Summary
# Fundamental concepts of return and risk
This section explores the fundamental concepts of investment, focusing on the relationship between return and risk, the principles of market efficiency, asset pricing, and the crucial role of diversification.
## 1. Fundamental concepts of return and risk
Investments inherently differ in their potential for risk and return, forming the bedrock of financial decision-making. Understanding these concepts is essential for navigating the investment landscape and making informed choices [4](#page=4) [6](#page=6).
### 1.1 Return and risk
* **Return:** The primary motivation for investing is to earn a return on capital that would otherwise be held as cash. Holding cash itself incurs an opportunity cost, as it could have been invested to generate returns [6](#page=6).
* **Expected return:** This refers to the anticipated gain an investor foresees from an investment over a future period [6](#page=6).
* **Realized return:** This is the actual return that an investment generates over a past period. It is important to note that the realized return is not always the same as the expected return [6](#page=6).
* **Risk:** Risk is defined as the possibility that the realized return will deviate from the expected return. It represents the uncertainty surrounding the actual return an investment will yield [6](#page=6).
> **Tip:** The fundamental principle is that greater risk should be compensated with higher expected returns [6](#page=6).
### 1.2 Trade-off between return and risk
The financial markets are competitive, operating under a "no-free-lunch" rule, which implies that to earn a return above the risk-free rate, one must take on risk. Investors are generally risk-averse, meaning they dislike bearing risk unless they are adequately compensated for it. Consequently, investors will only take on risk if they anticipate a corresponding return [6](#page=6) [7](#page=7).
The core of investment decisions lies in balancing the potential for expected return against the level of risk an investor is willing to tolerate. This trade-off is influenced by an investor's time horizon and their individual risk tolerance [7](#page=7).
* **General principle:** Lower-risk assets, such as corporate bonds, typically offer lower expected returns. Conversely, higher-risk assets, including stocks, warrants, options, and futures, have the potential for greater returns [8](#page=8).
* **Historical evidence:** Long-term data (1980-2021) demonstrates that equity markets have delivered substantially higher returns than inflation and government bonds, especially when accounting for reinvested dividends and coupons. This highlights the long-term rewards associated with bearing higher risk. For instance, over this period, equities generally provided significantly higher realized returns with greater associated risk compared to low-risk assets like short-term government bonds [8](#page=8).
> **Example:** From 1980 to 2021, equities (e.g., MSCI US, MSCI Europe) significantly outperformed inflation and government bonds, demonstrating the long-term benefits of higher risk-taking [8](#page=8).
* **Short-term complexities:** The risk-return trade-off may not always hold true over shorter periods. Market behavior can be complex, with asset class performance varying widely. For example, between 1999 and 2011, equities experienced substantial volatility and downturns, while some traditionally safe assets like Greek government bonds suffered severe losses during a debt crisis. This illustrates that high risk does not guarantee higher realized returns in the short term [8](#page=8).
> **Tip:** While the realized risk-return trade-off can break down in the short term, the trade-off between risk and *expected* return still holds. This is because higher-risk assets must be priced lower to offer a higher expected return compared to lower-risk assets [8](#page=8).
* **Shifting perceptions of safety:** Recent trends indicate that assets previously considered safe may no longer be as secure, highlighting the dynamic nature of investment risk [8](#page=8).
### 1.3 Efficient markets
The concept of market efficiency suggests that asset prices quickly incorporate all relevant information. There are two key types of expected returns considered in this context [9](#page=9):
* **E(r):** The expected return based on fundamental and forecasted information [9](#page=9).
* **Required r:** The return an investor demands as compensation for bearing risk, typically comprising a real rate, inflation premium, and a risk premium [9](#page=9).
In an efficient market, the expected return on an asset equals the required return, meaning there are no easy bargains or "free lunches" to exploit. The market price is considered the fair price [9](#page=9).
**Implications of the Efficient Market Hypothesis (EMH):**
* The return an investor receives is compensation for the risk they undertake [9](#page=9).
* It is not consistently possible to outperform the market through active management strategies [9](#page=9).
* Passive investment management is often preferred over active management due to market efficiency [9](#page=9).
> **Example:** If an asset's risk increases in an efficient market, investors will demand a higher required return. To provide this higher return, the asset's price must decrease, reflecting a lower upfront payment relative to its future expected cash flows [9](#page=9).
### 1.4 Pricing of assets
Asset prices are fundamentally determined by discounting their expected future cash flows at a required rate of return. An increase in the required rate of return leads to a decrease in the present value of these cash flows, thereby reducing the asset's price. In efficient markets, this calculated market price aligns with the fair value derived from the discounted cash flow approach [10](#page=10).
### 1.5 Diversification
Diversification is a key strategy to manage risk in investment portfolios. It involves holding a variety of different assets, so that the exposure to any single asset is limited. The core principle is encapsulated by the adage, "Don't put all your eggs in one bucket" [10](#page=10).
* **Objective:** The primary goal of diversification is to reduce the overall risk and volatility of a portfolio [10](#page=10).
* **Mechanism:** By spreading investments across various assets, diversification smooths out returns and diminishes the impact of substantial losses that could arise from a single underperforming investment [10](#page=10).
* **Impact on returns:** While diversification helps lower portfolio risk and volatility, it does not necessarily increase the overall expected returns [10](#page=10).
---
# Investment process and decision making
The investment process provides a structured, five-step framework for making and managing investment decisions, encompassing policy setting, analysis, portfolio construction, revision, and performance evaluation [11](#page=11) [13](#page=13).
### 2.1 The five-step investment process
The investment process is a systematic approach for investors to make decisions about which marketable securities to invest in, the extent of these investments, and when to make them. It can be broken down into five distinct steps [11](#page=11):
#### 2.1.1 Set investment policy
This initial step involves identifying the investor's unique objectives and the amount of wealth available for investment. It requires stating objectives in terms of risk and return and identifying potential investment categories. A robust investment policy should address [12](#page=12):
* A mission statement and long-run financial goals [12](#page=12).
* Risk tolerance, defining the amount of risk an investor is willing to bear [12](#page=12).
* The policy asset mix, which is the long-run allocation to broad asset classes and is considered the most important decision [12](#page=12).
* The choice between active or passive management strategies [12](#page=12).
#### 2.1.2 Perform security analysis and valuation
This step requires understanding the characteristics of specific securities and the factors that influence them. A valuation model is then applied to estimate the security's fair price by comparing it to its current market price, with the goal of identifying undervalued securities. This is a time-consuming but necessary process [12](#page=12).
#### 2.1.3 Construct a portfolio
Portfolio construction involves identifying the specific assets and asset classes in which to invest, utilizing the results from security analysis. A crucial principle here is that a portfolio taken as a whole is not simply the sum of its individual parts; this highlights the importance of diversification, often summarized by the adage "don't put all your eggs into one basket" [12](#page=12) [13](#page=13).
#### 2.1.4 Revise a portfolio
Portfolio revision addresses how and when the portfolio should be updated. This step involves periodically repeating the portfolio construction process (Step 3). Revisions may be necessary due to changing market conditions or investor circumstances, and can involve increasing or decreasing existing holdings, deleting certain securities, or adding new ones [13](#page=13).
#### 2.1.5 Evaluate performance
The final step involves measuring how portfolio performance should be assessed. This entails the periodic determination of the portfolio's performance relative to its risk and return objectives, requiring appropriate risk and return measures [13](#page=13).
> **Tip:** The investment process is hierarchical, guiding decisions from broad categories like stocks and bonds down to individual securities, ensuring consistency with overall policy and risk tolerance [13](#page=13).
### 2.2 Key investment decisions
Investors typically face two fundamental decision steps when constructing a portfolio [13](#page=13):
#### 2.2.1 Asset allocation
Asset allocation is the choice among broad asset classes such as money, stocks, bonds, real estate, commodities, and derivatives. Top-down investment strategies begin with asset allocation. This decision involves tailoring the mix of equities, bonds, and short-term assets to an investor's time horizon and risk tolerance. Allocation matrices and guidelines often suggest a higher allocation to equities for younger investors, gradually increasing allocations to bonds and short-term assets with age, reflecting different risk/return profiles and investment horizons [13](#page=13) [14](#page=14).
#### 2.2.2 Security selection
Security selection is the choice of particular securities within each asset class, for example, choosing between specific stocks like GM or IBM. This requires a valuation process for individual securities, also known as "security analysis". Bottom-up investment strategies start with security selection [14](#page=14).
> **Tip:** The decision of asset allocation versus security selection is a central debate in investment management, with active asset allocation involving market timing and active security selection focusing on individual stock picking, while passive strategies rely on fixed allocations or index tracking [15](#page=15).
### 2.3 Investment management strategies
Investment management strategies can be broadly categorized as active or passive, and they can focus on asset allocation or security selection [15](#page=15).
* **Active management:** Involves making specific investment choices to try and outperform a benchmark. This can include market timing (active asset allocation) or picking individual stocks believed to be undervalued (active security selection) [15](#page=15).
* **Passive management:** Relies on a predetermined allocation strategy or tracking a specific market index.
The philosophy often adopted is a **top-down approach**, which begins with broad macroeconomic insights to guide asset allocation, followed by security selection and performance evaluation. This contrasts with a **bottom-up approach**, which prioritizes identifying promising individual securities first and then aggregating them into a portfolio [15](#page=15).
> **Tip:** In a fully efficient market where prices reflect all available information, active attempts to beat the market are unlikely to succeed, making a passive, index-tracking approach generally the best strategy [15](#page=15).
### 2.4 Factors affecting investment decisions
Investment decisions are significantly influenced by the inherent uncertainty of future returns and the inevitability of making mistakes [16](#page=16).
* **Uncertainty:** The future is inherently uncertain, and all investors, both individual and professional, must make estimates and can make investing errors [16](#page=16).
* **Global Investment Area:** Adding foreign financial assets can provide opportunities to enhance returns or reduce risk through diversification. However, this introduces currency risk [16](#page=16).
* **Market Efficiency:** The belief in whether markets are efficient or inefficient plays a critical role in shaping investment strategies. If markets are believed to be efficient, passive indexing is often preferred. If markets are considered inefficient, active strategies like stock picking and market timing might offer opportunities for outperformance [16](#page=16).
> **Example:** When investing internationally, it is crucial to express returns in the investor's home currency (e.g., Euros) to accurately measure performance, as currency fluctuations can significantly impact the final outcome [17](#page=17).
### 2.5 How securities are traded
Securities are traded in financial markets, which can be categorized as primary or secondary [17](#page=17).
* **Primary markets:** Where issuers of securities raise funds by selling new securities, often with the assistance of investment bankers. Initial Public Offerings (IPOs) are a key example. While European and global IPOs historically show positive first-day returns due to underpricing, their long-term performance often underperforms comparable, already-listed companies [17](#page=17) [18](#page=18).
* **Secondary markets:** Where investors trade previously issued securities among themselves, often with the help of brokers or dealers. These markets are essential for providing liquidity and making securities attractive for investment. Examples include the NYSE, Nasdaq, and LSE [17](#page=17) [18](#page=18).
**Trading costs** in secondary markets include explicit costs like broker commissions and implicit costs such as the bid-ask spread [19](#page=19).
* **Bid price:** The price at which a dealer will buy a security from an investor [19](#page=19).
* **Ask price:** The price at which a dealer will sell a security to an investor [19](#page=19).
* The **bid-ask spread** is the difference between these two prices, representing the dealer's profit [19](#page=19).
**Types of Orders:**
* **Market orders:** Execute immediately at the best available price. They ensure fast trades but offer no control over the exact transaction price [19](#page=19).
* **Price-contingent orders:** Execute only if certain price conditions are met.
* **Limit orders:** An order to buy or sell at a specified price or better [20](#page=20).
* **Stop orders:** An order that becomes active as a market order once a specified trigger price is reached [20](#page=20).
**Market depth** refers to the total number of shares offered for trading at the best bid and ask prices. Greater market depth indicates a more liquid market, allowing larger trades without significantly impacting prices [19](#page=19) [20](#page=20).
### 2.6 Buying on margin
Buying on margin involves purchasing securities with money borrowed in part from a broker. Key terms include [20](#page=20):
* **Initial Margin Requirement (IMR):** The minimum percentage of initial investor equity required, currently set at 50% for stocks by the Federal Reserve's Regulation T [20](#page=20).
* **Maintenance Margin Requirement (MMR):** The minimum equity level an account must maintain. Exchanges mandate a minimum of 25% [20](#page=20).
* **Margin Call:** A notification from a broker requiring the investor to deposit additional funds or face liquidation of their positions if the equity falls below the maintenance margin level [20](#page=20).
---
# Portfolio theory and capital allocation
Portfolio theory provides a framework for understanding how investors construct diversified portfolios to balance expected return and risk, and how they decide to allocate their capital between risky assets and risk-free assets based on their individual risk aversion [58](#page=58).
### 3.1 Expected return and risk
Investors are primarily concerned with expected returns, which represent the most likely outcome from a distribution of possibilities. This distribution is constructed by considering various economic states, their associated probabilities, and the estimated returns for each state. Historically, expected returns can be estimated by calculating the arithmetic average of past returns, assuming past performance is indicative of future results [49](#page=49) [50](#page=50).
Risk is defined as the possibility that the realized return will differ from the expected return, or the uncertainty about the actual return to be earned. A common measure of risk is standard deviation ($\sigma$), which quantifies the dispersion of expected returns around the mean. Variance ($\sigma^2$) is the square of the standard deviation and is also used to measure risk [51](#page=51) [52](#page=52).
#### 3.1.1 Measuring risk
Risk can be measured using historical data, by calculating the variance or standard deviation of past returns. However, this historical approach assumes that past performance will repeat itself, which is not always true. Additionally, asset returns may not follow a normal (symmetric) distribution, which can lead to understating the true risk of negative returns when using standard deviation alone [52](#page=52) [53](#page=53).
#### 3.1.2 Required rate of return
The required rate of return links expected return and expected risk, reflecting the trade-off investors demand for taking on risk. It comprises three components [54](#page=54):
1. **Real interest rate:** Compensation for postponing consumption [54](#page=54).
2. **Expected inflation:** Compensation for maintaining purchasing power [54](#page=54).
3. **Risk premium:** Additional return for bearing risk, which is a function of investment risk and the investor's risk aversion [56](#page=56) [59](#page=59).
The nominal interest rate is the sum of the real interest rate and expected inflation. The Fisher equation describes this relationship: $1 + \text{Nominal Rate} = (1 + \text{Real Rate})(1 + \text{Inflation Rate})$ [54](#page=54).
#### 3.1.3 Risk premiums
A risk premium is the expected return in excess of the risk-free rate. It compensates investors for the uncertainty associated with an investment. Empirically, historical risk premiums have been considerable and generally correlate with higher levels of risk. For example, small stocks typically exhibit higher risk premiums than large stocks [56](#page=56) [57](#page=57).
In an efficient market where investors are rational, the expected return of an asset should equal its required rate of return. Thus, the expected return can be approximated as [58](#page=58):
$$ \text{Expected Return} \approx \text{Nominal Risk-Free Interest Rate} + \text{Risk Premium} $$
### 3.2 Portfolio theory
Portfolio theory, pioneered by Harry Markowitz, focuses on selecting optimal portfolios of assets. It involves two main stages [58](#page=58):
1. **Selection of a portfolio of risky assets:** This can often be delegated to professional managers [58](#page=58).
2. **Combination of the risky portfolio with a risk-free asset:** This decision depends on the investor's personal risk aversion and preferences for risk versus expected return. This is known as capital allocation [58](#page=58).
#### 3.2.1 Risk aversion and the price of risk
Risk aversion describes the degree to which investors are willing to hold risky assets; they avoid risk unless adequately compensated. A higher degree of risk aversion leads investors to demand a higher risk premium for taking on risk. The risk premium is influenced by both the investment risk ($\sigma^2$) and the investor's risk aversion (A). The "price of risk" represents the risk premium an investor requires per unit of risk [58](#page=58) [59](#page=59).
### 3.3 Capital allocation
Capital allocation is the process of deciding how to divide an investment portfolio between a risk-free asset and a risky portfolio [60](#page=60).
#### 3.3.1 Portfolio expected return and risk
For a complete portfolio (C) comprising a proportion $y$ invested in a risky portfolio (P) and $(1-y)$ in a risk-free asset ($\text{R}_F$), the expected return and risk are calculated as follows:
**Expected Return:**
The expected return of the complete portfolio $\text{E}(\text{R}_C)$ is a weighted average of the expected returns of the risk-free asset and the risky portfolio:
$$ \text{E}(\text{R}_C) = (1-y)\text{R}_F + y\text{E}(\text{R}_P) $$
This can be decomposed into the risk-free rate plus a risk premium that depends on the allocation to the risky portfolio and its excess return:
$$ \text{E}(\text{R}_C) = \text{R}_F + y[\text{E}(\text{R}_P) - \text{R}_F $$
**Risk (Variance and Standard Deviation):**
The variance of the complete portfolio $\sigma^2_C$ is:
$$ \sigma^2_C = y^2\sigma^2_P $$
Where $\sigma^2_P$ is the variance of the risky portfolio. The standard deviation of the complete portfolio $\sigma_C$ is:
$$ \sigma_C = y\sigma_P $$
This indicates that the total risk of the complete portfolio is directly proportional to the proportion invested in the risky portfolio and the risk of that portfolio [61](#page=61).
#### 3.3.2 Capital allocation line (CAL)
The Capital Allocation Line (CAL) graphically represents all possible risk-return combinations achievable by combining a specific risky portfolio with a risk-free asset. It is a straight line with [62](#page=62):
* **Intercept:** The risk-free rate ($\text{R}_F$).
* **Slope:** The Sharpe ratio of the risky portfolio, defined as $\frac{\text{E}(\text{R}_P) - \text{R}_F}{\sigma_P}$. The Sharpe ratio measures the excess return per unit of risk [62](#page=62).
The optimal allocation weight $y$ for the risky portfolio is determined by the investor's risk aversion (A) and the risk and expected return of the risky portfolio:
$$ y = \frac{\text{E}(\text{R}_P) - \text{R}_F}{\sigma^2_P} \div A $$
**Key insights from the CAL:**
* An investor's risk aversion determines their position along the CAL; higher aversion leads to a point closer to the risk-free asset.
* A higher Sharpe ratio for the risky portfolio makes it more attractive.
* The decision to invest in the risky asset versus the risk-free asset depends on whether the expected return adequately compensates for the risk.
* Leverage can be used by borrowing at the risk-free rate to invest more than 100% in the risky portfolio (i.e., $y>1$), which extends the CAL beyond the risk-free asset's return [64](#page=64).
#### 3.3.3 CAL in practice
* **Risk-free asset:** Typically represented by government bonds, though inflation-indexed bonds (e.g., TIPS) are preferred for inflation protection. Treasury bills are often used as a proxy due to their low interest rate sensitivity [65](#page=65).
* **Risky asset/portfolio:** Commonly a well-diversified portfolio of common stocks, such as an index fund tracking the S&P 500. Passive investment strategies ("indexing") are often favored for their cost-effectiveness and historical performance relative to active management [65](#page=65).
### 3.4 Diversification and portfolio risk
Diversification is the strategy of combining different risky assets in a portfolio to reduce overall risk, adhering to the principle of "not putting all your eggs in one basket" [67](#page=67).
#### 3.4.1 Sources of risk
The risk of a financial asset can be divided into two components [67](#page=67):
1. **Market (systematic) risk:** Arises from general economic uncertainty (e.g., business cycles, inflation, interest rates) and affects all firms. This risk cannot be diversified away [67](#page=67).
2. **Firm-specific (idiosyncratic) risk:** Stems from factors unique to an individual company (e.g., R&D success, management decisions). This risk can be eliminated through diversification [67](#page=67).
#### 3.4.2 Diversification and portfolio risk reduction
As the number of assets in a portfolio increases, the total risk (measured by standard deviation) declines. This reduction is primarily due to the elimination of firm-specific risk. Beyond a certain point (e.g., around 20 stocks), further diversification yields diminishing returns in risk reduction, and the portfolio's risk approaches the level of market (systematic) risk [68](#page=68) [69](#page=69).
#### 3.4.3 Correlation and diversification
The correlation between asset returns significantly impacts diversification benefits [70](#page=70).
* **Perfect positive correlation ($\rho = 1$):** No diversification benefit; portfolio risk is a linear sum of individual risks [71](#page=71).
* **No correlation ($\rho = 0$):** Provides some diversification benefits, reducing risk below a simple weighted average [72](#page=72).
* **Perfect negative correlation ($\rho = -1$):** Allows for complete elimination of portfolio risk by choosing appropriate weights [70](#page=70) [71](#page=71).
Lower correlations between assets lead to greater risk reduction through diversification. The investment opportunity set, which represents all attainable risk-return combinations, bows inward for correlations less than 1, demonstrating diversification benefits [72](#page=72) [73](#page=73).
### 3.5 Optimal portfolio construction
#### 3.5.1 Optimal risky portfolio (tangency portfolio)
When a risk-free asset is available, the optimal risky portfolio is the one that maximizes the Sharpe ratio when combined with the risk-free asset. This portfolio is known as the **tangency portfolio**. It forms the highest possible Capital Allocation Line (CAL) when plotted against the risk-free rate [75](#page=75) [76](#page=76).
#### 3.5.2 Efficient frontier
For a set of risky assets, the **efficient frontier** represents the set of portfolios offering the highest possible expected return for each level of portfolio standard deviation [80](#page=80).
* Portfolios on the efficient frontier are considered "efficient" because no other attainable portfolio offers a higher return for the same level of risk, or lower risk for the same level of return [80](#page=80).
* Individual assets typically lie inside the efficient frontier, as combinations of assets through diversification yield better risk-return profiles [80](#page=80).
* Portfolios below the efficient frontier are inefficient and can be improved by rebalancing to a frontier portfolio.
* The **mean-variance criterion** is used to construct the efficient frontier, favoring portfolios that are "northwest" (higher return, lower risk) [80](#page=80).
#### 3.5.3 Combining risky assets and the risk-free asset
The **separation property** in portfolio theory states that the optimal risky portfolio (tangency portfolio) is the same for all investors, regardless of their risk aversion. The investor's individual risk aversion then determines how they combine this optimal risky portfolio with the risk-free asset to construct their **complete portfolio** [81](#page=81) [82](#page=82).
* **Asset Allocation:** Involves deciding on the allocation to broad asset classes (e.g., stocks, bonds, real estate). This decision is crucial and explains a significant portion of a portfolio's performance [93](#page=93) [94](#page=94).
* **Security Selection:** Involves choosing individual securities within those asset classes [93](#page=93).
#### 3.5.4 Index models and practical implementation
Estimating expected returns, volatilities, and correlations for a large number of assets to construct an efficient frontier can be computationally intensive and prone to estimation errors ("Garbage In – Garbage Out"). **Index models**, particularly the single-index model, simplify this by assuming that a common market factor drives most of the co-variability between asset returns [82](#page=82) [86](#page=86) [87](#page=87).
The single-index model for an asset's return ($R_i$) is given by:
$$ R_i = \alpha_i + \beta_i R_m + \epsilon_i $$
where:
* $\alpha_i$ is the asset's alpha (constant term) [88](#page=88).
* $\beta_i$ is the asset's beta, measuring its sensitivity to market movements [88](#page=88).
* $R_m$ is the market's excess return over the risk-free rate [88](#page=88).
* $\epsilon_i$ is the firm-specific residual, representing unsystematic risk [88](#page=88).
This model breaks down risk into systematic (market) risk ($\beta_i^2 \sigma_m^2$) and unsystematic (firm-specific) risk ($\sigma^2(\epsilon_i)$). By estimating $\alpha_i$, $\beta_i$, and $\sigma^2(\epsilon_i)$ for each asset (often via regression analysis on historical data, known as the Security Characteristic Line or SCL), and estimating the market's variance ($\sigma_m^2$), the covariance between assets can be simplified, significantly reducing the number of parameters needed for portfolio optimization [89](#page=89) [90](#page=90) [91](#page=91).
International diversification can further reduce risk due to lower cross-country correlations, although globalization has increased these correlations over time. Asset classes like bonds and real estate often exhibit low or negative correlations with equities, offering valuable diversification benefits [92](#page=92) [93](#page=93) [94](#page=94).
---
# Capital asset pricing model (CAPM) and multifactor models
The CAPM and multifactor models provide frameworks for understanding how systematic risk influences expected asset returns, moving beyond single-factor explanations to incorporate multiple sources of risk [97](#page=97).
### 4.1 Capital asset pricing model (CAPM)
The Capital Asset Pricing Model (CAPM) is an equilibrium pricing model used to determine the fair price of a stock by predicting the relationship between risk and expected return. It extends the single-index model by focusing on how systematic risk drives asset pricing in equilibrium, providing a benchmark for investment evaluation [97](#page=97) [98](#page=98).
#### 4.1.1 Market portfolio
A core implication of CAPM is that all investors will hold the same portfolio of risky assets, known as the market portfolio (M). This portfolio is composed of all available securities in proportion to their market values. This homogeneity of investor expectations and optimization behavior, along with shared access to information and identical time horizons, leads to a single optimal risky portfolio for everyone. If any asset were excluded from this portfolio, no one would demand it, causing its price to fall to zero [100](#page=100) [98](#page=98) [99](#page=99).
The market portfolio must lie on the efficient frontier, representing the optimal mean-variance portfolio. The Mutual Fund Theorem supports this, suggesting that only one efficient mutual fund of risky assets (the market portfolio) is needed to satisfy all investment demands. This leads to the separation property, where investors first choose the market portfolio (investment decision) and then decide how much to allocate between this portfolio and the risk-free asset based on their risk aversion (financing decision) [100](#page=100) [99](#page=99).
#### 4.1.2 Assumptions of CAPM
CAPM relies on several simplifying assumptions to be theoretically derived [98](#page=98):
1. **Rational Investors:** All investors are rational, mean-variance optimizers aiming to maximize their Sharpe ratio [98](#page=98).
2. **Common Time Period:** Investors operate within the same investment time horizon [98](#page=98).
3. **Homogeneous Expectations:** Given the same information, all investors form identical expectations about expected returns, variances, and covariances [98](#page=98).
4. **Market Structure:** Assumptions regarding market structure are considered less critical [98](#page=98).
These assumptions imply that investors are "homogeneous," with identical characteristics like risk aversion (though this only affects their position on the Capital Market Line, not the line itself), investment horizons, and access to information. While some assumptions are unrealistic, the model's strength lies in its explanatory and predictive power rather than the strict realism of its assumptions [98](#page=98) [99](#page=99).
#### 4.1.3 Implications of CAPM
The CAPM has significant implications for asset pricing and portfolio management [99](#page=99):
* **Market Portfolio on Efficient Frontier:** The market portfolio is the tangency portfolio on the efficient frontier, meaning it's the mean-variance efficient portfolio of all risky assets [100](#page=100).
* **Market Risk Premium:** The risk premium of the market portfolio, $E(R_m) - R_f$, is determined by the average level of risk aversion ($\bar{A}$) and the total market risk ($\sigma_m^2$). It is proportional to both average investor risk aversion and the economy's total market risk [99](#page=99).
$$ E(R_m) - R_f = \bar{A} \sigma_m^2 $$ .
* **Individual Asset Risk Premium:** The risk premium for individual assets is determined by their systematic risk, measured by beta ($\beta_i$), and the market risk premium .
$$ E(R_i) - R_f = \beta_i (E(R_m) - R_f) $$ .
* **Systematic Risk is Key:** Investors are compensated only for bearing systematic (market-related) risk, not firm-specific (non-systematic) risk, which can be diversified away .
* **Same Reward-to-Risk Ratio:** In equilibrium, all assets and portfolios must offer the same reward-to-risk ratio .
* **Expected Return-Beta Relationship:** CAPM posits a linear and positive relationship between an asset's expected return and its beta. A security with a higher beta commands a higher expected return .
#### 4.1.4 Capital market line (CML) and security market line (SML)
* **Capital Market Line (CML):** The CML depicts the risk-return trade-off for efficient portfolios, connecting the risk-free asset to the market portfolio. It plots portfolio risk premium against portfolio standard deviation and is only applicable to fully diversified (efficient) portfolios [100](#page=100) .
$$ E(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m} \sigma_p $$ .
* **Security Market Line (SML):** The SML illustrates the CAPM's core prediction: the relationship between an individual asset's expected return and its systematic risk (beta). It starts at the risk-free rate (intercept) and slopes upward with the market risk premium as the rate of increase. The SML applies to all assets and portfolios, unlike the CML .
$$ E(R_i) = R_f + \beta_i (E(R_m) - R_f) $$ .
* Assets plotting **above** the SML are considered **undervalued** (positive alpha), while those **below** are **overvalued** (negative alpha) .
* If investors become more risk-averse, the market risk premium increases, making the SML steeper, leading to higher required returns and lower stock prices .
#### 4.1.5 Alpha and active management
Alpha ($\alpha$) represents the excess return of an asset or portfolio relative to its expected return predicted by CAPM for its level of systematic risk .
* **Ex ante alpha:** Forecasted return minus the CAPM-required return .
* **Ex post alpha:** Actual return minus the CAPM-required return .
In efficient markets, alpha should theoretically be zero. Empirical evidence suggests most mutual funds have near-zero alpha on average, indicating that persistent, large positive alphas are rare .
> **Tip:** Active management strategies aim to generate positive alpha by identifying mispriced securities or predicting market movements. CAPM provides a benchmark to evaluate the success of these strategies.
#### 4.1.6 CAPM and the single-index model (SIM)
The CAPM uses an unobservable market portfolio and theoretical expected returns, while the SIM uses an observable market index and focuses on realized returns. The SIM regression model, $R_{i,t} = \alpha_i + \beta_i R_{m,t} + \epsilon_{i,t}$, can be used to estimate an asset's beta and alpha, making CAPM more empirically testable. The estimated alpha from SIM is compared to CAPM's prediction that alpha should be zero .
* **Aggressive investments** have betas greater than 1 ($\beta > 1$) .
* **Defensive investments** have betas less than 1 ($\beta < 1$) .
#### 4.1.7 CAPM in the real world
CAPM has practical applications in:
* **Investment Decisions:** Determining the required rate of return to evaluate investment opportunities and manage portfolio beta .
* **Security Valuation:** Pricing new listings and determining the cost of capital for capital budgeting .
Estimating beta is typically done through linear regression, but faces challenges like estimation errors and the potential for betas to change over time. Betas often exhibit mean reversion, meaning historical betas may need adjustment to predict future betas more accurately .
> **Tip:** Adjusting historical beta estimates towards 1 can improve prediction accuracy, as seen in Merrill Lynch's approach, which uses a weighted average of 1 and the historical beta .
#### 4.1.8 Critique and empirical evidence
Roll's critique suggests CAPM is untestable because the true market portfolio cannot be observed. Empirical studies often find the SML to be "too flat," meaning high-beta stocks do not consistently earn proportionally higher returns than predicted. Anomalies, such as the superior performance of small-cap stocks and stocks with high book-to-market ratios, contradict the CAPM's single-factor explanation .
### 4.2 Multifactor models
Multifactor models extend CAPM by proposing that systematic risk, and consequently expected returns, are influenced by more than just the market factor. This recognition stems from empirical observations that other factors besides market beta help explain asset returns .
#### 4.2.1 Motivation for multifactor models
The limitations of CAPM that motivate multifactor models include:
* The unobservable nature of the market portfolio .
* The assumption that market return captures all macroeconomic factor impacts is often insufficient .
* Empirical research indicates that other factors significantly affect asset returns .
Multifactor models aim to provide a more comprehensive explanation of systematic risk and expected returns by incorporating multiple sources of economy-wide uncertainty .
#### 4.2.2 Identifying factors
Factors for multifactor models can be identified through two primary approaches:
1. **Theoretical Grounds (Macroeconomic-Based):** Factors are chosen based on economic theory, identifying major systematic risks that affect investors. These could include factors correlated with important consumption goods, future investment opportunities, or the general state of the economy (e.g., interest rates, industrial production). Examples include models by Chen, Roll, and Ross .
2. **Empirical Grounds (Microeconomic-Based):** Factors are identified by analyzing historical data to find variables that have empirically predicted high average returns. These are often based on firm characteristics. Examples include the Fama-French models .
Each priced risk factor is expected to carry its own risk premium .
#### 4.2.3 Fama and French three-factor model
The Fama-French three-factor model is a prominent example of a multifactor model that posits stock returns are related to factors beyond just the market index. It includes :
1. **Market index:** The excess return of the market portfolio over the risk-free rate ($R_m - R_f$) .
2. **Firm size (SMB):** The difference in returns between portfolios of small-cap stocks and large-cap stocks (Small Minus Big). This factor proxies for exposure to systematic risk related to company size .
3. **Firm book-to-market ratio (HML):** The difference in returns between portfolios of stocks with high book-to-market ratios (value stocks) and low book-to-market ratios (growth stocks) (High Minus Low). This factor captures systematic risk related to value characteristics .
The model's equation is:
$$ E(R_i) = R_f + \beta_i (E(R_m) - R_f) + s_i \cdot E(SMB) + h_i \cdot E(HML) $$
Where:
* $E(R_i)$ is the expected return of asset $i$.
* $R_f$ is the risk-free rate.
* $\beta_i$ is the sensitivity of asset $i$ to the market factor.
* $E(R_m) - R_f$ is the market risk premium.
* $s_i$ is the sensitivity of asset $i$ to the size factor (SMB).
* $E(SMB)$ is the expected premium for the size factor.
* $h_i$ is the sensitivity of asset $i$ to the book-to-market factor (HML).
* $E(HML)$ is the expected premium for the book-to-market factor.
The Fama-French model was motivated by empirical findings that small firms and value stocks historically generated higher average returns than predicted by CAPM, suggesting these are priced risk factors .
> **Tip:** The SMB factor represents the excess return from investing in smaller companies compared to larger ones, and HML represents the excess return from investing in value stocks (high book-to-market) compared to growth stocks (low book-to-market).
#### 4.2.4 Applying the three-factor model
Using the Fama-French three-factor model to estimate the required rate of return involves:
1. Estimating the asset's sensitivities (betas) to the market, SMB, and HML factors .
2. Forecasting the expected returns for the market factor, SMB, and HML .
3. Combining these with the risk-free rate using the model's equation .
While more complex due to the need for multiple factor forecasts, this model can provide a more nuanced and potentially more accurate estimate of required returns than CAPM alone, especially when these factors are found to be significant .
---
# Market efficiency and behavioral finance
Market efficiency posits that asset prices reflect all available information, influencing investment strategies and creating a debate with behavioral finance, which explains anomalies by incorporating psychological factors into investor decision-making .
### 6.1 Market efficiency
The efficient market hypothesis (EMH) suggests that security prices rapidly adjust to new information, implying that current prices reflect all available knowledge about a security and follow a random walk, making them unpredictable. The core of EMH is that expected returns should equal required returns, meaning there are no "bargains" or opportunities for a "no-free-lunch" scenario, and market prices equal fair prices. This implies that it is not possible to consistently achieve returns better than the market's risk-adjusted compensation [9](#page=9).
#### 6.1.1 Active versus passive management
In a perfectly efficient market where all securities are fairly priced, passive management, such as investing in a market index fund, is considered appropriate as it requires no analysis. Active management, on the other hand, aims to outperform the market through asset allocation and security selection, seeking abnormal returns (positive alpha). To achieve abnormal returns, an active manager's information or skill must surpass that of the overall market, which is considered impossible in an efficient market where everyone has access to the same information and draws the same conclusions .
> **Tip:** The debate between active and passive management hinges on one's belief about the degree of market efficiency. If markets are truly efficient, passive investing is generally favored.
Market efficiency implies that searching for undervalued stocks is often not worthwhile. If prices fully reflect all current information, investors' time spent analyzing for undervaluation would be unproductive. If prices do not fully reflect information, then an investor who finds and uses that information might be able to beat the market .
Examples of market efficiency testing include:
* **Cumulative abnormal returns surrounding takeover attempts:** Target stock prices jump immediately upon announcement, reflecting new information. However, abnormal returns observed in the days preceding an announcement can suggest insider trading or rumors, pointing towards market inefficiency .
* **Returns following CNBC report announcements:** Stock prices adjust quickly to news, but the observation of buying or selling minutes before an announcement can indicate insider information or front-running. A lagged second response to negative news also suggests opportunities for exploitation, indicating inefficiency .
* **Market response to COVID-19 vaccine announcement:** A significant positive news event leads to an immediate and sustained price increase, with a lack of gradual drift suggesting rapid incorporation of information, consistent with market efficiency .
> **Tip:** Even in efficient markets, risk still matters, and understanding the risk-return properties of securities is crucial. Decisions about how much risk to take, based on risk aversion and investment horizon, remain important .
##### 6.1.1.1 Ingredients of an informationally efficient market
Three key conditions are necessary for an "informationally" efficient market :
1. **Competition:** A large number of profit-maximizing participants analyze and value securities, constantly seeking overlooked information and aiming to outperform others .
2. **Randomness of new information:** New information arrives in the market unpredictably, meaning prices themselves will also be unpredictable .
3. **Rapid price adjustment:** Profit-maximizing investors adjust security prices quickly to reflect new information as it arrives .
The degree of efficiency can differ across markets, with larger stocks and developed markets generally being more efficient than smaller stocks or emerging markets due to differences in information availability and investor attention .
#### 6.1.2 Why does it matter?
The implications of market efficiency are significant:
* **If prices fully reflect information:** It is not worth an investor's time to use information to find undervalued securities .
* **If prices do NOT fully reflect information:** Finding and using that information might lead to beating the market .
#### 6.1.3 Why should capital markets be efficient?
The conditions for an efficient market are driven by competition among many investors who analyze securities and react rapidly to unpredictable new information .
#### 6.1.4 Alternative efficient market hypotheses
The EMH is typically presented in three forms, differing in the type of information that security prices are assumed to reflect :
1. **Weak-form EMH:** Current stock prices fully reflect all information contained in the history of past trading (past prices, returns, and trading volume) .
2. **Semi-strong-form EMH:** Current stock prices reflect all publicly available information, including past market data and non-market information about a firm's prospects (e.g., management quality, earnings forecasts, product lines) .
3. **Strong-form EMH:** Stock prices reflect all information, public and private, relevant to the firm .
Weak-form EMH implies that technical analysis, which uses past market data to predict future prices, is without merit. If markets are weak-form efficient, examining past price and market data to predict future price changes would be futile .
Semi-strong-form EMH implies that all publicly available information, such as that found in newspapers or company reports, is already fully reflected in prices. This means that decisions based on new information after it becomes public should not lead to above-average risk-adjusted profits. Consequently, fundamental analysis based solely on publicly available information is generally considered ineffective for gaining a consistent competitive advantage. However, superior prediction of future conditions that are not yet fully anticipated can still create value .
Strong-form EMH is the most stringent, suggesting that even insiders with private information cannot consistently beat the market. This would require perfect markets where all information is free and universally available, which is unrealistic .
> **Example:** Evidence contradicting weak-form EMH could be a known seasonal pattern (e.g., in January) that consistently yields abnormal returns, suggesting past information can predict future performance. Evidence contradicting semi-strong-form EMH could be if widely known public indicators, like low P/E ratios, consistently lead to abnormal returns, challenging the idea that all public information is priced in .
##### 6.1.4.1 Weak-form EMH
Under weak-form efficiency, past stock price data is fully incorporated into current prices. This means historical price trends and patterns cannot be used to predict future returns, and technical analysis has no value .
##### 6.1.4.2 Semistrong-form EMH
Semi-strong-form efficiency means that stock prices incorporate all publicly available information. This includes not only past market data but also fundamental information about the firm. As a result, publicly released news, reports, and analyses are immediately reflected in prices, making it impossible to earn excess returns using only public data .
##### 6.1.4.3 Strong-form EMH
Strong-form efficiency states that stock prices reflect all information, including private or insider information. This is the most extreme form and implies that even individuals with privileged access to information cannot consistently achieve superior returns .
#### 6.1.5 Implications of markets efficiency
Market efficiency has implications for various investment practices:
* **Technical Analysis:** EMH (weak-form) suggests technical analysis is without merit because historical price and volume data are already priced in .
* **Fundamental Analysis:** EMH (semi-strong-form) suggests that fundamental analysis based solely on publicly available information is also without merit for consistently outperforming the market. However, superior prediction of future events not yet fully priced in can still offer opportunities .
* **Portfolio Management:** Proponents of EMH argue that active management is largely a waste of effort and unlikely to justify its costs. They advocate for passive strategies, like buying and holding index funds, as active portfolio management is predicted to be without merit in efficient markets .
> **Tip:** Even if markets are efficient, portfolio management still plays a role, focusing on tailoring portfolios to individual needs rather than trying to beat the market.
#### 6.1.6 Role of portfolio management in efficient markets
In efficient markets, proponents of EMH suggest that active management is unlikely to justify its costs and that passive strategies, such as holding a broad-based index fund, are preferable. Research indicates that most money managers do not consistently outperform the market, and even when they do, the performance is often eroded by transaction costs and fees. This strongly supports market efficiency and highlights the value of low-cost index funds .
However, even in perfectly efficient markets, portfolio management has a role:
1. **Portfolio management without superior analysts:** Focuses on passive strategies, determining and quantifying risk preferences, constructing appropriate portfolios (balancing risk-free and risky assets), rebalancing to maintain risk levels, diversifying globally to eliminate unsystematic risk, and minimizing total transaction costs (taxes, trading turnover, liquidity costs) .
2. **Portfolio management with superior analysts:** If a portfolio manager has access to superior analysts who can better predict the future or identify less efficient market segments, active management can be profitable .
Index funds are a direct consequence of market efficiency, as they replicate market performance, minimize costs, and match market returns, avoiding the drawbacks of costly and often underperforming active management .
#### 6.1.7 Are markets efficient?
Empirical evidence on market efficiency is mixed, with numerous anomalies challenging the EMH.
* **Weak-form tests:**
* **Momentum effect:** A tendency for poorly or well-performing stocks to continue their performance in subsequent periods, suggesting short-term predictability .
* **Reversal effect:** A tendency for poorly or well-performing stocks to experience reversals in later periods, suggesting long-term predictability .
* **Predictors of broad market performance:** Factors like dividend yield, earnings yield, and bond market data have been found to predict aggregate market returns .
* **Semi-strong tests (Market Anomalies):**
* **P/E effect:** Portfolios of low P/E stocks tend to have higher risk-adjusted returns than high P/E stocks .
* **Small-firm effect:** Stocks of small firms have historically earned abnormal returns, often concentrated in January .
* **Neglected-firm effect:** Stocks of less-known firms can generate abnormal returns .
* **Book-to-market effect:** Shares of firms with high book-to-market ratios can generate abnormal returns .
* **Post-earnings announcement price drift:** Stock prices exhibit a sluggish response to earnings announcements, with momentum continuing past the announcement day, suggesting inefficiencies .
* **Bubbles and market efficiency:** Speculative bubbles can cause prices to rise above intrinsic value for extended periods, challenging immediate price adjustment to fundamental value .
> **Tip:** When interpreting anomalies, it's crucial to consider whether they represent true market inefficiencies or simply compensation for previously unidentified systematic risks. Some anomalies have shown less staying power after being reported, suggesting markets adapt or the findings were due to data mining .
Research on mutual fund performance generally shows that apparent skill is often eliminated by fees, competition, and known risk factors, resulting in negative net alphas after adjustments. While some managers may possess stock-pricing ability sufficient to cover costs, sustained outperformance is rare .
Overall, while numerous anomalies exist, intense competition limits persistent profits from them, leading to the conclusion that markets are "near-efficient" .
### 6.2 Behavioral finance
Behavioral finance is a growing field that moves beyond the assumption of perfect rationality, incorporating psychological factors and how real people make decisions to explain observed market behavior and anomalies. It recognizes that investors do not always process information correctly and often make inconsistent or suboptimal decisions .
#### 6.2.1 Using psychological biases to explain behavior
Behavioral finance offers explanations for several observed investor behaviors:
* **Riding losers and selling winners:** This can be explained by prospect theory, where people are more sensitive to losses than to gains .
* **Overconfidence in forecasts:** This can be attributed to confirmation bias, where individuals selectively seek information that confirms their existing beliefs .
* **Investing more in failing investments:** This is an example of escalation bias, where the decision to invest more might be driven by a desire to recoup past losses rather than a rational assessment of future prospects .
#### 6.2.2 Insights from behavioral finance for portfolio management
Behavioral finance suggests that persistent investor biases and "herd mentality" can create trading opportunities. This supports contrarian investment strategies, where investors take positions opposite to the prevailing market sentiment. Some mutual funds employ behavioral finance strategies to exploit investor biases and achieve above-normal returns .
---
# Equity and bond valuation and portfolio management
This topic explores the fundamental methods for valuing stocks and bonds, the impact of interest rate risk on bond portfolios, and strategies for managing these investments.
### 6.1 Equity valuation
The primary goal of equity valuation is to determine a stock's intrinsic value and compare it to its current market price to identify investment opportunities. Investors typically follow a two-step investment process: first, asset allocation, which involves choosing broad asset classes, and second, security selection, which focuses on identifying undervalued securities within those classes through valuation analysis .
#### 6.1.1 Market efficiency and security analysis
An efficient market is one where security prices fully and quickly reflect all available information. In such markets, the expected return is consistent with the level of risk, and market prices equal fair intrinsic values, making it difficult to find undervalued securities. Despite this, infrequent discoveries of minor mispricings can justify the work of security analysts .
* **Active vs. Passive Management**:
* **Passive management** assumes market efficiency and aims to match market returns through strategies like investing in index funds, with a focus on low costs .
* **Active management** seeks to "beat the market" by finding undervalued securities through superior analysis or timing, aiming to generate positive alpha .
Security analysis is a top-down process that begins with analyzing the macroeconomy and industries, then moves to individual companies and their financial statements to estimate a fair stock price .
#### 6.1.2 Valuation models
The core of equity valuation involves comparing a stock's current market price ($P_0$) to its estimated intrinsic value ($V_0$) .
* **Investment Opportunity Identification**: An investment opportunity exists if:
1. The "fair" price ($V_0$) is higher than the market price ($P_0$) .
2. The expected return is higher than the required rate of return .
3. The company's valuation ratio is below that of its industry .
* **Expected Return vs. Required Return**:
* **Expected Return**: The return anticipated based on available market information, including expected dividends and price changes. It is a forecast and does not guarantee compensation for risk (#page=136, 137) .
* **Required Rate of Return**: The return demanded by investors as compensation for taking on risk, often calculated using models like CAPM or APT. If the expected return exceeds the required return, the stock is considered undervalued and offers positive alpha .
* **Intrinsic Value**: The present value of all expected future cash payments, including dividends and the proceeds from the eventual sale of the stock, discounted at an appropriate risk-adjusted rate .
$$V_0 = \frac{E(D_1) + E(P_1)}{1+k}$$
where $V_0$ is the intrinsic value at time 0, $E(D_1)$ is the expected dividend at time 1, $E(P_1)$ is the expected price at time 1, and $k$ is the required rate of return .
> **Tip:** The intrinsic value ($V_0$) represents your estimate of the correct price, while the market price ($P_0$) is the consensus value from all traders. If $V_0 > P_0$, the stock is considered undervalued and attractive for purchase .
#### 6.1.3 Dividend Discount Models (DDM)
DDMs value a stock as the present value of all future expected dividends. This approach avoids the need to separately forecast the future stock price, which can be highly uncertain .
* **Constant Growth DDM (Gordon Growth Model)**: This model assumes dividends grow at a constant rate ($g$) indefinitely.
$$V_0 = \frac{D_1}{k - g}$$
where $D_1$ is the expected dividend in the next period, $k$ is the required rate of return, and $g$ is the constant dividend growth rate (#page=141, 143). A critical condition for this model is that $k > g$ .
> **Tip:** The intrinsic value ($V_0$) is positively related to expected dividends ($D_1$) and the expected growth rate ($g$), and negatively related to the required rate of return ($k$).
If the market is efficient, the required rate of return ($k$) can be expressed as the sum of the dividend yield and the growth rate:
$$k = \frac{D_1}{P_0} + g$$
where $P_0$ is the current stock price .
* **Multistage Dividend Growth Models**: These models account for periods where dividend growth rates may differ, such as an initial period of high growth followed by a stable growth phase .
#### 6.1.4 Other Discounted Cash Flow (DCF) Approaches
Beyond dividends, other DCF methods include valuing based on:
* Free cash flow to equity (FCFE) .
* Free cash flow to the firm (FCFF) .
These models are useful when dividends do not fully capture the firm's economic value or when a company has low dividend payouts but significant reinvestment .
#### 6.1.5 Price-Earnings (P/E) Ratio
The P/E ratio relates a stock's current market price to its earnings per share. It's a popular valuation metric due to its simplicity (#page=148, 155) .
$$P/E \text{ ratio} = \frac{\text{Price per share}}{\text{Earnings per share}}$$
* **Valuation by Comparables**: Comparing a firm's P/E ratio to the average P/E of its industry can help estimate intrinsic value (#page=148, 149). If a company's P/E is lower than the industry average, and its fundamentals are sound, it may be undervalued (#page=148, 149) .
If $V_0 > P_0$, the stock is attractive, which often implies its P/E ratio is lower than the industry P/E .
* **P/E Ratio and Growth Opportunities**: The P/E ratio can be derived from the DDM and is influenced by payout ratio, growth rate ($g$), and required return ($k$) (#page=149, 150) .
The P/E ratio can be decomposed into a no-growth component ($1/k$) and a premium for the present value of growth opportunities (PVGO). Firms with strong growth prospects typically have higher P/E ratios (#page=150, 151, 154) .
> **Tip:** Peter Lynch's PEG ratio (P/E divided by growth rate $g$) suggests a fairly priced stock has a PEG of 1. A PEG less than 1 may indicate a bargain .
* **Cyclically Adjusted P/E (CAPE) Ratio**: Proposed by Shiller, this ratio uses the average inflation-adjusted earnings over an extended period (e.g., 10 years) to provide a more stable P/E measure, smoothing out short-term earnings volatility and helping identify long-term market mispricing .
* **P/E and Expected Returns**: There is generally a negative relationship between expected returns ($E(r)$) and the P/E ratio. A low P/E ratio suggests a high earnings yield ($E_1/P_0$), potentially leading to an expected return higher than the required return, indicating an attractive, undervalued stock (#page=154, 155) .
#### 6.1.6 Other Valuation Ratios
* **Price-to-Book Value (PB) Ratio**: A low PB ratio might suggest that the market underestimates a company's future potential relative to its historical book value .
* **Price-to-Sales (PS) Ratio**: Useful for valuing companies that may not yet have available earnings, such as early-stage growth companies .
#### 6.1.7 Best Approach
No single valuation method is perfect; DCF and P/E ratios have their merits, and using both can lead to more robust estimates. All methods are subject to estimation errors due to the inherent uncertainty of the future .
### 6.2 Bond valuation and portfolio management
Bond valuation involves determining the present value of a bond's future cash flows (coupon payments and principal repayment) discounted at the appropriate market interest rate .
#### 6.2.1 Types of Bonds
* **Government Bonds**: Typically considered low-risk, including Treasury bills and bonds .
* **Corporate Bonds**: Issued by companies, carrying default risk (#page=156, 157, 160) .
* **Zero-Coupon Bonds**: Sold at a discount to face value, with no intermediate coupon payments; investors receive the face value at maturity (#page=156, 162) .
* **Inflation-Indexed Bonds**: Principal and coupon payments are adjusted for inflation, preserving real returns .
#### 6.2.2 Bond Pricing
The value of a bond is the sum of the present values of its promised cash flows, discounted at the required rate of return (market interest rate) .
$$ \text{Bond Value} = \sum_{t=1}^{T} \frac{\text{Coupon Payment}}{(1+r)^t} + \frac{\text{Par Value}}{(1+r)^T} $$
where $r$ is the market interest rate, $T$ is the number of periods to maturity, and coupon payments are made periodically .
#### 6.2.3 Market Interest Rate (Yield)
The market interest rate ($r$) for a bond, often referred to as the yield to maturity (YTM), reflects the required rate of return for investors, considering the bond's characteristics and market conditions. It comprises several components :
* Real risk-free rate .
* Compensation for expected inflation .
* Premiums for specific bond risks: default risk, liquidity risk, and call risk .
#### 6.2.4 Inverse Relationship Between Bond Prices and Yields
Bond prices and market interest rates move inversely. When market interest rates rise, the present value of a bond's fixed future cash flows decreases, leading to a lower bond price. Conversely, when market rates fall, bond prices increase (#page=159, 160). This relationship is non-linear and convex. Longer-maturity bonds are generally more sensitive to interest rate fluctuations than shorter-maturity bonds (#page=159, 168, 169) .
#### 6.2.5 Default Risk (Credit Risk)
Default risk refers to the possibility that the bond issuer will fail to make promised payments. Rating agencies like Moody's, S&P, and Fitch assess this risk, categorizing bonds into investment grade (AAA to BBB) and speculative grade (BB and below) (#page=156, 160). Higher default risk leads to higher yield spreads (the difference between the bond's yield and a comparable risk-free bond's yield) (#page=160, 161) .
#### 6.2.6 Yield to Maturity (YTM)
YTM is the discount rate that equates the present value of all future cash flows to the bond's current market price; it represents the bond's internal rate of return if held to maturity (#page=162, 163). It reflects real interest rates, inflation expectations, and risk premiums .
#### 6.2.7 Yield Curve
The yield curve plots the YTM of bonds against their time to maturity, typically using risk-free government zero-coupon bonds. Its shape provides insights into market expectations of future interest rates and inflation (#page=164, 166) .
* **Theories of the Term Structure**:
1. **Expectations Theory**: Suggests that long-term yields are determined by current and expected future short-term interest rates. The yield curve's shape reflects market expectations of future rate changes (#page=164, 165) .
2. **Liquidity Preference Theory**: Argues that investors demand a premium for holding longer-term, less liquid bonds due to increased interest rate risk and potentially lower liquidity. This premium can cause the yield curve to slope upward even if future short rates are not expected to rise (#page=165, 166) .
* **Yield Curve in Practice**: The term spread (difference between long- and short-term yields) can be an economic indicator. An inverted yield curve (short-term yields higher than long-term yields) often signals an impending recession .
### 6.3 Bond portfolio management
Managing bond portfolios involves balancing risk and return, primarily focusing on interest rate risk and credit risk.
#### 6.3.1 Interest Rate Risk and Duration
Interest rate risk is the risk that a bond's price will change due to fluctuations in market interest rates. This risk is captured by the concept of **duration**, which measures a bond's price sensitivity to changes in interest rates (#page=168, 169) .
* **Duration**: A weighted average measure of the time until a bond's cash flows are received.
$$ \text{Duration (D)} = \sum_{t=1}^{T} t \times \frac{CF_t}{(1+y)^t} / \text{Bond Price} $$
where $CF_t$ is the cash flow at time $t$, and $y$ is the yield to maturity .
* **Modified Duration**: Directly relates percentage price changes to yield changes:
$$ \text{% Change in Bond Price} \approx -\text{Modified Duration} \times \Delta y $$
where Modified Duration $= D / (1+y)$ .
* **Factors Affecting Duration**:
* **Maturity**: Longer maturity generally means higher duration (#page=169, 170) .
* **Coupon Rate**: Higher coupon rates lead to lower duration, as more cash is received earlier .
* **Yield to Maturity**: Higher yields generally lead to lower duration .
* **Zero-Coupon Bonds**: Duration equals maturity .
> **Tip:** For coupon bonds, duration is always less than maturity. The longer the maturity or the lower the coupon rate, the higher the bond's duration and its sensitivity to interest rate changes.
#### 6.3.2 Passive Bond Management: Immunization
Immunization strategies aim to protect a portfolio from interest rate risk by matching the duration of assets with the duration of liabilities (#page=172, 173) .
* **Banks**: Immunize by matching the duration of their assets (loans) with their liabilities (deposits) to protect the firm's current value against interest rate fluctuations .
* **Pension Funds**: Match the duration of their assets with the present value of their future liabilities (promises to retirees) to ensure sufficient funds are available (#page=172, 173) .
Immunization aims to offset price risk (impact of rate changes on bond values) with reinvestment rate risk (impact of rate changes on the return from reinvesting coupons) .
> **Tip:** Immunization is most effective for small changes in interest rates and requires periodic rebalancing as durations change over time .
#### 6.3.3 Convexity
Convexity measures the curvature of the price-yield relationship, providing a more accurate estimate of price changes than duration alone, especially for larger interest rate movements. Higher convexity is desirable because it leads to smaller price declines when interest rates rise and larger price increases when interest rates fall, offering an asymmetrical payoff .
#### 6.3.4 Active Bond Management
Active strategies seek to add value by:
1. **Interest Rate Forecasting**: Anticipating movements on the yield curve and adjusting portfolio duration accordingly. For example, expecting falling rates might lead to increasing portfolio duration, while expecting rising rates would lead to decreasing duration .
2. **Identifying Relative Mispricings**: Uncovering bonds that are undervalued due to factors like overly large default premiums .
Success in active management depends on having superior information or insights compared to the market consensus .
---
# Portfolio performance evaluation
The evaluation of investment performance is crucial for determining whether active managers genuinely outperform their benchmarks and for understanding returns in relation to the risks taken .
### 9.1 Investment performance evaluation
Active managers operate under the assumption that markets are not always perfectly efficient, allowing for potential opportunities to find undervalued securities. The fundamental goal of the investment process is to evaluate and understand portfolio performance. Performance evaluation aims to ascertain if active managers deliver superior returns .
#### Average rates of return
Different methods exist for calculating average returns, each with its own nuances:
* **Arithmetic average:** This calculates the simple mean of returns and represents the expected yearly return on average, but it does not account for the effect of compounding .
* **Geometric average:** This method incorporates the effect of compounding, providing a truer average return for growth over multiple periods. However, it does not consider the impact of additional investments made during the period. It is a time-weighted average where each period's return has equal weight .
* **Dollar-weighted average (Internal Rate of Return - IRR):** This method accounts for both the compounding effect and the timing and amounts of additional investments or withdrawals. It is calculated by finding the discount rate that equates the present value of all cash flows (inflows and outflows) to zero .
> **Tip:** The geometric average is generally considered more representative of long-term growth rates than the arithmetic average, while the dollar-weighted average (IRR) best reflects an individual investor's actual experience. The time-weighted return (TWR) is preferred for evaluating a fund manager's skill independent of investor cash flows .
#### Key questions in performance measurement
Three primary questions guide investment performance evaluation :
1. Was the portfolio's return adequate, at least preserving wealth ?
2. Did the returns sufficiently compensate for the risk taken by the investor or portfolio manager ?
3. How did the performance compare to an appropriate benchmark ?
Performance can be decomposed to determine if it stems from asset allocation, security selection, or market timing .
#### Comparison universe and benchmarks
* **Comparison Universe:** Comparing a fund's performance against other managers with similar investment styles (e.g., using Morningstar's style box) can be insightful. However, this method can be misleading if risk profiles differ significantly among peers .
* **Benchmark Comparison:** A more robust approach is to compare performance against a passive index portfolio with similar risk characteristics, such as the S&P 500 or Euro Stoxx 50. A key challenge here is ensuring that the chosen benchmark accurately reflects the manager's investment style, as a mismatch can attribute performance differences to factors other than genuine skill .
> **Tip:** Accurate performance appraisal requires confirming that both the manager's portfolio and the chosen benchmark operate under comparable risk conditions .
### 9.2 How to account for risk?
Simply comparing average returns is insufficient; returns must be adjusted for the risk taken to provide a meaningful evaluation. This is achieved through various risk-adjusted performance measures, often derived from mean-variance analysis and the Capital Asset Pricing Model (CAPM) .
#### Methods of Risk-Adjusted Performance
Key risk-adjusted performance measures include :
* Sharpe Ratio .
* Treynor Measure .
* Jensen’s Alpha .
* M² Measure (Modigliani and Modigliani, 1997) .
* Information Ratio .
These metrics help determine if returns are adequate compensation for the total risk, systematic risk, or diversifiable risk undertaken .
#### 9.2.1 Sharpe Ratio
The Sharpe Ratio measures the portfolio's excess return (return above the risk-free rate) per unit of total risk (standard deviation). A higher Sharpe Ratio indicates better risk-adjusted performance. Portfolios with Sharpe Ratios greater than that of the market portfolio are considered to have superior performance and would plot above the Capital Market Line (CML) on an expected return-standard deviation graph. In efficient markets with no transaction costs, the Sharpe Ratio of any efficient portfolio ($S_P$) should be less than or equal to the Sharpe Ratio of the market portfolio ($S_M$). Calculations for the Sharpe Ratio typically use realized (ex-post) variables .
The formula for the Sharpe Ratio is:
$$ SR_P = \frac{R_P - R_f}{\sigma_P} $$
where:
* $R_P$ is the portfolio's realized return .
* $R_f$ is the realized risk-free rate .
* $\sigma_P$ is the portfolio's total risk (standard deviation) .
> **Example:** If Portfolio A has a return of 15%, a risk-free rate of 5%, and a standard deviation of 10%, its Sharpe Ratio is $\frac{0.15 - 0.05}{0.10} = 1.0$. If Portfolio B has a return of 18%, a risk-free rate of 5%, and a standard deviation of 15%, its Sharpe Ratio is $\frac{0.18 - 0.05}{0.15} = 0.867$. Portfolio A has superior risk-adjusted performance according to the Sharpe Ratio .
#### 9.2.2 Treynor Measure
The Treynor Measure, derived from CAPM, assesses the portfolio's excess return per unit of systematic risk (beta). It implicitly assumes the portfolio is fully diversified, meaning firm-specific risk is eliminated. A Treynor Ratio higher than the market's risk premium suggests superior performance, and portfolios plotting above the Security Market Line (SML) are considered to have out- or underperformed relative to their systematic risk. In efficient markets with no transaction costs, the Treynor measure for any efficient portfolio ($T_P$) should be greater than or equal to the Treynor measure for the market portfolio ($T_M$) .
The formula for the Treynor Measure is:
$$ T_P = \frac{R_P - R_f}{\beta_P} $$
where:
* $R_P$ is the portfolio's realized return .
* $R_f$ is the realized risk-free rate .
* $\beta_P$ is the portfolio's systematic risk (beta) .
> **Tip:** The Treynor Measure is most appropriate for evaluating portfolios that are part of a larger, well-diversified investment strategy, as it focuses solely on systematic risk .
#### 9.2.3 Jensen’s Alpha
Jensen's Alpha measures the difference between a portfolio's realized return and its required return as predicted by CAPM. It quantifies the excess return generated by the portfolio manager beyond what would be expected given the portfolio's systematic risk. Superior managers generate a statistically significant positive alpha ($\alpha > 0$), while inferior managers produce a negative alpha ($\alpha < 0$). Sources of superior performance can be attributed to consistent selection of undervalued securities or effective market timing .
The formula for Jensen's Alpha is:
$$ \alpha_P = R_P - [R_f + \beta_P (R_M - R_f)] $$
where:
* $R_P$ is the portfolio's realized return .
* $R_f$ is the realized risk-free rate .
* $\beta_P$ is the portfolio's systematic risk (beta) .
* $R_M$ is the realized market return .
> **Note:** The Treynor Measure and Jensen's Alpha will always yield consistent results; if one indicates superior performance, the other will too. Alpha is often considered more straightforward as it directly quantifies the relative excess return .
#### 9.2.4 M2 Measure
The M² Measure, developed by Modigliani and Modigliani, addresses a perceived limitation of the Sharpe and Treynor ratios by converting them into a percentage return format that is more directly interpretable. It essentially adjusts the portfolio's return to match the market's total risk level, allowing for a direct comparison of hypothetical returns under identical risk conditions. This allows for ranking portfolios based on their adjusted returns, similar to how the Sharpe ratio ranks based on slope .
The calculation involves:
1. Calculating the portfolio's Sharpe Ratio.
2. Determining the hypothetical return ($r_P^*$) if the portfolio's risk were adjusted to match the market's standard deviation ($\sigma_M$).
$$ r_P^* = R_f + SR_P \times \sigma_M $$
The M² value is then the difference between this adjusted portfolio return and the market return:
$$ M^2 = r_P^* - R_M $$
> **Example:** If a portfolio has a Sharpe Ratio of 0.398 and the market has a standard deviation of 18.5% and a return of 18.5%, and the risk-free rate is 0%, the adjusted portfolio return would be $0.398 \times 18.5\% = 7.36\%$. The M² would then be $7.36\% - 18.5\% = -11.14\%$. However, if the market risk were 15% and return 18.5%, and the portfolio risk was adjusted to 15%, the M2 would represent the excess return. The M² measure calculates the return earned if the portfolio's total risk were adjusted to match the benchmark's total risk .
#### 9.2.5 Information Ratio
The Information Ratio (IR) is used to measure active management performance by assessing the excess return over a benchmark relative to the portfolio's tracking error (the standard deviation of those excess returns). It quantifies how much additional return a manager generates per unit of active risk taken. A higher IR indicates more consistent and efficient outperformance .
The formula for the Information Ratio is:
$$ IR = \frac{R_P - R_B}{\sigma_{P-B}} $$
where:
* $R_P$ is the portfolio's realized return .
* $R_B$ is the benchmark's realized return .
* $\sigma_{P-B}$ is the tracking error (standard deviation of the difference between portfolio and benchmark returns) .
> **Tip:** The Information Ratio is particularly useful when evaluating the addition of active positions to an already well-diversified portfolio, as it focuses on the consistency of active return generation relative to benchmark deviation .
### 9.3 Appropriate performance measures
The choice of the appropriate performance measure depends on the context of the portfolio and its role within a larger investment strategy :
* **Sharpe Ratio (or M² Measure):** Best suited for evaluating stand-alone portfolios that represent the entirety of an individual's investments, as it considers total risk .
* **Treynor Measure or Jensen’s Alpha:** Ideal for evaluating the performance of individual managers within a larger, diversified fund or pension plan, as they focus on systematic risk .
* **Information Ratio:** Appropriate for assessing an active position being added to a largely passive and well-diversified portfolio, focusing on the consistency of active returns relative to the benchmark .
### 9.4 Multi-index models and style analysis
#### Performance evaluation with a multi-index model
Multi-index models, such as the Fama-French 3-Factor model, enhance performance evaluation by accounting for multiple sources of systematic risk beyond just market beta. These models decompose returns into factors like market risk, size (SMB - Small Minus Big), and value (HML - High Minus Low). By explaining a larger portion of return variation, multi-factor models provide a more accurate assessment of whether a manager's alpha is due to genuine skill or simply exposure to other risk factors .
The Fama-French 3-Factor model expresses returns as:
$$ R_t - R_{f,t} = \alpha + \beta_1 (R_{M,t} - R_{f,t}) + \beta_2 SMB_t + \beta_3 HML_t + \epsilon_t $$
where:
* $R_t$ is the portfolio return at time t .
* $R_{f,t}$ is the risk-free rate at time t .
* $\alpha$ is the abnormal return or Jensen's alpha .
* $R_{M,t}$ is the market return at time t .
* $SMB_t$ is the return on the size factor at time t .
* $HML_t$ is the return on the value factor at time t .
* $\beta_1, \beta_2, \beta_3$ are the factor sensitivities .
* $\epsilon_t$ is the error term .
#### Style analysis
Introduced by William Sharpe style analysis evaluates performance by regressing a fund's returns against returns of various asset classes or style indices. Studies have shown that a significant portion (often over 90%) of a fund's return variation can be explained by its allocation to broad asset classes like stocks, bonds, and bills. This suggests that asset allocation plays a more dominant role in performance than individual security selection .
The regression model in style analysis takes the form:
$$ R_P = \alpha + \beta_{Style1} R_{Style1} + \beta_{Style2} R_{Style2} + ... + \beta_{StyleN} R_{StyleN} + \epsilon $$
where the sum of the betas ($\sum \beta_i$) typically equals 1, representing the total exposure to different asset classes. The R² value indicates how much of the portfolio's return variation is explained by the style factors .
> **Example:** In Sharpe’s analysis of the Fidelity Magellan Fund, over 97.5% of the return variation was explained by allocations to medium caps, large caps, and high P/E firms. While the active management component (security selection and market timing) explained a smaller portion (2.5%), the fund still generated a substantial cumulative abnormal return due to its intercept (alpha) .
On average, studies of mutual funds show a slightly negative intercept across many funds, supporting the argument for investing in passive funds .
#### Morningstar's risk-adjusted rating
Morningstar categorizes funds based on style definitions and then ranks them according to risk-adjusted performance, assigning a star rating from 1 to 5. This system aligns with measures like the Sharpe ratio, where higher percentiles in risk-adjusted performance lead to higher star ratings, indicating superior performance among peers .
### 9.5 Market timing
Market timing involves adjusting asset allocation between different asset classes, typically stocks and risk-free instruments, based on anticipated market movements. Theoretically, a perfect market timer could achieve exceptional returns by shifting resources optimally. However, there is little empirical evidence to suggest that most managers possess consistent market-timing ability .
A perfect market timer's payoff profile would be nonlinear. Testing for market timing ability often involves using regression models that allow the portfolio's beta to vary with market conditions .
#### Testing for market timing
Two common regression models used to test for market timing are:
* **Treynor and Mazuy Model:** This model adds a quadratic term to the standard single-index model regression. A positive and statistically significant coefficient ($c$) for this term suggests market timing ability .
$$ R_{P,t} - R_{f,t} = \alpha + \beta (R_{M,t} - R_{f,t}) + c (R_{M,t} - R_{f,t})^2 + \epsilon_t $$
* **Hendriksson and Merton Model:** This model uses an interaction term that captures changes in beta based on whether the market return exceeds the risk-free rate. A positive and significant interaction coefficient indicates timing skill. Let $D = 1$ if $R_{M,t} > R_{f,t}$ and $D = 0$ otherwise .
$$ R_{P,t} - R_{f,t} = \alpha + \beta (R_{M,t} - R_{f,t}) + c D (R_{M,t} - R_{f,t}) + \epsilon_t $$
> **Example:** In testing two portfolios (P and Q), portfolio Q exhibited market timing ability due to a positive timing coefficient ($c=0.10$), indicating it increased its market exposure when the market performed better than the risk-free asset. When market timing is accounted for, the alpha attributable to security selection can be more accurately isolated .
The potential value of market timing is enormous over long periods, as demonstrated by comparing returns of T-bills, equities, and a perfect market timer, where the perfect timer generates vastly superior returns due to consistent compounding gains .
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## Common mistakes to avoid
- Review all topics thoroughly before exams
- Pay attention to formulas and key definitions
- Practice with examples provided in each section
- Don't memorize without understanding the underlying concepts
Glossary
| Term | Definition |
|------|------------|
| Real Assets | Tangible assets that have intrinsic value due to their substance and properties, such as property, equipment, and commodities. |
| Financial Assets | Claims on real assets or the income generated by them, typically represented by securities like stocks, bonds, and derivatives. |
| Investment Policy | A formal statement outlining an investor's objectives, risk tolerance, constraints, and preferences, guiding all subsequent investment decisions. |
| Security Analysis | The process of evaluating individual securities to estimate their fair value by analyzing factors affecting their price and determining whether they are undervalued, fairly valued, or overvalued. |
| Portfolio Construction | The process of selecting and combining various assets into a portfolio to achieve specific investment objectives while managing risk and return. |
| Diversification | An investment strategy of spreading investments across various asset classes, industries, and geographic regions to reduce overall portfolio risk by minimizing the impact of any single investment's poor performance. |
| Risk | The possibility that an investment's realized return will differ from its expected return, or the uncertainty surrounding the actual return that will be earned. |
| Return | The gain or loss on an investment over a specific period, typically expressed as a percentage of the initial investment. |
| Efficient Market Hypothesis (EMH) | A theory stating that asset prices fully reflect all available information, making it impossible to consistently achieve risk-adjusted excess returns. |
| Passive Management | An investment strategy that aims to match the performance of a market index by holding a diversified portfolio of securities that replicates the index, typically involving lower costs and less active decision-making. |
| Active Management | An investment strategy that seeks to outperform a benchmark index by actively selecting securities, timing the market, or adjusting asset allocation based on research and forecasts. |
| Asset Allocation | The strategy of dividing an investment portfolio among different asset categories, such as stocks, bonds, and cash, based on an investor's risk tolerance, time horizon, and financial goals. |
| Security Selection | The process of choosing specific investments within asset classes, aiming to identify undervalued securities that are expected to outperform the market. |
| Capital Asset Pricing Model (CAPM) | A financial model that describes the relationship between systematic risk (beta) and expected return for assets, particularly stocks, stating that expected return equals the risk-free rate plus a risk premium proportional to beta. |
| Beta ($\beta$) | A measure of a security's volatility or systematic risk in relation to the overall market. A beta of 1 indicates that the security's price will move with the market, while a beta greater than 1 suggests higher volatility and a beta less than 1 suggests lower volatility. |
| Market Risk Premium | The excess return that investing in the stock market provides over the risk-free rate as compensation for investors bearing the market's risk. It is calculated as the expected market return minus the risk-free rate. |
| Systematic Risk | Market risk that affects the entire market or a large segment of it, stemming from macroeconomic factors such as changes in interest rates, inflation, or economic recessions. This risk cannot be eliminated through diversification. |
| Unsystematic Risk | Risk that is specific to an individual company or industry and can be diversified away by holding a portfolio of different assets. Also known as firm-specific or idiosyncratic risk. |
| Holding Period Return (HPR) | The total return received from holding an investment for a specific period, including capital gains and income distributions, expressed as a percentage of the initial investment. |
| Arithmetic Mean | A simple average of a series of returns over multiple periods, calculated by summing the returns and dividing by the number of periods. It is often used for forecasting but ignores the effect of compounding. |
| Geometric Mean | The compound average rate of return over multiple periods, calculated by multiplying the returns for each period, taking the n-th root (where n is the number of periods), and subtracting 1. It accurately reflects the actual growth of an investment over time. |
| Dollar-Weighted Return | An investment return measure that accounts for the timing and size of cash flows into and out of an investment, effectively calculating the internal rate of return (IRR). |
| Expected Return | The anticipated return on an investment, often calculated based on historical data, scenario analysis, or statistical models, reflecting the weighted average of possible returns weighted by their probabilities. |
| Required Rate of Return | The minimum return an investor expects to receive for taking on a specific level of investment risk, typically determined by factors like the risk-free rate, inflation expectations, and a risk premium. |
| Dividend Discount Model (DDM) | A method of valuing a stock by calculating the present value of all expected future dividend payments, discounted at the required rate of return. |
| Price-Earnings Ratio (P/E Ratio) | A valuation metric calculated by dividing a company's stock price by its earnings per share, indicating how much investors are willing to pay for each dollar of earnings. |
| Yield to Maturity (YTM) | The total return anticipated on a bond if it is held until it matures, accounting for all coupon payments and the difference between the purchase price and par value, expressed as an annual rate. |
| Duration | A measure of a bond's sensitivity to changes in interest rates, representing the weighted average time until the bond's cash flows are received. Longer duration implies greater price volatility. |
| Immunization | A bond portfolio strategy designed to protect the portfolio's value against interest rate fluctuations by matching the duration of assets and liabilities. |
| Convexity | A measure of the curvature of the relationship between a bond's price and its yield. It refines duration's approximation of price sensitivity, especially for larger interest rate changes. |
| Market Efficiency | The degree to which market prices reflect all available information. Markets are considered efficient if prices adjust rapidly to new information, making it difficult to consistently earn excess returns. |
| Behavioral Finance | A field of study that incorporates psychological factors and cognitive biases to explain how investors make decisions and how these behaviors influence market outcomes, often explaining anomalies not predicted by traditional financial models. |
| Alpha ($\alpha$) | A measure of a portfolio manager's ability to generate returns in excess of what is predicted by a market model (like CAPM), after accounting for the risk taken. Positive alpha suggests outperformance. |
| Sharpe Ratio | A risk-adjusted performance measure that calculates the excess return per unit of total risk (standard deviation). Higher Sharpe ratios indicate better risk-adjusted performance. |
| Treynor Measure | A risk-adjusted performance measure that calculates the excess return per unit of systematic risk (beta). It is used to compare portfolios that are part of a larger, diversified investment. |
| Jensen's Alpha | A measure of a portfolio's risk-adjusted performance relative to the expected return predicted by CAPM. It represents the average return of the portfolio over and above that predicted by CAPM. |
| Information Ratio | A measure of a portfolio manager's ability to generate excess returns relative to a benchmark per unit of active risk (tracking error). It assesses the consistency of an active manager's performance. |
| Market Timing | An investment strategy that aims to increase returns by shifting between risky assets (like stocks) and risk-free assets (like cash or short-term bonds) based on predictions of market movements. |
| Growth Stocks | Stocks of companies that are expected to grow at an above-average rate compared to other companies in the market, often reinvesting earnings rather than paying dividends. |
| Value Stocks | Stocks that appear to be trading for less than their intrinsic or fundamental value, often characterized by low price-to-earnings (P/E) or price-to-book (P/B) ratios, and sometimes higher dividend yields. |
| Present Value of Growth Opportunities (PVGO) | The portion of a stock's price that is attributed to the company's future growth prospects, rather than its current earnings or assets. |
| Cyclically Adjusted P/E Ratio (CAPE) | A valuation metric developed by Robert Shiller that divides a company's current stock price by the average inflation-adjusted earnings over the past 10 years, used to smooth out short-term earnings volatility and assess long-term market valuation. |