Portfolio Management

Study Guide

Portfolio Risk and Return: Part I

This section introduces the fundamental tools for quantifying investment risk and return.

Measures of Return

Holding Period Return (HPR): Total return for a single period. $HPR = \frac{P_1 - P_0 + D_1}{P_0}$ Where $P_1$ = Ending Price, $P_0$ = Beginning Price, $D_1$ = Cash Flow/Dividend.
Arithmetic Mean Return: Simple average; best for forecasting a single period's return.
Geometric Mean Return: Time-weighted compounded rate of return; best for measuring past performance over multiple periods. $G = \sqrt[n]{(1+R_1)(1+R_2)...(1+R_n)} - 1$
Money-Weighted Rate of Return (MWRR): An internal rate of return (IRR) calculation that is sensitive to the timing and size of cash flows.
Time-Weighted Rate of Return (TWRR): Measures compound growth, removing the effects of cash flow timing. It is the standard for evaluating portfolio manager performance.

Measures of Risk

Variance ( $\sigma^2$ ) and Standard Deviation ( $\sigma$ ): Measure the dispersion of returns around the mean. Standard deviation is the square root of variance.
- Sample Variance: $s^2 = \frac{\sum_{i=1}^{n}(R_i - \bar{R})^2}{n-1}$
Covariance: Measures how two assets move together.
- Positive: Assets tend to move in the same direction.
- Negative: Assets tend to move in opposite directions.
Correlation ( $\rho$ ): A standardized measure of co-movement, ranging from -1 to +1. $\rho_{AB} = \frac{Cov(R_A, R_B)}{\sigma_A \sigma_B}$ $ρ_{A B} = \frac{C o v ( R _{A} , R _{B} )}{σ _{A} σ _{B}}$
- $\rho = +1$ : Perfect positive correlation.
- $\rho = -1$ : Perfect negative correlation (maximum diversification benefit).
- $\rho = 0$ : No linear relationship.

Portfolio Risk and Return: Part II

This section builds on Part I, focusing on how combining assets into a portfolio affects its risk and return characteristics.

Portfolio Return and Risk

Portfolio Expected Return: A weighted average of the expected returns of the individual assets. $E(R_p) = \sum_{i=1}^{n} w_i E(R_i)$
Portfolio Variance (Two-Asset Case): $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_A w_B \rho_{AB} \sigma_A \sigma_B$ Note: The primary benefit of diversification comes from the correlation term. The lower the correlation, the lower the portfolio risk.

Modern Portfolio Theory (MPT)

Efficient Frontier: A plot of portfolios that provide the highest expected return for each level of risk (standard deviation). Rational, risk-averse investors choose portfolios on this frontier.
Global Minimum-Variance Portfolio: The single portfolio on the efficient frontier with the lowest possible risk.
Capital Allocation Line (CAL): Represents the risk-return combinations available by combining a risk-free asset with any risky portfolio.
Capital Market Line (CML): The optimal CAL, where the risky portfolio is the market portfolio. It represents the most efficient portfolios. $E(R_p) = R_f + (\frac{E(R_m) - R_f}{\sigma_m})\sigma_p$

Systematic vs. Unsystematic Risk

Systematic (Market) Risk: Non-diversifiable risk affecting the entire market (e.g., interest rate changes, economic cycles). Measured by Beta ( $\beta$ ).
Unsystematic (Firm-Specific) Risk: Diversifiable risk unique to a specific company or industry. Can be eliminated through diversification.
Total Risk = Systematic Risk + Unsystematic Risk

The Capital Asset Pricing Model (CAPM)

CAPM Equation: Defines the expected return on an asset based on its systematic risk. $E(R_i) = R_f + \beta_i [E(R_m) - R_f]$ Where $[E(R_m) - R_f]$ is the market risk premium.
Beta ( $\beta$ ): Measures an asset's volatility relative to the overall market. $\beta_i = \frac{Cov(R_i, R_m)}{\sigma_m^2}$ $β_{i} = \frac{C o v ( R _{i} , R _{m} )}{σ _{m}^{2}}$
- $\beta > 1$ : More volatile than the market.
- $\beta < 1$ : Less volatile than the market.
Security Market Line (SML): A graphical representation of the CAPM. Plots expected return vs. beta.
- Assets plotting above the SML are considered undervalued.
- Assets plotting below the SML are considered overvalued.

Performance Evaluation Measures

Ratio	Formula	Measures	Best For
Sharpe Ratio	$\frac{E(R_p) - R_f}{\sigma_p}$	Reward per unit of total risk	Evaluating diversified portfolios
Treynor Ratio	$\frac{E(R_p) - R_f}{\beta_p}$	Reward per unit of systematic risk	Evaluating individual stocks or portfolios in a broader context
Jensen's Alpha	$\alpha_p = R_p - E(R_i)$	Excess return over CAPM's prediction	Measuring performance against a benchmark (CAPM)

Portfolio Management: An Overview

This section describes the structure and process of managing investment portfolios for different types of clients.

The Portfolio Management Process

Planning Step: Analyzing investor objectives and constraints. This culminates in the creation of the Investment Policy Statement (IPS).
Execution Step: Asset allocation and security selection. The portfolio is constructed based on the IPS.
Feedback Step: Monitoring and rebalancing the portfolio and reviewing the IPS.

Types of Investors

Individual Investors: Investing for personal goals (e.g., retirement, education).
Institutional Investors: Large entities managing funds on behalf of others.
- Defined Benefit (DB) Pension Plans: Employer promises a defined retirement benefit; employer bears the investment risk.
- Defined Contribution (DC) Pension Plans: Employee and/or employer make defined contributions; employee bears the investment risk (e.g., 401(k)).
- Endowments & Foundations: Long time horizons, focused on funding ongoing activities.
- Banks & Insurance Companies: Typically conservative due to high liquidity needs and regulatory oversight.

Basics of Portfolio Planning and Construction

This section details the creation of the Investment Policy Statement (IPS), the foundational document for portfolio management.

The Investment Policy Statement (IPS) The IPS is a formal document that governs all investment decisions. It ensures a disciplined approach.

Components of an IPS

Objectives:
- Return Objective: The level of return needed or desired. Can be stated as an absolute (e.g., 9%) or relative (e.g., S&P 500 + 2%) goal.
- Risk Tolerance: The ability and willingness of the investor to take on risk.
Constraints (T-L-L-U-R):
- Time Horizon: The period over which investments will be managed. Longer horizons generally allow for higher risk tolerance.
- Liquidity: The need for cash or assets that can be quickly converted to cash with minimal loss of value.
- Legal and Regulatory: Rules and regulations that may restrict investment choices (e.g., ERISA for pension funds).
- Taxes: The tax status of the investor and the tax treatment of various investment accounts and asset types.
- Unique Circumstances: Any other specific investor preferences or restrictions (e.g., ESG considerations, restrictions on holding certain securities).

Asset Allocation

Strategic Asset Allocation (SAA): Sets the long-term target weights for different asset classes based on the IPS.
Tactical Asset Allocation (TAA): Short-term, active deviations from the SAA to capitalize on market opportunities.

The Behavioral Biases of Individuals

This section explores how psychological biases can lead to irrational and suboptimal investment decisions.

Two Main Categories of Biases

Cognitive Errors: Errors in reasoning due to faulty information processing or memory. Can be corrected with better information and education.
Emotional Biases: Decisions driven by feelings, impulses, or intuition rather than facts. Harder to correct; must be adapted to.

Category	Bias	Description
Cognitive Errors	Conservatism	Under-reacting to new information; maintaining prior views.
	Confirmation	Seeking out information that confirms one's existing beliefs.
	Representativeness	Classifying new information based on past experiences (stereotyping).
	Illusion of Control	Believing one can control or influence outcomes when they cannot.
	Hindsight	Believing past events were predictable and obvious in retrospect.
	Anchoring & Adjustment	"Anchoring" on an initial piece of information, even if irrelevant.
	Mental Accounting	Treating different sums of money differently based on their source or intended use.
	Framing	Answering a question differently based on how it is asked or "framed".
Emotional Biases	Loss Aversion	Feeling the pain of a loss more strongly than the pleasure of an equal gain.
	Overconfidence	Unwarranted faith in one's own abilities and judgment.
	Self-Control	Failing to act in pursuit of long-term goals due to a lack of self-discipline.
	Status Quo	A preference for doing nothing and maintaining one's current portfolio.
	Endowment	Valuing an asset more highly simply because one owns it.
	Regret Aversion	Avoiding making decisions out of fear that the decision will be wrong in hindsight.

Introduction to Risk Management

This section provides a framework for identifying, measuring, and managing various risks.

Risk Management Framework

Risk Governance: The top-down system that sets risk tolerance and provides guidance for risk management activities.
Risk Identification and Measurement: Identifying all relevant risks and quantifying them (e.g., using VaR, scenario analysis).
Risk Budgeting: Allocating the firm's total risk tolerance to different business units or asset classes.
Risk Modification: Using strategies to alter the firm's risk exposure.

Key Risk Types

Market Risk: Risk from changes in market prices (e.g., equity risk, interest rate risk, currency risk).
Credit Risk: Risk that a counterparty will fail to meet its obligations.
Liquidity Risk: Risk of being unable to sell an asset quickly at a fair price.
Operational Risk: Risk of loss from failed internal processes, people, or systems.
Legal and Regulatory Risk: Risk that changes in laws or regulations will negatively impact an investment.

Methods of Risk Modification

Risk Avoidance: Not engaging in activities with a particular risk.
Risk Acceptance: Bearing the risk and its potential consequences (self-insurance).
Risk Transfer: Shifting the risk to another party (e.g., buying insurance).
Risk Shifting: Changing the distribution of risk outcomes (e.g., using derivatives like futures or options to hedge).

Risk Measures

Value at Risk (VaR): The minimum loss that would be expected over a given period at a given probability level (e.g., "a 5% daily VaR of $1 million" means there is a 5% chance of losing at least$ 1 million on any given day).
Conditional VaR (CVaR): The expected loss, given that the loss exceeds the VaR level. It answers the question: "If things go bad, how bad are they expected to be?"