A century ago, life expectancy was roughly 40 years, hence saving for retirement was not an important economic issue. Nowadays, life expectancy exceeds 80 years in many countries, and one should expect to live and consume many years after retirement. Thus, optimizing investments and savings for retirement, with a long planned investment horizon, has become an issue of crucial importance. How should individuals invest for the long run? Does the optimal asset allocation between risky stocks and relatively safer bonds change with age? Is the idiom "stocks for the long run" backed by scientific evidence? This book provides a comprehensive in-depth review of these issues, bridging theory and practice of investments for various horizons.
Author(s): Haim Levy
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 493
City: Singapore
Contents
Preface
Introduction
Chapter 1. Asset Allocation and the Horizon: The Ongoing Disputes
1.1. The “Planned” and the “Actual” Investment Horizons
1.2. Dispute Number 1: Do Stocks Become Riskier or Safer with the Horizon?
1.2.1. Employing the Black and Scholes option model to evaluate stock risk
1.2.2. Measuring stock volatility for various horizons: Risk and stock/bond relative attractiveness
1.3. Dispute Number 2: The Geometric Mean and Expected Utility Conflict for Investment for the Very Long Run
1.3.1. Definition of the GM
1.3.2. The MGM rule and diversification
1.3.3. Example: A case where the MGM implies diversification
1.4. The Probability that a Stock’s Terminal Wealth will be Larger than a Bond’s Terminal Wealth for Various Finite Long Investment Horizons
1.5. Diversification Across Time — The Case for Target-Date Funds
1.6. What is the Rationale for the Glide Path Investment Strategy?
1.7. Summary
Chapter 2. The Distribution of Returns and the Horizon
2.1. The Investment Horizon and Skewness: The Theory
2.1.1. The importance of skewness for expected utility maximizers
2.1.2. The multi-period skewness built up: The theoretical analyses
2.2. The Investment Horizon and Skewness: The Best Fit Theoretical Distribution
2.3. Skewness and the Best Fit Distribution: Autocorrelation is Incorporated
2.4. Summary
Chapter 3. Mean–Variance, Stochastic Dominance, and the Investment Horizon
3.1. The M–V Rule and the Horizon
3.2. The Multi-Period Variance as a Function of the One-Period Parameters
3.3. Stochastic Dominance Rules
3.3.1. First-degree SD
3.3.2. Second-degree SD
3.4. The One-Period and Multi-Period FSD
3.5. Risk-Aversion: The Multi-Period SSD
3.6. The FSD, SSD, and the M–V Multi-Period Efficient Sets
3.7. The SSD Rule with One-Period Normal Distribution and the Horizon Effect
3.8. The SSD Rule with One-Period Log-Normal Distributions and the Horizon Effect
3.9. The Economic Loss Induced by Employing the M–V Rule for Long Horizons
3.10. The Arrow–Pratt Risk Premium and the Investment Horizon
3.11. Diversification: Markowitz’s Efficient Frontier in the Multi-Period Case
3.12. The Multi-Period Variance Where Returns May be Dependent Over Time: The Empirical Evidence
3.13. Discussion
3.14. Summary
Chapter 4. Performance Indices and the Investment Horizon
4.1. The Three Main Performance Indices: Definitions
4.2. The Employment of the SR in Practice
4.3. The One-Period and Multi-Period SRs: Mathematical Analysis of Horizon Mismatch
4.4. The Changes in the Beta, Treynor Ration (TR), and Jensen’s Alpha (JA) with the Increase in the Horizon
4.4.1. The horizon effect on the beta
4.4.2. The effect of an increase in the horizon on the multi-period TR
4.4.3. The effect of an increase in the horizon on the multi-period JA
4.5. The Changes in the Beta and the Performance Indices with a Horizon Shorter than the CAPM Horizon
4.5.1. The effect of a decrease in the horizon on the beta
4.5.2. The effect of a decrease in the horizon on the TR
4.5.3. The effect of a decrease in the horizon on JA
4.6. The Empirical Evidence: Relaxing the i.i.d. and the CAPM Assumptions: The Implication for the Reported SFE
4.7. Summary
Chapter 5. Stocks Versus Bonds: Mean–Variance and Expected Utility Paradigms
5.1. Changes in the Correlation with the Horizon
5.1.1. The definition of the one-period and multi-period correlations
5.1.2. With i.i.d., the multi-period correlation converges to zero
5.2. The Optimal M–V Diversification and the Horizon
5.3. Asset Allocation and the Horizon in the M–V Framework: The Empirical Evidence
5.3.1. The stock–bond M–V optimal allocation and the horizon with i.i.d. assumption
5.3.2. The stock–bond optimal M–V asset allocation where autocorrelations are considered
5.4. Optimal M–V Diversification with 10 Portfolios of Risky Assets
5.5. Contrasting the M–V and Expected Utility Results for Various Horizons
5.6. Diversification: The Optimal Stock–Bond Investment Weights for Various Utility Functions
5.7. Summary
Chapter 6. Risk and the Horizon: The Discounting Cash-Flows Approach with Rothschild and Stiglitz’s Definition of Risk
6.1. The Annualized Volatility and the Horizon
6.2. Discussion
6.3. Does the Annualized Volatility Measure Risk?
6.4. The DCM with Rothschild and Stiglitz’s Definition of Risk
6.4.1. The suggested three stages for risk comparison faced on different dates
6.4.2. Example: The discounting methods with Rothschild and Stiglitz’s definition of risks
6.4.3. Discussion
6.4.4. Generalization of Rothschild and Stiglitz’s approach
6.5. Are Stocks Riskier or Safer with the Horizon? The Empirical Evidence
6.6. Summary
Chapter 7. Stock Risk: Do Historical Crashes Tell the Whole Story? The Black Swan Hypothesis
7.1. Contrasting Views Regarding the Association Between Historical and Future Stock Risk
7.2. Case A: Adding to the Historical Distribution an Event with a −70% Annual Rate of Return with a Probability P(X)
7.2.1. The M–V optimal portfolio with −70% crashes in the stock market
7.2.2. The expected utility optimal investment weights with additional annual −70% crashes
7.2.3. Discussion
7.3. Change in the Economic Regime: A Reduction of a Constant From All Rates of Returns
7.4. Summary
Chapter 8. Discrete and Continuous Returns and the Investment Horizon
8.1. The Continuous and Discrete Rates of Returns and the Investment Horizon
8.2. Employing the Continuous Rate of Return to Mitigate the Positive Asymmetry of the Distribution
8.3. Ranking Prospects by the Continuous and Discrete Distributions of the Rate of Return
8.3.1. Numerical counterexample
8.4. The Log-Normal and Normal Distributions
8.5. Summary
Chapter 9. Almost Stochastic Dominance Rules and the Horizon
9.1. Almost First-Degree Stochastic Dominance
9.1.1. The change in the FSD violation area of stocks and bonds with the horizon
9.1.2. An example of an FSD paradox and of a pathological preference
9.1.3. The AFSD rule and non-pathological preferences
9.2. Determining the Allowed FSD Violation Area by Experiments
9.2.1. Discussion
9.3. Comparing Stocks and Bonds by AFSD for Various Horizons, N
9.4. Stocks, Bonds, and the Horizon: The Set of Non-Pathological Preferences
9.4.1. The negative exponential utility function
9.4.2. The myopic preferences
9.5. Summary
Chapter 10. Prospect Theory and the Horizon
10.1. The Basic Ingredients of Prospect Theory
10.2. The Reference Point, the Distributions of Return, and the Horizon
10.3. The Change in the Expected Value Function of Stocks and Bonds with the Horizon with Two Reference Points
10.4. Stock–Bond Diversification with Prospect Theory
10.5. Summary
Chapter 11. The Change in the Relative Attractiveness of Stocks and Bonds with the Horizon with a Riskless Asset
11.1. The Return on a Portfolio of Stocks and the Riskless Asset
11.2. The FSDR Criterion
11.3. Numerical Illustration of the FSDR Rule
11.4. Stocks versus Bonds and the Horizon: The FSDR Empirical Evidence
11.5. Stock–Bond Portfolios with the Riskless Asset
11.6. Summary
Name Index
Subject Index
