# Portfolio Theory Concept Card: [Your course, case or portfolio] > A concept card for the investments elective: mean-variance portfolio theory beyond the core. It covers portfolio > return and risk in matrix form, the global minimum-variance portfolio, the efficient frontier, the tangency portfolio > and the capital market line, two-fund separation, the choice of a complete portfolio from risk aversion, the limits of > diversification and the estimation error that makes textbook optimisation fragile in practice. It builds on > mba/mba-capm-wacc-card, which introduces diversification, beta and the CAPM in the core course; read its "idea" > section first, because this card does not re-teach it. It is for second-year MBA students in investments or asset > management electives, students preparing for buy-side interviews, and members of investment committees. Worked > example A builds the frontier for a fictional hospital foundation, Worked example B (half solved) works a two-fund > problem, and Exercise C is left for you, with a separate answer key. Source basis: Markowitz (1952), Tobin (1958), > Sharpe (1964), Merton (1972) and the later work on estimation error, written up in original words. Every capital > market assumption is an invented *Example* value, not a forecast. Replace every [bracketed] value and every row > marked *Example*. This is a learning aid, not investment advice. ## Document control | Field | Value | |---|---| | Course, case or portfolio | [Investments elective, session 3 / foundation pool review] | | Learner or analyst | [Your name] | | Analysis date | YYYY-MM-DD | | Base currency and horizon | [USD, 10 years] | | Capital market assumptions source and date | [Named source, method, date] | | Risk-free rate used and date | [Instrument, maturity, yield, date] | | Constraints applied | [Long-only, asset-class limits, liquidity reserve] | | Reviewed by | [Study partner, tutor or investment committee member] | | Version | [0.1] | | Last updated | YYYY-MM-DD | ## Contents - [When to use it](#when-to-use-it) - [The idea in plain language](#the-idea-in-plain-language) - [Formulas and notation](#formulas-and-notation) - [Assumptions and limits](#assumptions-and-limits) - [Worked example A (fully solved): Tallow Creek foundation frontier](#worked-example-a-fully-solved-tallow-creek-foundation-frontier) - [Worked example B (half solved): two funds](#worked-example-b-half-solved-two-funds) - [Exercise C (unsolved): diversification and the complete portfolio](#exercise-c-unsolved-diversification-and-the-complete-portfolio) - [Process diagram](#process-diagram) - [How investment managers use it](#how-investment-managers-use-it) - [Connections to other cards](#connections-to-other-cards) - [Common mistakes and misconceptions](#common-mistakes-and-misconceptions) - [Self-test questions](#self-test-questions) - [My notes and next steps](#my-notes-and-next-steps) - [Sources and further reading](#sources-and-further-reading) - [Answer key](#answer-key) - [Revision history](#revision-history) ## When to use it **One-sentence rule:** combine assets so that no other combination offers a higher expected return for the same risk, then choose how much of that risky mix to hold against a risk-free asset according to how much risk the owner can bear. | Use when | Use something else, or add to it, when | |---|---| | Setting a strategic asset allocation across a handful of asset classes | Selecting hundreds of individual securities (use a factor risk model, mba/mba-factor-models-card) | | Explaining why adding a volatile asset can lower portfolio risk | Liabilities drive the decision, as for a pension plan (use a surplus or liability-relative optimisation, mba/mba-fixed-income-card) | | Comparing a proposed allocation with the efficient frontier | Return distributions are strongly skewed or option-like (add downside measures, mba/mba-value-at-risk-card) | | Teaching the logic behind index funds and the CAPM | Inputs are pure guesses with no defensible source (use simple heuristics or a Black-Litterman prior) | ## The idea in plain language **Risk depends on co-movement.** Markowitz (1952) treated a portfolio as a choice of means, variances and covariances. An asset's contribution to portfolio risk depends on how it moves with everything else you hold. Two assets that are not perfectly correlated produce a portfolio whose volatility is below the weighted average of their volatilities. The lower the correlation, the larger the gain. At a correlation of minus one, a riskless combination exists. **The efficient frontier.** Plot every possible portfolio of the risky assets on a chart of expected return against standard deviation. The set has a left edge: for each expected return, the lowest attainable risk. The leftmost point is the **global minimum-variance (GMV) portfolio**. The upper half of the edge, above the GMV portfolio, is the **efficient frontier**: a rational mean-variance investor never holds a risky portfolio below it or to its right. **Adding a risk-free asset changes everything.** Tobin (1958) showed that once investors can lend and borrow at a risk-free rate, the best risky portfolio is the same for everyone with the same inputs: the **tangency portfolio**, where a straight line from the risk-free rate touches the frontier. That line, the **capital allocation line** through the tangency portfolio, has the steepest possible slope, which is the highest **Sharpe ratio** (excess return per unit of volatility). Every investor holds a mix of the risk-free asset and the tangency portfolio; only the mix differs. This is **two-fund separation**: the "what to hold" decision is separate from the "how much risk" decision. **From separation to the CAPM.** If every investor holds the same tangency portfolio, in equilibrium it must be the market portfolio, and the line becomes the **capital market line**. Sharpe (1964) and Lintner (1965) derived from this that an asset's expected return depends on its beta with the market, which is where mba/mba-capm-wacc-card starts. **Why practitioners are cautious.** The optimiser treats the inputs as known. They are estimates, and expected returns in particular are very imprecise. The optimiser then overweights assets whose returns were overestimated and whose risks were underestimated: Michaud (1989) called optimisers "error maximisers", and Chopra and Ziemba (1993) showed that errors in means matter roughly an order of magnitude more than errors in variances. Constraints, shrinkage and the Black and Litterman (1992) approach are the practical responses. ## Formulas and notation Portfolio expected return and variance for $n$ assets: $$ \mu_p = \mathbf{w}^\top \boldsymbol{\mu} = \sum_{i=1}^{n} w_i \mu_i \qquad \sigma_p^2 = \mathbf{w}^\top \Sigma \, \mathbf{w} = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{ij}, \quad \sigma_{ij} = \rho_{ij} \sigma_i \sigma_j $$ Two assets, and the minimum-variance weight on asset 1: $$ \sigma_p^2 = w^2 \sigma_1^2 + (1-w)^2 \sigma_2^2 + 2 w (1-w) \rho_{12} \sigma_1 \sigma_2 \qquad w_{\min} = \frac{\sigma_2^2 - \rho_{12} \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho_{12} \sigma_1 \sigma_2} $$ Global minimum-variance and tangency portfolios (no short-sale constraint): $$ \mathbf{w}_{GMV} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^\top \Sigma^{-1} \mathbf{1}} \qquad \mathbf{w}_{T} = \frac{\Sigma^{-1} (\boldsymbol{\mu} - r_f \mathbf{1})}{\mathbf{1}^\top \Sigma^{-1} (\boldsymbol{\mu} - r_f \mathbf{1})} $$ The frontier in closed form (Merton 1972), with $a = \mathbf{1}^\top \Sigma^{-1} \mathbf{1}$, $b = \mathbf{1}^\top \Sigma^{-1} \boldsymbol{\mu}$, $c = \boldsymbol{\mu}^\top \Sigma^{-1} \boldsymbol{\mu}$ and $d = ac - b^2$: $$ \sigma^2(\mu_p) = \frac{a \mu_p^2 - 2 b \mu_p + c}{d}, \qquad \mu_{GMV} = \frac{b}{a}, \qquad \sigma^2_{GMV} = \frac{1}{a} $$ Sharpe ratio, capital allocation line, and the optimal share $y^*$ of wealth in the tangency portfolio for an investor with mean-variance utility $U = E[r] - \tfrac{1}{2} A \sigma^2$: $$ S_p = \frac{\mu_p - r_f}{\sigma_p} \qquad E[r_C] = r_f + S_T \, \sigma_C \qquad y^* = \frac{\mu_T - r_f}{A \, \sigma_T^2} $$ Diversification with $N$ equally weighted assets, average variance $\bar{\sigma}^2$ and average covariance $\overline{\operatorname{cov}}$: $$ \sigma_p^2 = \frac{1}{N} \bar{\sigma}^2 + \left( 1 - \frac{1}{N} \right) \overline{\operatorname{cov}} \;\longrightarrow\; \overline{\operatorname{cov}} \text{ as } N \to \infty $$ | Symbol | Meaning | Unit | |---|---|---| | $\mathbf{w}$, $w_i$ | Vector of portfolio weights, weight of asset $i$ (sum to 1) | Fraction | | $\boldsymbol{\mu}$, $\mu_i$ | Expected returns of the assets | Rate per year | | $\Sigma$, $\sigma_{ij}$ | Covariance matrix, covariance of assets $i$ and $j$ | Rate squared | | $\sigma_i$, $\rho_{ij}$ | Standard deviation (volatility) of asset $i$, correlation of $i$ and $j$ | Rate per year, number | | $\mu_p$, $\sigma_p$ | Expected return and standard deviation of the portfolio | Rate per year | | $\mathbf{1}$ | Vector of ones | | | $r_f$ | Risk-free rate for the horizon and currency | Rate per year | | $S_p$, $S_T$ | Sharpe ratio of portfolio $p$, of the tangency portfolio | Number | | $A$ | Coefficient of risk aversion in the utility function | Number | | $y^*$ | Optimal share of wealth in the tangency portfolio ($y^* > 1$ means borrowing) | Fraction | | $C$ | Complete portfolio: risk-free asset plus tangency portfolio | | ## Assumptions and limits 1. **Mean and variance are all that matter.** True if returns are normal or utility is quadratic. Skew, fat tails and drawdowns matter to real owners; check downside measures alongside volatility. 2. **One period.** The model chooses once for a single horizon. Rebalancing, spending rules and changing opportunities need multi-period thinking (Merton's later work on intertemporal choice; the practical answer is a rebalancing policy). 3. **Inputs are known.** They are estimated. Expected returns from short histories are nearly useless: with an annual volatility of 16%, the standard error of a mean estimated from 20 years of data is about $16\% / \sqrt{20} = 3.6$ percentage points, larger than most risk premiums. 4. **Unlimited borrowing and short selling at one rate.** Most institutional investors are long-only and cannot borrow freely. Constraints move the optimum inside the theoretical frontier. 5. **Correlations are stable.** In crises correlations between risky assets tend to rise, so diversification is least available when it is most wanted. 6. **No costs, taxes or liquidity limits.** Private assets, transaction costs and taxes can reverse the ranking of allocations that look close on paper. 7. **Naive diversification is a tough benchmark.** DeMiguel, Garlappi and Uppal (2009) found that an equally weighted portfolio was hard to beat out of sample with many optimised strategies, because estimation error ate the theoretical gains. ## Worked example A (fully solved): Tallow Creek foundation frontier **Situation (fictional).** The foundation that supports Tallow Creek Medical Center invests a long-term pool in three asset classes. The investment committee wants to see the efficient frontier and the tangency portfolio before it reviews the policy allocation (the full case is in mba/mba-investments-guide). Capital market assumptions are *Example* values for teaching only. | *Example* asset class | Expected return | Volatility | Correlation with equity | with bonds | with real assets | |---|---|---|---|---|---| | Global equity | 7.5% | 16% | 1.0 | 0.2 | 0.6 | | Investment-grade bonds | 4.5% | 6% | 0.2 | 1.0 | 0.3 | | Real assets | 6.0% | 12% | 0.6 | 0.3 | 1.0 | | Risk-free rate | 3.5% | 0% | | | | **Step 1: covariance matrix.** $\sigma_{ij} = \rho_{ij} \sigma_i \sigma_j$. With volatilities in a row range `sd` (1 x 3) and correlations in `Corr` (3 x 3), Excel 365: `=MMULT(TRANSPOSE(sd),sd)*Corr`. | Covariance | Equity | Bonds | Real assets | |---|---|---|---| | Equity | 0.025600 | 0.001920 | 0.011520 | | Bonds | 0.001920 | 0.003600 | 0.002160 | | Real assets | 0.011520 | 0.002160 | 0.014400 | **Step 2: global minimum-variance portfolio.** `=MMULT(MINVERSE(Cov),{1;1;1})/SUM(MMULT(MINVERSE(Cov),{1;1;1}))` gives weights equity 3.15%, bonds 88.87%, real assets 7.99%. Expected return 4.71%, volatility 5.86%: lower than the bond class alone (6%), because a little equity and real assets diversify the bonds. **Step 3: tangency portfolio.** With excess returns in a column `Ex` (4.0%, 1.0%, 2.5%): `=MMULT(MINVERSE(Cov),Ex)/SUM(MMULT(MINVERSE(Cov),Ex))` gives equity **33.90%**, bonds **51.94%**, real assets **14.17%**. Its expected return is 5.73%, volatility 7.94% and Sharpe ratio **0.2808**. All weights are positive, so Solver with a long-only constraint (maximise `=(MMULT(TRANSPOSE(w),mu)-rf)/SQRT(MMULT(TRANSPOSE(w),MMULT(Cov,w)))` subject to `SUM(w)=1`, `w>=0`) returns the same weights, up to Solver's convergence tolerance (tighten the convergence setting if the third decimal differs). **Step 4: compare with simple allocations.** | *Example* portfolio | Equity | Bonds | Real assets | Expected return | Volatility | Sharpe ratio | |---|---|---|---|---|---|---| | Global minimum variance | 3.15% | 88.87% | 7.99% | 4.71% | 5.86% | 0.2073 | | Tangency | 33.90% | 51.94% | 14.17% | 5.73% | 7.94% | 0.2808 | | Equal weight | 33.33% | 33.33% | 33.33% | 6.00% | 9.12% | 0.2742 | | 60/40 equity and bonds | 60% | 40% | 0% | 6.30% | 10.35% | 0.2705 | | Equity alone | 100% | 0% | 0% | 7.50% | 16.00% | 0.2500 | **Step 5: the capital market line.** Holding the tangency portfolio with leverage $y = 10.35\% / 7.94\% = 1.3039$ gives the 60/40 volatility with an expected return of 6.41%, about 0.11 of a point more. The gain is small because the 60/40 portfolio is already close to the line; the committee cannot borrow anyway, which is why it uses the constrained frontier. **Step 6: the complete portfolio.** For three risk-aversion levels, $y^* = (5.73\% - 3.5\%) / (A \times 0.0794^2)$: | *Example* risk aversion $A$ | $y^*$ in tangency | Expected return | Volatility | Reading | |---|---|---|---|---| | 2 | 1.7689 | 7.44% | 14.04% | Borrows 76.89% of wealth (not allowed for the foundation) | | 4 | 0.8845 | 5.47% | 7.02% | Holds 11.55% in the risk-free asset | | 6 | 0.5896 | 4.81% | 4.68% | Holds 41.04% in the risk-free asset | **Chart.** The efficient frontier (computed with Merton's closed form; $a = 291.3654$, $b = 13.7356$, $c = 0.6834$), the capital market line and the portfolios above. ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Efficient frontier, capital market line and Example portfolios for the Tallow Creek foundation (Example inputs)", "width": 440, "height": 260, "layer": [ { "data": {"values": [ {"sd": 5.86, "ret": 4.71}, {"sd": 6.05, "ret": 5.0}, {"sd": 7.18, "ret": 5.5}, {"sd": 8.96, "ret": 6.0}, {"sd": 11.10, "ret": 6.5}, {"sd": 13.41, "ret": 7.0}, {"sd": 15.82, "ret": 7.5}, {"sd": 18.30, "ret": 8.0} ]}, "mark": {"type": "line", "interpolate": "monotone"}, "encoding": { "x": {"field": "sd", "type": "quantitative", "title": "Volatility (% a year)", "scale": {"domain": [0, 19]}}, "y": {"field": "ret", "type": "quantitative", "title": "Expected return (% a year)", "scale": {"domain": [3, 8.5]}} } }, { "data": {"values": [ {"sd": 0, "ret": 3.5}, {"sd": 5, "ret": 4.90}, {"sd": 10, "ret": 6.31}, {"sd": 15, "ret": 7.71} ]}, "mark": {"type": "line", "strokeDash": [6, 4]}, "encoding": { "x": {"field": "sd", "type": "quantitative"}, "y": {"field": "ret", "type": "quantitative"} } }, { "data": {"values": [ {"name": "Global equity", "sd": 16.0, "ret": 7.5}, {"name": "IG bonds", "sd": 6.0, "ret": 4.5}, {"name": "Real assets", "sd": 12.0, "ret": 6.0}, {"name": "Tangency", "sd": 7.94, "ret": 5.73}, {"name": "GMV", "sd": 5.86, "ret": 4.71}, {"name": "60/40", "sd": 10.35, "ret": 6.30}, {"name": "Equal weight", "sd": 9.12, "ret": 6.00} ]}, "mark": {"type": "point", "filled": true, "size": 80}, "encoding": { "x": {"field": "sd", "type": "quantitative"}, "y": {"field": "ret", "type": "quantitative"}, "color": {"field": "name", "type": "nominal", "title": "Portfolio"} } } ] } ``` **Conclusion.** With these *Example* inputs the best risky mix is bond-heavy and has a modest expected return. An investor who needs more return than 5.73% should, in theory, lever the tangency portfolio; a foundation that cannot borrow must instead move up the frontier towards equity, accepting a lower Sharpe ratio. Notice too how sensitive the answer is: raising the equity expected return from 7.5% to 8.0% changes the tangency weights noticeably (try it), which is the estimation-error problem in one cell. ## Worked example B (half solved): two funds **Situation (fictional).** A Northfield Outdoor Co. employee chooses between an equity index fund (expected return 8.0%, volatility 18%) and a bond index fund (4.0%, 7%) in the company savings plan. The correlation is -0.1 and the risk-free rate is 3.5% (*Example* values). | *Example* weight on equity fund | Expected return | Volatility | Sharpe ratio | |---|---|---|---| | 100% | 8.00% | 18.00% | 0.2500 | | 80% | 7.20% | 14.33% | 0.2582 | | 60% | 6.40% | [ ] | [ ] | | 40% | 5.60% | 7.96% | 0.2637 | | 20% | 4.80% | 6.35% | 0.2048 | | 0% | 4.00% | 7.00% | 0.0714 | Excel for the volatility column, weight in A2: `=SQRT(A2^2*0.18^2+(1-A2)^2*0.07^2+2*A2*(1-A2)*-0.1*0.18*0.07)`. 1. Complete the 60% row. 2. The weighted average of the two volatilities at 60/40 is [ ]. Why is the portfolio volatility lower? 3. The minimum-variance weight on the equity fund is $w_{\min} =$ [ ], with expected return 4.62% and volatility [ ]. 4. The tangency weight on the equity fund (two-asset version of the formula above) is 50.91%, giving expected return 6.04%, volatility 9.46% and Sharpe ratio [ ]. 5. Why does the 100% bond portfolio have a higher volatility than the 20% equity portfolio? ## Exercise C (unsolved): diversification and the complete portfolio **Part 1.** Statman (1987) asked how many stocks make a diversified portfolio. A stock picker holds equal weights in $N$ stocks. Each stock has a volatility of 40% and every pair has a correlation of 0.25 (*Example* values). 1. Compute the portfolio volatility for $N$ = 1, 10, 30 and 100. 2. What is the limit as $N$ grows, and what share of a single stock's variance can be diversified away? 3. What does the answer imply about the risk premium a stock with high company-specific risk should earn? **Part 2.** The best risky portfolio available to an investor has an expected return of 8% and a volatility of 15%; the risk-free rate is 3.5%. 4. Compute $y^*$, the expected return, the volatility and the utility $U$ for $A$ = 2, 3 and 5. 5. Which investor holds exactly the risky portfolio? Which would need to borrow if $A$ were 1.5? ## Process diagram The flow below is the order in which a strategic allocation study uses mean-variance analysis, with the robustness loop that keeps estimation error under control. ```mermaid flowchart TD A["Objectives, horizon, liabilities and constraints (IPS)"] --> B["Capital market assumptions: returns, volatilities, correlations"] B --> C["Covariance matrix"] C --> D{"Short sales and borrowing allowed?"} D -->|Yes| E["Closed form: GMV, tangency, frontier"] D -->|No| F["Solver or quadratic program with constraints"] E --> G["Pick the point on the line or frontier that fits risk tolerance"] F --> G G --> H{"Robust to plausible input changes?"} H -->|No| I["Shrink inputs, add limits, or use a Black-Litterman prior"] I --> C H -->|Yes| J["Policy weights, ranges and rebalancing rule"] ``` ## How investment managers use it - **Strategic asset allocation.** Endowments, foundations, pensions and sovereign funds set policy weights with a constrained optimisation over 5 to 15 asset classes, then add ranges and a rebalancing rule (finance/strategic-asset-allocation-study, finance/investment-policy-statement-institutional). Brinson, Hood and Beebower (1986) found that the policy allocation explained most of the variation of a pension fund's returns over time, which is why committees spend their time here. - **Risk budgeting.** Each asset's contribution to portfolio variance is $w_i (\Sigma \mathbf{w})_i$; the shares add up to 100%. A 60/40 portfolio often takes most of its risk from equities even though equities are 60% of the capital. - **Robust optimisation.** Practitioners shrink the covariance matrix towards a structured target (Ledoit and Wolf 2004), start from market-implied returns and tilt them with explicit views (Black and Litterman 1992), resample, or simply cap weights. - **Wealth management.** Advisers map a client's risk tolerance to a point on a model-portfolio line, which is the complete-portfolio decision made practical (finance/investment-policy-statement-individual, finance/client-risk-profile). - **Performance and risk review.** The Sharpe ratio, the capital allocation line and the frontier are the reference points for performance evaluation (mba/mba-performance-evaluation-card) and for risk limits (mba/mba-value-at-risk-card). ## Connections to other cards - mba/mba-capm-wacc-card: the core-course foundation (diversification, beta, the security market line). - mba/mba-factor-models-card: replaces the full covariance matrix with a few factors, the only practical way to handle thousands of securities. - mba/mba-fixed-income-card: duration and liability matching, the other half of an allocation study. - mba/mba-performance-evaluation-card: Sharpe, Treynor, Jensen's alpha and the information ratio. - mba/mba-value-at-risk-card: downside measures that complement volatility. - mba/mba-investments-guide: places this card in sessions 3 and 4 and uses it in the Tallow Creek mini-case. - academic/stats-correlation: estimating and interpreting correlations, and their sampling error. ## Common mistakes and misconceptions 1. **Averaging volatilities.** Portfolio volatility is not the weighted average of volatilities unless every correlation is 1. 2. **Calling the GMV portfolio "optimal".** It minimises risk; it ignores expected returns. The tangency portfolio maximises the Sharpe ratio. 3. **Putting inefficient portfolios on the frontier.** The lower half of the minimum-variance curve is dominated. 4. **Mixing variances and standard deviations.** $y^*$ divides by variance; the Sharpe ratio divides by standard deviation. Annualise both consistently (variance scales with time, volatility with its square root). 5. **Trusting optimiser output to two decimals.** Small input changes swing the weights; report ranges and test sensitivity. 6. **Using historical average returns as expected returns.** Twenty years of data leave a standard error of several percentage points on each mean. 7. **Forgetting constraints.** The unconstrained tangency portfolio may short assets or require borrowing that the investor cannot do. 8. **Believing more holdings always reduce risk.** Only diversifiable risk falls; the average covariance sets a floor. 9. **Assuming correlations hold in a crisis.** Stress-test with higher correlations. 10. **Ignoring liabilities.** For a pension, the risk that matters is the surplus, not the asset volatility. 11. **Confusing the capital allocation line with the security market line.** The CAL plots return against total volatility for efficient portfolios; the SML plots return against beta for every asset. ## Self-test questions 1. In one sentence, why can adding a risky asset lower a portfolio's volatility? 2. Define the efficient frontier and the global minimum-variance portfolio. 3. Two assets each have volatility 20% and zero correlation. What is the volatility of a 50/50 portfolio? 4. A portfolio has expected return 9% and volatility 15%; the risk-free rate is 3%. What is its Sharpe ratio? 5. For that portfolio as the tangency portfolio, what share $y^*$ does an investor with $A = 3$ hold in it? 6. Two assets have volatilities 20% and 10% and correlation -1. What weight on the first asset gives zero risk? 7. State two-fund separation and explain why it supports index investing. 8. Why do optimised weights often look extreme, and name two remedies. 9. What is the limit of portfolio variance as the number of equally weighted assets grows? 10. Interleaving: the capital market line and the security market line. What is on each axis, and which assets lie on each line in equilibrium? ## My notes and next steps | Item | My entry | |---|---| | Confidence with frontier, tangency and complete portfolio (1 = lost, 5 = could teach it) | [ ] | | Capital market assumptions source I will use, and its date | [ ] | | The input I trust least in my allocation, and how I tested it | [ ] | | Mistakes logged in mba/mba-error-log | [Date and short description] | | Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD | | Next card to study | [mba/mba-factor-models-card / mba/mba-performance-evaluation-card] | ## Sources and further reading - Markowitz, H. (1952). Portfolio selection. *The Journal of Finance*, 7(1), 77-91. - Tobin, J. (1958). Liquidity preference as behavior towards risk. *The Review of Economic Studies*, 25(2), 65-86. - Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. *The Journal of Finance*, 19(3), 425-442. - Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. *The Review of Economics and Statistics*, 47(1), 13-37. - Merton, R. C. (1972). An analytic derivation of the efficient portfolio frontier. *Journal of Financial and Quantitative Analysis*, 7(4), 1851-1872. - Brinson, G. P., Hood, L. R., and Beebower, G. L. (1986). Determinants of portfolio performance. *Financial Analysts Journal*, 42(4), 39-44. - Statman, M. (1987). How many stocks make a diversified portfolio? *Journal of Financial and Quantitative Analysis*, 22(3), 353-363. - Michaud, R. O. (1989). The Markowitz optimization enigma: is "optimized" optimal? *Financial Analysts Journal*, 45(1), 31-42. - Black, F., and Litterman, R. (1992). Global portfolio optimization. *Financial Analysts Journal*, 48(5), 28-43. - Chopra, V. K., and Ziemba, W. T. (1993). The effect of errors in means, variances, and covariances on optimal portfolio choice. *The Journal of Portfolio Management*, 19(2), 6-11. - Ledoit, O., and Wolf, M. (2004). Honey, I shrunk the sample covariance matrix. *The Journal of Portfolio Management*, 30(4), 110-119. - DeMiguel, V., Garlappi, L., and Uppal, R. (2009). Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? *The Review of Financial Studies*, 22(5), 1915-1953. - Textbooks (by name only): Bodie, Z., Kane, A., and Marcus, A. J., *Investments*; Elton, E. J., Gruber, M. J., Brown, S. J., and Goetzmann, W. N., *Modern Portfolio Theory and Investment Analysis*; Ang, A., *Asset Management: A Systematic Approach to Factor Investing*. - Open resource: OpenStax, *Principles of Finance* (CC BY-NC-SA 4.0), chapters on risk and return and on portfolios. ## Answer key **Worked example B.** (1) 60% row: volatility 10.88% (10.8826%), Sharpe ratio $(6.40\% - 3.5\%) / 10.88\% = 0.2665$. (2) $0.6 \times 18\% + 0.4 \times 7\% = 13.60\%$; the portfolio is at 10.88% because the correlation (-0.1) is below 1, so the funds' deviations partly offset. (3) $w_{\min} = (0.0049 + 0.00126) / (0.0324 + 0.0049 + 0.00252) = 15.47\%$ (0.1547), volatility 6.28% (6.2826%). (4) $(6.04\% - 3.5\%) / 9.46\% = 0.2681$, slightly above the 60% row, as it must be: no other mix has a higher Sharpe ratio. (5) Moving from 0% to 20% equity adds an asset with negative correlation to the bonds, which lowers variance until the minimum at 15.47%; beyond that point equity's own variance dominates. **Exercise C.** (1) Average covariance $= 0.25 \times 0.40^2 = 0.04$. $N = 1$: 40.00%; $N = 10$: $\sqrt{0.16/10 + 0.9 \times 0.04} = 22.80\%$; $N = 30$: 20.98%; $N = 100$: 20.30%. (2) The limit is $\sqrt{0.04} = 20.00\%$. The share of a single stock's variance that can be removed is $1 - 0.04/0.16 = 75\%$. (3) None for the company-specific part: it can be diversified away at almost no cost, so in equilibrium only the systematic part earns a premium (the CAPM argument in mba/mba-capm-wacc-card). (4) $y^* = 0.045 / (A \times 0.0225)$: $A = 2$: $y^* = 1.0000$, expected return 8.00%, volatility 15.00%, $U = 8.00\% - 0.5 \times 2 \times 0.0225 = 5.75\%$. $A = 3$: $y^* = 0.6667$, 6.50%, 10.00%, $U = 5.00\%$. $A = 5$: $y^* = 0.4000$, 5.30%, 6.00%, $U = 4.40\%$. Excel: `=(8%-3.5%)/(A*15%^2)`. (5) The investor with $A = 2$; with $A = 1.5$, $y^* = 1.3333$, so that investor borrows a third of wealth. **Self-test.** 1. Because its returns are not perfectly correlated with the rest of the portfolio, part of its movements offset movements elsewhere. 2. The efficient frontier is the set of risky portfolios with the highest expected return for each level of volatility (the upper edge of the feasible set); the GMV portfolio is the single portfolio with the lowest volatility. 3. $\sqrt{0.25 \times 0.04 + 0.25 \times 0.04} = 14.14\%$. 4. $(9\% - 3\%) / 15\% = 0.40$. 5. $y^* = 0.06 / (3 \times 0.0225) = 0.8889$: 88.89% in the risky portfolio and 11.11% in the risk-free asset. 6. $w = \sigma_2 / (\sigma_1 + \sigma_2) = 10 / 30 = 33.33\%$. 7. All investors with the same inputs hold the same risky portfolio and differ only in how much they combine it with the risk-free asset. In equilibrium that portfolio is the market, and a low-cost index fund is the closest practical version. 8. Optimisers load on assets whose returns are overestimated and whose risks or correlations are underestimated. Remedies: weight limits, shrinkage of the covariance matrix, Black-Litterman returns, resampling. 9. The average covariance between the assets. 10. The CML has total volatility on the horizontal axis and expected return on the vertical; only efficient portfolios (combinations of the risk-free asset and the market) lie on it. The SML has beta on the horizontal axis; in equilibrium every asset and portfolio lies on it. ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First pass through the card | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Example capital market assumptions with a dated, named source | ## Standard and licence note Based on mean-variance portfolio theory in the primary works listed above (Markowitz 1952, Tobin 1958, Sharpe 1964, Lintner 1965, Merton 1972) and the later literature on estimation error. The text, examples, numbers and questions are original; no textbook text is copied or paraphrased, and Tallow Creek Medical Center, its foundation and Northfield Outdoor Co. are fictional. All capital market assumptions are *Example* values, not forecasts or current data. The template is independent and not affiliated with or endorsed by any business school, publisher or data provider. Check current data before you rely on a figure. Template licence: MIT.