# Fixed Income Portfolio Measures of Risk, Return and Correlation

Macaulay duration represents the weighted average time to receive a bond's promised payments. It serves as both a sensitivity and time measurement.

Modified duration is derived from the Macaulay duration statistic divided by one plus the yield per period. This formula estimates the percentage change in a bond's price given a change in its yield to maturity.

Effective duration measures a bond's price sensitivity to changes in the yield curve using a benchmark or relevant index's yield curve. Complex bonds with uncertain future cash flows must assess interest rate risk.

Duration of the key rate (or partial duration) indicates a bond's sensitivity to changes in the benchmark yield curve at a specified maturity point or segment. Bond or portfolio durations offer insights into the “shaping risk” – sensitivity to changes in the shape of the benchmark yield curve.

Empirical duration, determined from market data through regression analysis, quantifies interest rate sensitivity. It calculates the slope between interest rate changes and bond price changes.

Money duration measures the price change in the bond's denominated currency. It can be expressed per unit or per 100 of its par value. The term “dollar duration” is commonly used in the United States.

The price value of a basis point (PVBP) estimates a bond's price change for a 1-basis point (bps) yield change. Scaling money duration helps interpret PVBP as the money gained or lost for each 1-basis point change in the reference interest rate. It is also known as “basis point value” (BPV), calculated as money duration times 0.0001 (1 bp).

Convexity describes a bond's price behavior as a second-order effect, primarily relevant for more significant yield movements. It captures deviations from the linear relationship between yield and price. Bonds with positive convexity will have a higher rate of return than those with lower convexity and the same duration if interest rates change. Investors benefit from convexity but must ‘purchase’ it, usually through lower yield to maturity. The nominal convexity calculation assumes constant cash flows when yields to maturity change.

Effective convexity is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve. It determines new prices using a pricing model with upward (PV+) and downward (PV$$^\ast$$) shifts of the benchmark curve by the same amount ($$^\ast$$Curve) while holding other factors constant.

## Portfolio Measures of Risk and Return

Bond portfolio duration measures how sensitive a bond portfolio is to changes in interest rates. It's noteworthy that it may not fully capture convexity effects. To calculate it, one averages the duration of individual bonds in the portfolio or the time to receipt of the aggregate cash flows.

Modified duration of a bond portfolio gauges the percentage change in the market value of a bond for a 1% change in yield to maturity. For example, if the modified duration of a portfolio is 20, a 100 basis point (bp) increase or decrease in yield to maturity would lead to an expected 20% decrease or increase in the portfolio's market value.

The convexity of a bond portfolio represents the second-order effect of the rate of change in bond duration, which measures the expected change in bond price due to shifting interest rates. Although convexity changes are smaller than those caused by duration, they can be valuable when positioning a portfolio. Particularly, this happens in times of more volatile yields to maturity.

Incorporating convexity into a portfolio comes at a cost. Higher convexity results in lower yields to maturity for investors, providing a safety cushion due to positive convexity. The value of convexity becomes more significant when yields to maturity are more volatile.

Effective duration and convexity comprehensively assess a bond portfolio's expected response to changing interest rates, ensuring greater accuracy than duration alone.

Spread duration quantifies a portfolio's sensitivity to fluctuations in credit spreads. Note that interest rate change duration gauges the impact of interest rate fluctuations on bond prices. Spread duration, however, focuses on the difference between the yield spread and interest rate and its effect on bond prices. Spread duration indicates the approximate percentage increase in bond price for a 1% decrease in credit spread.

Duration Time Spread (DTS) is a modified version of spread duration that accounts for the percentage-based changes across the credit spectrum. DTS enhances the magnitude of price changes linked to spread adjustments by incorporating the current credit spread as a weighting factor. Further discussion on DTS will follow later in the lesson.

Portfolio dispersion characterizes the variability in times to receipt of cash flows compared to duration. It is a crucial measure for assessing interest rate immunization for liabilities. Unlike Macaulay duration, dispersion considers the weighted average of time variance until cash flow receipt, providing insight into the spreading of payments over time. Dispersion also influences convexity, as cash flow dispersion leads to increased convexity.

## Correlations between Fixed-income Sectors

Correlation characteristics between fixed-income sectors refer to the interactions among benchmark rates, spreads, and factors such as currencies. Fixed-income sectors may have higher correlations within a market due to country-specific factors such as central bank policies and economic conditions.

For instance, in developed economies, interest rate changes in the sovereign yield curve are highly correlated with investment-grade securities with low default probability. On the other hand, spread changes have a more substantial impact on below-investment-grade securities and often show stronger correlations with equity markets.

Additionally, interest rates and spreads may exhibit negative correlations. In such an instance, interest rates fall, and spreads widen during economic downturns and vice versa during economic improvements. Changes can influence correlations between local currency exchange rates and global government bonds in interest rates and spreads.

## Portfolio Duration in Total Return Mandates

Actively managed total return mandates often utilize both top-down and bottom-up approaches in portfolio management. The top-down analysis identifies significant risk factors, while bottom-up selection focuses on individual securities.

Portfolio managers incorporate economic forecasts, evaluate the current business and regulatory landscape, and create investment themes that align with these insights. They also adjust portfolio duration based on expectations for interest rate changes and the yield curve shape.

For instance, if they anticipate rising interest rates and a steeper yield curve, they may reduce exposure to longer-dated bonds relative to the benchmark, decreasing portfolio duration. When their expectations align with market conditions, this active management can lead to outperformance, resulting in active excess returns.

## Managing Credit Exposure Using Spread Duration

Portfolio managers often use spread duration measures to assess their portfolios' sensitivity to changes in credit spreads.
Increasing the spread duration is advisable if a portfolio manager expects credit spreads to narrow (decrease). This means positioning the portfolio to benefit from narrowing credit spreads.

Conversely, reducing the spread duration is prudent if credit spreads are anticipated to widen (increase). This positioning helps mitigate the negative impact of widening credit spreads on the portfolio's value.

However, the manager may encounter certain constraints, including duration limits, rating-based restrictions, and limitations on derivatives. These factors can restrict the portfolio manager's ability to effectively adjust spread duration to capture alpha.

An alternative method to enhance the portfolio's credit exposure is to select bonds with different credit qualities. The portfolio's overall credit exposure increases by incorporating more bonds with speculative or lower credit ratings. This approach may introduce additional risk due to the lower credit quality. In addition, it can be a way to adjust the portfolio's risk and return profile.

## Relative Value

Relative value is crucial in managing fixed-income portfolios, helping determine which securities to include. It involves systematically ranking and comparing securities to identify the most profitable ones for inclusion in the portfolio. Portfolio managers assess and rank securities based on valuation, issuer fundamentals, and market technical patterns. This analysis is applied to various bonds to make informed investment decisions.

## Question

Which of the following is least likely a result of adding convexity to a bond portfolio?

1. Bond portfolios with higher convexity gain value faster when interest rates fall than those with lower convexity.
2. Bond portfolios with higher convexity lose value faster when interest rates fall than those with lower convexity.
3. When interest rates rise, bond portfolios with higher convexity lose value slower than those with lower convexity.

Solution

Answer choices A and C are correct, while B is incorrect. Convexity positively impacts bond portfolios in falling rate scenarios and has a protective effect in rising rate scenarios.

A and C are incorrect. Answers A and C are correct.

Portfolio Construction: Learning Module 2: Overview of Fixed-Income Portfolio Management; Los 2(b) Describe fixed-income portfolio measures of risk and return as well as correlation characteristics

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