Key Rate Durations Research Articles

Practical Applications of A Key Rate Approach to Replicating Annuities with US Treasury Funds

In <b><i>A Key Rate Approach to Replicating Annuities with US Treasury Funds</i></b>, from the Summer 2021 issue of <b><i>The Journal of Fixed Income</i></b>, <b>Joseph Prendergast</b> (of the <b>Kogod School of Business at American University</b>) illustrates how retail investors, or their advisors, can replicate an annuity that will save money by avoiding large fees charged by annuity providers. A general approach to achieving this objective is to match the duration of a reference annuity with a replicating portfolio of funds consisting of bonds of various maturities. However, a single duration measure will not adequately capture an asset’s price sensitivity to interest rate changes unless there are only parallel shifts in the yield curve. Prendergast illustrates the use of key rate durations, which correspond to specific points along the yield curve within a constrained optimization, to achieve far more accurate results. Periodic optimization provides replicating portfolio weights for rebalancing so that cash flows closely mimic an annuity.

A Key Rate Approach to Replicating Annuities with US Treasury Funds

Prendergast (2021) develops a methodology that enables retail investors to structure annuities using commonly available US Treasury Exchange Traded Funds (ETFs). This article extends that methodology through the use of key rate durations. Back tests and stress tests show that the use of key rate durations substantially enhances the ability of the portfolio of ETFs to replicate an annuity in an environment where yield curves undergo a variety of slope and curvature changes over time. <b>TOPICS:</b>Fixed income and structured finance, exchange-traded funds and applications, statistical methods <b>Key Findings</b> ▪ Relative to using single durations, the use of key rate durations dramatically improves the ability of an annuity replicating portfolio to provide the desired annuity payments while achieving a zero-ending balance. ▪ This article provides a means for retail investors, or their advisors, to design an annuity that is robust to interest rate changes without paying sizable fees to life insurance companies and other annuity providers.

Replicating Maximum-Yield Annuities with US Treasury Funds

Many individuals rely on annuity purchases to provide a steady stream of income during their post-retirement years. Life insurance companies provide a variety of annuity contracts, but they often come with sizable fees that are extremely detrimental to the purchaser in a low-interest-rate environment. This article develops a methodology that enables retail investors to structure annuities using commonly available US Treasury exchange-traded funds (ETFs) or mutual funds. Only historical price or NAV data and Treasury yield data are required to implement the strategy. The funds compose a dynamically managed portfolio requiring only an initial investment. Each period, the portfolio is rebalanced to match the modified duration of a reference annuity. This minimizes interest rate risk from parallel shifts in future yield curves. The weights are optimized to provide the highest yield possible. The objective is for the portfolio to provide the required periodic annuity payments and have a zero balance at annuity maturity under a variety of interest rate scenarios, while providing the maximum possible yield. Scenario tests show that the strategy is effective under parallel yield curve shifts, but may have shortfalls for curve steepening and gains for curve flattening. Investors may choose to add to their initial investment to reduce the risk of shortfalls if the curve steepens. The article concludes with an implementation of the strategy using actual Treasury ETFs. <b>TOPICS:</b>Retirement, exchange-traded funds and applications, fixed income and structured finance, performance measurement <b>Key Findings</b> ▪ US Treasury ETFs can be used to replicate an annuity, thus enabling retail investors to avoid the sizable fees charged by life insurance companies. ▪ The replication performs well for parallel yield curve shifts. ▪ Future work using key-rate durations may enhance the performance of the replication for non-parallel yield curve shifts.

A Key Rate Approach to Replicating Annuities with U.S. Treasury Funds

Prendergast (2021) develops a methodology that enables retail investors to structure annuities using commonly available U.S. Treasury Exchange Traded Funds (ETFs). This paper extends that methodology through the use of key rate durations. Back tests and stress tests show that the use of key rate durations substantially enhances the ability of the portfolio of ETFs to replicate an annuity in an environment where yield curves undergo a variety of slope and curvature changes over time.

A Nonparametric Approach to Bond Portfolio Immunization

We consider the problem of short term immunization of a bond-like obligation with respect to changes in interest rates using a portfolio of bonds. In the case that the zero-coupon yield curve belongs to a fixed low-dimensional manifold, the problem is widely known as parametric immunization. Parametric immunization seeks to make the sensitivities of the hedged portfolio price with respect to all model parameters equal to zero. However, within a popular approach of nonparametric (smoothing spline) term structure estimation, parametric hedging is not applicable right away. We present a nonparametric approach to hedging a bond-like obligation allowing for a general form of the term structure estimator with possible smoothing. We show that our approach yields the standard duration based immunization in the limit when the amount of smoothing goes to infinity. We also recover the industry best practice approach of hedging based on key rate durations as another particular case. The hedging portfolio is straightforward to calculate using only basic linear algebra operations.

A\xa0duration approach for the measurement of biometric risks in life insurance

Duration concepts are standard methods for measuring interest rate risks of portfolios, liabilities or other cash flows. Macaulay duration, effective duration and key-rate duration are widely used in practice according to different types of yield curves. In this paper, we will present a formulation for a forward rate duration measure by using multi-dimensional Taylor series approximation. It allows to measure the interest rate risk based on arbitrary forward rates. This approach can easily be adopted for biometric risk management, allowing the definition of a biometric duration. The biometric duration will be applied to actuarial present values of premiums and benefits as well as the actuarial reserve to quantify the corresponding biometric risk. It proves to be an easy-to-use tool in the actuarial practise.

The Key Rate Durations of Municipal Bonds

Effective duration, calculated by parallel shifts of the yield curve, is the standard measure of portfolio-based interest rate risk. Key rate durations, obtained by shifting individual key rates, describe how the risk is distributed along the term structure. Ordinarily the sum of the key rate durations equals the effective duration. However, in the case of tax-exempt municipal bonds the sum and the effective duration may differ. The reason is that the prices of discount munis are tax-affected, and the applicable tax rate depends on the size of the discount. The authors’ result has ramifications for the hedging of muni portfolios, and for the measurement of interest rate risk under the recently introduced SEC N-Port regulation. <b>TOPICS:</b>Project finance, statistical methods, credit risk management <b>Key Findings</b> • The tax effect on the price of a discount muni depends on the size of the discount (applicable tax rate around 20% if <i>de minimis</i>, 40% if not). • Because bumping individual key rates has smaller effects on price than shifting the entire yield curve, the applicable tax rates may differ. • Consequently key rate durations for munis may not add up to effective duration, creating challenges for portfolio hedging and SEC risk reporting.

Issues with the Smith–Wilson method

We analyse various features of the Smith–Wilson method used for discounting under the EU regulation Solvency II, with special attention to hedging. In particular, we show that all key rate duration hedges of liabilities beyond the Last Liquid Point will be peculiar. Moreover, we show that there is a connection between the occurrence of negative discount factors and singularities in the convergence criterion used to calibrate the model. The main tool used for analysing hedges is a novel stochastic representation of the Smith–Wilson method.

Mind the Solvency II Gap: A Coherent Measure of Market Consistent Embedded Value to Interest Rate Risk in ALM

This paper derives Asset and Liability Management ('ALM') interest rate risk measures (i.e. duration and convexity gaps) applicable to Life insurance companies that are adequate such as asset and liability-driven strategies can be used within a mix of tactics and instruments to achieve financial objectives, for a given set of ALM interest rate risk indicator tolerances. The approach adopted relies on a half-way approach between immunization (i.e. duration match) and dedication (i.e. cash flows match) that consists to match both the duration and convexity of the market value of assets and best estimate liabilities as well as the inclusion of partial and key-rate duration and convexity matches. The recent changes in regulation through the adoption of a market consistent framework (i.e. MCEV) combined to distressed economic conditions with failing equity markets, brutal changes in interest rates and spreads further increase the motivation for a rigorous and coherent definition of ALM risk measures in order to manage appropriately interest rate risk of Life insurance companies. Through this paper, effective and key-rate duration (and convexity) gaps are derived based on the interpretation of ALM risk indicators as approximation of interest rate sensitivities within a market consistent framework. Those concepts are discussed with respect to their usefulness for ALM of Life insurance companies. This paper is of high interest in the context of Solvency II given the lack of guidelines provided by EIOPA regarding the treatment and management of interest rate risk from an ALM perspective.

Alternative models for hedging yield curve risk: An empirical comparison

We test alternative models of yield curve risk by hedging US Treasury bond portfolios through note/bond futures. We show that traditional implementations of models based on principal component analysis, duration vectors and key rate duration lead to high exposure to model errors and to sizable transaction costs, thus lowering the hedging quality. Also, this quality randomly varies from one model and hedging problem to the other. We show that accounting for the variance of modeling errors substantially reduces both hedging errors and transaction costs for all considered models. Additionally, it leads to much more stable weights in the hedging portfolios and – as a result – to more homogeneous hedging quality. On this basis, error-adjusted principal component analysis is found to systematically and significantly outperform alternative models.

A Guide to Duration, DV01, and Yield Curve Risk Transformations

Duration and DV01 (dollar duration) measure price sensitivity and provide the basic risk measure for bonds, swaps, and other fixed income instruments. When valuing instruments off a yield curve, duration and DV01 naturally extend to a vector of partial DV01s or durations (key rate durations) and these are widely used in the finance industry. But partial DV01s or durations can be measured with respect to different rates: forwards, par rates, zero yields, or others. This paper reviews the concepts of partial DV01 and duration and then discusses a simple method for transforming partial DV01s between different rate bases and provides examples. The benefit of this transformation method is that it only requires calculating the risk of a small set of alternate instrument and does not require re-calculating the original portfolio risk. (This paper is also available in an interactive version with enhanced digital content - see references.)

Alternative Models for Hedging Yield Curve Risk: An Empirical Comparison

We develop alternative models for hedging yield curve risk and test them by hedging US Treasury bond portfolios through note/bond futures. We show that traditional implementations of models based on principal component analysis, duration vectors and key rate duration lead to high exposure to model errors and to sizable transaction costs, thus lowering the hedging quality. Also, this quality varies from one test case to the other, so that a clear ranking of the models is not possible. We show that accounting for the variance of modeling errors substantially reduces both hedging errors and transaction costs for all considered models. Also, this allows to clearly rank these models: error‐adjusted principal component analysis systematically and significantly outperforms alternative models.

Assessing Interest Rate Risk Beyond Duration – Shift, Twist, Butterfly

High fiscal deficits, a sharp rise in the issuance of sovereign debt from major developed economies, rising inflation expectations, and possible changes in central bank rates could cause the yield curves from around the world to change significantly. In the US, the yield curve has already experienced significant steepening in the past twelve months. This environment poses challenges for Fixed Income professionals who need to address possible non-parallel changes to the term structure. This paper illustrates the capabilities of Shift-Twist-Butterfly (STB) factor models to help address these challenges. We provide a longer-term perspective on term structure changes in the Euro zone, US and Japan. Using four portfolio case studies, we show that the use of risk measures such as duration, convexity or key rate durations have some limitations and may not be very efficient. Therefore we show how these limitations can be efficiently overcome by the complementary use of advanced fixed income risk models based on STB interest rate risk factors.

Duration and Bond Return Approximation: The Quasi-Convexity Effect

Duration is often applied to relate bond price changes to changes in the yield to maturity (or key interest rates). As the relationship between bond price and yield is non-linear, convexity characteristics can be used to improve the linear first order approximation. In this paper, we show that knowledge of a bond’s duration (or key rate durations) allows a better price return approximation than is suggested in the literature. The proposed approximations may be helpful in Value-at-Risk analyses where duration (and convexity) approximations are used as fast alternatives for full revaluation. Our main approximation formula is based on only duration but incorporates quasi-convexity characteristics. This signifies a substantial improvement in approximation accuracy, even for substantial yield changes. The approximants based on duration and convexity are virtually exact, even for extreme yield changes.

Managing Interest Rate Risk: The Next Challenge?

Are the managers of financial institutions ready for the small but increasingly significant risk of inflation in the near future, due to the unprecedented fiscal and monetary responses of the U.S. government to prevent an economic collapse? This paper addresses this important issue by reviewing important findings in the area of interest rate risk management. We discuss five classes of models in the fixed income literature that deal with hedging the risk of large, non-parallel yield curve shifts. These models are given as M-Absolute/M-Square models, duration vector models, key rate duration models, principal component duration models, and extensions of these models for fixed income derivatives, for valuing and hedging bonds, loans, demand deposits, and other fixed income instruments. These models can be used for designing various hedging strategies such as portfolio immunization, bond index replication, duration gap management, and contingent immunization, to protect against changes in the height, slope, and curvature of the yield curve. We argue that the current regulatory models proposed by the U.S. Federal Reserve, the Office of Thrift Supervision, and the Bank of International Settlements, may understate the true interest rate risk exposure of financial institutions, if sharp increases in interest rates lead to higher default risk and quickening of the pace of deposit withdrawals.

Multifactor Models for Managing Interest Rate Risk

How do the managers of financial institutions hedge against the effects of non-parallel yield curve shifts? This paper addresses this important issue by reviewing the important findings in the area of interest rate risk management over the past two decades. We discuss four classes of models in the fixed income literature that deal with hedging the risk of large, non-parallel yield curve shifts. These models are given as M-Absolute/M-Square models, duration vector models, key rate duration models, and principal component duration models. These models can be used for designing various passive strategies such as portfolio immunization, bond index replication, and duration gap management; and hybrid strategies (i.e., active/passive) such as targeted yield curve shifts speculation (based on change in either the height, and/or the slope, and/or the curvature of the yield curve) and contingent immunization.

Generalized Ho-Lee Model

This article presents a multifactor arbitrage-free interest rate lattice model. The model uses a state and time dependent implied volatility function. In calibrating this function from the set of swaption prices over a range of tenors and expiration, the interest rate model can capture surface not just the perceived volatility but the distributional form, in terms of the mix of lognormal and normal distributions, and the correlations of the rates. Swaptions are actively traded in the market that reflects the market views on the yield curve stochastic movements. Interest rate models tend to fix a distributional form or the correlation matrix of the rates, and therefore, they fail to subtract the market views from the swaption prices. This article provides a model that overcomes these crucial assumptions and it can be shown to provide significant explanatory power and computational efficiency. This model interest rate process has a number of desirable properties: (1) Consistent with historical rate experiences, with a mean reversion process, non-explosive rates, and truncated low rates. (2) A binomial recombining lattice provides efficient algorithms for American options and stratified sampling in valuing path dependent securities. (3) Straightforward to specify the multi-factor model and to specify the key rate durations and key rate vegas for risk management. The model is tested in this article empirically using monthly swaption prices over a one year sample period. The results show that the model can be calibrated to the observed swaption prices quite accurately.

Managing Interest Rate Volatility Risk

Interest rate contingent claims such as swaptions, caps and floors, callable bonds, mortgage-backed securities and many balance sheet items are subject to vega risk. Vega risk is defined as the interest rate volatility risk of an interest rate contingent claim, the uncertain change in volatility affecting the embedded option value. Since these contingent claims9 values are affected by the interest rate volatilities and these volatilities are stochastic, any hedging strategies, or more generally risk management, of these interest rate contingent claims should take the vega risk into account. However, based on market conventions, the interest rate volatilities are represented by a set of approximately 120 implied volatilities of the at-the-money swaptions over a range of tenors and expiration dates, called the volatility surface. And this volatility surface is stochastic. Controlling for these vega risks poses a significant practical challenge to managing these large number of risk sources. Thus far, there is a lack of solutions. This article introduces the key rate vega measures to manage the vega risks. Furthermore, the article shows how the key rate vega measures can depict the option risks embedded in an interest rate contingent claim and how they can be used for hedging a mortgage interest-only strip using key rate vega measures and key rate duration measures together, as an illustration of broader applications of the key rate vega measures.

Convexity Analysis of Option Embedded Bonds

Interest rate risk hedge strategies usually assume that term structure of interest rates in moving under a specific way. If a real term structure is moving differently from the assumed term structure movement, the interest rate risk hedge strategies assuming the special term structure movement may incur unexpected large loss for a bond portfolio manager. Hence an interest rate risk hedge strategy which could be effective under various types of term structure movements has been strongly needed by bond portfolio managers. Duration vector strategies have been developed to satisfy this practical need. To allow various types of term structure movements, duration vector strategies assume multi-factor models for the term structure movement. When a duration vector strategy is considered as a generalization of a duration strategy which is a single factor model for the term structure movement, there will be a generalized concept which measures convexity of a bond under the duration vector model. This study identifies the convexity property of an option embedded bond portfolio under ‘key rate duration model‘ which is a kind of duration vector model suggested by Ho (1992).

A Note on Common Interest Rate Risk Measures

Portfolio managers, traders, and risk managers need measures to quantify exposure to changes in interest rates. Two measures beyond duration that take into account interest rate exposure due to a change in the yield curve are partial durations and key rate durations. Par curve changes are used to compute partial durations, and the spot curve is used to compute key rate durations. Hence, in the case of partial durations, trading strategies and risk analysis are based on par yield changes and only indirectly on the spot curve; users of key rate durations base their strategies directly on the spot curve. Users of these sensitivity measures must understand the relationships between the two types of measures, as the results for the approaches often differ significantly. The analysis illustrates the differences between computing level, slope, and curvature durations using the par and the spot curve, and briefly examines the impact of these methodologies on implied forward rates.