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Expected return

Expected return is the anticipated profit or loss on an investment, computed as the weighted average of all possible returns, where each return is multiplied by its associated probability of occurrence.^[1]^[2] This measure, rooted in probability theory, provides investors with a long-term average outcome based on historical data or estimated scenarios, expressed typically as a percentage of the initial investment.^[3] For a single asset, the expected return E(R) is calculated as E(R) = \sum (P_i \times R_i), where P_i represents the probability of each possible return R_i, and the probabilities sum to 1.^[1]^[2] In portfolio management, the expected return of a diversified portfolio is the weighted sum of the expected returns of its individual assets, weighted by their respective proportions in the portfolio.^[2] This concept is central to modern portfolio theory (MPT), pioneered by Harry Markowitz in 1952, which emphasizes constructing efficient portfolios that maximize expected return for a given level of risk, measured by return variance or standard deviation.^[4]^[5] MPT demonstrates that through diversification, investors can achieve higher expected returns without proportionally increasing risk, as correlations between asset returns influence overall portfolio volatility.^[5] Expected return also underpins models like the Capital Asset Pricing Model (CAPM), where an asset's expected return is estimated as the risk-free rate plus beta multiplied by the market risk premium: E(R_i) = R_f + \beta_i (E(R_m) - R_f).^[1] This framework helps assess whether an investment compensates adequately for its systematic risk relative to the market.^[1] However, expected return calculations rely on probabilistic estimates that may not accurately predict future outcomes, as they are based on historical data and assumptions about probability distributions, necessitating pairing with risk metrics like standard deviation for informed decision-making.^[1]^[2]

Fundamentals

Definition

Expected return is the long-run average value of returns that can be anticipated from an investment over repeated trials under identical conditions, or equivalently, the probability-weighted sum of all possible returns from a single trial.^[2] This concept serves as a foundational measure in decision-making under uncertainty, quantifying the anticipated profitability of an investment based on the likelihood of various outcomes.^[1] The general formula for expected return, denoted as E[R], is given by

E[R] = \sum_{i} p_i r_i,

where p_i is the probability of outcome i and r_i is the return associated with that outcome, with the probabilities summing to 1.^[2] This formulation originates from 17th-century probability theory, developed through the correspondence between Blaise Pascal and Pierre de Fermat in 1654, who addressed the "problem of points" in gambling and introduced the principle of mathematical expectation as a fair division of stakes based on winning probabilities.^[6] The concept was later adapted to finance in the 20th century, particularly in Harry Markowitz's 1952 portfolio selection theory, where expected returns form the basis for optimizing investment combinations under risk. Expected return is typically estimated using the arithmetic mean for forward-looking projections, as this provides an unbiased estimate of the average return per period across independent trials.^[7] In contrast, the geometric mean, which incorporates compounding effects, is more suitable for measuring historical performance over multiple periods but understates forward-looking expectations due to volatility drag.^[8]

Mathematical Foundations

The expected value of a random variable, often termed expected return in financial contexts, finds its rigorous foundation in the axiomatic framework of probability theory developed by Andrey Kolmogorov. These axioms define a probability measure on a sample space as a function that is non-negative, assigns probability 1 to the entire space, and satisfies countable additivity: for any countable collection of pairwise disjoint events A_1, A_2, \dots, the probability of their union equals the sum of their individual probabilities, P\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty P(A_i). This additivity axiom underpins the summation form of the expected value for discrete random variables, enabling the weighted average over outcomes to be expressed as a limit of finite sums consistent with the probability measure. A fundamental property of the expected value operator E[\cdot] is its linearity, which states that for any constants a, b \in \mathbb{R} and random variables X, Y,

E[aX + bY] = a E[X] + b E[Y].

This holds irrespective of any dependence between X and Y. To outline the proof for the discrete case, assume X and Y take values x_i and y_j with joint probabilities p_{ij}. Then,

E[aX + bY] = \sum_i \sum_j (a x_i + b y_j) p_{ij} = a \sum_i x_i \sum_j p_{ij} + b \sum_j y_j \sum_i p_{ij} = a E[X] + b E[Y],

where the inner sums yield the marginal expectations. This linearity extends to finite linear combinations and follows directly from the additivity of the probability measure. The expected value possesses several important properties arising from the non-negativity and normalization of probabilities. Specifically, if a random variable X attains values bounded between a minimum m and maximum M, then m \leq E[X] \leq M, as E[X] is a convex combination of the possible outcomes weighted by probabilities that sum to 1. Additionally, the expected value is affine, meaning it preserves weighted averages: for probabilities p_i \geq 0 with \sum p_i = 1, E\left[\sum p_i X_i\right] = \sum p_i E[X_i]. These properties ensure the expected value serves as a coherent measure of central tendency within the probabilistic framework. In statistical inference, the sample mean \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i, where the X_i are independent and identically distributed with expected value \mu, acts as an unbiased estimator of \mu. Unbiasedness means E[\bar{X}] = \mu, which follows from linearity: E[\bar{X}] = \frac{1}{n} \sum_{i=1}^n E[X_i] = \frac{1}{n} \cdot n \mu = \mu. This property highlights the sample mean's reliability for estimating the population expected value without systematic error.

Calculation Approaches

Discrete Probability Distributions

In discrete probability distributions, the expected return is calculated for random variables that take on a finite or countable number of distinct values, each associated with a specific probability from the probability mass function (PMF). The PMF, denoted p(x) = \Pr(X = x), assigns probabilities to each outcome x in the support set S_X, satisfying p(x) \geq 0, \sum_{x \in S_X} p(x) = 1, and p(x) = 0 outside S_X. These outcomes represent mutually exclusive and exhaustive events, ensuring all possible scenarios are covered without overlap.^[9] The expected return E[R] is the probability-weighted average of these outcomes, given by the formula

E[R] = \sum_{x \in S_X} x \cdot p(x),

where the summation occurs over all possible returns x. This step-by-step process involves identifying each possible return, its corresponding probability from the PMF, multiplying them pairwise, and summing the results. For instance, consider a stock like MassAir with three possible returns in different economic states: -10% with probability 0.20, 10% with probability 0.60, and 40% with probability 0.20. The calculation yields

E[R] = (-0.10)(0.20) + (0.10)(0.60) + (0.40)(0.20) = -0.02 + 0.06 + 0.08 = 0.12

or 12%.^[10] The binomial distribution extends this to scenarios involving repeated independent trials, each with binary outcomes, such as modeling investment returns as a series of "success" (e.g., positive return) or "failure" (e.g., negative return) akin to fair coin flips. In a single trial following a Bernoulli distribution with success probability \pi, the expected return is \pi \cdot r_s + (1 - \pi) \cdot r_f, where r_s and r_f are the success and failure returns. For n trials under the binomial distribution, the expected number of successes is n\pi, and the overall expected return scales linearly with n. An example is a binomial pricing model where each period's return is +5% (up) with probability 0.5 or -2.5% (down) with probability 0.5, giving a per-period expected return of (0.05)(0.5) + (-0.025)(0.5) = 0.0125 or 1.25%; for multiple periods, the total expected return is the sum of individual expectations.^[9]^[10] These calculations can be performed using computational tools like spreadsheets, where functions such as SUMPRODUCT multiply returns by probabilities and sum them automatically, facilitating analysis of discrete distributions without manual computation.^[11]

Continuous Probability Distributions

In continuous probability distributions, the expected return E[R] for a random return variable R is computed using an integral over its probability density function f(r), reflecting the uncountable set of possible outcomes. Specifically, E[R] = \int_{-\infty}^{\infty} r f(r) \, dr, where the integral is taken over the support of f(r). This formula arises as the limit of the discrete summation approach when the number of outcomes increases indefinitely and the probability mass between points approaches a density; by approximating the continuous variable with finer discrete partitions, the sum \sum r_i P(R = r_i) converges to the Riemann integral as the partition width \Delta r \to 0.^[12] A prominent example in finance involves the lognormal distribution for asset returns, which models the multiplicative nature of price changes. If the continuously compounded (log) returns follow a normal distribution with mean \mu and variance \sigma^2, the simple return R follows a lognormal distribution, and its expected value is E[R] = e^{\mu + \sigma^2/2} - 1, where \mu represents the expected log-return and \sigma the volatility.^[13] This closed-form expression derives from the moment-generating function of the normal distribution, M(t) = e^{t\mu + t^2 \sigma^2 / 2}, such that E[e^Z] = M(1) for Z \sim N(\mu, \sigma^2), yielding the adjustment for the exponential transformation.^[13] The lognormal distribution is favored for asset returns because it ensures strictly positive values (preventing negative prices) and captures the right-skewness observed in empirical financial data, where large upward movements are more probable than symmetric downside risks. This skewness aligns with the compounded growth process of investments, as log-returns are approximately normally distributed under models like geometric Brownian motion.^[13] When analytical solutions like the lognormal formula are unavailable for complex distributions, numerical methods such as Monte Carlo simulation approximate the expected return by generating thousands of random draws from the density f(r) and averaging the outcomes. In finance, this involves sampling returns (often assuming lognormality) across multiple paths to estimate E[R] as the sample mean, providing probabilistic forecasts while incorporating parameters like mean, volatility, and correlations.^[14]

Financial Applications

Single Asset Evaluation

In evaluating a single asset, expected return serves as a primary metric for investment selection, particularly under risk-neutral assumptions where investors prioritize maximizing anticipated gains without regard for volatility. A risk-neutral investor selects the asset offering the highest expected return, as their utility is linear in wealth, rendering them indifferent to risk differences across options.^[15] This approach contrasts with risk-averse strategies but provides a baseline for comparison; in practice, it often precedes the incorporation of risk-adjusted metrics like the Sharpe ratio to refine choices.^[16] Historical estimation of expected return relies on the sample mean from past realized returns, approximating the ex-ante value as the arithmetic average over a period. For a single asset with observed returns r_1, r_2, \dots, r_n, the estimator is given by:

\hat{E}[R] = \frac{1}{n} \sum_{t=1}^{n} r_t

This method, commonly used for equities and bonds, assumes stationarity in the return-generating process but can introduce bias if historical data reflects atypical conditions, such as varying interest rates or market risks.^[17]^[18] Refinements like Bayesian adjustments improve precision by accounting for estimation error. Forward-looking estimates enhance historical methods by integrating prospective data, such as analyst earnings forecasts to derive implied costs of capital as proxies for expected returns. These estimates correct for predictable biases in forecasts, yielding more accurate single-asset valuations than relying solely on past data.^[19] Scenario analysis complements this by assigning probabilities to discrete outcomes—e.g., optimistic, base, and pessimistic cases—to compute a probability-weighted expected cash flow or return, thereby capturing uncertainty in future performance.^[20] In capital budgeting, the hurdle rate is the minimum required rate of return for an investment to be considered viable, often equivalent to the cost of capital. It serves as a benchmark; a project is accepted if its internal rate of return (IRR)—the discount rate at which net present value (NPV) equals zero—exceeds the hurdle rate, ensuring it meets or exceeds opportunity costs. For instance, if the hurdle rate is 10%, only projects with an IRR of at least 10% would proceed.^[21]

Portfolio Context

In the context of portfolio theory, the expected return of a multi-asset portfolio is determined by aggregating the expected returns of its individual components according to their respective weights. This relationship stems from the linearity of expectation, a fundamental property in probability theory, which allows the expected value of a linear combination of random variables to be expressed as the linear combination of their individual expected values. Specifically, for a portfolio consisting of n assets with returns R_1, R_2, \dots, R_n and weights w_1, w_2, \dots, w_n (where \sum_{i=1}^n w_i = 1), the portfolio return is R_p = \sum_{i=1}^n w_i R_i. Applying linearity yields the expected portfolio return as

E[R_p] = E\left[ \sum_{i=1}^n w_i R_i \right] = \sum_{i=1}^n w_i E[R_i],

demonstrating that E[R_p] is simply the weighted average of the individual assets' expected returns. This formulation was central to the development of modern portfolio theory, as outlined by Harry Markowitz, who emphasized its role in enabling investors to construct portfolios with desired return profiles through asset allocation.^[22] A key implication of this aggregation is its behavior under diversification. Unlike portfolio variance, which depends on the covariances between asset returns and thus benefits from diversification to reduce overall risk, the expected return remains unaffected by these correlations. The linearity ensures that correlations influence only the dispersion of returns, not their central tendency; the portfolio's expected return is always the weight-based average, regardless of how assets co-move. This distinction highlights why diversification primarily impacts risk management rather than return generation in expectation, allowing investors to achieve targeted returns through weighting without concern for interdependence in means. Markowitz illustrated this in his analysis of efficient portfolios, where expected return lines form straight, parallel boundaries independent of covariance structures.^[22] Over time, however, realized returns cause asset weights to drift from their initial allocations, altering the portfolio's composition and thereby its expected return. For instance, if a higher-return asset outperforms, its weight increases naturally, elevating the overall E[R_p] toward that asset's higher expectation; conversely, underperformance reduces its weight and pulls E[R_p] downward. Rebalancing involves periodically adjusting holdings to restore target weights, which directly resets E[R_p] to the intended weighted average and prevents unintended shifts in return exposure. This process maintains alignment with the investor's strategic objectives but may involve transaction costs and tax implications that indirectly affect net returns. Research on rebalancing strategies confirms that buy-and-hold approaches can lead to higher expected returns due to drift toward outperforming assets, while constant reweighting enforces the original target but potentially sacrifices some upside.^[23] To illustrate, consider a two-asset portfolio with 60% allocated to stocks (E[R_s] = 10\%) and 40% to bonds (E[R_b] = 5\%). The initial expected return is E[R_p] = 0.6 \times 10\% + 0.4 \times 5\% = 8\%. If stocks return 15% and bonds 3% in a period, the new weights become approximately 62.5% stocks and 37.5% bonds, shifting E[R_p] to $0.625 \times 10\% + 0.375 \times 5\% \approx 8.125\%. Rebalancing back to 60/40 restores E[R_p] to 8%, preserving the original return profile.^[22]

Extensions and Limitations

Relation to Risk Measures

In modern portfolio theory, the expected return is intrinsically linked to risk measures, particularly variance, which quantifies the dispersion of possible returns around the mean. Variance is defined as the expected value of the squared deviation from the expected return, mathematically expressed as \operatorname{Var}(R) = E[(R - E[R])^2], serving as a proxy for total risk in investment decisions.^[22] This relationship underpins the risk-return tradeoff, where investors demand higher expected returns to compensate for bearing greater variance, as higher-risk assets exhibit wider return fluctuations that increase the potential for both gains and losses.^[24] Harry Markowitz formalized this integration in his seminal 1952 work on portfolio selection, establishing mean-variance analysis as the foundation for balancing expected return against risk in diversified portfolios.^[24] Building on this, the Capital Asset Pricing Model (CAPM) further refines the connection by relating an asset's expected return to its systematic risk, measured by beta (\beta). In CAPM, the expected return of asset i is given by E[R_i] = R_f + \beta_i (E[R_m] - R_f), where R_f is the risk-free rate, E[R_m] is the expected market return, and \beta_i captures the asset's sensitivity to market movements, representing non-diversifiable risk that commands a premium in expected returns.^[25] To evaluate performance on a risk-adjusted basis, William Sharpe introduced the Sharpe ratio in 1966, defined as (E[R] - R_f) / \sigma, where \sigma is the standard deviation of returns (the square root of variance). This metric assesses excess return per unit of total risk, enabling comparisons across investments and highlighting how expected return must exceed the risk-free rate to justify volatility exposure.^[26]

Assumptions and Real-World Deviations

The calculation of expected return in finance relies on several foundational assumptions that idealize investor behavior and market conditions. Central to these is the rational expectations hypothesis, which posits that investors form unbiased forecasts of future returns using all available information optimally, leading to expectations that are correct on average. ^[27] Another key assumption is the stationarity of probability distributions, implying that the underlying probabilities of returns remain constant over time, allowing historical data to reliably inform future expectations. ^[28] Additionally, models typically assume frictionless markets with no transaction costs, enabling investors to adjust portfolios costlessly in response to new information. These assumptions are frequently violated in practice, as highlighted by behavioral finance research, which documents systematic deviations from rationality due to cognitive biases and emotional factors. For instance, investors often exhibit overconfidence, leading to overly optimistic return expectations that diverge from objective probabilities, or loss aversion, which causes them to overweight negative outcomes irrationally. ^[29] ^[30] Such biases challenge the rational expectations framework, resulting in mispriced assets and inefficient markets, as evidenced by persistent anomalies like the equity premium puzzle where actual returns exceed what rational models predict. ^[31] ^[32] Empirical evidence further reveals deviations through the prevalence of fat-tailed return distributions, characterized by excess kurtosis far beyond the normal distribution's value of zero, which standard expected return models assume. ^[33] This leptokurtosis implies a higher likelihood of extreme events than predicted, leading to systematic underestimation of downside risks in portfolio evaluations. ^[34] A stark illustration is the 1987 stock market crash, where the Dow Jones Industrial Average plummeted 22.6% in a single day, an event with probability near zero under normal assumptions but reflective of fat tails in real financial data. ^[35] Estimation of expected returns often compounds these issues by overrelying on historical data, which ignores structural breaks—sudden shifts in market regimes that invalidate past patterns. ^[36] The 2008 global financial crisis exemplifies this, as it introduced profound changes in volatility, liquidity, and intermarket correlations, rendering pre-crisis historical estimates unreliable for post-crisis forecasting and contributing to widespread underestimation of risks. ^[37] ^[38] Such breaks can bias expected return calculations upward during stable periods and fail to anticipate crises, amplifying estimation errors in long-horizon predictions. ^[39] To address these limitations, robust alternatives such as Bayesian updating methods incorporate prior beliefs with new data to dynamically revise expected return estimates, providing greater flexibility in non-stationary environments. ^[40] These approaches mitigate overreliance on historical averages by shrinking estimates toward a global mean and accounting for uncertainty, as demonstrated in hierarchical models for asset allocation that improve out-of-sample performance amid regime shifts. ^[41] Bayesian techniques thus offer a principled way to handle fat tails and structural breaks, enhancing the reliability of expected return forecasts in real-world applications. ^[42]

References

[1]
Understanding Expected Return: A Guide to Investment Profitability
Expected return is the profit or loss an investor expects based on historical rates of return and calculated as a weighted average of potential outcomes.Calculating Expected Return · Limitations of Expected Return
[2]
How to Calculate a Portfolio's Expected Return
The expected return on an investment is the expected value of the probability distribution of possible returns it can provide to investors.What is Expected Return? · Calculating the Expected...
[3]
Expected Return | Formula + Calculator - Wall Street Prep
The expected return measures the anticipated return on an investment or portfolio of securities, expressed in the form of a percentage.
[4]
Portfolio Selection - jstor
PORTFOLIO SELECTION*. HARRY MARKOWITZ. The Rand Corporation. THE PROCESS OF SELECTING a portfolio may be divided into two stages. The first stage starts with ...
[5]
[PDF] Modern Portfolio Theory
It is an investment theory based on the idea that risk-averse investors can construct portfolios to optimize or maximize expected return based on a given.
[6]
July 1654: Pascal's Letters to Fermat on the "Problem of Points"
Jul 1, 2009 · In the mid-17th century, an exchange of letters between two prominent mathematicians–Blaise Pascal and Pierre de Fermat–laid the foundation for ...
[7]
[PDF] The Misuse of Expected Returns - University of Colorado Boulder
Much textbook emphasis is placed on the mathematical notion of expected return and its historical estimate via an arithmetic average of past returns.
[8]
Geometric vs Arithmetic Return | CFA Level 1 - AnalystPrep
The geometric return is always a better measure of investment performance compared to the arithmetic return, unless there is no volatility of returns.
[9]
[PDF] Introduction to Computational Finance and Financial Econometrics ...
Jun 28, 2011 · Table 1: Discrete Distribution for Annual Return. Example: Probability Distribution for Annual Return on Microsoft. Example: Bernouli ...
[10]
[PDF] Chapter 6 Introduction to Return and Risk
probability of each outcome (state). Example. The S&P 500 index and the stock of MassAir, a regional airline company, give the following returns:.<|separator|>
[11]
[PDF] 1 PROBABILITY MODELS FOR ECONOMIC DECISIONS Chapter 2
The standard deviation is the square root of the variance, and it is returned in cell D13 by the formula. =SUMPRODUCT(D5:D9,C5:C9)^0.5. To simplify these ...
[12]
[PDF] Topic 8: The Expected Value - Arizona Math
The formula for continuous random variables is obtained ... Thus, expected values for continuous random variables are determined by computing an integral.
[13]
[PDF] Lognormal Model for Stock Prices - UCSD Math
We'll say that a random variable X = exp(σZ + µ), where Z ∼ N (0, 1), is lognormal(µ,σ2). Note that the parameters µ and σ are the mean and standard deviation ...
[14]
[PDF] Statistical Analysis of the Log Returns of Financial Assets
Apr 9, 2008 · A log-normal distribution has a short lower tail and a fatter upper tail. If simple returns are independent and identically distributed ...
[15]
None
### Summary of Monte Carlo Simulation for Expected Returns in Finance
[16]
[PDF] Risk Neutral Investors Do Not Acquire Information∗
Apr 7, 2008 · The reason is that a risk neutral investor is indifferent between a risky asset or a safe bond in optimum and expects the same return to her ...Missing: maximization | Show results with:maximization
[17]
Portfolio Risk and Return Part I | IFT World
Note the following: Risk-neutral investor: For a risk-neutral investor, the utility is the same irrespective of risk as the investor is concerned only about ...Missing: maximization single
[18]
[PDF] ON ESTIMATING THE EXPECTED RETURN ON THE MARKET An ...
Ibbotson and. Sinquefield. (1976, 1979) have carefully cataloged the historical average returns on stocks and bonds from 1926 to 1978. However, they provide no.
[19]
Mean-Variance Optimization Using Forward-Looking Return Estimates
Oct 1, 2017 · We argue that investors are better off using the implied cost of capital based on analysts' earnings forecasts as a forward-looking return estimate.
[20]
[PDF] SCENARIO ANALYSIS, DECISION TREES AND SIMULATIONS
In scenario analysis, we estimate expected cash flows and asset value under various scenarios, with the intent of getting a better sense of the effect of ...
[21]
[PDF] Portfolio Selection Harry Markowitz The Journal of Finance, Vol. 7 ...
Sep 3, 2007 · We illustrate geometrically relations between beliefs and choice of portfolio accord- ing to the "expected returns-variance of returns" rule.
[22]
Rebalancing with transaction costs: theory, simulations, and actual ...
Sep 1, 2022 · A buy-and-hold strategy theoretically generates higher expected returns than a fixed-weight strategy, where the portfolio weights are regularly readjusted/ ...<|control11|><|separator|>
[23]
PORTFOLIO SELECTION* - Markowitz - 1952 - The Journal of Finance
This paper is based on work done by the author while at the Cowles Commission for Research in Economics and with the financial assistance of the Social ...
[24]
CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM ...
CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* - Sharpe - 1964 - The Journal of Finance - Wiley Online Library.Introduction · II. Optimal Investment Policy... · III. Equilibrium in the Capital...
[25]
[PDF] Mutual Fund Performance - William F. Sharpe - Finance
Oct 19, 2006 · We define this as the reward-to-variability ratio: For Boston Fund the ratio is equal to the distance XP on Figure 1 divided by the distance XV ...
[26]
ECON 251 - Lecture 15 - Uncertainty and the Rational Expectations ...
According to the rational expectations hypothesis, traders know the probabilities of future events, and value uncertain future payoffs by discounting their ...
[27]
[PDF] Rational Expectations and Asset Prices - Karl Whelan
A second assumption that we will make for the moment is that the expected return on assets equals some constant value for all future periods, unrelated to ...
[28]
Behavioral Finance: Biases, Emotions and Financial Behavior
Investors are found to systematically hold on to losing investments longer than rational expectations would predict, and they also sell winners too early. This ...
[29]
(PDF) Behavioral Finance: The Impact of Investor Expectation on ...
Behavioral finance challenges traditional financial theories by questioning the assumption of investor rationality, highlighting the influence of cognitive ...
[30]
Departures from Rational Expectations and Asset Pricing Anomalies
Dec 7, 2009 · However, when taken alone, these deviations from rational expectations cannot resolve the equity premium and risk-free rate puzzles.
[31]
[PDF] A SURVEY OF BEHAVIORAL FINANCE° - Nicholas Barberis
Behavioral finance argues that some financial phenomena can plausibly be understood using models in which some agents are not fully rational.
[32]
[PDF] Fat Tails in Financial Return Distributions Revisited - arXiv
Previous studies have considered market crashes and volatility clustering as important factors affecting the existence of fat tails in return distributions.
[33]
[PDF] PDF - Crashes, Fat Tails, and Efficient Frontiers - white paper
We show that returns models that use a lognormal distribution underestimate the downside risk of a portfolio. Models using a log-TLF distribution are ...
[34]
Fat Tail Risk | Avenue Investment Management
Financial markets typically exhibit excess kurtosis, confirming the presence of fat tails. ... The 1987 crash, 2000-2002 tech bubble collapse, 2008 financial ...
[35]
[PDF] VaR-x: Fat tails in financial risk management
The bi-weekly data excludes the crash of October 1987, so that we can estimate the value-at-risk consistent with normal market conditions. Summary statistics ...
[36]
Testing and estimating structural breaks in time series and panel ...
Aug 26, 2025 · Identifying structural change is a crucial step when analyzing time series and panel data. The longer the time span, the higher the ...
[37]
Full article: The global financial crisis impact on stock market efficiency
Sep 1, 2024 · This study investigates how the Global Financial Crisis has affected the weak-form Efficient Market Hypothesis (EMH) on the stock prices of sixteen nations ...
[38]
[PDF] Can Forecast Errors Predict Financial Crises? Exploring the ...
May 15, 2020 · In many cases, however, these data are subject to large structural breaks due to definition changes following the transition of the country ...
[39]
(PDF) Impact of Structural Break Location on Forecasting Accuracy
This paper focuses on how structural breaks in conditional heteroscedastic financial and macroeconomic data affect forecasting accuracy on short and long-term ...
[40]
Factor investing: A Bayesian hierarchical approach - ScienceDirect
This paper investigates the asset allocation problem when returns are predictable. We introduce a market-timing Bayesian hierarchical (BH) approach.
[41]
Using Bayesian Solutions to Resolve the Factor Zoo - Alpha Architect
Jul 22, 2024 · We propose a novel framework for analyzing linear asset pricing models: simple, robust, and applicable to high-dimensional problems.<|control11|><|separator|>
[42]
[PDF] BAYESIAN AND ROBUST OPTIMISATION FOR EFFICIENT ...
The constructed Bayesian efficient frontier allows us to develop an intelligent technique by performing interval forecasting of the impending updates of the.