Portfolio Theory: Markowitz Optimization, CAPM, Beta, Alpha, and Capital Budgeting

Portfolio theory begins with a choice: how should an investor combine uncertain investments? CAPM adds an equilibrium question: if investors make those choices together, what expected returns should market prices imply? Corporate finance then uses those required returns to evaluate projects with comparable risks.

Learning goals and modeling assumptions

You should finish able to construct a mean–variance portfolio, distinguish the CML from the SML, derive beta’s role in expected return, interpret a regression alpha, and choose a discount rate appropriate to a project’s cash flows.

Use simple total returns over a common horizon, consistent currencies, and fixed beginning-of-period weights. Numerical inputs are hypothetical, not live return forecasts. Ignore taxes and trading costs in the basic portfolio model. Add financing and tax assumptions explicitly in capital budgeting.

Symbol	Meaning
$\mathbf w$	Vector of portfolio wealth weights
$\boldsymbol\mu$	Vector of expected asset returns
$\Sigma$	Return covariance matrix
$r_f$	Risk-free return over the same horizon
$R_M$	Market portfolio return
$\beta_i$	Asset $i$‘s exposure to market return
$\alpha_i$	Intercept relative to the specified return benchmark

1. Markowitz portfolio theory: optimize the whole portfolio

For a fully invested portfolio, $\sum_iw_i=1$ and

$$R_P=\sum_iw_iR_i,\qquad \mu_P=\mathbf w^{\mathsf T}\boldsymbol\mu.$$

Negative weights represent short positions. Borrowing is represented by a negative financing weight; risky holdings can then exceed 100% of net wealth.

The variance is

$$\sigma_P^2=\sum_iw_i^2\sigma_i^2 +\sum_i\sum_{j\ne i}w_iw_j\operatorname{Cov}(R_i,R_j) =\mathbf w^{\mathsf T}\Sigma\mathbf w.$$

The double sum counts each distinct covariance twice. Equivalently use $2\sum_{i<j}$, but do not apply both counting conventions at once.

For two assets, with weight $w$ in A:

$$\sigma_P^2=w^2\sigma_A^2+(1-w)^2\sigma_B^2 +2w(1-w)\rho_{AB}\sigma_A\sigma_B.$$

The relevant risk question is how an asset contributes to the entire portfolio, rather than its standalone volatility alone.

Worked example: a diversified pair

Let A have expected return 12% and volatility 20%, B have expected return 6% and volatility 10%, and their correlation be zero. A 50/50 mix has expected return 9% and

$$\sigma_P=\sqrt{0.5^2(0.20)^2+0.5^2(0.10)^2} \approx11.1803\%.$$

The weighted average standalone volatility is 15%. Diversification lowers portfolio volatility because the assets do not move perfectly together.

For nonnegative weights summing to one, $\sigma_P\le\sum_iw_i\sigma_i$, with a strict reduction when positive-volatility exposures are genuinely mixed and not all perfectly positively correlated. Short positions require a bound using absolute weights; the long-only statement does not transfer unchanged.

2. Diversification and the covariance floor

For $N$ equal-weight assets with common volatility $\sigma$ and common pairwise correlation $\rho$:

$$\sigma_P^2=\sigma^2\left[\frac1N+\frac{N-1}{N}\rho\right].$$

With fixed nonnegative correlation, the standalone variance contribution shrinks with $N$, while the common covariance contribution remains. At volatility 25% and correlation 0.2, 25 assets have portfolio volatility

$$0.25\sqrt{0.2+0.8/25}\approx12.0416\%,$$

approaching $0.25\sqrt{0.2}\approx11.1803\%$ as the number becomes very large. The result explains why additional names do little if they share the same dominant exposures.

A one-factor representation makes the distinction explicit:

$$R_i=\alpha_i+\beta_iR_M+\varepsilon_i.$$

If residuals are mutually uncorrelated and uncorrelated with the market, broadly spreading small positions can make their aggregate variance small. Market exposure remains. Finite portfolios, correlated residuals, concentrated industries, and changing relationships can leave substantial residual risk.

Thus idiosyncratic risk can be diversified away under suitable assumptions, not automatically eliminated by owning a large number of securities. Systematic risk is exposure to common priced factors; the one-market-factor description belongs to CAPM, while other models allow multiple factors.

3. The minimum-variance boundary and efficient frontier

For a target mean $\mu^*$, solve

$$\min_{\mathbf w}\ \mathbf w^{\mathsf T}\Sigma\mathbf w \quad\text{subject to}\quad \mathbf1^{\mathsf T}\mathbf w=1,\qquad \boldsymbol\mu^{\mathsf T}\mathbf w=\mu^*.$$

Varying the target traces the minimum-variance boundary. The boundary is the edge of the feasible opportunity set, not the entire set. Its upper branch from the global minimum-variance point is efficient: no feasible portfolio offers a higher mean at the same or lower volatility.

The familiar bullet-shaped region describes a standard nondegenerate multi-asset model. Two fully invested risky assets trace a curve instead of filling a region. Short-sale prohibitions and other constraints alter the feasible set.

Finding the global minimum-variance portfolio

When $\Sigma$ is invertible and short sales are unrestricted:

$$\mathbf w_{GMV}=\frac{\Sigma^{-1}\mathbf1} {\mathbf1^{\mathsf T}\Sigma^{-1}\mathbf1}.$$

For two uncorrelated assets, $w_A^{GMV}=\sigma_B^2/(\sigma_A^2+\sigma_B^2)$. In the A/B example this gives 20% A and 80% B, expected return 7.2%, and volatility 8.9443%.

This is the least volatile fully invested risky mix, not necessarily the investor’s preferred allocation. A higher-return efficient portfolio can be preferable for someone willing to bear more risk.

4. Tangency and two-fund separation

Introduce a risk-free return of 3%. A mix of fraction $x$ in a risky portfolio T and $1-x$ in the risk-free asset has

$$\mu_C=r_f+x(\mu_T-r_f),\qquad \sigma_C=|x|\sigma_T.$$

For $x\ge0$, the line’s slope is the risky portfolio’s Sharpe ratio:

$$\operatorname{SR}_T=\frac{\mu_T-r_f}{\sigma_T}.$$

The tangency portfolio maximizes that slope. In the well-posed unconstrained model with invertible covariance and positive normalization:

$$\mathbf w_T=\frac{\Sigma^{-1}(\boldsymbol\mu-r_f\mathbf1)} {\mathbf1^{\mathsf T}\Sigma^{-1}(\boldsymbol\mu-r_f\mathbf1)}.$$

For uncorrelated A and B, unnormalized weights are $0.09/0.04=2.25$ and $0.03/0.01=3$. Normalizing gives

$$w_A=\frac37,\qquad w_B=\frac47.$$

The tangency mix has expected return 8.5714%, volatility 10.3016%, and Sharpe ratio approximately 0.5408.

Portfolio	A weight	Expected return	Volatility	Sharpe ratio at 3%
Global minimum variance	20.00%	7.20%	8.94%	0.4696
Tangency	42.86%	8.57%	10.30%	0.5408
Equal weight	50.00%	9.00%	11.18%	0.5367

Holding half in T and half risk-free gives expected return 5.7857% and volatility 5.1508%. Borrowing 50% at the same 3% rate to hold 150% in T gives mean 11.3571% and volatility 15.4524%.

What two-fund separation actually assumes

Investors with mean–variance preferences, identical forecasts and opportunity sets, and access to the same unrestricted risk-free borrowing/lending rate choose the same risky tangency portfolio. Their risk aversion changes $x$.

This is a conditional theorem. Different beliefs, liabilities, taxes, horizons, constraints, or preferences beyond mean and variance can change the optimal risky holdings. A higher borrowing rate also breaks the simple single-line extension. It is too strong to say that every rational investor must hold identical risky assets in practice.

5. From individual choice to market equilibrium

A tangency portfolio is an optimal portfolio for specified inputs. Identifying it with the market portfolio requires additional equilibrium assumptions.

In the standard CAPM, investors face a common single-period opportunity set, share beliefs, use mean–variance analysis, trade frictionlessly, and can borrow and lend at the common risk-free rate. If all hold the same risky portfolio, aggregate demand must equal the supply of risky assets. The shared portfolio must therefore be market-value weighted:

$$w_i^M=\frac{\text{Market value of asset }i} {\text{Total market value of risky assets}}.$$

The theoretical market includes the modeled universe of risky wealth, not automatically a single stock index. A broad equity index is an empirical proxy; excluded assets and human capital complicate testing.

The capital market line

When T is identified with M, efficient combinations of the risk-free asset and M obey

$$\mathbb E[R_P]=r_f+ \frac{\mathbb E[R_M]-r_f}{\sigma_M}\sigma_P.$$

This is the capital market line (CML) on its efficient positive-market-exposure branch. It relates expected return to total volatility for efficient portfolios, not arbitrary individual stocks.

If $r_f=3\%$, market expected return is 9%, and market volatility is 15%, an efficient portfolio with 10% volatility has expected return $3\%+(6\%/15\%)(10\%)=7\%$. It holds two-thirds in the market and one-third in the risk-free asset under these assumptions.

6. Why beta appears in the security market line

The derivative of portfolio variance with respect to an asset’s weight is governed by its covariance with the current portfolio. At a tangency optimum, expected excess returns must be proportional to those covariance contributions. In vector form:

$$\boldsymbol\mu-r_f\mathbf1=\lambda\Sigma\mathbf w_M.$$

For asset $i$ this says

$$\mu_i-r_f=\lambda\operatorname{Cov}(R_i,R_M).$$

Multiply by market weights and sum. Because $\sum_iw_i^M=1$:

$$\mu_M-r_f=\lambda\sigma_M^2.$$

Substituting back yields the CAPM:

$$\beta_i=\frac{\operatorname{Cov}(R_i,R_M)}{\operatorname{Var}(R_M)}, \qquad \mu_i=r_f+\beta_i(\mu_M-r_f).$$

This is the security market line (SML). Within the model’s equilibrium, it applies to individual assets and arbitrary portfolios, not only efficient ones. It is an expected-return relation, not a promise about realized returns.

Beta is not standalone volatility

Using covariance $\rho_{iM}\sigma_i\sigma_M$:

$$\beta_i=\rho_{iM}\frac{\sigma_i}{\sigma_M}.$$

An asset with volatility 30%, market correlation 0.25, and market volatility 15% has beta 0.5. With risk-free rate 3% and market premium 6%, its CAPM required return is 6%, despite its high standalone volatility.

A second asset with volatility 15% and correlation 0.9 has beta 0.9 and required return 8.4%. The lower-volatility asset has higher market exposure. CAPM prices that covariance exposure rather than rewarding all forms of risk.

Feature	CML	SML
Horizontal axis	Total volatility $\sigma$	Market beta $\beta$
Applies to	Efficient market/risk-free combinations	All assets and portfolios in CAPM equilibrium
Slope	Market Sharpe ratio	Expected market risk premium
Does not establish	An arbitrary stock’s return from its volatility	A guaranteed realized return

Portfolio beta is linear: $\beta_P=\sum_iw_i\beta_i$. The market has beta one; a risk-free holding has beta zero.

7. Estimating beta and Jensen’s alpha

Use consistently timed total returns and subtract the matching period’s risk-free return:

$$R_{i,t}-R_{f,t}=\alpha_i+ \beta_i(R_{M,t}-R_{f,t})+\varepsilon_{i,t}.$$

With an intercept, the ordinary least-squares slope is sample covariance of asset and market excess returns divided by sample market-excess-return variance. The estimated intercept is

$$\widehat\alpha_i=\overline{(R_i-R_f)}- \widehat\beta_i\overline{(R_M-R_f)}.$$

The simpler covariance formula in Section 6 treats $r_f$ as deterministic over the modeled horizon. With a varying historical risk-free series, use excess returns consistently in the regression.

Worked regression interpretation

Suppose estimated monthly beta is 1.2, average monthly market excess return is 0.50%, and the fund’s average monthly excess return is 0.80%. Then

$$\widehat\alpha=0.80\%-1.2(0.50\%)=0.20\%\text{ per month}.$$

An estimated intercept of 0.20% with standard error 0.15% has a $t$-statistic of about 1.33. It is positive, but weak statistical evidence against zero under conventional inference. Serial correlation or changing volatility may require robust standard errors.

CAPM implies a zero population alpha relative to the true market benchmark. An estimated positive alpha is historical performance unexplained by the fitted benchmark, not automatically skill or a future trading opportunity. Missing factors, fees, stale pricing, selection effects, and chance can affect it. State whether returns are gross or net of fees.

Choose a sample window, return frequency, and market proxy deliberately. Betas can change with business mix, leverage, and market conditions. A stock’s total risk can remain high even when its estimated beta is low.

8. Capital budgeting: match the rate to the claim

CAPM provides a model-based opportunity cost for an exposure with beta $\beta$:

$$r=r_f+\beta\bigl(\mathbb E[R_M]-r_f\bigr).$$

The project beta should reflect the project’s operating and financing exposure, rather than automatically copying the parent company’s equity beta. A firm applying one company-wide hurdle rate to every project can accept overly risky projects and reject valuable safer ones.

Worked unlevered project

A project costs 100 today and has expected cash flows of 60 at each of the next two year-ends. Assume these are unlevered operating cash flows, project asset beta is 0.8, the risk-free rate is 3%, and expected market premium is 5%. With no financing or tax complications:

$$r_A=3\%+0.8(5\%)=7\%.$$

Then

$$NPV=-100+\frac{60}{1.07}+\frac{60}{1.07^2}\approx8.48.$$

The project adds value under the forecast and risk assumptions. That is not a guarantee that realized cash flows will be favorable. The constant beta and discount rate are modeling assumptions; changing risks or maturity-specific rates can require a richer valuation.

Equity, assets, and WACC

For a levered firm, the equity beta generally exceeds its asset beta because debt concentrates residual risk in equity. Under a common simplified tax-adjusted leverage relation, assuming debt beta approximately zero:

$$\beta_A\approx\frac{\beta_E}{1+(1-\tau_c)D/E}.$$

For a comparable firm’s equity beta 1.4, debt/equity ratio 0.5, and corporate tax rate 25%, estimated asset beta is $1.4/[1+0.75(0.5)]\approx1.0182$. Relevering that exposure at a target debt/equity ratio of 0.25 gives equity beta approximately $1.0182[1+0.75(0.25)]=1.2091$.

This leverage formula has assumptions about debt risk and tax shields; it is not universal. Use market-value capital structure and comparable business risks, and account for nontrivial debt beta when necessary.

Cash flows being valued	Consistent discount-rate approach
Cash flows to equity after debt financing effects	Cost of equity at the relevant equity risk
Free cash flow to the firm before financing payments	Appropriate WACC under consistent financing assumptions
Unlevered operating cash flows, with financing effects valued separately	Unlevered asset cost plus a separate financing valuation

A standard WACC expression is

$$WACC=\frac{E}{D+E}r_E+\frac{D}{D+E}r_D(1-\tau_c).$$

For debt/equity 0.25, weights are 20% debt and 80% equity. With the relevered beta above, $r_E=3\%+1.2091(5\%)\approx9.0455\%$. Assuming debt cost 4% and tax rate 25%, WACC is approximately 7.8364%. Use this only with the corresponding business risk, financing policy, tax treatment, and free-cash-flow definition.

9. Historical use and empirical limitations

Graham and Harvey’s 2001 corporate-finance survey collected responses in 1999 from 392 CFOs. It reports that 73.5% of respondents always or almost always used CAPM to estimate the cost of equity. The outline’s roughly 74% is therefore a historical survey result, not a current worldwide adoption rate or evidence that CAPM is empirically exact.

CAPM’s usefulness is its disciplined link between priced market exposure and opportunity cost. Its limitations include uncertain expected market premia, unstable beta estimates, an imperfectly observed market portfolio, restrictive investor assumptions, and incomplete empirical explanations of returns.

Black’s zero-beta CAPM relaxes access to the risk-free borrowing/lending asset in a particular equilibrium setting. Its expected-return relation uses the expected return of an appropriate zero-beta portfolio:

$$\mathbb E[R_i]=\mathbb E[R_Z]+\beta_i\bigl(\mathbb E[R_M]-\mathbb E[R_Z]\bigr).$$

Zero beta means zero market covariance, not zero volatility or guaranteed cash flows. Multifactor approaches instead allow several systematic exposures, schematically

$$\mathbb E[R_i]-r_f=\sum_k\beta_{ik}\lambda_k.$$

Factor choice, economic interpretation, and estimation still matter. A fitted regression model and a no-arbitrage or equilibrium pricing theory are related but distinct claims. CAPM remains an important benchmark; using it responsibly includes sensitivity analysis and comparison with alternatives.

10. Practice questions

Use annual effective return inputs unless the question explicitly uses monthly data.

1. Two assets have volatilities 25% and 15% and correlation 0.2. Find covariance and the volatility of a 50/50 portfolio.
2. A portfolio has expected return 10%, volatility 16%, and risk-free rate 2%. Find Sharpe ratio. Find mean and volatility for a 75% risky / 25% risk-free combination.
3. A stock has market covariance 0.018 and the market variance is 0.0225. Find beta and CAPM return when $r_f=3\%$ and market expected return is 9%.
4. A portfolio holds 40% in beta-0.5 equity and 60% in beta-1.5 equity. Find beta and CAPM expected return using question 3’s market inputs.
5. A manager’s average monthly excess return is 0.7%, estimated beta is 0.9, and average monthly market excess return is 0.6%. Find estimated alpha. If its standard error is 0.10%, find the $t$-statistic.
6. A project costs 100 and pays expected 115 in one year. Project asset beta is 1.2, $r_f=4\%$, and the market premium is 5%. Ignoring financing and taxes, find its hurdle rate and NPV.
7. A comparable has equity beta 1.2, debt/equity 0.4, tax rate 25%, and negligible debt beta. Estimate asset beta using the simplified leverage formula.
8. Why does the SML apply to inefficient portfolios within CAPM while the CML does not? Why is the survey’s 73.5% not a current adoption estimate?

11. Answers and reasoning

1. Covariance is $0.2(0.25)(0.15)=0.0075$. Variance is $0.25(0.0625)+0.25(0.0225)+0.5(0.0075)=0.025$, so volatility is approximately 15.8114%.
2. Sharpe ratio is $(0.10-0.02)/0.16=0.5$. Mixed mean is $0.02+0.75(0.08)=8\%$ and volatility is 12%.
3. Beta is $0.018/0.0225=0.8$. Required return is $3\%+0.8(6\%)=7.8\%$.
4. Beta is $0.4(0.5)+0.6(1.5)=1.1$. Expected return is $3\%+1.1(6\%)=9.6\%$.
5. Alpha is $0.7\%-0.9(0.6\%)=0.16\%$ per month. The $t$-statistic is 1.6; a positive point estimate alone does not establish statistically reliable skill.
6. Required return is $4\%+1.2(5\%)=10\%$. NPV is $-100+115/1.10\approx4.55$.
7. $\beta_A\approx1.2/[1+0.75(0.4)]\approx0.9231$. Check whether the debt-risk and tax-shield assumptions fit before applying this in practice.
8. SML prices covariance exposure and is linear in portfolio weights; an inefficient portfolio can still satisfy CAPM while bearing unnecessary diversifiable volatility. CML describes efficient total-volatility combinations only. The survey reports a particular historical sample and question, with responses collected in 1999.

Further reading

• Andrew W. Lo, MIT OpenCourseWare, Portfolio Theory: optimization and the tangency portfolio.
• MIT OpenCourseWare, The CAPM and APT slides: equilibrium, risk exposure, and implementation.
• MIT OpenCourseWare, Capital Budgeting: project evaluation and cost-of-capital applications.
• John R. Graham and Campbell R. Harvey (2001), The Theory and Practice of Corporate Finance: Evidence from the Field, Journal of Financial Economics 60, 187–243: the historical CFO survey.

Feynman study lab

Explain it simply

Portfolio risk depends on how assets move together, not just on each asset’s individual volatility.

Concrete example. Two equal-weight assets each with 10% volatility and zero correlation produce portfolio volatility about 7.07%.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Why does this benefit disappear if their correlation rises to one?

Check your explanation after you try

With perfect positive correlation their fluctuations move together; the equal-weight portfolio’s volatility becomes 10%.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.