When fixed-income managers discuss interest rate risk, two concepts dominate the conversation: duration and volatility. While bonds are commonly described by their maturity dates, this metric tells only part of the story. A 30-year bond with a 10% coupon behaves quite differently from a 30-year zero-coupon bond when interest rates shift, despite sharing identical maturities.
Frederick Macaulay introduced the concept of duration in 1938 to address this limitation. Macaulay duration measures the weighted average time until a bond's cash flows are received, where weights are the present values of those cash flows. For a zero-coupon bond, duration equals maturity since there is only one cash flow. For coupon-bearing bonds, duration is always less than maturity because investors receive periodic payments before the principal is returned.
This 29-page September 1985 Salomon Brothers manual presents duration through an elegant visual analogy: imagine cash flows as containers of different sizes resting on a seesaw, with each container filled to its present value. The balance point where you could place a fulcrum to keep the system level represents the duration. This visualization immediately reveals why higher coupon bonds have shorter durations: more weight sits closer to the present.
Modified duration, which equals Macaulay duration divided by $$1 + \frac{y}{f},$$ serves as the more practical measure for volatility estimation. Modified duration was developed by Hicks in 1939 without reference to Macaulay's work. It quantifies the approximate percentage change in a bond's full price (including accrued interest) for a 100 basis point change in yield. A bond with modified duration of 7.5 years should experience roughly a 7.5% price change for a 1% parallel shift in yields.
The relationship between price sensitivity and yield changes introduces convexity, the curvature in the price-yield relationship. When yields fall, bond prices rise more than duration alone predicts. When yields rise, prices fall less than expected. This asymmetry favors bondholders and explains why longer-duration bonds with dispersed cash flows command premium valuations in certain market environments.
Written by R.W. Kopprasch, the manual systematically develops duration from first principles through advanced applications. The document begins by acknowledging duration's long dormancy—introduced in 1938 but largely ignored until being "rediscovered" in the 1970s. This rediscovery coincided with increased interest rate volatility, which made quantitative risk management essential rather than optional.
Section II develops Macaulay duration both formulaically and graphically. The manual demonstrates how duration varies with maturity, yield level, and coupon rate. For par and premium bonds, duration increases with maturity but approaches an asymptotic limit: the duration of a perpetual annuity, given by $$\frac{1+r}{r}.$$ Discount bonds exhibit more complex behavior—for very long maturities, their duration can actually decrease as maturity extends further, ultimately converging to the perpetual annuity duration from above.
The manual carefully traces duration's behavior through time. Between coupon dates, duration declines linearly. One day of elapsed time reduces duration by precisely one day. On coupon payment dates, duration jumps upward (though typically by less than the coupon period). This creates a sawtooth pattern. Critically, the manual clarifies that bond price volatility follows a smooth path despite duration's jagged pattern.
Section III introduces modified duration and establishes its role in volatility measurement. The distinction is vital: Macaulay duration suits immunization strategies, while modified duration quantifies price sensitivity to interest rate changes.
Section IV (the manual's most practical contribution) demonstrates volatility weighting for hedging, bond swaps, and arbitrage. The manual proves the equivalence of three weighting methods:
Price Value of a Basis Point (PVBP): The dollar price change for a one basis point yield change. This measure provides direct indication of dollar volatility.
Yield Value of 1/32: The yield change required to move price by 1/32. An inverse measure of volatility (higher values indicate lower price sensitivity).
Duration-based weighting: Using modified duration, full prices, and yield betas. The hedge ratio formula is
$$\text{HR} = \frac{\text{PVBP}_{\text{target}}}{\text{PVBP}_{\text{hedge}}} \times \beta_{\text{yield}}$$
where $$\beta_{\text{yield}}$$ captures the historical relationship between yield changes in the two securities.
The manual emphasizes that duration itself does not provide the hedge ratio—the ratio of durations is incorrect. Proper weighting requires multiplying modified duration by full price to obtain dollar volatility, then adjusting for relative yield movements.
A particularly instructive example compares the following Treasury bonds. Bond B (13.25% coupon, 5/15/11 maturity) has similar duration to Bond D (8.75% coupon, 5/15/11 maturity), yet the proper hedge ratio is 0.79 rather than 1.0. Duration measures percentage volatility; hedging requires matching dollar volatility.
Section V covers convexity, explaining why it exists through the seesaw analogy. When yields decline, longer cash flows gain more value proportionally, shifting the balance point (duration) rightward. This lengthening duration amplifies the next price increase. Conversely, when yields rise, duration shortens, cushioning further price declines. The manual illustrates these dynamics through extreme "barbell" portfolios (combining short and long zero-coupon bonds) which exhibit high convexity but potentially come at a yield cost.
Section VI extends duration concepts to non-standard instruments:
The manual concludes by noting that while it explains proper duration usage, it deliberately avoids discussing hedging nuances, tax effects, or statistical estimation of yield betas—all practical concerns that can dominate real-world applications.
Four decades after publication, this manual's relevance remains undiminished. The core mathematics of present value and interest rate sensitivity are immutable. However, the market context has evolved dramatically.
In September 1985, when the manual was published, 10-year Treasury yields ranged between 9.7% and 10.8%. As of late October 2025, the 10-year yield trades around 4.11%. This lower rate environment affects duration in predictable ways: for a given maturity and coupon, duration increases as yields decline. A 10-year par bond at 4% yield has longer duration than an equivalent bond priced at 10% yield.
The manual's emphasis on convexity has become more critical in recent years. During the extended low-rate regime following the 2008 financial crisis and the 2020 pandemic response, convexity commanded substantial premiums. Investors recognized that in a low-rate world, asymmetric outcomes favor bond owners—limited upside from further rate declines but meaningful cushion if rates rise. The rapid rate increases of 2022-2023 tested these relationships, with longer-duration bonds experiencing losses that exceeded simple duration-based predictions due to parallel shift assumptions breaking down.
The manual's treatment of callable bonds presaged an analytical challenge that intensified in the 1990s and 2000s. Mortgage-backed securities and structured products proliferated, requiring sophisticated option-adjusted analytics. The basic insight (that embedded options create path-dependent, non-linear cash flows) remains central to modern fixed-income analysis.
Modern portfolio management has largely adopted the manual's terminology and methodology. Price value of a basis point (DV01 in current parlance) serves as the standard risk metric for trading desks. Pension funds structure liability-driven investments around duration matching. Central banks discuss their quantitative easing programs' impact on portfolio duration extraction. The Federal Reserve's balance sheet management explicitly considers the duration risk transferred between the public and private sectors.
The Salomon manual provides foundational tools for constructing systematic fixed-income strategies. Several approaches emerge directly from its framework:
Duration-Neutral Spread Trading: By matching the PVBP of long and short positions (adjusted for yield beta), systematic managers can isolate spread movements from level changes. This allows systematic capture of credit risk premiums, yield curve shape bets, or liquidity premia without directional rate exposure. The manual's hedge ratio formulas enable precise construction of these positions.
Convexity Harvesting: Understanding that convexity's value fluctuates with volatility regimes suggests systematic rebalancing rules. When implied volatility is low, overweight high-convexity bonds (long maturities, low coupons). When volatility spikes and convexity becomes expensive, rotate toward lower-convexity positions. The manual's explanation of why barbell structures exhibit high convexity informs this rotation.
Curve Steepener/Flattener Trades: The manual's demonstration that duration varies predictably with maturity enables systematic yield curve trades. Duration-weight positions to eliminate parallel shift risk, then position for curve reshaping based on economic cycle indicators or relative value signals.
Roll-Down Strategies: The manual's treatment of duration changes through time suggests systematic approaches. As bonds age, they roll down the yield curve (assuming an upward-sloping curve). Duration decreases in a predictable sawtooth pattern. Systematic strategies can exploit this rolldown by continuously rebalancing to maintain target duration while capturing the yield give-up as bonds approach maturity.
Factor-Based Duration Targeting: Modern quantitative approaches combine the manual's duration framework with factor models. Construct portfolios that maintain duration exposure to specific factors (level, slope, curvature of yield curve) while neutralizing others. The manual's emphasis on separate measurement of dollar volatility and yield relationships provides the mathematical foundation.
Volatility-Adjusted Position Sizing: Rather than equal-weight bonds in a portfolio, use PVBP to allocate capital such that each position contributes equally to portfolio volatility. This approach, implicit in the manual's hedging section, ensures diversification in risk terms rather than nominal terms.
The manual's most enduring lesson for systematic strategies is the primacy of measurement precision. Successful quantitative fixed-income management requires accurate calculation of duration, PVBP, and convexity; understanding of basis relationships between securities; and realistic modeling of how these parameters evolve. Qualitative judgment about rate direction matters less than rigorous quantification of the risks being assumed.
For today's quantitative portfolio managers, the 1985 Salomon manual serves as both foundation and reminder. The foundation: duration-based analytics provide the essential toolkit for measuring and managing interest rate risk. The reminder: elegant frameworks require disciplined implementation. Hedge ratios must account for basis risk. Convexity models must reflect realistic yield curve dynamics. Statistical estimation of yield betas must incorporate sufficient history while recognizing regime changes.
The manual's enduring value lies not in predicting market movements but in providing the precise analytical tools needed to quantify and manage the risks inherent in fixed-income portfolios. These tools remain as relevant today as they were in 1985, even as the market environment has transformed completely.