Risk, Speculation, Gambling and Risk Aversion

Risk, Speculation and Gambling

• Bearing risk should be compensated by a risk premium.
• Speculation is taking considerable investment risk for commensurate gain.
• Considerable risk means the risk is sufficient to affect a decision.
• Commensurate gain means a positive risk premium, exceeding the risk-free alternative.
• Gambling is betting on an uncertain outcome.
• A gamble is assuming risk for the sake of the risk itself.
• Speculation is undertaken despite the risk because of a favorable risk-return outlook.
• A risky investment with zero risk premium is considered a fair game, which is essentially a gamble.

Risk Aversion and Utility

• Risk-averse investors avoid fair games or worse investments.
• These investors will only consider speculative prospects that have positive risk premiums, or risk-free alternatives
• Investors quantify their willingness to trade off return against risk through a utility score.
• Utility scores rank investment portfolios based on expected return and risk.
• Higher utility values represent more attractive risk-return profiles.
• Higher expected returns increase utility scores, while higher volatility decreases them.
• Utility is enhanced by high expected returns but diminished by high risk.
• Risk-free portfolios' utility score equals their (known) rate of return, with no penalty for risk.

Risk Aversion and Investment Choices

• The variance of risky portfolios lowers utility, depending on an investor's risk aversion (A).
• More risk-averse investors (higher A) penalize risky investments more severely.
• Investors select the investment portfolio with the highest utility score when comparing options.
• Utility score can be seen as a certainty equivalent rate of return.
• Certainty equivalent rate of return: the rate a risk-free investment would need to offer to match the utility of a risky portfolio.
• This allows for comparison of utility values across different portfolios.
• Risk-neutral investors (A=0) only consider expected rates of return, ignoring the level of risk.
• Risk is irrelevant to the risk neutral investor, there is no risk penalty.
• For risk neutral investors, a portfolio's certainty equivalent rate of return is simply its expected rate of return.
• Risk lovers (A<0) enjoy fair games and gambles.
• These investors adjust the expected return upwards to account for the “fun” of confronting risk.
• Investors are attracted to portfolios with high risk and higher expected returns compared to portfolios with lower risk and lower expected return.
• Equally preferred portfolios lie on the indifference curve.
• Indifference curve: connects all portfolio points with the same utility value in the mean-standard deviation plane.

Portfolio Allocation

• The standard deviation of a complete portfolio (risky and risk-free assets) is the risky asset's standard deviation multiplied by its weight in the portfolio.
• Capital Allocation Line (CAL): A graph showing all feasible risk-return combinations of a risky and a risk-free asset.
• CAL illustrates all risk-return combinations available to investors.
• Reward-to-volatility ratio (Sharpe ratio) formula: S= (ER-r)/ σ

Optimal Risky Asset Weight

• The formula for calculating the optimal weight of a risky asset (y*) in a portfolio:
• y* = (ER-r)/(Aσ²)
• The optimal investment in a risky asset is inversely proportional to risk aversion and risk level, and directly proportional to the risk premium.
• More risk-averse investors have steeper indifference curves.
• Steeper indifference curves indicate investors need a larger increase in expected return to compensate for additional portfolio risk reflecting their risk-averse behavior.
• Higher indifference curves mean higher utility levels.
• Investors seek to find the complete portfolio along the highest possible indifference curve.
• Identify the highest possible indifference curve via superimposing indifference curves on the investment opportunity set represented by the CAL.
• The tangency point corresponds to the standard deviation and expected return of the optimal complete portfolio.
• Choice of y* (allocation to risky vs. risk-free assets) mainly depends on risk aversion.

Investment Strategies

• The CAL is derived using the risk-free asset and "the" risky portfolio, P.
• Selection of assets for risky portfolio P can be done through passive or active strategies.
• Passive strategy: Portfolio decisions without direct or indirect security analysis.
• The capital market line (CML) is the capital allocation line using 1-month T-bills and an index of common stocks.
• Passive strategy results in an investment opportunity set shown by the CML.

Explanation of risk, speculation, and gambling, highlighting the importance of a risk premium for bearing risk. It distinguishes speculation from gambling based on the risk-return outlook. Risk-averse investors avoid investments without positive risk premiums and quantify their investment preferences with a utility score.