Portfolio theory Study Guide

📖 Core Concepts Modern Portfolio Theory (MPT) – A mathematical framework that chooses asset weights to maximise expected return for a given risk (or minimise risk for a given return). Risk measure – Portfolio risk is captured by variance $\sigma^{2}$ (or its square‑root standard deviation $\sigma$); variance is tractable because it combines linearly with covariances. Diversification – Holding assets that are not perfectly positively correlated lowers portfolio $\sigma$; the risk reduction comes mainly from covariances, not individual variances. Efficient frontier – The set of portfolios that deliver the lowest risk for each level of expected return; the upper hyperbolic part of the feasible region. Global Minimum‑Variance Portfolio (GMVP) – The left‑most point of the frontier; the lowest‑risk portfolio among all risky assets. Risk‑free asset – An asset with a certain return $Rf$ and zero variance (e.g., Treasury bills). Adding it turns the frontier into a straight Capital Allocation Line (CAL) tangent to the efficient frontier. Capital Asset Pricing Model (CAPM) – In equilibrium, the required return on asset $i$ is $$E(Ri)=Rf+\betai\bigl(E(Rm)-Rf\bigr)$$ where $\betai$ is the asset’s covariance with the market portfolio divided by the market variance. Systematic vs. Specific risk – Systematic (market) risk cannot be diversified away; specific (idiosyncratic) risk can. Only systematic risk earns a risk premium. Two‑Fund (Two‑Mutual‑Fund) Theorem – Any efficient‑frontier portfolio can be built as a linear combination of any two distinct efficient portfolios. One‑Fund Separation Theorem – With a risk‑free asset, all investors need only the risk‑free asset and the tangency (market) portfolio to achieve any optimal risk‑return mix. --- 📌 Must Remember Portfolio return: $Rp=\sumi wi \mui$ Portfolio variance: $\sigmap^{2}= \sumi\sumj wi wj \sigma{ij}$ Efficient frontier = upper hyperbola; GMVP = vertex of the hyperbola. CAL slope = Sharpe ratio of the tangency portfolio: $$\text{Sharpe} = \frac{E(Rt)-Rf}{\sigmat}$$ CAPM equation (see above). Beta definition: $\betai = \frac{\operatorname{Cov}(Ri,Rm)}{\operatorname{Var}(Rm)}$ Key theorems: Two‑Fund → only two efficient funds needed; One‑Fund → only risk‑free + tangency fund needed. Diversification rule: Any two assets with correlation $<+1$ produce a portfolio lower than the weighted‑average σ. --- 🔄 Key Processes Mean‑Variance Optimization (MVO) Set up: minimize $w^{\top}\Sigma w$ subject to $w^{\top}\mathbf{1}=1$ and $w^{\top}\mu = R{\text{target}}$. Solve via Lagrange multipliers → obtain efficient‑frontier weights. Constructing the CAL Identify the tangency portfolio (max Sharpe). Draw a straight line from $(\sigma=0,\,R=Rf)$ through the tangency point. Lending (points between intercept and tangency) = allocate part to $Rf$; borrowing (beyond tangency) = leverage the risky portfolio. Applying CAPM Estimate $\betai$ from historical covariances. Plug into CAPM to get required return $E(Ri)$. Use $E(Ri)$ as discount rate for valuation. Two‑Fund Construction Choose any two efficient portfolios $A$ and $B$. Any other efficient portfolio $C$ can be written as $C = \lambda A + (1-\lambda) B$. --- 🔍 Key Comparisons Risk‑free vs. Risky asset Risk‑free: $ \sigma=0$, known $Rf$; Risky: $\sigma>0$, uncertain return. Systematic risk vs. Specific risk Systematic: market‑wide, non‑diversifiable, measured by $\beta$; Specific: asset‑specific, can be eliminated by diversification. Variance vs. Standard deviation Variance: $\sigma^{2}$, easier algebraically; Standard deviation: $\sigma$, intuitive “units of return”. Efficient frontier (no $Rf$) vs. CAL (with $Rf$) Frontier: curved hyperbola, all risky combos; CAL: straight line, all combos of $Rf$ + tangency portfolio. --- ⚠️ Common Misunderstandings “Higher variance = higher return.” Variance is a risk measure, not a return driver; the efficient frontier shows the best return for a given variance, not that every high‑variance asset outperforms. “CAPM eliminates all risk.” CAPM only prices systematic risk; specific risk remains diversifiable. “Historical variance = future risk.” Historical estimates are proxies; they can be biased if market conditions change. “Diversification eliminates all risk.” Only specific risk is removed; systematic risk persists. --- 🧠 Mental Models / Intuition “Risk‑return trade‑off as a slope.” Think of the CAL slope = Sharpe ratio; a steeper slope means you get more return per unit of risk. “Portfolio as a weighted average of points.” In the mean‑variance plane, a portfolio is the midpoint of its constituent assets weighted by $wi$. “All roads lead to the tangency point.” No matter how many assets you start with, the optimal risky mix (in the presence of a risk‑free asset) collapses to one point – the market portfolio. --- 🚩 Exceptions & Edge Cases Perfect positive correlation ($\rho=+1$) → diversification offers no σ reduction; portfolio σ = weighted average of individual σ. Non‑normal return distributions – MPT’s variance‑based risk measure may mis‑price assets with skewness or fat tails. Parameter uncertainty – If expected returns $\mu$ or covariances $\Sigma$ are estimated poorly, the calculated efficient frontier can be unstable. Absence of a true risk‑free asset – In practice, Treasury bills are only an approximation; any proxy introduces a small variance. --- 📍 When to Use Which Use MVO (quadratic programming) when you have reliable estimates of $\mu$ and $\Sigma$ and need the optimal risky mix without a risk‑free asset. Switch to CAL (tangency + $Rf$) when a risk‑free proxy exists and you want to incorporate borrowing/lending decisions. Apply CAPM to compute a discount rate for valuation or to assess whether an asset is fairly priced given its $\beta$. Employ Two‑Fund theorem when constructing mutual‑fund style products – you only need two efficient funds to span the whole frontier. Consider Black–Litterman (or other extensions) when historical inputs are noisy and you want to blend them with subjective views. --- 👀 Patterns to Recognize Hyperbola → efficient frontier; tangent point → highest Sharpe. Any efficient portfolio = λ·Portfolio A + (1‑λ)·Portfolio B (Two‑Fund). CAPM line (SML) slope = market risk premium; assets above the line are underpriced, below are overpriced. Correlation < 1 → σ reduction; the lower the correlation, the greater the diversification benefit. --- 🗂️ Exam Traps Choosing variance instead of standard deviation for “risk” when the question asks for “volatility.” Confusing the SML (beta vs. return) with the CAL (σ vs. return). Both are straight lines but have different axes. Assuming the efficient frontier is a straight line – it’s a hyperbola unless a risk‑free asset is added. Treating specific risk as rewarded – CAPM rewards only systematic risk; any extra return attributed to idiosyncratic risk is noise. Using historical returns as exact future expectations – exam questions may test understanding of parameter uncertainty. ---