Risk Aversion and Optimal Portfolios

Lecture Objectives

• Familiarize with the concept of risk aversion.
• Learn about the properties of the quadratic utility function and indifference curves.
• Derive and define the Capital Allocation Line.
• Compute expected portfolio returns and standard deviation.
• Learn how to choose the optimal portfolio of risky assets.
• Understand the value and the sources of portfolio diversification.
• Familiarize with the concepts of minimum-variance frontier, global minimum-variance portfolio, and Markowitz efficient frontier.
• Learn about the two-fund separation theorem.

Risk-Return Trade-Off

• Historically, a higher risk premium is associated with higher levels of risk (e.g., standard deviation of returns), reflecting risk aversion among investors.
• The optimal mix of assets in a portfolio depends on preferences towards risk and return.

Choice Under Uncertainty

• Investors consider not only the level of expected gains but also their degree of risk when selecting among alternative investments.
• Example:
  • Option 1: Win £100 with 50% probability or get nothing with 50% probability.
  • Option 2: Get £50 without participating in the game (certainty).
  • Both options yield an expected gain of £50.

Behaviour Towards Risk

• Risk Averse Investor:
  • Dislikes risk and would turn down a risky investment for a lower but guaranteed return.
  • Aims to minimize risk for a given level of expected return or maximize expected return for a given amount of risk.
• Risk Neutral Investor:
  • Indifferent about risk and only cares about the level of expected return.
  • Chooses the investment with the highest level of expected return.
• Risk Seeking Investor:
  • Likes risk and would prefer a risky investment even if its expected return is lower than the guaranteed one.
• Risk Tolerance:
  • The willingness of an investor to take risk.
  • Negatively related to risk aversion.

Utility Function

• A metric that captures investor behavior towards risk and expected return.
• Quadratic Utility Function:
  • U = E(r) -
    A/2 * σ^2
  • Where:
    • $U$: Utility derived from the investment.
    • $E(r)$: Expected Return.
    • $σ^2$: Variance (Risk) of the investment returns.
    • $A$: Degree of Risk Aversion.
      • A > 0: Risk Aversion (leads to risk-return trade-off).
      • $A = 0$: Risk Neutrality.
      • A < 0: Risk Seeking.

Mean-Variance Criterion

• Investors with a quadratic utility function care only about the mean and the variance of returns.
• This leads to the mean-variance portfolio theory.
• Mean-Variance Criterion:
  • For a given level of expected return, investors prefer the portfolio with the lowest possible level of variance.
  • For a given level of variance, investors prefer the portfolio with the highest possible level of expected return.

Indifference Curves

• Plots the combinations of risk-return pairs that an investor would accept to maintain a given level of utility.
• The investor is indifferent about the combinations on any one curve because they yield the same level of utility.
• Indifference curves are convex for a risk-averse investor:
  • Higher return contributes to a higher utility level.
  • Higher variance (risk) reduces investor utility.
• The slope of the indifference curve is related to the degree of investor’s risk aversion.

Quadratic Utility

• Investors should choose the investment that maximizes their utility.
• The relative ranking of utility scores matters (absolute score is not meaningful per se).

Expected Portfolio Return and Risk

One Risky Asset and the Risk-free Asset

• Consider a portfolio consisting of:
  1. The risk-free asset.
  2. A risky asset.
• Risk-free asset:
  • Return (known): $R_f$
  • Risk = 0
• Risky asset:
  • Expected Return: $E(R<em>i)$, which is assumed to be > $R</em>f$
  • Risk: $σ_i$
• Assign weight (% total portfolio value) $w<em>1$ to the risk-free asset and weight $(1 - w</em>1)$ to the risky asset.
• Expected Portfolio Return:
  • $E(R<em>p) = w</em>1R<em>f + (1 - w</em>1)E(R_i)$
• Variance of Portfolio Returns:
  • $σ<em>p^2 = (1 - w</em>1)^2σ_i^2$
• Standard Deviation of Portfolio Returns:
  • $σ<em>p = (1 - w</em>1)σ_i$
  • Which implies that:
    • $w<em>1 = 1 - \frac{σ</em>p}{σ_i}$

Capital Allocation Line (CAL)

• If we replace the expression $w<em>1 = 1 - \frac{σ</em>p}{σ_i}$ into the formula for expected portfolio return, we get:
  • $E(R<em>p) = (1 - \frac{σ</em>p}{σ<em>i})R</em>f + \frac{σ<em>p}{σ</em>i}E(R_i)$
  • $E(R<em>p) = R</em>f + \frac{E(R<em>i) - R</em>f}{σ<em>i}σ</em>p$
• This is the Capital Allocation Line (CAL), representing all feasible portfolio combinations between the risky and the risk-free asset.
• CAL depicts the feasible risk-return combinations, and hence the investment opportunity set.
• Intercept of CAL:
  • Risk-free rate, $R<em>f$ (where $w</em>1 = 100\%$ and $σ_p = 0$)
• Slope of CAL:
  • Sharpe Ratio of the risky asset
  • Additional required return for every increment in risk
• We can achieve $E(R<em>p) > E(R</em>i)$, if $(1 - w<em>1) > 100\%$, so w1 < 0\%.
  • Borrow at the risk-free rate to invest more than one’s own funds!
  • Leveraged position or "Buy the risky asset on margin"

Optimal Portfolio

• Optimal feasible portfolio is the one that maximizes investor utility.
• The optimal portfolio is the point of tangency between the CAL and the indifference curve of the investor.
• Different degrees of risk aversion lead to different tangency points on CAL.

Expected Portfolio Return

• Portfolio Return is the weighted average of the constituent assets’ returns.
• Expected Portfolio Return with multiple (N) assets:
  • $E(R<em>p) = \sum</em>{i=1}^{N} w<em>iE(R</em>i)$
  • Where $w<em>i$ is the portfolio weight (% of total portfolio value) of asset i, with $\sum</em>{i=1}^{N} w_i = 1$ (i.e., 100%).
• Expected Portfolio Return with two assets:
  • $E(R<em>p) = w</em>1E(R<em>1) + (1 - w</em>1)E(R_2)$
  • Because $w<em>1 + w</em>2 = 1$.

Portfolio Risk

• Variance of Portfolio Return with multiple (N) assets:
  • $Var(R<em>p) = σ</em>p^2 = Var(\sum<em>{i=1}^{N} w</em>iR_i) =$
  • $= \sum<em>{i=1}^{N} w</em>i^2Var(R<em>i) + \sum</em>{i,j=1, i\neq j}^{N} w<em>iw</em>jCov(R<em>i, R</em>j)$
  • Where $Cov(R<em>i, R</em>j)$ is the covariance between the returns of asset i and asset j.
  • Recall that: $Cov(R<em>i, R</em>j) = σ<em>iσ</em>jρ_{ij}$
  • Where $ρ_{ij}$ denotes the correlation coefficient between the returns of asset i and asset j.
• Variance of Portfolio Return with two assets:
  • $σ<em>p^2 = w</em>1^2σ<em>1^2 + w</em>2^2σ<em>2^2 + 2w</em>1w<em>2Cov(R</em>1, R_2) =$
  • $= w<em>1^2σ</em>1^2 + w<em>2^2σ</em>2^2 + 2w<em>1w</em>2σ<em>1σ</em>2ρ_{12}$
  • Where $ρ_{12}$ is the correlation coefficient between the returns of asset 1 and asset 2.
• Standard Deviation of Portfolio Return with two assets:
  • $σ<em>p = \sqrt{w</em>1^2σ<em>1^2 + w</em>2^2σ<em>2^2 + 2w</em>1w<em>2σ</em>1σ<em>2ρ</em>{12}}$

The Effect of Correlation

• The correlation coefficient, $ρ_{12}$, provides a standardized measure of the co-movement of two assets’ returns.
• Recall that $ρ_{12}$ takes values between $-1$ and $+1$:
  • $ρ_{12} = 1$: returns are perfectly positively correlated.
  • $ρ_{12} = -1$: returns are perfectly negatively correlated.
  • $ρ_{12} = 0$: returns are uncorrelated.
• If $ρ<em>{12} = 1$, then $σ</em>p = w<em>1σ</em>1 + w<em>2σ</em>2$
• If $ρ<em>{12} < 1$, then σp < w1σ1 + w2σ2
• Portfolio Risk is lower than the weighted average of the constituent assets’ risks, unless their returns are perfectly positively correlated.

Portfolios of Two Risky Assets

• Assume Stock 1 with $E(R<em>1) = 7\%$ and $σ</em>1 = 12\%$ & Stock 2 with $E(R<em>2) = 15\%$ and $σ</em>2 = 25\%$. Portfolio Risk is crucially affected by the value of $ρ_{12}$.
• When $ρ_{12} ∈ (-1, 1)$, the feasible portfolio risk-return combinations (i.e., the investment opportunity set) form a curve.

Portfolio Diversification

• Diversification involves spreading a portfolio over many assets to avoid excessive exposure to any one source of risk.
• Diversification can reduce portfolio risk without reducing the expected portfolio return.
• Diversification can also reduce the chance of catastrophic losses (value-at-risk and expected shortfall) due to individual assets.
• The key in diversifying risk is the correlation among the risky assets’ returns.
• Challenge: Find assets whose returns have a correlation coefficient much lower than 1, or even negative correlation (hedge assets).

Diversification: Avoiding Catastrophic Losses

• Holding one’s wealth in a single or few stocks: exposure to firm-specific risk.

Diversification: How Does it Work?

• Recall that the variance of portfolio returns is given by:
  • $σ<em>p^2 = \sum</em>{i=1}^{N} w<em>i^2σ</em>i^2 + \sum<em>{i,j=1, i\neq j}^{N} w</em>iw<em>jCov(R</em>i, R_j)$
• Assume a portfolio with equal weights $(\frac{1}{N})$ among N assets, with $σ^2$ the average variance of asset returns and $Cov$ the average covariance among asset returns. Then:
  • $σ_p^2 = \frac{σ^2}{N} + \frac{N-1}{N}Cov$
• As $N → ∞$, then $σ_p^2 → Cov$.
• If we further assume that all assets have the same $σ^2$ and $ρ$, then
  • $σ_p^2 = \frac{σ^2}{N} + \frac{N-1}{N}ρσ^2$
• As $N → ∞$, then $ρ$ is the key determinant of $σ_p^2$.
• Substantial reduction in portfolio risk as N increases
• Lower limit of diversification as N sufficiently large

Diversification: How to Achieve it?

• Diversify across different asset classes and industries.
  • Correlations across asset classes are lower than correlations of securities within each asset class. Similar for industries.
• Diversify among countries (International Diversification).
  • Correlations are lower across countries with different economic structures, policies and cycles, natural resources, political characteristics, and currency regimes.
• Diversify with index funds
  • Investing directly in a large number of assets can be prohibitively costly to trade and track for the typical investor.
  • ETFs or index funds enable low-cost access to already well-diversified portfolios.
• Evaluate each asset before adding to a portfolio
  • Rule: If the new asset improves the risk-return combination of the existing portfolio, it should be included
• Buy insurance for risky portfolios
  • Insurance and hedge assets have negative correlation with the returns of existing portfolio assets!
  • Valuable for portfolio risk reduction.
  • Hedge assets: Gold and other commodities
  • Derivative assets: Put options, currency or index futures
• Take into account background risk
  • Example: Employees are already exposed to their firm risk (human capital and income risk).
  • Therefore, inappropriate to invest retirement savings in own company stock.
  • More generally, diversify away from existing background risk exposure (e.g., country/region, profession, sector)

Investment Opportunity Set

Multiple Risky Assets

• As the number of available risky assets increases, the number of their possible combinations (portfolios) increases
• These feasible portfolios constitute the investment opportunity set
• These feasible portfolios are located within a "frontier" (i.e., to the right of the curve)
• The addition of less than perfectly correlated risky assets can expand out the frontier to the NW, providing a superior risk-return trade-off
• Risk-return combinations to the left of the frontier are infeasible

Efficient Frontier of Risky Assets

• Considering all investable assets, we can construct the minimum-variance frontier, which is consisted of the minimum-variance portfolios
  • Minimum-Variance Portfolio is the feasible portfolio of risky assets with the lowest level of variance for a given level of expected return
• The portfolio with the minimum level of variance among all portfolios of risky assets is called the Global Minimum-Variance Portfolio.
  • This is the left-most point on the minimum-variance frontier.
• The Markowitz Efficient Frontier is the upper segment of the minimum-variance frontier, above the global minimum-variance portfolio.
  • It is called the efficient frontier because it contains the portfolios that yield the highest level of expected return for a given level of variance.

Introducing the Risk-Free Asset

• We have now identified the set of portfolios of risky assets, among which a risk-averse investor should choose!
• Efficient portfolios on the Markowitz efficient frontier
• If investors can also invest in the risk-free asset, we can determine their Capital Allocation Line
• The optimal CAL is tangent to the Markowitz efficient frontier at a single point!
• Optimal risky portfolio
• The optimal (i.e., dominant) CAL represents the combinations of the optimal portfolio of risky assets with the risk-free asset, which yield the highest possible reward-to-risk (i.e., Sharpe) ratio
  • Intercept of CAL: Risk-free rate ($R_f$)
  • Slope of CAL: Sharpe Ratio = $\frac{E(R<em>p) - R</em>f}{σ_p}$

Optimal Capital Allocation Line

• Point Rf: 100% invested in the risk-free asset
• Point P: 100% invested in the optimal portfolio of risky assets
• Point above P: > 100% invested in the optimal portfolio of risky assets

Two-Fund Separation Theorem

• The optimal CAL leads to a very powerful conclusion:
• Two-Fund Separation Theorem
  • All risk-averse investors will hold a combination of two portfolios (or funds) only regardless of their degree of risk aversion and initial wealth:
    • a risk-free asset
    • an optimal portfolio of risky assets.
• This theorem divides the investment problem into two steps:
  1. Investment Decision: Identify the optimal portfolio of risky assets
     • Based on the risky assets’ expected returns, variances, and correlations; not affected by investor’s risk aversion
  2. Financing Decision: Allocate investor’s wealth between the risk-free asset and the optimal portfolio of risky assets
     • Decide how much to lend or borrow at the risk-free rate
     • Depends on investor’s degree of risk aversion

Optimal Investor Portfolio

• The optimal combination of the risk-free asset and the risky assets’ portfolio is the point where the investor’s indifference curve is tangent to the CAL.

Reading for this Topic

• CFA Program Curriculum, 2025, Level I, Volume 9: Portfolio Management
  • Learning Module 1: Sections 4-12
  • Learning Module 3: Sections 1-3
• Investments (13th edition) by Bodie, Kane and Marcus
  • Chapter 6: Capital Allocation to Risky Assets
  • Chapter 7: Efficient Diversification