dollars, equalling four thousand seven hundred and fifty dollars. This means the bond's dollar value will change by approximately four thousand seven hundred and fifty dollars for a one percentage point change in yield.

The DV01, also called the dollar value of a basis point or PVBP for price value of a basis point, is the dollar change in a bond's value for a one basis point change in yield. It is calculated by multiplying the modified duration by the price and dividing by ten thousand, or equivalently by dividing the dollar duration by one hundred. For the bond above, the DV01 is four thousand seven hundred and fifty dollars divided by one hundred, equalling forty-seven dollars and fifty cents. This means the bond's value changes by approximately forty-seven dollars and fifty cents for each one basis point change in yield.

DV01 is the most commonly used risk metric in professional fixed income portfolio management because it expresses interest rate risk in concrete dollar terms that can be aggregated across all positions in a portfolio to determine the total dollar sensitivity of the portfolio to interest rate changes. A portfolio with a total DV01 of one hundred thousand dollars will gain approximately one hundred thousand dollars in value if yields fall by one basis point across all maturities and lose approximately one hundred thousand dollars if yields rise by one basis point.

Key Determinants of Duration

Understanding the factors that determine a bond's duration is essential for anticipating how different bonds will respond to interest rate changes and for constructing portfolios with desired duration characteristics.

Maturity is the most intuitive driver of duration. Longer-maturity bonds have higher durations than shorter-maturity bonds, all else being equal, because more of their total cash flows are received further in the future. A thirty-year bond has a much longer weighted average time to cash flow receipt than a two-year bond, making it far more sensitive to interest rate changes. This is why long-duration bonds experience much larger price swings for a given change in interest rates than short-duration bonds.

Coupon rate is the second major determinant of duration. Lower-coupon bonds have higher durations than higher-coupon bonds of the same maturity and yield, because lower coupons mean that a smaller proportion of total cash flows is received through early coupon payments and a larger proportion is received at maturity. A zero coupon bond, which pays no coupons at all, has a duration exactly equal to its maturity because all of its cash flows are received at the single maturity date. A high-coupon bond pays large cash flows early in its life, reducing the weighted average time to receipt and therefore its duration.

Yield to maturity affects duration because higher yields reduce the present values of distant cash flows more than near-term cash flows, shifting the weight of the calculation toward earlier payments and reducing duration. Conversely, lower yields increase the relative weight of distant cash flows and increase duration. This relationship means that bonds experience higher durations in low-yield environments, making them more sensitive to interest rate changes precisely when yields are lowest and the potential for rate increases may be greatest.

Call features and other embedded options affect the effective duration of a bond in more complex ways. Callable bonds, which give the issuer the right to redeem the bond before maturity when interest rates fall, have lower effective duration than equivalent non-callable bonds because the issuer is likely to exercise the call option when rates decline, limiting the price appreciation the bondholder can realise. This option-adjusted duration, which accounts for the effect of embedded options on the expected timing of cash flows, is a more accurate measure of price sensitivity for callable bonds than the simple modified duration calculation that ignores the call feature.

Duration of a Portfolio

One of the most practically important properties of duration is that the duration of a portfolio of bonds is simply the weighted average of the durations of the individual bonds in the portfolio, where the weights are the proportional market values of each bond.

This additive property makes portfolio duration straightforward to calculate and to manage. A portfolio that holds fifty percent of its value in a bond with a duration of two years and fifty percent in a bond with a duration of eight years has a portfolio duration of five years. Adding a bond with a ten-year duration to a portfolio with a four-year duration will increase the portfolio duration toward ten years in proportion to the weight of the new bond.

Portfolio managers use this additive property to actively manage the duration of their portfolios in response to their interest rate views and the risk parameters of their mandates. A manager who believes interest rates will rise will shorten portfolio duration by selling longer-duration bonds and buying shorter-duration bonds or by using derivatives such as Treasury futures or interest rate swaps to reduce effective duration without changing the underlying bond holdings. A manager who believes rates will fall will lengthen portfolio duration to maximise price appreciation from the anticipated rate decline.

The target duration of a portfolio is typically specified in the investment policy statement or portfolio mandate, establishing the permissible range within which the manager may allow portfolio duration to vary. A portfolio benchmarked to the Bloomberg US Aggregate Bond Index, which has historically had a duration of approximately five to six years, might have a mandate allowing the manager to operate within a range of plus or minus two years from the benchmark duration, permitting modest active duration management while limiting the risk of large deviations from benchmark performance.

Duration and Yield Curve Risk

Standard modified duration assumes that all yields move by the same amount at all maturities simultaneously, a parallel shift in the yield curve. In reality, interest rate changes frequently involve non-parallel shifts in which different maturities move by different amounts, with the short end and long end of the curve sometimes moving in opposite directions.

Key rate duration, also called partial duration, measures the sensitivity of a bond or portfolio to changes in yields at specific maturity points on the yield curve rather than to a parallel shift in all yields simultaneously. By calculating key rate durations at the two-year, five-year, ten-year, and thirty-year points for example, a portfolio manager can understand not only the total interest rate sensitivity of the portfolio but also which parts of the yield curve the portfolio is most exposed to. This granular understanding is essential for managing yield curve risk in sophisticated fixed income portfolios where the curvature and slope of the yield curve can change independently of the general level of rates.

Bullet portfolios concentrate their duration exposure at a single maturity point on the yield curve, maximising sensitivity to changes at that specific maturity. Barbell portfolios concentrate exposure at two widely separated maturity points, combining short-term and long-term holdings with minimal intermediate exposure. Ladder portfolios spread duration exposure evenly across multiple maturity points. The choice among these structural approaches reflects the manager's views on yield curve shape and the risk management requirements of the mandate.

Duration in Liability-Driven Investment

Duration matching is the foundational technique of liability-driven investment, in which the objective is to manage a portfolio of fixed income assets to match the characteristics of a corresponding set of future liabilities such as pension obligations, insurance liabilities, or other defined future cash flow requirements.

In a pure duration matching strategy, the portfolio manager sets the duration of the asset portfolio equal to the duration of the liability portfolio, ensuring that changes in interest rates affect the values of assets and liabilities by approximately the same amount. When rates rise, both the asset portfolio and the liability portfolio decline in value by similar magnitudes, leaving the funded status of the pension plan or insurance reserve approximately unchanged. When rates fall, both assets and liabilities increase in value by similar magnitudes, again leaving the funded status approximately unchanged.

Duration matching alone is an imperfect hedge because it protects against only parallel shifts in the yield curve and because convexity differences between assets and liabilities create residual interest rate risk for large changes in rates. Cash flow matching, in which specific bond cash flows are matched to specific liability payment dates, provides a more complete but also more expensive and less flexible approach to liability-driven investment. Most sophisticated liability-driven investment programmes combine duration and key rate duration matching with cash flow matching for near-term liabilities and active duration management for longer-dated obligations.

Convexity: Refining the Duration Approximation

Duration provides a linear approximation of the relationship between bond price changes and yield changes, but the actual price-yield relationship is curved rather than linear, a property called convexity. The linear approximation provided by duration becomes less accurate as yield changes become larger, and convexity measures the degree of curvature in the price-yield relationship that duration ignores.

Convexity is always positive for standard fixed income instruments, meaning that the actual price increase from a yield decline is larger than the duration approximation predicts, and the actual price decrease from a yield increase is smaller than the duration approximation predicts. A bond with higher convexity will outperform a bond with the same duration but lower convexity in both rising and falling rate environments, because it benefits more from rate declines and suffers less from rate increases. This positive asymmetry makes convexity a desirable property in a bond portfolio, and bonds with higher convexity command higher prices reflecting the value of this asymmetry.

The refined approximation incorporating both duration and convexity is: percentage price change equals negative modified duration multiplied by the change in yield plus one half multiplied by convexity multiplied by the square of the change in yield. For large yield changes this two-term approximation is significantly more accurate than the duration-only approximation, making convexity an important supplementary risk measure for portfolios with significant exposure to large rate movements.

Zero coupon bonds have the highest duration for any given maturity but also relatively high convexity compared to coupon-bearing bonds of the same maturity. Callable bonds have negative convexity in the range of yields where the call option is likely to be exercised, meaning that their price appreciation as yields fall is limited by the call price, creating an unfavourable asymmetry that is reflected in the higher yields callable bonds must offer to attract investors.

Practical Applications of Duration in Portfolio Management

Duration permeates virtually every aspect of professional fixed income portfolio management, from the most basic decisions about bond selection to the most sophisticated hedging and risk management strategies.

Interest rate positioning involves deliberately managing portfolio duration above or below the benchmark duration to express a view on the future direction of interest rate changes. A portfolio manager who expects rates to rise will shorten portfolio duration below the benchmark, reducing the portfolio's sensitivity to the anticipated rate increase. A manager who expects rates to fall will lengthen portfolio duration above the benchmark, positioning the portfolio to benefit from price appreciation as yields decline. The accuracy of the manager's interest rate views and the consistency of their duration positioning decisions are the primary determinants of whether active duration management adds value relative to maintaining a neutral benchmark duration.

Immunisation strategies use duration matching to protect the value of a fixed income portfolio from the effects of interest rate changes, ensuring that the portfolio can meet a specified future obligation regardless of how rates move between now and the obligation's due date. A portfolio is immunised against a single liability if its duration equals the time to the liability payment date and its present value equals the present value of the liability. Rate changes that would reduce the market value of the portfolio are offset by reinvestment of coupons at higher rates, and vice versa, so that the accumulated value of the portfolio at the liability date is protected.

Hedging interest rate risk using derivatives involves constructing an offsetting position in Treasury futures, interest rate swaps, or other interest rate derivatives that has a DV01 equal and opposite to the DV01 of the position being hedged, neutralising the net interest rate sensitivity of the combined position. A portfolio manager who holds a long position in corporate bonds with a total DV01 of five hundred thousand dollars can hedge the interest rate component of that risk by selling Treasury futures with an equivalent DV01 of five hundred thousand dollars, leaving only the credit spread risk of the corporate bonds unhedged.

Examination Relevance and Key Takeaways

Duration is one of the most heavily and comprehensively tested topics on the Series 65 examination and is also tested on the Series 7. Candidates must understand the conceptual definition of Macaulay duration as the weighted average time to receipt of cash flows, the practical interpretation of modified duration as the percentage price change for a one percentage point change in yield, the DV01 as the dollar price change for a one basis point change in yield, the key determinants of duration including maturity, coupon rate, and yield, the additive property of portfolio duration as the weighted average of individual bond durations, the relationship between duration and convexity and the direction of the convexity adjustment for standard fixed income instruments, and the practical applications of duration in interest rate positioning, immunisation, and derivative hedging strategies.

The core points to retain are these: duration measures both the weighted average time to receipt of a bond's cash flows and its price sensitivity to interest rate changes; modified duration equals Macaulay duration divided by one plus the periodic yield and represents the percentage price change for a one percentage point yield change; the DV01 is the dollar price change for one basis point change in yield and equals modified duration multiplied by price divided by ten thousand; longer maturity, lower coupon, and lower yield all increase duration; zero coupon bonds have duration equal to their maturity, the maximum possible duration for any bond of that term; portfolio duration is the weighted average of individual bond durations; convexity is always positive for standard bonds meaning actual price changes exceed the linear duration approximation with asymmetric benefits in both rising and falling rate environments; and duration is the primary tool for immunisation strategies, interest rate hedging using derivatives, and active interest rate positioning in fixed income portfolio management.