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Z-spread
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Z-spread
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The Z-spread, ZSPRD, zero-volatility spread, or yield curve spread of a bond is the parallel shift or spread over the zero-coupon Treasury yield curve required for discounting a predetermined cash flow schedule to arrive at its present market price. The Z-spread is also widely used in the credit default swap (CDS) market as a measure of credit spread that is relatively insensitive to the particulars of specific corporate or government bonds.

Since the Z-spread uses the entire yield curve to value the individual cash flows of a bond, it provides a more realistic valuation than an interpolated yield spread based on a single point of the curve, such as the bond's final maturity date or weighted-average life. However, the Z-spread does not incorporate variability in cash flows, so a fuller valuation of an interest-rate-dependent security often requires the more realistic (and more complicated) option-adjusted spread (OAS).

Definition

[edit]

The Z-spread of a bond is the number of basis points (bp, or 0.01%) that one needs to add to the Treasury yield curve (or technically to Treasury forward rates) so that the Net present value of the bond cash flows (using the adjusted yield curve) equals the market price of the bond (including accrued interest). The spread is calculated iteratively.

For a mortgage-backed security, a projected prepayment rate tends to be stated; for example, the PSA assumption for a particular MBS might equate a particular group of mortgages to an 8-year amortizing bond with 6% mortality per annum. This gives a single series of nominal cash flows as if the MBS were a riskless bond. If these payments are discounted to net present value (NPV) with a riskless zero-coupon Treasury yield curve, the sum of their values will tend to overestimate the market price of the MBS. This difference arises because the MBS market price incorporates additional factors such as liquidity and credit risk and embedded option cost.

Benchmark for CDS basis

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The Z-spread is widely used as the "cash" benchmark for calculating the CDS basis. The CDS basis is commonly the CDS fee minus the Z-spread for a fixed-rate cash bond of the same issuer and maturity. For instance, if a corporation's 10-year CDS is trading at 200 bp and the Z-spread for the corporation's 10-year cash bond is 287 bp, then its 10-year CDS basis is –87 bp.

Example

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Assume that on 7/1/2008:

  • A bond has three future cash flows: $5 on 7/1/2009; $5 on 7/1/2010; $105 on 7/1/2011.
  • The corresponding zero-coupon Treasury rates (compounded semi-annually) are: 4.5% for 7/1/2009; 4.7% for 7/1/2010; 5.0% for 7/1/2011.
  • The bond's accrued interest is 0.
  • The Z-spread is 50 bp.

Then the price P of this bond on 7/1/2008 is given by:

where (for simplicity) the calculation has ignored the slight difference between parallel shifts of spot rates and forward rates.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Z-spread

View on Grokipedia
from Grokipedia
The zero-volatility spread (Z-spread), also known as the static spread, is a financial metric used in analysis to measure the constant spread that must be added to each point on the risk-free zero-coupon —typically the spot rate curve—such that the of a bond's expected flows equals its observed market . This spread quantifies the additional yield compensation demanded by investors for risks such as credit default, constraints, and other non- factors, assuming no changes in volatility. Unlike simpler yield measures, the Z-spread accounts for the entire term structure of s by applying the same parallel shift to every maturity along the curve, making it particularly useful for bonds with non-standard patterns, such as callable securities or mortgage-backed securities (MBS). To calculate the , analysts solve for the constant spread ZZ in the pricing equation where the bond's price PP is the discounted value of its cash flows CFtCF_t using adjusted spot rates rt+Zr_t + Z, often with semi-annual :
P=∑t=1TCFt(1+rt+Zm)m⋅tP = \sum_{t=1}^{T} \frac{CF_t}{(1 + \frac{r_t + Z}{m})^{m \cdot t}}
Here, rtr_t is the spot rate for period tt, mm is the frequency (e.g., 2 for semi-annual), and TT is the bond's maturity. This typically requires an iterative , such as Newton-Raphson, to find ZZ that equates the model's price to the market price, as no closed-form solution exists for most bonds. For instance, a trading at a premium might exhibit a lower or even negative Z-spread if its covenants or provide advantages over Treasuries, while riskier issuers show higher spreads to reflect elevated default probabilities. The Z-spread plays a critical role in bond valuation and relative value analysis, enabling investors to compare securities across different maturities and credit qualities on an apples-to-apples basis against a benchmark curve. It is especially valuable for assessing credit spreads in corporate and structured finance markets, where it serves as a proxy for the market's pricing of default risk adjusted for recovery rates and term structure effects—approximately equating to the hazard rate times (1 - recovery rate) under simplified models. However, its "zero-volatility" assumption ignores embedded options like prepayment or call features, which can lead to mispricing for volatile instruments; in such cases, the option-adjusted spread (OAS) is preferred as it incorporates stochastic interest rate paths. Limitations include sensitivity to the choice of benchmark curve (e.g., Treasury vs. SOFR swap curve) and compounding conventions, which can cause discrepancies of several basis points in reported values. Overall, the Z-spread remains a foundational tool in portfolio management and risk assessment, widely computed by platforms like Bloomberg for real-time trading decisions.

Definition and Basics

Definition

The Z-spread, also known as the zero-volatility spread or static spread, is a key metric in analysis that quantifies the additional yield a bond offers over the to account for various risks. To understand it, one must first grasp foundational concepts: spot rates represent the market discount rates for default-risk-free zero-coupon bonds, derived from the yields on Treasury securities of varying maturities, forming the benchmark . discounting, in turn, calculates the current worth of future cash flows by adjusting them using these spot rates, reflecting the and the of capital. At its core, the Z-spread is defined as the constant spread added to each point along the risk-free spot rate curve—typically the U.S. Treasury spot curve—to make the of a bond's expected cash flows (including coupons and principal) equal to its observed market price. This parallel shift ensures that the discounting process incorporates the bond's full term structure, providing a more accurate reflection of its pricing relative to the benchmark than simpler measures like . By capturing this uniform premium across all maturities, the Z-spread effectively measures the compensation investors demand for bearing the bond's (the potential for issuer default) and (the ease of trading the security without price impact), as well as other factors like embedded options in more complex bonds. Unlike maturity-specific spreads, it accounts for the shape of the , offering a comprehensive view of the bond's risk-adjusted yield advantage. The Z-spread emerged in the late and as part of the of advanced analytics, developed to overcome the shortcomings of basic yield-based measures that ignored the term structure and volatility effects in bond pricing. This period saw growing market complexity in corporate and structured debt, prompting the need for precise tools to evaluate relative value and risk premia in portfolios.

Importance in Fixed Income Analysis

The Z-spread serves as a fundamental tool in analysis, allowing portfolio managers to conduct apples-to-apples comparisons of bonds exhibiting varying maturities, structures, and embedded options against a non-flat , thereby providing a more accurate measure of relative value and than simpler yield-based metrics. By incorporating the entire spot rate curve, it quantifies the constant parallel shift required to equate a bond's of cash flows to its market price, enabling better-informed decisions on risk-adjusted returns in diverse market environments. This approach is particularly valuable for institutional investors seeking to optimize bond selections amid fluctuating term structures. Since the late , the Z-spread has seen widespread adoption among institutional investors, portfolio managers, and analysts for enhanced precision in assessment, reflecting the maturation of term structure models that moved beyond rudimentary parallel shift assumptions. Its integration into major market platforms, such as Bloomberg terminals, has further solidified its role, with dedicated functions like the Asset Swap Spread (ASW) calculator facilitating real-time Z-spread computations alongside other spread measures for bond evaluation and hedging. Similarly, management systems, including Bloomberg's Trade Order Management System (TOMS), decompose credit curves into Z-spread components to isolate and track risky elements, aiding in comprehensive portfolio . This transition underscored its utility in measuring premiums over the benchmark , supporting applications in portfolio and duration matching without oversimplifying shapes.

Calculation

Methodology

The methodology for computing the Z-spread involves a structured that relies on the spot rate to discount a bond's flows, adjusting for a constant spread until the matches the observed market price. This approach assumes zero volatility, meaning no fluctuations in rates are considered during the , and a parallel shift in the , where the same spread is added uniformly to each spot rate across all maturities. The first step is to obtain the spot rate curve for the maturities corresponding to the bond's timings. This curve is typically derived from U.S. STRIPS, which provide zero-coupon yields directly applicable as spot rates, and can be sourced from official U.S. Department of the data releases. For practical implementation, interpolated spot rate curves are often accessed via financial data terminals such as Bloomberg or , ensuring alignment with current market conditions. The second step requires listing the bond's projected cash flows, including periodic payments and repayment at maturity, along with their exact timings in years from the settlement date. These cash flows are determined based on the bond's rate, , and payment schedule, providing the foundational inputs for discounting. The third step employs iterative numerical methods to determine the constant spread, denoted as z, that equates the discounted value of these cash flows to the bond's current market . Common techniques include the Newton-Raphson method, which converges efficiently by successively refining an initial guess for z through root-finding iterations on the pricing function. This process is typically performed using financial software or spreadsheets to handle the non-linear solving requirements.

Mathematical Formulation

The Z-spread, or zero-volatility spread, is mathematically defined through the bond pricing equation that incorporates a constant parallel shift to the benchmark spot rate . For a bond with cash flows CFtCF_t at times t=1,2,…,Nt = 1, 2, \dots, N (where tt is in years), the theoretical price PP is given by P=∑t=1NCFt(1+st+zm)mt,P = \sum_{t=1}^{N} \frac{CF_t}{\left(1 + \frac{s_t + z}{m}\right)^{m t}}, where sts_t is the spot rate for time tt derived from the benchmark (typically the ), zz is the constant Z-spread in the same units as the spot rates (decimal form), and mm is the compounding frequency per year (e.g., m=2m=2 for semi-annual ). This formulation assumes discrete and fixed cash flows, with the Z-spread added uniformly to each spot rate to reflect and premia over the risk-free . To derive the Z-spread, is set equal to the observed market price PmP_m of the bond: Pm=∑t=1NCFt(1+st+zm)mt.P_m = \sum_{t=1}^{N} \frac{CF_t}{\left(1 + \frac{s_t + z}{m}\right)^{m t}}. Solving for zz requires numerical iteration because is nonlinear in zz; the discount factors depend on powers of (1+st+zm)mt\left(1 + \frac{s_t + z}{m}\right)^{m t}, precluding a closed-form solution. Common methods include the Newton-Raphson , which uses the (related to the bond's duration) to converge quickly, or for robustness, typically achieving precision within basis points in few iterations. The plain Z-spread formulation ignores embedded options in the bond, such as call or put features, by assuming deterministic cash flows and zero volatility; this can overstate the spread for option-embedded securities. In contrast, the option-adjusted spread (OAS) extends this model by incorporating s to value the option component separately. Changes in the Z-spread influence the bond's sensitivity measures. The spread duration, which quantifies the percentage price change for a 100 parallel shift in the Z-spread, approximates the modified duration computed using the shifted curve (st+z)(s_t + z). An increase in zz effectively raises discount rates, shortening the Macaulay duration and reducing convexity, as later cash flows are discounted more heavily relative to earlier ones.

Applications

Bond Valuation

The Z-spread serves as a key tool in valuing non- bonds, such as corporate and municipal securities, by incorporating a constant spread over the spot rate to discount the bond's expected cash flows to their , thereby estimating the bond's fair market price. This method ensures that the of coupons and principal equals the observed market price, capturing the additional yield required for credit and liquidity risks beyond the . Unlike simpler yield spreads, the Z-spread applies uniformly to each spot rate along the , providing a more precise reflection of the bond's pricing in relation to the entire term structure. In practical scenarios, such as valuing corporate bonds where intensifies with longer maturities due to heightened default probabilities over time, the Z-spread adjusts the process to account for this varying profile across dates. For instance, a with staggered cash flows might exhibit a Z-spread that highlights how market-implied credit premia evolve, enabling investors to assess whether the bond's price adequately compensates for maturity-specific uncertainties. This application is particularly valuable for fixed-rate instruments, where the Z-spread's zero-volatility assumption simplifies valuation while aligning with observed market dynamics. Z-spreads facilitate relative value analysis by allowing comparisons of spreads across issuers within the same sector or rating category, helping to pinpoint overvalued bonds (with narrower spreads relative to peers) or undervalued opportunities (with wider spreads indicating potential mispricing). Traders and portfolio managers use these comparisons to construct strategies that exploit discrepancies, such as favoring issuers with historically tight Z-spreads that may signal undervaluation amid similar profiles. This approach enhances decision-making in fixed-income markets by emphasizing the term structure of spreads over aggregate metrics. Post-2008 , credit spreads including Z-spreads have informed frameworks for bond portfolios by helping to model valuation adjustments under hypothetical scenarios involving distorted yield curves and expanded credit spreads, such as sharp widenings in yields relative to Treasuries. This approach aids in assessing resilience to economic downturns and strains observed in past crises.

Credit Risk Assessment

The Z-spread serves as a key metric for decomposing the total yield premium on a bond into components related to , premium, and tax effects, enabling investors to isolate the portion attributable to the issuer's default probability and recovery expectations from other market frictions. A analysis demonstrates that the non- elements, encompassing and regulatory or tax influences, tend to rise alongside the component, amplifying overall spreads during periods of heightened uncertainty. This decomposition is particularly valuable for attributing spread changes to fundamental credit deterioration versus transient strains, as liquidity premia can account for 25-40% of observed spreads in structural models of . Monitoring Z-spread dynamics provides a real-time indicator of evolving credit quality, with widening spreads signaling potential deterioration in issuer fundamentals or market perceptions of default risk, while narrowing reflects improving conditions. For instance, during the 2020 market stress, Z-spreads on ineligible corporate bonds issued by affected firms spiked by over 100 basis points in early 2020, reflecting acute concerns over evaporation and heightened default probabilities amid economic lockdowns. Such movements underscore the Z-spread's sensitivity to systemic shocks, allowing portfolio managers to adjust exposures proactively based on spread trajectories. In sector applications, Z-spreads vary significantly by , with high-yield bonds exhibiting substantially higher levels than investment-grade counterparts to compensate for elevated default risks. Empirical observations indicate that average Z-spreads for BBB-rated investment-grade corporate bonds hovered around 150 basis points in 2023, compared to 300-500 basis points for high-yield issues, highlighting the gradient across the spectrum. This differentiation aids in sector allocation, as higher spreads in non-investment-grade segments capture both greater expected losses and during volatile periods. Regulatory frameworks leverage Z-spreads to inform provisioning by deriving implied probabilities of default from spread levels, adjusting for recovery assumptions to estimate expected losses. Under , Z-spreads contribute to spread risk modeling and the calculation of fundamental spreads, which isolate credit components for capital requirements on bond holdings. Similarly, in implementations, Z-spread-implied default probabilities support forward-looking expected credit loss provisions, enhancing the accuracy of reserves for financial institutions.

CDS Basis Trades

The CDS-bond basis measures the difference between the premium on a credit default swap (CDS) and the Z-spread of the underlying with matching maturity, serving as an indicator of relative mispricing between the cash bond and synthetic CDS markets. This basis is typically calculated as the CDS spread minus the bond's Z-spread, where a negative value implies the CDS is trading "cheap" relative to the bond, often due to differences in , risk, or costs. Over the long term, the median CDS-bond basis for investment-grade bonds has been around -19 basis points, reflecting persistent structural frictions in the markets. In CDS basis trades, arbitrageurs exploit this differential by taking offsetting positions in the bond and CDS. For a negative basis (Z-spread > CDS spread), traders go long the basis by purchasing the cash bond—earning the higher Z-spread—and selling CDS protection, receiving the lower CDS premium; the net carry is positive, with profits realized upon basis convergence through maturity or unwinding. Conversely, for a positive basis (CDS spread > Z-spread), traders go short the basis by shorting the bond (or using a bond future) and buying CDS protection, profiting if the mispricing narrows. These trades are hedged against via swaps and against migration via the matched reference entity, though execution requires capacity due to regulatory constraints on leverage. Historical episodes highlight the basis's sensitivity to market stress. During the , the CDS-bond basis for investment-grade bonds turned persistently negative, widening to extremes of around -250 basis points by late 2008 amid liquidity shortages, counterparty fears, and forced unwinds by leveraged funds. Similar, though less severe, blowouts occurred during periods of heightened volatility, such as the 2020 shocks and 2022 rate hikes amid inflation pressures, where funding strains amplified divergences; for instance, investment-grade bases averaged approximately -50 basis points in 2024 under normalized conditions. The Z-spread plays a central role as the benchmark for the "" leg in basis calculations, providing a zero-volatility adjustment to isolate from curve effects in the bond pricing. In practice, trade profitability incorporates adjustments for funding costs, such as repo rates for the bond leg versus collateralized CDS posting, which can widen the effective basis by 20-50 basis points post-crisis due to regulations like the Supplementary Leverage Ratio.

Comparisons with Other Spreads

Versus Nominal Spread

The nominal spread is defined as the simple arithmetic difference between the yield to maturity (YTM) of a fixed-income and the YTM of a benchmark with matching maturity. This measure uses a single benchmark yield for comparison, ignoring the term structure of spot rates across the bond's cash flow timings, which simplifies calculations but introduces inaccuracies when the is sloped. In comparison, the Z-spread represents a constant parallel shift added to every point along the spot rate to ensure the of the bond's cash flows equals its observed market price. The primary distinction arises in handling non-flat yield : the nominal spread ignores by relying on a single YTM point, whereas the Z-spread incorporates the full term structure, providing a more precise reflection of credit and liquidity premia. For premium bonds (trading above par) in an upward-sloping environment, the Z-spread exceeds the nominal spread because earlier coupon payments are discounted at lower short-term spot rates, necessitating a larger constant adjustment to replicate the bond's price. This discrepancy highlights the nominal spread's tendency to understate risk in sloped markets, as seen in practitioner examples where Z-spreads are 1-10 basis points higher than nominal spreads for intermediate maturities, with divergences amplifying for longer durations due to greater exposure to shape. Consequently, the nominal spread suits rapid initial screenings of relative value, while the Z-spread is essential for detailed portfolio analysis and valuation where dynamics matter.

Versus G-Spread

The G-spread, a form of nominal spread over an interpolated benchmark, measures the difference between a bond's and the interpolated yield corresponding to the bond's exact maturity date, relying on a single reference point on the . is used when no security matches the bond's maturity exactly. In contrast to the basic nominal spread, this approach accounts for non-standard maturities but still overlooks variations in the across the bond's timings. A key limitation of the G-spread arises when the is not flat, as it fails to account for the intra-maturity shape, resulting in potential mispricing for bonds with uneven or distributed cash flows, such as amortizing securities or those trading away from par. For instance, in an upward-sloping curve, the G-spread tends to understate the true credit premium because early cash flows are discounted implicitly at lower rates without adjustment, whereas the Z-spread ensures consistent risk compensation across all periods. The G-spread proves adequate for short-term instruments or par bonds where cash flows align closely with a single maturity point and curve distortions are minimal, facilitating quick relative value assessments against Treasuries. However, for more intricate fixed-income products like long-dated corporate bonds spanning 10 to 30 years, the Z-spread is favored due to its superior handling of curve dynamics and provision of a more reliable indicator of embedded and risks. Empirical examples illustrate the quantitative divergence: in one analyzed case, the G-spread measured 82 s while the Z-spread was 99 basis points, highlighting a 17 basis point gap attributable to curve shape effects. Such discrepancies can range from 10 to 50 basis points in steep yield environments, underscoring the G-spread's tendency to understate spreads relative to the Z-spread when short- and long-term rates diverge significantly.

Versus Option-Adjusted Spread

The option-adjusted spread (OAS) represents the Z-spread adjusted to account for the value of embedded options in a bond, such as call or put features, by incorporating volatility through modeling techniques like binomial trees or simulations. Unlike the Z-spread, which assumes zero volatility and treats embedded options as having fixed cash flows, the OAS isolates the option's cost, resulting in a lower spread value for the ; for callable bonds, this difference—known as the option cost—can range from 50 to 100 basis points or more in high-volatility environments, as the Z-spread embeds compensation for the option risk. OAS is essential for securities with significant optionality, such as mortgage-backed securities (MBS) where prepayment risk alters cash flows or corporate bonds with call provisions that allow early redemption, enabling better relative value comparisons; in contrast, the Z-spread suffices for option-free straight bonds where no such adjustments are needed. Building on the Z-spread framework, OAS emerged in the early for valuing complex instruments like collateralized obligations (CMOs), with its application expanding post-2008 to enhance risk assessment in structured products amid heightened awareness of option-related vulnerabilities.

Advantages and Limitations

Advantages

The Z-spread accounts for the term structure of interest rates by adding a constant spread to each spot rate on the yield curve, enabling more accurate discounting of a bond's cash flows across varying maturities compared to nominal spreads, which rely on a single benchmark yield. A key advantage of the Z-spread is its ability to standardize comparisons across non-callable bonds, regardless of maturity differences, by accounting for spot rate variations along the curve rather than assuming a flat yield environment. This comparability facilitates consistent relative value assessments in fixed income portfolios, particularly for bonds with irregular cash flow timings.

Limitations

The Z-spread's zero-volatility assumption posits that rates remain constant over the bond's , failing to incorporate potential changes in rates that could affect cash flow timing, particularly for bonds with embedded options like callables or putables. This limitation can lead to inflated spread estimates, as it does not account for the volatility-driven value of these options, making the measure less suitable for such securities where the option-adjusted spread (OAS) provides a more accurate alternative by modeling rate paths. Additionally, the Z-spread incorporates a parallel shift by adding a constant spread across the entire spot rate curve, which assumes uniform movement in yields regardless of maturity. This assumption proves inaccurate in non-parallel yield curve environments, where differential shifts occur, such as steeper short-end increases relative to long-end rates, potentially misrepresenting the bond's true premium. The calculation of the Z-spread demands significant computational resources, involving iterative of the full spot rate curve and repeated discounting of s until the bond's price matches the . This complexity renders it less accessible for retail investors or those without advanced financial software, as it requires precise input like spot rates and cash flow schedules. In illiquid markets, the Z-spread's effectiveness diminishes due to unreliable spot rate data and sparse bond pricing, which underpin the benchmark curve. For instance, in emerging markets prior to 2020, where trading volumes were low and government yield curves often lacked depth, the measure could produce distorted spreads that overstate or understate owing to incomplete market information.

Examples

Basic Bond Pricing Example

Consider a hypothetical 5-year corporate bond with a 5% annual coupon rate and a par value of 100, currently priced at 98 per 100 par. The benchmark Treasury spot rates for the cash flow maturities are assumed to be 2% for year 1, 2.5% for year 2, 3% for year 3, 3.5% for year 4, and 4% for year 5. The bond's cash flows are $5 at the end of each of the first four years and $105 at maturity in year 5. The Z-spread is the constant basis point addition ss to each spot rate that discounts these cash flows to the market price: 98=∑t=145(1+zt+s)t+105(1+z5+s)598 = \sum_{t=1}^{4} \frac{5}{(1 + z_t + s)^t} + \frac{105}{(1 + z_5 + s)^5} where ztz_t denotes the spot rates for each period tt. Solving this equation iteratively—for instance, via trial and error or optimization algorithms—yields a Z-spread of approximately 157 basis points. With s=0.0157s = 0.0157, the adjusted discount rates are 3.57% for year 1, 4.07% for year 2, 4.57% for year 3, 5.07% for year 4, and 5.57% for year 5, resulting in a present value of approximately 98. This 157 basis point Z-spread represents the parallel shift over the Treasury curve required to price the bond, reflecting an embedded premium for credit risk and reduced liquidity relative to Treasuries. The computation can be replicated in using the Goal Seek tool to vary ss until the discounted cash flows equal 98, or in Python via libraries like SciPy's optimize module for root-finding.

CDS Basis Example

Consider a hypothetical involving a 5-year where the Z-spread measures 200 basis points, while the CDS premium for the same issuer and maturity stands at 150 basis points, producing a CDS basis of +50 basis points (calculated as the Z-spread minus the CDS premium). This deviation signals a potential trading opportunity, as the bond appears relatively undervalued compared to the CDS-implied . The basis trade entails purchasing the bond on a notional amount (e.g., $10 million) and simultaneously buying CDS protection for the same notional to hedge default risk, effectively creating a synthetic risk-free position. The position is financed through repo borrowing against the bond collateral, typically at a rate near the risk-free benchmark plus a small premium. Assuming a risk-free rate of 3%, the bond's effective yield becomes approximately 5% (3% + 200 bps), offset by the 150 bps CDS premium payment, yielding a gross carry of 50 bps before financing costs. To enhance realism, adjustments for carry costs are essential, including the repo rate differential (often 5–20 bps above the due to and factors) and transaction fees (around 5–10 bps annually). If the repo rate is 3.10%, the net carry drops to about 39.9 bps ($39,900 annually on $10 million notional), providing a low-risk stream while awaiting basis convergence. Should the basis normalize to zero—through the bond's Z-spread tightening to 150 bps—the realizes additional profit from bond price appreciation (estimated at roughly 2.3% for a 5-year duration under parallel shift assumptions). In conditions akin to 2023, marked by initial corporate credit spread widening amid monetary tightening and banking stresses followed by subsequent tightening as markets stabilized, this convergence could amplify returns, combining carry with capital gains of 1–3% over the holding period.

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  42. https://www.investopedia.com/ask/answers/052115/what-difference-between-optionadjusted-spread-and-zspread-reference-mortgagebacked-securities-mbs.asp
  43. https://analystprep.com/study-notes/cfa-level-2/explain-the-calculation-and-use-of-option-adjusted-spreads/
  44. https://quant.stackexchange.com/questions/3319/what-is-the-difference-between-option-adjusted-spread-oas-and-z-spread
  45. https://thetradinganalyst.com/option-adjusted-spread/
  46. https://business.purdue.edu/faculty/mcconnell/publications/The-Origins-and-Evolution-of-the-Market.pdf
  47. https://www.soa.org/globalassets/assets/files/newsroom/research-2009-fin-crisis.pdf
  48. https://analystprep.com/study-notes/cfa-level-iii/merits-and-demerits-of-credit-spread-measures/
  49. https://fastercapital.com/topics/limitations-and-challenges-of-yield-spread-analysis.html
  50. https://www.[investopedia](/page/Investopedia).com/terms/z/zspread.asp
  51. https://www.bis.org/publ/work631.pdf
  52. https://personal.lse.ac.uk/shang/job_market_paper_qi_shang.pdf
  53. https://arxiv.org/pdf/0912.4618
  54. https://content.naic.org/sites/default/files/capital-markets-special-report-ye2023wrapup.pdf
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