IA 7-11

Comprehensive vocabulary flashcards covering Modern Portfolio Theory, mean-variance optimization, exogenous risk hedging, CAPM equilibrium, and empirical multi-factor asset pricing models.

Mean-variance utility

A mathematical representation of investor preferences that balances expected return against the cost of risk according to the equation $U = E(r) - \frac{1}{2} A \bold{\sigma}^2$. Higher expected return $E(r)$ increases utility, whereas higher variance reduces utility. The parameter $A$ quantifies risk aversion, where a larger $A$ indicates a stronger dislike for risk.

Optimal allocation with one risky asset and a risk-free asset

The optimal proportion invested in the risky portfolio is:

$$y^* = \frac{E(r_p) - r_f}{A \sigma_p^2}$$

$$w_{rf} = 1 - y^*$$

The complete portfolio's expected return and standard deviation is:

$$E(r_c) = r_f + y^*[E(r_p) - r_f]$$

$$\sigma_c = y^* \sigma_p$$

Investors lend if $y^* < 1$, hold only risky assets if $y^* = 1$, and borrow on leverage if $y^* > 1$.

Capital Allocation Line (CAL)

The straight line depicting all feasible risk-return combinations obtained by pairing a specific risky portfolio with the risk-free asset: $E(r_c) = r_f + \left[\frac{E(r_p) - r_f}{\sigma_p}\right] \sigma_c$

It features an intercept at $r_f$ and a slope equal to the Sharpe ratio, where a steeper slope indicates a superior risk-return trade-off.

Sharpe ratio

A performance metric that measures expected excess return earned per unit of total risk: $SR = \frac{E(r_p) - r_f}{\sigma_p}$

Higher Sharpe ratios represent more attractive risk-return trade-offs.

Covariance and correlation formulas

$$\sigma_{ij} = \rho_{ij} \sigma_i \sigma_j$$

$$\rho_{ij} = \frac{\sigma_{ij}}{\sigma_i \sigma_j}$$

Two-asset portfolio variance formula via covariance or correlation

The variance of a two-asset portfolio expressed using covariance as: $\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_{12}$,

or correlation as:

$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$.

or perfect correlation:

$$\sigma_p = w_1 \sigma_1 + w_2 \sigma_2$$

Diversification

The reduction of portfolio risk achieved by combining assets that are not perfectly positively correlated. Portfolio variance depends on both individual asset variances and pairwise covariances via $\sigma_p^2 = \mathbf{w}' \mathbf{\Sigma} \mathbf{w}$, yielding greater risk reduction benefits as asset correlation decreases.

Perfect negative correlation ($\rho_{12} = -1$) zero-variance weights

The condition where two asset returns move in exactly opposite directions, allowing the construction of a zero-variance portfolio ($\sigma_p = 0$) using weights $w_1 = \frac{\sigma_2}{\sigma_1 + \sigma_2}$ and $w_2 = \frac{\sigma_1}{\sigma_1 + \sigma_2}$.

Global Minimum Variance (GMV) portfolio weights

The unique risky portfolio that achieves the lowest possible return variance across all feasible portfolios of risky assets, located at the leftmost point of the mean-variance frontier.

The asset proportions that minimize total variance in a two-asset portfolio:

$w_1^{GMV} = \frac{\sigma_2^2 - \sigma_{12}}{\sigma_1^2 + \sigma_2^2 - 2 \sigma_{12}}$ and $w_2^{GMV} = 1 - w_1^{GMV}$.

Optimal portfolio: two risky assets without a risk-free asset

$$\sigma_{DE} = \rho_{DE} \sigma_D \sigma_E$$

The optimal weights:

$$w_D^* = \frac{E(r_D) - E(r_E) + A(\sigma_E^2 - \sigma_{DE})}{A(\sigma_D^2 + \sigma_E^2 - 2 \sigma_{DE})}$$

$w_E^* = 1 - w_D^*$.

The resulting expected return and standard deviation: $E(r_p) = w_D^* E(r_D) + w_E^* E(r_E)$ $\sigma_p = \sqrt{(w_D^*)^2 \sigma_D^2 + (w_E^*)^2 \sigma_E^2 + 2 w_D^* w_E^* \sigma_{DE}}$ .

Efficient portfolio vs. optimal portfolio

An efficient portfolio delivers the highest expected return for a given level of risk or the lowest risk for a specified expected return.

An optimal portfolio is the specific feasible portfolio that maximizes an individual investor's mean-variance utility $U = E(r) - \frac{1}{2} A \sigma^2$, which directly depends on their risk aversion coefficient $A$.

Tangency portfolio

The portfolio consisting solely of risky assets that maximizes the Sharpe ratio: $\max \frac{E(r_p) - r_f}{\sigma_p}$. Its Capital Allocation Line is tangent to the efficient frontier, establishing it as the optimal risky portfolio whenever a risk-free asset is available.

Optimal portfolio: two risky assets with a risk-free asset

A two-step portfolio selection process that first determines the tangency weights:

$$w_D^T = \frac{[E(r_D) - r_f]\sigma_E^2 - [E(r_E) - r_f]\sigma_{DE}}{[E(r_D) - r_f](\sigma_E^2 - \sigma_{DE}) + [E(r_E) - r_f](\sigma_D^2 - \sigma_{DE})}$$

$$w_E^T = 1 - w_D^T$$

Calculate tangency expected return and variance

$$E\left(r_{T}\right)=w_{D}^{T}E\left(r_{D}\right)_{}+w_{E}^{T}E\left(r_{E}\right)$$

$$\sigma_{T}^2=w_{D}^{T2}\sigma_{D}^2+w_{E}^{T2}\sigma_{E}^2+2w_{D}^{T}w_{E}^{T}\sigma_{DE}$$

And subsequently scales them by the total risky allocation $y^* = \frac{E(r_T) - r_f}{A \sigma_T^2}$

$w_D^* = y^* w_D^T$ and $w_E^* = y^* w_E^T$ and $w_{rf}=1-y^{\star}$

Optimal portfolio expected return and variance:

$$E(r_{c})=r_{f}+y^{*}[E(r_{T})-r_{f}]$$

$$\sigma_{c}=y^{*}\sigma_{T}$$

Two-fund separation theorem

The Two-Fund Separation Theorem states that when a risk-free asset is available, all mean-variance investors hold the same optimal risky portfolio (tangency portfolio), regardless of their risk aversion.

Investors differ only in how much they allocate between:

1. The risk-free asset

2. The tangency portfolio

3. ($w_i^* = y w_i^T$)

Effect of risk aversion on asset allocation

Since the risky portfolio allocation follows $y^* = \frac{E(r_T) - r_f}{A \sigma_T^2}$, an increase in risk aversion $A$ reduces $y^*$. Consequently, more risk-averse investors hold a smaller allocation in the tangency portfolio and allocate more capital to $r_f$, leaving the internal composition of the tangency portfolio unchanged.

Recovering risk aversion ($A$)

The formula to calculate an investor's implied risk aversion: $A = \frac{E(r_T) - r_f}{y^* \sigma_T^2}$. For any two investors facing identical tangency portfolios, the invariant relation $A_1 y_1^* = A_2 y_2^*$ applies, allowing one to recover $A_2 = A_1 \left(\frac{y_1^*}{y_2^*}\right)$.

Portfolio weights from invested amounts

Asset weights computed as $w_i = \frac{\text{amount invested in asset } i}{\text{total amount invested}}$, whereas weights within the risky sub-portfolio alone are calculated as $w_i^{Risky} = \frac{V_i}{\sum \text{risky } V_j}$, highlighting the distinction between complete-portfolio and risky-portfolio weights.

$K$-asset weight vector

A $K \times 1$ column vector $\mathbf{w}$ of risky-asset proportions. With a $K \times 1$ summation vector of ones $\mathbf{1}$, the transpose product $\iota^{\prime}\mathbf{w}=w_1+\dots+w_{K}$ must equal $1$ for a fully risky portfolio, while with a risk-free asset, $w_{rf}=1-\iota^{\prime}\mathbf{w}$ .

Excess return vector

The vector measuring returns above the risk-free rate, where individual asset excess returns are $r_i^e = r_i - r_f$. For $K$ assets, the expected excess return vector is $\bm{\mu}^{e}=E(\mathbf{r}^{e})=\bm{\mu}-r_{f}\iota$ , with $\mathbf{\mu}$ representing the raw expected return vector.

Covariance matrix ($\mathbf{\Sigma}$)

A symmetric $K \times K$ matrix containing individual asset variances along its main diagonal and pairwise covariances in off-diagonal positions. It encapsulates total risk and asset co-movements, determining portfolio variance via $\sigma_p^2 = \mathbf{w}' \mathbf{\Sigma} \mathbf{w}$.

Expected return and variance in matrix form

Matrix representations of portfolio parameters: expected risky return is $E(r_p) = \mathbf{w}' \mathbf{\mu}$, expected complete-portfolio return with a risk-free asset is $E(r_c) = r_f + \mathbf{w}' \mathbf{\mu}^e$, portfolio variance is $\sigma_p^2 = \mathbf{w}' \mathbf{\Sigma} \mathbf{w}$, and standard deviation is $\sigma_p = \sqrt{\mathbf{w}' \mathbf{\Sigma} \mathbf{w}}$.

Inverse of a $2 \times 2$ matrix

For a matrix $\mathbf{\Sigma} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, its matrix inverse is defined as $\mathbf{\Sigma}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, where the quantity $ad - bc$ is the determinant.

Efficient frontier optimization

The constrained optimization problem that minimizes variance for a target expected return $\mu_p$: $\min_{\mathbf{w}} \mathbf{w}' \mathbf{\Sigma} \mathbf{w}$ subject to $\mathbf{w}' \mathbf{\mu} = \mu_p$ and $\mathbf{w}^{\prime}\iota=1$ . Tracing this across varying $\mu_p$ traces out the mean-variance frontier, the upper segment of which constitutes the efficient frontier.

Mean-variance frontier

The parabolic locus representing the lowest achievable variance portfolios for every level of expected return, with the GMV portfolio marking the leftmost boundary. Its lower half is strictly inefficient, while its upper half represents the efficient frontier offering maximum return for given risk.

Optimization problem ($K$ risky assets + risk-free asset)

The optimization problem where an investor selects tradable weights to maximize total utility: $\max_{\mathbf{w}} \left[r_f + \mathbf{w}' \mathbf{\mu}^e - \frac{1}{2} A \mathbf{w}' \mathbf{\Sigma} \mathbf{w}\right]$, systematically balancing expected excess return against aggregate portfolio variance.

First-order condition for $K$-asset mean-variance optimization

The mathematical optimality condition derived by setting the gradient with respect to $\mathbf{w}$ to zero: $\mathbf{\mu}^e - A \mathbf{\Sigma} \mathbf{w} = \mathbf{0}$, or equivalently $A \mathbf{\Sigma} \mathbf{w} = \mathbf{\mu}^e$.

Optimal portfolio: $K$ risky assets with a risk-free asset

The vector of investor-specific risky holdings obtained from the first-order condition:

$\mathbf{w}^* = \frac{1}{A} \mathbf{\Sigma}^{-1} \mathbf{\mu}^e$ with $\mu^{e}=\mu-r_{f}\iota$

$$w_{rf}^{*}=1-\iota^{\prime}\mathbf{w}^{*}$$

Optimal portfolio expected return and variance:

$$E(r_{c})=r_{f}+(\mathbf{w}^{*})^{\prime}\bm{\mu}^{e}$$

$$\sigma_c^2 = (\mathbf{w}^*)' \mathbf{\Sigma} \mathbf{w}^*$$

These risky allocations do not need to sum to one.

$K$-asset tangency portfolio

The Sharpe ratio-maximizing risky portfolio given in matrix notation by

$$\mathbf{w}^{T}=\frac{\bm{\Sigma}^{-1}\bm{\mu}^{e}}{\iota^{\prime}\bm{\Sigma}^{-1}\bm{\mu}^{e}}$$

The denominator normalizes portfolio weights such that $\iota^{\prime}\mathbf{w}^{T}=1$ , distinguishing composition $\mathbf{w}^T$ from investor-specific position sizes $\mathbf{w}^*$.

Exogenous risk

An untradable or illiquid risky asset already embedded in an investor's overall wealth (such as residential real estate). Because this exposure cannot be directly liquidated, the investor adjusts their tradable portfolio to hedge against its return fluctuations.

Covariance vector with exogenous risk

A $K \times 1$ vector $\mathbf{\Sigma}_{rq} = [\text{Cov}(r_1, r_q), \dots, \text{Cov}(r_K, r_q)]'$ defining covariances between each liquid risky asset and the non-tradable exogenous exposure. When correlations are provided, each element is $\text{Cov}(r_i, r_q) = \rho_{iq} \sigma_i \sigma_q$.

Optimal portfolio with exogenous risk

The tradable risky portfolio vector accounting for background risk: $\mathbf{w}^* = \frac{1}{A} \mathbf{\Sigma}^{-1} \mathbf{\mu}^e - q \mathbf{\Sigma}^{-1} \mathbf{\Sigma}_{rq}$

$$w_{rf}=1-\iota^{\prime}\mathbf{w}^{*}$$

The complete portfolio expected return and variance is:

$$E\left(r_{c}\right)=r_{f}+w^{\star}`\mu^{e}+q\mu_{q}$$

$$\sigma_c^2 = (\mathbf{w}^*)' \mathbf{\Sigma} \mathbf{w}^* + q^2 \sigma_q^2 + 2 q (\mathbf{w}^*)' \mathbf{\Sigma}_{rq}$$

Speculative demand

The component of optimal risky asset holdings driven purely by risk-adjusted expected excess returns: $\frac{1}{A} \mathbf{\Sigma}^{-1} \mathbf{\mu}^e$. It matches the risky portfolio the investor would select in the absence of exogenous exposure.

Hedging demand

The portfolio adjustment $q \mathbf{\Sigma}^{-1} \mathbf{\Sigma}_{rq}$

Hedging demand is the demand for tradable assets that help offset exposure to exogenous, untradable risks.

In the optimal risky weight formula, it is subtracted from speculative demand.

Estimating hedging demand using regression

Regress exogenous, untradable risk returns on tradable asset returns:

$r_{q,t} = \alpha + \mathbf{\beta}' \mathbf{r}_{p,t} + \varepsilon_t$.

The estimated coefficients show how tradable assets move with the untradable risk and can be used to construct a hedging portfolio.

Systematic risk vs. firm-specific risk

Systematic risk is pervasive market-wide volatility that cannot be eliminated through diversification. Firm-specific risk is idiosyncratic volatility unique to a single company that washes out in large portfolios. Under CAPM, investors are compensated solely for systematic risk.

Total risk ($\sigma$) vs. systematic risk ($\beta$)

Total risk ($\sigma$) captures overall volatility and governs mean-variance efficiency. Systematic risk ($\beta$) measures non-diversifiable sensitivity to market fluctuations. Because rational diversification eliminates idiosyncratic noise, CAPM dictates that expected asset returns depend solely on $\beta$, not $\sigma$.

Market portfolio ($M$)

The aggregate value-weighted portfolio encompassing all risky assets in the economy, where each asset's weight is $w_i = \frac{\text{market value of asset } i}{\text{total market value of risky assets}}$. Under CAPM equilibrium, $M$ is the identical tangency portfolio chosen by all investors.

Why the market portfolio is the tangency portfolio

Because all investors have identical expectations about returns, risks, and the risk-free rate, they choose the same tangency portfolio. In equilibrium, asset prices adjust until investors collectively hold all available risky assets, meaning the tangency portfolio must equal the market portfolio.

Beta ($\beta_i$)

A standardized sensitivity coefficient measuring an asset's exposure to systematic market risk: $\beta_i = \frac{\text{Cov}(r_i, r_M)}{\text{Var}(r_M)} = \frac{\text{Cov}(r_i, r_M)}{\sigma_M^2}$, or alternatively $\beta_i = \frac{\rho_{iM} \sigma_i}{\sigma_M}$.

Interpretation of beta

A baseline of $\beta = 1$ reflects systematic risk equal to the market average; $\beta > 1$ denotes an aggressive asset with heightened market sensitivity; $0 < \beta < 1$ reflects a defensive asset less volatile than the market; and $\beta < 0$ signifies an asset whose returns move inversely to the market.

Portfolio beta ($\beta_p$)

The systematic risk metric of an aggregate portfolio, calculated as the linear market-weighted average of the individual component asset betas: $\beta_p = \sum_{i=1}^K w_i \beta_i$.

Capital Asset Pricing Model (CAPM)

An equilibrium asset pricing model postulating that an asset's expected return is determined exclusively by its systematic market risk: $E(r_i) = r_f + \beta_i[E(r_M) - r_f]$. Firm-specific risks offer no risk premium because they can be diversified away.

Market price of risk in CAPM

The equilibrium reward per unit of aggregate market risk, measured as the market risk premium divided by market variance: $\frac{E(r_M) - r_f}{\sigma_M^2}$.

Security Market Line (SML)

The graphical line representing the CAPM equation $E(r_i) = r_f + \beta_i[E(r_M) - r_f]$, plotting expected return on the y-axis against systematic risk $\beta$ on the x-axis. It possesses a y-intercept of $r_f$ and a slope equal to the market risk premium, applying to both individual securities and portfolios.

Capital Market Line (CML) vs. Security Market Line (SML)

The CML graphs expected return against total risk $\sigma$ and applies strictly to efficient portfolios formed from $M$ and $r_f$. The SML graphs expected return against systematic risk $\beta$ and applies to all individual assets as well as inefficient or efficient portfolios (memory aid: CML relates to $\sigma$, SML relates to $\beta$).

Comparing risk aversion on the same CAL

When investors face the identical tangency portfolio, the invariant relation $A_1 y_1^* = A_2 y_2^*$ holds; an investor choosing a smaller risky allocation $y^*$ is strictly more risk-averse, locating their complete portfolio closer to $r_f$.

MPT diagram: efficient frontier and CAL

The blue curve is the mean-variance frontier. The upper part is the efficient frontier, while the lower part is inefficient. The GMV portfolio is the portfolio with the lowest possible variance.

The CAL starts at $r_{f}$ and is tangent to the efficient frontier at the tangency portfolio T.

Investor positions on the CAL:

y<1

Investor holds T plus the risk-free asset.

y=1

Investor holds only T.

y>1

Investor borrows at $r_{f}$ and invests more than 100% in T (leverage).

Higher risk aversion A → lower y → position closer to $r_{f}$

Capital Market Line (CML) diagram

The Capital Market Line (CML) shows all efficient portfolios formed by combining the risk-free asset with the market portfolio M.

Under CAPM:

T = M

The CML is tangent to the efficient frontier at M.

Its slope is the market portfolio's Sharpe ratio:

$$\frac{E\left(r_{M}\right)-r_{f}}{\sigma_{M}}$$

The CML relates expected return to total risk $\sigma$ and applies only to efficient portfolios.

Security Market Line (SML) diagram

The Security Market Line (SML) shows the CAPM relationship between expected return and systematic risk beta:

$$E\left(r_{i}\right)=r_{f}+\beta_{i}\left\lbrack E\left(r_{M}\right)-r_{f}\right\rbrack$$

• Intercept = $r_{f}$

• Slope = $E\left(r_{M}\right)-r_{f}$ , the market risk premium

• Market portfolio M has $\beta=1$

Unlike the CML, the SML applies to individual assets and portfolios.

A security above the SML has:

$$\alpha>0$$

A security below the SML has:

$$\alpha<0$$

Independence of tangency portfolio composition from $A$

The structural principle that the weights inside the tangency portfolio are dictated entirely by market investment opportunities ($\mathbf{\mu}$, $\mathbf{\Sigma}$, and $r_f$). Individual risk aversion $A$ only scales the magnitude of total funds invested in that portfolio via $y^* = \frac{E(r_T) - r_f}{A \sigma_T^2}$, without altering its internal asset weights.