Portfolio Theory

Basics of Financial Engineering | 2025.02.13

Portfolio Theory is a core area of financial engineering that deals with asset allocation methods for investors to achieve an optimal balance between risk and return.

Since the pioneering research of Harry Markowitz in 1952, modern portfolio theory has provided a theoretical foundation for investment decision-making and has brought revolutionary changes to asset management and risk management fields.

In this page, we will explore the key concepts of portfolio theory, mean-variance optimization, asset pricing models, multi-factor models, portfolio performance measurement, and practical asset allocation strategies.

1️⃣ Basic Concepts of Portfolio Theory

Portfolio theory is based on the principle that investment performance can be improved by diversifying investments across various assets. The key to this approach is the concept of risk diversification, which can be mathematically formalized as the old investment wisdom of "Don't put all your eggs in one basket."

Definition of Return and Risk

Return

Definition of Return
It is expressed as the ratio of profit or loss on an investment to the amount invested.
Simple Return
```
R_t = (P_t - P_{t-1} + D_t) / P_{t-1}
```
Here, P_t is the current price, P_{t-1} is the previous price, and D_t is the dividend.
Log Return
```
r_t = ln(P_t / P_{t-1})
```
Log returns, also known as continuously compounded returns, are mathematically convenient because they can be simply summed over multiple periods.
Expected Return
The expected return is the probabilistic expectation of future returns, which can be estimated through the average of past returns or projections about the future:
```
E(R) = Σ p_i R_i
```
Here, p_i represents the probability of each scenario, and R_i denotes the return in each scenario.

Risk

Definition of Risk
It refers to the uncertainty or volatility of investment returns. It is generally measured by the standard deviation of returns:
```
σ = √[Σ p_i (R_i - E(R))²]
```
Here, σ is the standard deviation, and E(R) is the expected return.
Variance
The square of the standard deviation, which indicates how far the returns are, on average, from the expected value:
```
σ² = Σ p_i (R_i - E(R))²
```
Various Aspects of Risk
In portfolio theory, risk is primarily measured using standard deviation; however, in practice, various risk measurement methods are employed:
1. Downside Risk
  This measurement method focuses solely on the possibility of loss, with examples including semi-variance, Value at Risk (VaR), and Conditional VaR (CVaR).
2. Beta
  This is an indicator that shows how correlated the returns of an individual asset are with the overall market returns.
3. Maximum Drawdown
  This measures the largest decline from the highest point to the lowest point during a specific period.

Covariance and Correlation

A key concept in understanding the risk of a portfolio is the covariance and correlation, which represent the relationship of returns between different assets.

Covariance and Correlation

Covariance
It is an indicator of how the returns of two assets move together:
```
Cov(R_A, R_B) = E[(R_A - E(R_A))(R_B - E(R_B))]
```
Positive covariance indicates that two assets tend to move in the same direction, while negative covariance indicates that they tend to move in opposite directions.
Correlation Coefficient
It is a measure that standardizes covariance to a value between -1 and 1:
```
ρ_AB = Cov(R_A, R_B) / (σ_A × σ_B)
```
- ρ = 1: Perfect positive correlation (the two assets always move in the same direction)
- ρ = 0: No correlation (the movements of the two assets are independent of each other)
- ρ = -1: Perfect negative correlation (the two assets always move in opposite directions)
Importance in Portfolio Diversification
A low correlation between the assets included in a portfolio maximizes the benefits of diversification. In particular, combining assets with negative correlation can significantly reduce the overall risk of the portfolio.

Portfolio Return and Risk

The return and risk of a portfolio composed of various assets is calculated as follows:

Portfolio Return and Risk

Expected Portfolio Return
It is the weighted average of the expected returns of each asset, weighted by the proportion of that asset within the portfolio:
```
E(R_p) = Σ w_i E(R_i)
```
Here, ( w_i ) represents the weight of asset ( i ) in the portfolio, and ( E(R_i) ) is the expected return of asset ( i ).
Portfolio Variance and Standard Deviation
The variance of a portfolio is not simply the weighted average of the individual asset variances; it must also consider the covariances between all pairs of assets:
```
σ_p² = ΣΣ w_i w_j Cov(R_i, R_j)
```
For a portfolio consisting of 2 assets:
```
σ_p² = w_1²σ_1² + w_2²σ_2² + 2w_1w_2Cov(R_1, R_2)
```
The standard deviation of a portfolio is the square root of the variance:
```
σ_p = √σ_p²
```

Benefits of Diversification

The key advantage of diversification is that it can reduce the portfolio's risk below the weighted average risk of the individual assets.

Benefits of Diversification

Mathematical Foundation of Diversification
This risk reduction effect occurs when combining assets with a correlation less than 1, and can be explained as follows:
When the correlation coefficient ρ_AB between two assets A and B is less than 1:
```
σ_p < w_A σ_A + w_B σ_B
```
In other words, the risk of the portfolio is less than the weighted average of the individual asset risks.
Example of complete hedge:
In an extreme case where the correlation coefficient of two assets is -1, investing in appropriate proportions can reduce the portfolio risk to 0:
```
w_A = σ_B / (σ_A + σ_B)  
w_B = σ_A / (σ_A + σ_B)
```
Limitations of Diversification
1. Market risk (systematic risk) cannot be eliminated through diversification.
2. In times of crisis, the correlation between assets tends to increase, weakening the effects of diversification.
3. As the number of assets in a portfolio increases, the additional benefits of diversification gradually decrease.

The basic concepts of portfolio theory provide essential foundations for investors to understand the relationship between risk and return, enabling them to make optimal investment decisions that align with their risk preferences.

2️⃣ Markowitz Mean-Variance Optimization

Markowitz Mean-Variance Optimization, proposed by Harry Markowitz in 1952, is a methodology for portfolio construction and forms the core of Modern Portfolio Theory (MPT). Markowitz received the Nobel Prize in Economic Sciences in 1990 for this research.

Principles of Optimal Portfolio Construction

Key Assumptions of Markowitz Theory

Investors are risk-averse: they prefer investments with lower risk for the same level of return.
Investors consider only the expected return and risk (variance or standard deviation) of the portfolio.
Asset returns follow a normal distribution, or the investor's utility function is quadratic.
There are no transaction costs or taxes.
All investors have the same investment horizon.

Efficient Frontier

Concept of Efficient Frontier

Definition of Efficient Frontier
It is a curve that represents the set of all portfolios that provide the maximum expected return for a given level of risk, or the minimum risk for a given expected return.
Efficient Portfolio
A portfolio on the Efficient Frontier that does not have any other portfolios with a higher return for the same risk or a lower risk for the same return.
Minimum Variance Portfolio
The portfolio with the lowest risk on the Efficient Frontier, located at the leftmost point of the frontier.

Mathematical Formulation of Mean-Variance Optimization

The Markowitz model can be expressed as the following optimization problem:

Formulation of the Optimization Problem

Pursuit of Maximum Return
Maximize expected return at a given risk level σ*:

Maximize:    E(R_p) = Σ w_i E(R_i)
Constraints: σ_p² = ΣΣ w_i w_j Cov(R_i, R_j) = (σ*)²
             Σ w_i = 1

Minimizing Risk
Minimize portfolio variance at the given expected return μ*:

Minimize:    σ_p² = ΣΣ w_i w_j Cov(R_i, R_j)
Constraints: E(R_p) = Σ w_i E(R_i) = μ*
             Σ w_i = 1

Additional Constraints
In actual application, the following additional constraints may be included:
- Short-selling restrictions (w_i ≥ 0)
- Maximum investment proportion limits (w_i ≤ c_i)
- Allocation constraints for specific asset classes

Dual Parameter Optimization and Utility Function

Dual Parameter Optimization

This approach maximizes a utility function that reflects the investor's risk aversion, considering both returns and risks simultaneously:

Maximize:    U = E(R_p) - (λ/2)σ_p²
Constraints: Σ w_i = 1

Here, λ represents the investor's risk aversion coefficient, where a higher value indicates a greater tendency for the investor to avoid risk.

Practical Challenges of Mean-Variance Optimization

Limitations of the Markowitz Model

Estimation Errors of Input Variables
There can be significant errors in estimating expected returns, variances, and covariances, which can greatly impact the optimization results.
Extreme Allocations
The optimization results often suggest allocations that are extremely concentrated in certain assets, making practical implementation difficult.
Assumption of Time Invariance
It is assumed that the statistical properties of asset returns do not change over time; however, in reality, they can vary depending on market conditions.
Non-Normal Distribution
Asset returns often exhibit fat-tailed distributions rather than normal distributions.

Improved Optimization Approaches

Improved Optimization Methods

Resampling
A method that extracts multiple samples from the input data to generate several efficient frontiers and averages them, resulting in more robust outcomes against estimation errors.
Black-Litterman Model
A Bayesian approach that combines investor views with prior expected returns reflecting market equilibrium, which mitigates extreme allocations.
Robust Optimization
Explicitly models the uncertainty of input variables and selects portfolios that perform well even in the worst-case scenarios.
Adding Constraints
In practice, various constraints are added, such as minimum/maximum weight limits for assets and sector allocation restrictions, to prevent extreme allocations.

Markowitz's mean-variance optimization provides the theoretical foundation for modern asset allocation and serves as a starting point for the various portfolio optimization methodologies developed thereafter. Despite its theoretical limitations, the fundamental principles still significantly influence asset management and investment decision-making today.

3️⃣ Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM), independently developed in the early 1960s by William Sharpe, John Lintner, and Jan Mossin, explains the relationship between the expected return of an asset and its systematic risk (market risk). Sharpe received the Nobel Prize in Economic Sciences in 1990 alongside Harry Markowitz and Merton Miller for this research.

Basic Assumptions of CAPM

Key Assumptions of CAPM

Assumptions about Investor Behavior
- Investors select portfolios according to Markowitz's mean-variance analysis.
- Investors are risk-averse and prefer lower risk for the same expected return.
- All investors have the same investment horizon.
Assumptions about the Market
- All investors have access to the same information.
- Investors can invest in both risk-free assets and all risky assets.
- Investors can borrow or lend at the risk-free rate.
- There are no transaction costs or taxes.
- All assets are completely divisible.
- There are no arbitrage opportunities in the market.

Capital Market Line (CML)

Concept of the Capital Market Line

Definition
It is the line that represents the set of all efficient portfolios formed by combining a risk-free asset and the market portfolio.
Mathematical Expression
```
E(R_p) = R_f + [(E(R_m) - R_f) / σ_m] × σ_p
```
Here:
- E(R_p) is the expected return of the portfolio
- R_f is the risk-free interest rate
- E(R_m) is the expected return of the market portfolio
- σ_m is the standard deviation of the market portfolio
- σ_p is the standard deviation of the portfolio
Meaning of the Slope
The slope of the Capital Market Line [(E(R_m) - R_f) / σ_m] can be interpreted as the excess return per unit of risk, which is known as the 'Sharpe Ratio'.
Efficient Portfolio
All efficient portfolios can be expressed as a combination of risk-free assets and the market portfolio, which is referred to as the 'Capital Allocation Problem' for investors.

Derivation of the CAPM Model

Derivation of the CAPM Equation

Required Return in Equilibrium
CAPM explains that in a market equilibrium state, the expected return of a specific asset is determined as follows:
```
E(R_i) = R_f + β_i [E(R_m) - R_f]
```
Here:
- E(R_i) is the expected return of asset i
- R_f is the risk-free interest rate
- β_i is the beta of asset i
- E(R_m) is the expected return of the market portfolio
- [E(R_m) - R_f] is the market risk premium
Definition of Beta
```
β_i = Cov(R_i, R_m) / σ_m²
```
Beta is an indicator that measures how much the asset's return moves in relation to the market return, assessing the systematic risk of the asset.
Interpretation of Beta
- β = 1: The asset moves in line with the market.
- β > 1: The asset moves more than the market (aggressive).
- β < 1: The asset moves less than the market (defensive).
- β = 0: The asset moves independently of the market.
- β < 0: The asset moves in the opposite direction of the market (rare case).

Security Market Line (SML)

Concept of Security Market Line

Definition
A line that represents the linear relationship between beta and expected return, graphically expressing the CAPM equation.
Interpretation of Equilibrium State
All assets and portfolios should be positioned on the Security Market Line in a state of equilibrium.
- Assets above the SML: Fairly priced
- Assets above the SML: Undervalued (positive alpha)
- Assets below the SML: Overvalued (negative alpha)
Definition of Alpha
The difference between the actual return of the asset and the return predicted by CAPM:
```
α_i = E(R_i) - [R_f + β_i (E(R_m) - R_f)]
```

Empirical Validation and Limitations of CAPM

Validation and Limitations of CAPM

Empirical Research Findings
Since the 1970s, various empirical studies have shown that the predictions of CAPM do not fully align with actual market data:
- The relationship between beta and returns is flatter than predicted.
- Factors beyond beta, such as firm size, value indicators (P/E, P/B), and momentum, influence returns.
- Abnormally high returns on risk-free assets (the risk-free asset puzzle).
Major Limitations of CAPM
- It is a single-period model, making it difficult to apply to investors with different investment horizons.
- The assumption that all investors have access to the market portfolio is unrealistic.
- Assumptions like the possibility of perfect short selling and unlimited borrowing are inconsistent with reality.
- Beta may not be stable over time.
- The assumption that investors share the same expectations is not realistic.
Practical Applications of CAPM
Despite these limitations, CAPM is still widely used in areas such as:
- Estimating the Cost of Capital.
- Evaluating investment performance.
- Risk management.
- Making asset allocation decisions.

Despite its simplicity, CAPM provides powerful insights into the relationship between risk and return, and it continues to hold an important position in financial theory and practice today. To overcome theoretical limitations, more sophisticated models, such as multi-factor models discussed in the next section, have been developed.

4️⃣ Multi-Factor Models and Extensions of CAPM

To overcome the limitations of CAPM, various extension models have been proposed, among which Multi-Factor Models consider multiple risk factors that influence asset returns, providing a more realistic explanation.

Basic Structure of Multi-Factor Models

General Form of Multi-Factor Models

Mathematical Expression
```
E(R_i) = R_f + β_i1 × RP_1 + β_i2 × RP_2 + ... + β_in × RP_n
```
Here:
- E(R_i) is the expected return of asset i
- R_f is the risk-free interest rate
- β_ij is the sensitivity of asset i to factor j
- RP_j is the risk premium for factor j
Estimation of Beta
The beta for each factor is estimated through time series regression analysis:
```
R_i - R_f = α_i + β_i1 × F_1 + β_i2 × F_2 + ... + β_in × F_n + ε_i
```
Here, ( F_j ) is the return of factor ( j ), and ( \epsilon_i ) is the error term.
Types of Factors
1. Macroeconomic Factors GDP growth rate, inflation, interest rates, industrial production index, etc.
2. Fundamental Factors Company size, value vs. growth, profitability, investment, leverage, etc.
3. Statistical Factors Factors extracted through statistical methods such as Principal Component Analysis (PCA).

Fama-French Three-Factor Model

Model Development
Developed by Eugene Fama and Kenneth French in 1992, this model extends the CAPM by adding size and value factors.
Model Structure
```
E(R_i) - R_f = β_i,MKT × (E(R_m) - R_f) 
               + β_i,SMB × SMB 
               + β_i,HML × HML
```
Here:
- MKT (Market): Market risk premium
- SMB (Small Minus Big): The return difference between small-cap and large-cap stocks
- HML (High Minus Low): The return difference between high-value (high book-to-market ratio) and low-value (low book-to-market ratio) stocks
Factor Interpretation
- SMB: Represents the size risk premium, reflecting the phenomenon where small-cap stocks tend to have higher returns compared to large-cap stocks.
- HML: Represents the value risk premium, reflecting the phenomenon where value stocks tend to have higher returns compared to growth stocks.
Empirical Success
The Fama-French model explains asset return variability better than the CAPM and has been validated in various countries and markets.

Carhart 4-Factor Model

Model Development
Developed in 1997 by Mark Carhart, who added a momentum factor to the Fama-French 3-factor model.

Model Structure

E(R_i) - R_f = β_i,MKT × (E(R_m) - R_f) 
               + β_i,SMB × SMB 
               + β_i,HML × HML 
               + β_i,MOM × MOM

Here, MOM (Momentum) refers to the return difference between stocks with high past returns (winners) and those with low returns (losers).

Momentum Effect
Momentum indicates the tendency of stocks that performed well in the past (over a short term of 3-12 months) to continue performing well. This is considered a challenge to the efficient market hypothesis.
Practical Applications
The Carhart four-factor model is widely used for evaluating the performance of mutual funds and hedge funds, risk analysis, and developing investment strategies.

Other Multi-Factor Models

Other Major Multi-Factor Models

Fama-French Five-Factor Model
This expanded model, released in 2015, adds profitability and investment factors to the existing three factors:
```
E(R_i) - R_f = β_i,MKT × (E(R_m) - R_f) 
               + β_i,SMB × SMB 
               + β_i,HML × HML 
               + β_i,RMW × RMW 
               + β_i,CMA × CMA
```
Here:
- RMW (Robust Minus Weak): The return difference between high and low profitability stocks.
- CMA (Conservative Minus Aggressive): The return difference between conservative and aggressive investment firms.
APT (Arbitrage Pricing Theory) Model
Developed by Stephen Ross in 1976, this theory assumes that several factors influence asset returns in the absence of arbitrage opportunities. Unlike CAPM, there are no specific limitations on the number or nature of the factors.
Macroeconomic Multi-Factor Model
A model that uses macroeconomic variables like inflation, industrial production, interest rate term structure, and fundamental risk premium as factors, with research by Chen, Roll, and Ross (1986) being representative.
BARRA Model
A commercially used risk model widely employed in practice, which includes various fundamental, technical, and macroeconomic factors.

Practical Applications of the Multifactor Model

Key Application Areas of the Multifactor Model

Performance Attribution
Analyzes which factors contributed to investment performance, assessing the actual skills of portfolio managers.
Risk Decomposition and Management
Breaks down the risks of a portfolio by various factors and manages excessive exposure to specific factors.
Smart Beta Strategies
Develops investment strategies that are strategically exposed to specific factors, forming the basis of factor investing.
Anomaly Verification
Verifies whether new anomalies can be explained by existing factors.
Asset Allocation
Improves portfolio diversification through risk factor-based allocation instead of traditional asset class allocation.

The multifactor model explains asset return fluctuations better than the simple single-factor approach of CAPM, enabling more nuanced risk management and investment decision-making in complex investment environments. However, debates continue over which factors are true risk factors and whether these factor premiums will persist.

5️⃣ Measuring and Evaluating Portfolio Performance

Objectively measuring and evaluating the performance of an investment portfolio is a crucial part of the investment decision-making process. Various metrics and methodologies are used to assess portfolio performance, each evaluating different aspects of performance.

Absolute Performance Measurement

Absolute Performance Measurement Metrics

Total Return
The most basic performance metric, representing the investment gains over a specific period as a ratio of the initial investment amount:
```
Total Return = (Ending Value - Starting Value + Cash Flows) 
                / Starting Value
```
Annualized Return
This standardizes returns on an annual basis to allow for comparison of performance over different periods:
```
Annualized Return = (1 + Total Return)^(365/days) - 1
```
Geometric Mean Return
A suitable metric for evaluating long-term investment performance, taking into account the effects of compounding:
```
Geometric Mean Return = [(1 + r₁) × (1 + r₂) × ... × (1 + rₙ)]^(1/n) - 1
```
Here, rᵢ represents the return for each period.
Time-Weighted Return
A measure that evaluates only the investment management capability by excluding the impact of cash flows, making it suitable for assessing portfolio managers' performance.
Money-Weighted Return
A measure that considers both the invested amount and time, which is the same as the Internal Rate of Return (IRR). It represents the actual experienced return from the investor's perspective.

Risk-Adjusted Performance Measurement

Risk-Adjusted Performance Indicators

Sharpe Ratio
A measure that assesses the excess return per unit of total risk (standard deviation):
```
Sharpe Ratio = (R_p - R_f) / σ_p
```
Here, R_p represents the portfolio return, R_f is the risk-free interest rate, and σ_p is the portfolio standard deviation.
Treynor Ratio
A metric that measures excess return per unit of systematic risk (beta):
```
Treynor Ratio = (R_p - R_f) / β_p
```
Here, β_p is the beta of the portfolio.
Information Ratio
The value obtained by dividing the excess return (alpha) over the benchmark by the tracking error:
```
Information Ratio = (R_p - R_b) / σ_(p-b)
```
Here, ( R_b ) is the benchmark return, and ( \sigma_{(p-b)} ) is the standard deviation of the difference between the portfolio and benchmark returns.
Jensen's Alpha
A performance measurement metric based on CAPM, representing the difference between the actual return of the portfolio and the return predicted by CAPM:
```
α_p = R_p - [R_f + β_p(R_m - R_f)]
```
Sortino Ratio
It is a variation of the Sharpe Ratio that considers only downside volatility instead of overall volatility:
```
Sortino Ratio = (R_p - R_f) / σ_downside
```
Here, σ_lower refers to the standard deviation of returns that fall below the target return.

Multi-Factor Model Based Performance Evaluation

Multi-Factor Performance Analysis

Multi-Factor Alpha
Risk-adjusted excess return using the multi-factor model:
```
α_multi = R_p - [R_f + Σ(β_i × F_i)]
```
Here, ( \beta_i ) represents the sensitivity to factor ( i ), and ( F_i ) is the return of factor ( i ).
Factor Contribution Analysis
Decomposing portfolio returns into contributions from each factor exposure:
```
R_p = α + Σ(β_i × F_i) + ε
```
Active Risk Decomposition
Active return is decomposed into factor exposure differences and stock selection effects compared to the benchmark to evaluate the portfolio manager's skill.
Style Analysis
A method developed by William Sharpe that estimates implicit exposure to various styles or asset classes based on the return patterns of the portfolio.

Drawdown and Risk Measurement

Drawdown and Loss Related Metrics

Maximum Drawdown
The largest drop in the portfolio's value from its peak to its trough during a specific period, measuring the portfolio's loss risk:
```
Maximum Drawdown = (Lowest Point Value - Highest Point Value) 
                    / Highest Point Value
```
Calmar Ratio
The value obtained by dividing the annualized return by the maximum drawdown, measuring the return per unit of drawdown risk:
```
Calmar Ratio = Annualized Return / |Maximum Drawdown|
```
Sterling Ratio
A variation of the Calmar Ratio, which often uses the average annual maximum drawdown instead of the maximum drawdown.

VaR (Value at Risk)
The expected maximum loss over a given period at a specified confidence level:

VaR(95%) = The value that predicts, with 95% probability, that the loss will not exceed this amount during a specific period.값

CVaR (Conditional VaR) or Expected Shortfall
The average loss expected when exceeding VaR, which better captures extreme loss scenarios.

Performance Persistence and Attribution Analysis

Performance Persistence
Analyzes the likelihood that past good performance will continue in the future. Methods used include the correlation of performance rankings over consecutive periods and winner/loser transition tables.
Performance Attribution
Methods to decompose and analyze the sources of portfolio performance:
1. Asset Allocation Effect
  Performance resulting from strategic allocation decisions across various asset classes.
2. Security Selection Effect
  Performance resulting from the selection of securities within each asset class.
3. Interaction Effect
  The interaction between asset allocation and security selection effects.
Brinson Model
One of the most widely used methods of performance attribution, decomposing portfolio performance into asset allocation, security selection, and interaction effects.

Performance measurement and evaluation go beyond merely recording past performance. They identify strengths and weaknesses in the investment process, improve investment strategies, and provide a foundation for effective communication with investors. It is important to comprehensively consider various performance indicators to evaluate portfolio performance from multiple perspectives.

6️⃣ Practical Portfolio Management Strategies

Theoretical portfolio models face various constraints and challenges in real investment environments. In this section, we will explore portfolio management strategies and asset allocation approaches used in practice.

Strategic Asset Allocation

Definition
An approach that sets target weights for major asset classes based on long-term investment goals and risk preferences.
Key Characteristics
- Long-term perspective (typically over 3-5 years)
- Based on long-term capital market expectations
- Reflects the investor's risk tolerance and investment objectives
- Maintains target weights through periodic rebalancing
Approach Methods
1. Mean-Variance Optimization
  Uses the Markowitz approach, reflecting long-term expectations and constraints.
2. Black-Litterman Model
  Combines market equilibrium with investor views to mitigate extreme allocations.
3. Risk Parity
  Allocates in such a way that each asset class contributes equally to the portfolio risk.
4. Asset-Liability Management (ALM)
  An asset allocation strategy considering the investor's liability structure, particularly suitable for pension funds and insurance companies.

Tactical Asset Allocation

Definition
An approach that temporarily deviates from strategic asset allocation based on short-term factors such as market conditions, economic outlook, and asset valuation.
Key Characteristics
- Short to medium-term perspective (typically a few months to 1-2 years)
- Focus on relative valuation and market opportunities
- Limited deviation range compared to strategic allocation
- Includes market timing elements
Approach
1. Macroeconomic Analysis
  Adjust asset allocation based on economic cycles, inflation, interest rate environment, etc.
2. Relative Value Analysis
  Evaluate the relative valuation of various asset classes
3. Technical Analysis
  Short-term tactical allocation considering price trends, momentum, market sentiment, etc.
4. Quant Signals
  Utilize asset allocation signals based on systematic models

Dynamic Asset Allocation

Definition
An approach that systematically adjusts asset allocation based on market conditions and portfolio performance.
Key Approaches
1. Constant-Mix
  A method that rebalances whenever asset weights deviate from target ratios, advantageous in volatile markets.
2. Constant Proportion Portfolio Insurance (CPPI)
  A strategy that invests a fixed proportion of the cushion (asset value - minimum preservation amount) in risky assets, allowing for upward participation while limiting downside risk.
3. Volatility Targeting
  Dynamically adjusts the allocation between risky and safe assets to maintain a target level of portfolio volatility.
4. Time Diversification
  An approach that adjusts the proportion of risky assets based on the investment horizon, typically increasing the allocation to risky assets as the investment period lengthens.

Factor Investing

Factor Investing Strategy

Definition
An investment approach that focuses on systematic factors influencing returns instead of asset classes.
Key Factors
1. Market: Overall market exposure
2. Size: Small-cap vs. large-cap
3. Value: Value vs. growth stocks
4. Momentum: Assets with recent high returns vs. those with low returns
5. Quality: High quality (profitability, stability) vs. low quality
6. Volatility: Low volatility vs. high volatility
7. Liquidity: High liquidity vs. low liquidity
Implementation Methods
1. Single Factor Portfolio
  Investing in securities exposed to a specific factor (e.g., value funds)
2. Multi-Factor Portfolio
  Investing in multiple factors simultaneously to seek diversification benefits among factors
3. Smart Beta ETFs
  ETFs that track indices exposed to specific factors or combinations of factors
4. Factor Timing
  Dynamically adjusting factor exposure based on economic cycles, valuations, etc.
Advantages and Challenges
- Advantages: Systematic approach, diversification, transparency, cost efficiency
- Challenges: Factor selection and weighting decisions, difficulties in factor timing, factor crowding

Alternative Investments and Portfolio Diversification

Alternative Investment Strategies

Alternative Asset Classes
Asset groups utilized for portfolio diversification beyond traditional stocks and bonds:
1. Private Equity
2. Hedge Funds
3. Real Estate
4. Infrastructure
5. Commodities
6. Natural Resources
7. Collectibles such as Art and Wine
Role within the Portfolio
- Diversification of Returns: Low correlation with traditional assets
- Risk Hedge: Protection against inflation, market downturns, etc.
- Enhanced Returns: Some alternative assets offer additional risk premiums
- Obtaining Illiquidity Premium: Leveraging the advantages of long-term investors
Considerations
- Liquidity Constraints: Many alternative assets have low liquidity
- Valuation Challenges: Often lack regular market pricing
- High Barriers to Entry: Significant minimum investment amounts and accredited investor requirements
- Complexity and Lack of Transparency: Complicated structures and limited information disclosure

Practical Considerations for Risk Management and Portfolio Construction

Practical Risk Management Techniques

Rebalancing Strategies
1. Periodic Rebalancing: Rebalancing at set intervals (monthly, quarterly, annually)
2. Range-Based Rebalancing: Rebalancing only when asset allocation deviates from the target by a certain range
3. Mixed Approach: Combining periodic monitoring with range-based triggers
Consideration of Constraints
Constraints to consider when constructing a portfolio:
1. Transaction costs and taxes
2. Liquidity requirements
3. Legal and regulatory constraints
4. Minimum/maximum allocation limits by asset
5. ESG (Environmental, Social, Governance) considerations
Tail Risk Management
Strategies to prepare for extreme market conditions:
1. Stress testing and scenario analysis
2. Portfolio insurance using options
3. Tail risk hedging strategies
4. Maintaining dry powder
Unconventional Optimization Approaches
1. Risk Parity: Equally distributing the risk contribution of assets
2. Maximum Diversification: Maximizing the diversification ratio
3. Minimum Variance: Minimizing risk without predicting expected returns
4. Conditional Value-at-Risk (CVaR) optimization: Minimizing the risk of extreme losses

Portfolio Strategies by Investor Type

Approaches by Investor Type

Individual Investors
- Lifecycle Investing: Adjusting asset allocation based on age and goals
- Goal-Based Investing: Portfolios aligned with specific financial objectives (retirement, education expenses, etc.)
- Customized allocations based on investor temperament
Institutional Investors
- Pension Funds: ALM-based asset allocation tailored to long-term liabilities
- University Endowments: Balancing long-term asset preservation and income generation based on the assumption of perpetuity
- Insurance Companies: Asset allocation considering liability characteristics and regulatory requirements
- Foundations: Balancing sustainable spending rates and asset growth
Asset Management Firms
- Benchmark-based active management
- Absolute return seeking strategies
- Multi-style/multi-manager approaches

Practical portfolio management should be grounded in theoretical models while also applying them pragmatically, taking into account the complexities and constraints of real market environments. A portfolio strategy that combines a systematic approach with rigorous risk management processes is more likely to succeed in the long run.

7️⃣ Behavioral Finance and Portfolio Decision-Making

Traditional portfolio theory assumes that investors are completely rational; however, psychological biases and behavioral patterns significantly influence investment decisions in reality. Behavioral Finance seeks to understand investor behavior and market inefficiencies by considering these psychological elements.

Investor Psychological Biases

Major Behavioral Biases

Loss Aversion
Investors are more sensitive to losses than to gains of the same magnitude. According to behavioral economists Kahneman and Tversky, people feel losses about 2 to 2.5 times more strongly than gains.
Overconfidence
Investors tend to overestimate their knowledge and predictive abilities, which can lead to excessive trading or a lack of diversification.
Hindsight Bias
This is the tendency to wrongly believe that one could have predicted an event after it has occurred, leading investors to overestimate their predictive capabilities.
Confirmation Bias
This is the tendency to accept information that supports one's existing beliefs while ignoring contradictory information.
Disposition Effect
Investors tend to hold onto losing positions for too long and quickly sell winning positions, driven by the psychological urge to avoid realizing losses.
Familiarity Bias
Investors have a tendency to invest excessively in assets they are familiar with, which can manifest as home bias or excessive holding of company stock.
Anchoring
This refers to the tendency to rely too heavily on specific reference points (anchors) when making decisions, such as past purchase prices or historical highs acting as anchors.

Behavioral Portfolio Theory

Prospect Theory
Developed by Kahneman and Tversky, this theory explains how investors evaluate losses and gains. Key features include:
- Reference Point Dependence: Gains and losses are evaluated relative to a reference point (usually the current state).
- Loss Aversion: Losses have a greater psychological impact than equivalent gains.
- Diminishing Sensitivity: As the size of gains or losses increases, sensitivity to additional changes decreases.
- Probability Weighting: Low probabilities are often overestimated, while high probabilities are underestimated.
Behavioral Portfolio Theory
Proposed by Hersh Shefrin and Meir Statman, this theory explains that investors construct portfolios as mental accounts for different goals:
- Safety Layer: Conservative investments for minimal financial safety.
- Potential Upside Layer: Aggressive investments aimed at wealth enhancement.
SP/A Theory (Security-Potential/Aspiration Theory)
Developed by Lopes and Oden, this theory describes how an investor's decision-making is influenced by desires for security (Safety) and potential (Potential), as well as levels of aspiration (Aspiration).

Market Inefficiencies and Behavioral Biases

Behavioral Explanations of Market Inefficiency

Overreaction and Underreaction
Investors sometimes overreact and sometimes underreact to news or events, causing price anomalies:
- Short-term Overreaction: Excessive reaction to extreme news
- Long-term Underreaction: Gradual incorporation of new information
Momentum and Reversal Effects
- Momentum Effect: The tendency for returns to persist over the medium term (3-12 months)
- Reversal Effect: The tendency for past performance to reverse over the long term (3-5 years)
Anomalies Market inefficiencies that can be explained by behavioral biases:
- Value Premium: Value stocks provide higher returns than growth stocks
- Size Effect: Small-cap stocks offer higher returns than large-cap stocks
- Calendar Effects: Return patterns based on specific times, such as the January effect, Monday effect, etc.

Practical Applications of Behavioral Finance

Strategies for Overcoming Behavioral Biases

Behavioral Coaching
An approach to help investors recognize and overcome their behavioral biases:
- Bias Recognition: Assisting investors in understanding their psychological biases
- Discipline Maintenance: Applying systematic approaches to prevent emotional decision-making
- Emphasizing Long-Term Perspective: Focusing on long-term goals rather than short-term market fluctuations
Rule-Based Investing
Minimizing emotional decision-making through predefined rules and procedures:
- Automated Rebalancing
- Predefined Trading Rules
- Systematic Investment Plans (SIPs) and Dollar-Cost Averaging
Behavioral Portfolio Construction
Designing portfolios that consider the psychological characteristics of investors:
- Mental Accounting: Separating portfolios by goals
- Adjustment of Safe Asset Proportions: Allocating based on loss aversion tendencies
- Complexity Management: Excessive complexity may increase behavioral errors

Robo-Advisors and Behavioral Finance

Robo-Advisors and Behavioral Biases

Behavioral Advantages of Robo-Advisors
Automated investment platforms can mitigate several behavioral biases:
- Emotional Exclusion: Elimination of emotional biases through algorithm-based decision-making
- Consistency: Application of consistent investment rules regardless of market conditions
- Automatic Rebalancing: Regular and disciplined portfolio adjustments
Robo-Advisors Integrating Behavioral Elements
Modern robo-advisors include features that take into account investors' behavioral biases:
- Investor Behavior Profiling
- Customized Risk Tolerance Assessment
- Behavioral Nudges: Designs that encourage desirable investment behaviors
- Optimized Investment Performance Reporting: Information provided considering loss aversion bias

Insights from behavioral finance contribute to complementing traditional portfolio theory, leading to the development of more realistic investment models and strategies. Investors and financial professionals can make more effective investment decisions by recognizing and managing psychological biases.

8️⃣ Conclusion: The Importance of Portfolio Theory

Portfolio theory is a core concept in modern finance, providing investors with the theoretical foundation essential for understanding the balance between risk and return and making optimal investment decisions. This field, which began with Markowitz's pioneering research, has continuously evolved to include CAPM, multifactor models, and behavioral finance, bringing revolutionary changes to financial markets and investment practices.

The importance of portfolio theory can be found in the following aspects:

Key Contributions of Portfolio Theory

Scientific Basis for Diversification
Portfolio theory offers a mathematical and statistical foundation for the old investment wisdom of "don't put all your eggs in one basket," allowing for systematic analysis of how asset combinations can minimize risk while maximizing returns.
Establishment of the Relationship Between Risk and Return
It clarifies the fundamental trade-off between risk and return and proposes methodologies for systematically measuring and managing risk.
Understanding Asset Pricing Mechanisms
CAPM and multifactor models enhance our understanding of how risk factors affect asset prices and expected returns, explaining the functioning of capital markets.
Providing a Framework for Investment Decision-Making
It offers a systematic framework that enables investors to make rational investment decisions considering their objectives and constraints.
Promoting Financial Innovation
It has served as a theoretical foundation for modern investment products and services such as ETFs, index funds, and robo-advisors, contributing to the development of the asset management industry.

As portfolio theory has developed, we have gained a better understanding of the complexities of financial markets and the irrational aspects of investor behavior. Although the original theoretical models are not perfect, their core principles remain valid today, continually improving through ongoing research and practical application.

In the upcoming series on the basics of financial engineering, we will cover more advanced topics, including risk management, derivatives, time series analysis, and financial engineering programming, based on these portfolio theories. A solid understanding of portfolio theory will serve as a crucial foundation for learning more complex financial engineering concepts.

9️⃣ References and Recommended Resources

For a deeper understanding of portfolio theory, the following resources may be helpful:

Recommended Books

“Portfolio Selection: Efficient Diversification Theory" (Author: Harry Markowitz, Publisher: Wiley)
"Modern Portfolio Theory and Investment Analysis" (Authors: Edwin Elton, Martin Gruber, Publisher: Wiley)
"Asset Allocation: A Practical Guide to Balanced Investment" (Author: David Swensen, Publisher: Free Press)
"Beyond CAPM: New Approaches to Asset Pricing" (Author: Campbell Harvey, Publisher: Princeton University Press)
"Behavioral Portfolio Management" (Author: Meir Statman, Publisher: Wiley)
"Foundations of Portfolio Construction and Risk Parity" (Author: Edward Quesnay, Publisher: Risk Books)
"Expected Returns: An Investor's Guide to Harvesting Market Rewards" (Author: Antti Ilmanen, Publisher: Wiley)

Online Resources

CFA Institute: https://www.cfainstitute.org/
Korea Financial Investment Association Education Center: https://www.kifin.or.kr/
Research Affiliates: https://www.researchaffiliates.com/
AQR Capital Management (Research): https://www.aqr.com/Insights/Research
Morningstar Portfolio Construction: https://www.morningstar.com/
BlackRock Investment Institute: https://www.blackrock.com/corporate/insights
Federal Reserve Bank of St. Louis (FRED): https://fred.stlouisfed.org/

Major Journal

Journal of Portfolio Management
Financial Analysts Journal
Journal of Finance
Journal of Financial Economics
Review of Financial Studies
Journal of Investment Management
Journal of Behavioral Finance
Journal of Asset Management

Free Software and Tool

Python: pandas, numpy, scipy, matplotlib, scikit-learn
R: quantmod, PerformanceAnalytics, PortfolioAnalytics
Excel: Solver, Analysis ToolPak
Portfolio Visualizer Tool: https://www.portfoliovisualizer.com/
Yahoo Finance: https://finance.yahoo.com/

Disclaimer

The content of this blog is written for educational and informational purposes and should not be considered an investment recommendation or a substitute for financial advice. For actual financial decisions, please seek the advice of a professional.

Portfolio Theory ​

1️⃣ Basic Concepts of Portfolio Theory ​

Definition of Return and Risk ​

Covariance and Correlation ​

Portfolio Return and Risk ​

Benefits of Diversification ​

2️⃣ Markowitz Mean-Variance Optimization ​

Principles of Optimal Portfolio Construction ​

Efficient Frontier ​

Mathematical Formulation of Mean-Variance Optimization ​

Dual Parameter Optimization and Utility Function ​

Practical Challenges of Mean-Variance Optimization ​

Improved Optimization Approaches ​

3️⃣ Capital Asset Pricing Model (CAPM) ​

Basic Assumptions of CAPM ​

Assumptions about Investor Behavior ​

Assumptions about the Market ​

Capital Market Line (CML) ​

Derivation of the CAPM Model ​

Security Market Line (SML) ​

Empirical Validation and Limitations of CAPM ​

4️⃣ Multi-Factor Models and Extensions of CAPM ​

Basic Structure of Multi-Factor Models ​

Fama-French Three-Factor Model ​

Carhart 4-Factor Model ​

Other Multi-Factor Models ​

Practical Applications of the Multifactor Model ​

5️⃣ Measuring and Evaluating Portfolio Performance ​

Absolute Performance Measurement ​

Risk-Adjusted Performance Measurement ​

Multi-Factor Model Based Performance Evaluation ​

Drawdown and Risk Measurement ​

Performance Persistence and Attribution Analysis ​

6️⃣ Practical Portfolio Management Strategies ​

Strategic Asset Allocation ​

Tactical Asset Allocation ​

Dynamic Asset Allocation ​

Factor Investing ​

Alternative Investments and Portfolio Diversification ​

Practical Considerations for Risk Management and Portfolio Construction ​

Portfolio Strategies by Investor Type ​

7️⃣ Behavioral Finance and Portfolio Decision-Making ​

Investor Psychological Biases ​

Behavioral Portfolio Theory ​

Market Inefficiencies and Behavioral Biases ​

Practical Applications of Behavioral Finance ​

Robo-Advisors and Behavioral Finance ​

8️⃣ Conclusion: The Importance of Portfolio Theory ​

9️⃣ References and Recommended Resources ​

Portfolio Theory

1️⃣ Basic Concepts of Portfolio Theory

Definition of Return and Risk

Covariance and Correlation

Portfolio Return and Risk

Benefits of Diversification

2️⃣ Markowitz Mean-Variance Optimization

Principles of Optimal Portfolio Construction

Efficient Frontier

Mathematical Formulation of Mean-Variance Optimization

Dual Parameter Optimization and Utility Function

Practical Challenges of Mean-Variance Optimization

Improved Optimization Approaches

3️⃣ Capital Asset Pricing Model (CAPM)

Basic Assumptions of CAPM

Assumptions about Investor Behavior

Assumptions about the Market

Capital Market Line (CML)

Derivation of the CAPM Model

Security Market Line (SML)

Empirical Validation and Limitations of CAPM

4️⃣ Multi-Factor Models and Extensions of CAPM

Basic Structure of Multi-Factor Models

Fama-French Three-Factor Model

Carhart 4-Factor Model

Other Multi-Factor Models

Practical Applications of the Multifactor Model

5️⃣ Measuring and Evaluating Portfolio Performance

Absolute Performance Measurement

Risk-Adjusted Performance Measurement

Multi-Factor Model Based Performance Evaluation

Drawdown and Risk Measurement

Performance Persistence and Attribution Analysis

6️⃣ Practical Portfolio Management Strategies

Strategic Asset Allocation

Tactical Asset Allocation

Dynamic Asset Allocation

Factor Investing

Alternative Investments and Portfolio Diversification

Practical Considerations for Risk Management and Portfolio Construction

Portfolio Strategies by Investor Type

7️⃣ Behavioral Finance and Portfolio Decision-Making

Investor Psychological Biases

Behavioral Portfolio Theory

Market Inefficiencies and Behavioral Biases

Practical Applications of Behavioral Finance

Robo-Advisors and Behavioral Finance

8️⃣ Conclusion: The Importance of Portfolio Theory

9️⃣ References and Recommended Resources