Lesson 10.2: Active Equity Construction and Fixed-Income Strategies

Introduction

In this lesson, we will delve into the concepts of active equity construction and fixed-income strategies within portfolio management. By the end of this lesson, students will be able to understand risk-budgeted active equity portfolio construction, liability-driven fixed-income across various interest rate scenarios, and active yield-curve strategies for excess return. We will also equip students with tools to build a risk-budgeted active equity portfolio and to manage liability-driven fixed income under different rate scenarios.

Learning Objectives

Understand risk-budgeted active equity portfolio construction.
Analyze liability-driven fixed income across interest rate scenarios.
Implement active yield-curve strategies for excess return.
Construct a risk-budgeted active equity portfolio.
Manage liability-driven fixed income under varying interest rate scenarios.

H2: Risk-Budgeted Active Equity Portfolio Construction

Active equity management aims to outperform a benchmark index through informed decision-making and strategic investing. One of the key aspects of active equity construction is risk budgeting. Risk budgeting refers to allocating risk across various portfolio segments or assets according to their expected contribution to the overall risk of the portfolio. Let’s break this down further.

Understanding Risk Budgeting

Risk budgeting is important because it helps investors understand the amount of risk they are taking on and allows them to optimize their portfolio accordingly. The goal is to minimize the total risk associated with the portfolio while aiming for the desired return. This ensures that an investor is not over-concentrating risk in a particular asset or sector.

Key Concepts in Risk Budgeting

Total Portfolio Risk: The overall risk of a portfolio is a function of the risk of individual assets and their correlations. Mathematically, this can be represented using the following formula:

$$\sigma_p = \sqrt{\sum_{i=1}^N w_i^2 \sigma_i^2 + \sum_{i=1}^N\sum_{j=1}^N w_i w_j \sigma_i \sigma_j

ho_{ij}}$$

where:

$w_i$ is the weight of asset $i$ in the portfolio.
$\sigma_i$ is the standard deviation of asset $i$.

ho_{ij} is the correlation coefficient between assets $i$ and $j$.

Risk Contribution: The contribution of each asset to the total portfolio risk can be analyzed using the risk parity approach, where the portfolio is constructed such that each asset contributes equally to the total risk.
Active Risk: Also known as tracking error, it is a measure of how much the active management deviates from the benchmark. It is calculated as:

$$\text{Active Risk} = \sigma_{P} - \sigma_{B}$$

where:

$\sigma_{P}$ is the standard deviation of the active portfolio.
$\sigma_{B}$ is the standard deviation of the benchmark.

Worked Example: Risk-Budgeted Equity Portfolio

Let’s consider an example where you want to construct a risk-budgeted equity portfolio with three stocks: A, B, and C. Let’s assume their weights ($w$), standard deviations ($\sigma$), and correlations are as follows:

Stock A: weight = 0.4, $\sigma$ = 0.2
Stock B: weight = 0.5, $\sigma$ = 0.3
Stock C: weight = 0.1, $\sigma$ = 0.25
Correlations:

ho_{AB} = 0.2$, $

ho_{AC} = 0.1$, $

ho_{BC} = 0.3.

Calculate the total portfolio risk:

Total risk can be calculated using the previously stated formula. We will perform the calculations step by step.
Calculate individual components:
For $A$: $0.4^2 \times 0.2^2 = 0.016$
For $B$: $0.5^2 \times 0.3^2 = 0.0225$
For $C$: $0.1^2 \times 0.25^2 = 0.000625$
Calculate the covariances:
$A$ and $B$: $0.4 \times 0.5 \times 0.2 \times 0.3 \times 0.2 = 0.006$
$A$ and $C$: $0.4 \times 0.1 \times 0.2 \times 0.25 \times 0.1 = 0.0005$
$B$ and $C$: $0.5 \times 0.1 \times 0.3 \times 0.25 \times 0.3 = 0.001125$

Total portfolio risk thus can be calculated as:

$$\sigma_p = \sqrt{0.016 + 0.0225 + 0.000625 + 0.006 + 0.0005 + 0.001125} = \sqrt{0.04675} \approx 0.2163$$

Now we know our total portfolio risk is $0.2163$.

H2: Liability-Driven Fixed Income Strategies

Liability-driven investment (LDI) strategies are designed to align a portfolio’s investments with future liabilities. LDI is particularly important for pension funds and insurance companies, where there are specific future payouts that need to be met. The goal is to effectively manage interest rate risk to ensure that there will be sufficient funds to cover these liabilities.

Interest Rate Scenarios and LDI

As interest rates fluctuate, the value of fixed income securities changes, and this can significantly impact an institution’s ability to meet its liabilities. The following factors should be considered in LDI:

Duration Matching: This process ensures that the interest rate sensitivity of assets matches that of liabilities. Duration is defined as the weighted average time until cash flows are received and can be calculated for a bond or portfolio as:

$$D = \frac{1}{P} \sum_{t=1}^{T} \frac{C_t}{(1+y)^t}$$

where:

$D$ is the duration.
$P$ is the price of the bond.
$C_t$ is the cash flow at time $t$.
$y$ is the yield.

Rate Scenario Stress Testing: This involves simulating how changes in interest rates affect the value of the portfolio. For instance, if interest rates increase by 1%, the price of existing bonds typically falls, and this should be accounted for in the LDI strategy.

Worked Example: LDI Strategy Implementation

Consider a pension fund that needs to meet a liability of $1 million in 10 years. The fund currently holds a bond portfolio with an average duration of 8 years and a total market value of $800,000.

Assuming the yield curves shift upwards by 1%:

Calculate the new bond portfolio value considering the price change from the duration effect. If the current yield on the bonds is 4%, the new yield will be 5%.

The percentage price change can be approximated using the bond’s duration:
Price Change ≈ - Duration × ΔYield = -8 × 0.01 = -0.08 (or -8%)
New Portfolio Value = Current Value × (1 - Price Change) = $800,000 × (1 - 0.08) = $800,000 × 0.92 = $736,000

Compare the new portfolio value to the liability:

Liability = $1,000,000
Current fund availability = $736,000
Shortfall = $1,000,000 - $736,000 = $264,000
This analysis indicates a shortfall that needs to be addressed.

H2: Active Yield-Curve Strategies for Excess Return

Active yield-curve strategies involve taking positions on the yield curve in an attempt to generate excess return. This is achieved by anticipating changes in interest rates and the shape of the yield curve.

Components of Yield-Curve Strategies

Shape of the Yield Curve: The yield curve represents the relationship between interest rates and different maturities of debt. It can take different shapes: normal, inverted, or flat. Understanding these shapes assists investors in predicting interest rate movements.
Strategic Investing on the Curve: Investors can adopt strategies that include:

Bull Steepener: This strategy is employed when the investor expects long-term rates to fall more significantly than short-term rates, resulting in a steeper yield curve.
Bear Flattener: This is the opposite of the bull steepener, where the investor expects short-term rates to rise more than long-term rates, leading to a flatter yield curve.

Worked Example: Implementing Yield-Curve Strategy

Assume an investor identifies that the yield curve is flattening. They currently hold:

Short-term bonds (1 year to maturity) yielding 3%
Long-term bonds (10 years to maturity) yielding 5%

To take advantage of the expected flattening, the investor can sell some of the long-term bonds and buy short-term bonds to lock in higher current yields.
If the investor reallocates $100,000 from the long-term to short-term, the new weighted average yield can be computed as:

Let’s say 40,000 remain in the long-term yielding 5% and now $60,000 is in short-term yielding 3%.

$ Weighted Average Yield = \$

$$\text{Weighted Avg Yield} = \frac{{(0.03 \times 60,000) + (0.05 \times 40,000)}}{{100,000}} = \frac{{1,800 + 2,000}}{{100,000}} = 0.038 = 3.8\%$$

Analyze the expected returns over a period of 5 years from this position:

Total income from short-term investment: $60,000 × 3.8% = $2,280
Total income from long-term investment: $40,000 × 5% = $2,000
This shows that while the total return focuses on current income, fluctuations in the curve will be crucial in strategy execution.

Conclusion

This lesson has explored key strategies in active equity construction and fixed-income strategies including risk-budgeting and liability-driven investment. Understanding these concepts is crucial for effectively managing portfolios and achieving investment goals. students should now be equipped not only to analyze and construct portfolios but also to anticipate and react to market scenarios.

Study Notes

Risk budgeting involves an optimal allocation of risk across portfolio assets.
The total risk of a portfolio can be derived from asset correlations and standard deviations.
Liability-driven investment focuses on matching asset durations with future liabilities.
Changes in interest rates impact the value of fixed incomes significantly.
Active yield-curve strategies can provide opportunities for excess returns based on interest rate expectations.