Risk and Return: Diversification, Portfolio Risk, and the Efficient Frontier

An investment’s average return is only part of its story. You also need to know how uncertain that return is and how it moves with the rest of your portfolio. An asset that looks risky in isolation may reduce portfolio risk when its returns offset other exposures.

Learning goals and assumptions

You should finish able to calculate return moments, build a covariance-based portfolio risk estimate, explain diversification limits, identify efficient portfolios, and derive a tangency portfolio with a risk-free asset.

Use simple total returns over the same investment horizon, consistent currency units, and fixed beginning-of-period weights. Ignore trading costs and taxes in the basic model. Portfolio means and covariance estimates are inputs, not known truths. Annual figures in worked examples are hypothetical annual moments; no monthly-to-annual conversion is implicit.

Symbol	Meaning
$R_i$	Random return of asset $i$
$\mu_i$	Expected return of asset $i$
$\sigma_i$	Return standard deviation
$\sigma_{ij}$	Covariance between returns $i$ and $j$
$\rho_{ij}$	Correlation between returns $i$ and $j$
$w_i$	Fraction of starting portfolio wealth allocated to asset $i$
$\Sigma$	Covariance matrix
$r_f$	Risk-free return for the matching horizon

1. Define the return before measuring risk

For a share with beginning price $P_0$, ending price $P_1$, and a dividend paid at the end of the period:

$$R=\frac{P_1-P_0+D_1}{P_0}.$$

A share bought at 100, paying 2 and ending at 106, earns 8%, not 6%. For longer histories with intermediate distributions, use a consistently constructed total-return series that accounts for reinvestment.

Expected return is a forecast over possible outcomes. Realized return is what actually occurs. A historical sample average is an estimator of expected return, not the expected return itself.

Arithmetic and compound growth are different

If annual returns are +20% and -20%, their arithmetic average is zero. Wealth instead changes from 100 to 120 to 96. The annual compound return is

$$g=\left[(1.20)(0.80)\right]^{1/2}-1\approx-2.0204\%.$$

The arithmetic mean describes the average one-period return in the sample. The geometric mean describes compound growth over that realized sequence. Do not substitute one for the other without specifying the question.

2. Lecture 12: expected return, variance, and volatility

For possible return outcomes $R_s$ with real-world probabilities $p_s$:

$$\mu=\mathbb{E}[R]=\sum_s p_sR_s, \qquad \sum_s p_s=1.$$

Variance and standard deviation measure dispersion:

$$\sigma^2=\mathbb{E}[(R-\mu)^2]=\sum_s p_s(R_s-\mu)^2, \qquad \sigma=\sqrt{\sigma^2}.$$

They treat equally sized upside and downside deviations symmetrically. Volatility is therefore a useful risk measure, but not a complete description of default, liquidity, drawdown, skewness, or extreme-tail risk.

Worked probability example

State	Probability	Return	Contribution to expected return
Expansion	25%	20%	5.00%
Normal	50%	8%	4.00%
Contraction	25%	-12%	-3.00%

$$\mu=0.25(0.20)+0.50(0.08)+0.25(-0.12)=0.06.$$

The deviations from 6% are 14, 2, and -18 percentage points:

$$\sigma^2=0.25(0.14)^2+0.50(0.02)^2+0.25(-0.18)^2=0.0132,$$

so $\sigma\approx11.4891\%$. Variance is in squared-return units; standard deviation returns to the same units as return.

Estimating moments from observations

For $n$ historical observations:

$$\bar R=\frac1n\sum_{t=1}^{n}R_t, \qquad s^2=\frac{1}{n-1}\sum_{t=1}^{n}(R_t-\bar R)^2.$$

The $n-1$ correction is the standard unbiased sample-variance formula under the usual sampling assumptions. It differs from the known-probability population calculation above. Expected returns are particularly noisy to estimate, and changing market regimes can make old observations less representative.

Multiplying periodic mean by the number of periods and volatility by its square root is a conventional approximation for additive, uncorrelated, stable-period returns. Serial dependence changes variance scaling, and exact compounded simple returns require separate treatment.

3. Covariance and correlation measure co-movement

$$\sigma_{ij}=\operatorname{Cov}(R_i,R_j) =\mathbb{E}[(R_i-\mu_i)(R_j-\mu_j)].$$

When both standard deviations are positive:

$$\rho_{ij}=\frac{\sigma_{ij}}{\sigma_i\sigma_j}, \qquad -1\le\rho_{ij}\le1.$$

Positive covariance means returns tend to be above or below their means together. Negative covariance means those deviations tend to offset. Zero correlation means no linear co-movement; it does not generally imply independence.

A risk-free asset has zero return variance over the chosen horizon. Its correlation with a risky asset is undefined because the denominator contains a zero standard deviation; its covariance with that asset is zero.

Worked covariance conversion

Let asset A have volatility 20%, asset B volatility 10%, and correlation 0.25. Their covariance is

$$\sigma_{AB}=0.25(0.20)(0.10)=0.005.$$

This covariance, rather than the correlation number alone, enters the portfolio variance calculation.

4. Historical returns and reported anomalies

Many long-run historical datasets show equities with higher average returns and volatility than short-term government bills. That observation does not establish that every risky security must have a higher expected return, or that equities will outperform over every horizon. Taking diversifiable risk alone need not earn a premium in standard equilibrium models.

A bill held to maturity can approximate a nominal risk-free investment in its currency over its own horizon. Inflation makes real purchasing power uncertain, early sale can introduce price risk, and rolling short bills creates reinvestment uncertainty.

When comparing historical assets, specify dates, market, currency, reinvested income, fees, inflation, and arithmetic versus geometric averages. Avoid mixing a price-only stock series with a total-return bond series.

The course also discusses historical patterns that motivate tests of simple asset-pricing models:

Pattern	Historical finding or hypothesis to investigate
Size	Smaller firms have outperformed larger firms in some samples
January	Some samples show unusually high January returns, particularly among smaller stocks
Value	High book-to-market, or low price-to-book, portfolios have outperformed in some samples
Momentum	Recent winners have tended to outperform recent losers over intermediate horizons
Accruals	High-accrual firms have shown lower subsequent returns than low-accrual firms in studied samples
Issuance	Equity-issuing firms have shown subsequent underperformance relative to selected benchmarks in some studies

These are not universal current facts or guaranteed opportunities. Risk compensation, behavior, transaction costs, illiquidity, selection bias, and data mining are possible explanations. Tests of efficiency also depend on the expected-return model used as a benchmark.

Momentum is often constructed from prior months 2–12, omitting the latest month rather than using every trailing-year definition interchangeably. The Kenneth French momentum documentation specifies this research convention. Research factors may involve long and short portfolios and may not be directly investable after costs.

5. Lecture 13: portfolio mean and variance

With weights fixed at the start of a single holding period:

$$R_P=\sum_{i=1}^{N}w_iR_i,\qquad \sum_iw_i=1.$$

A long-only fully invested portfolio has $w_i\ge0$. Short positions have negative weights; borrowing can finance risky weights above 100% if a corresponding negative cash weight is included. A zero-net-investment trade should be described with dollar positions rather than percentages of zero capital.

Linearity of expectation gives

$$\mu_P=\sum_iw_i\mu_i=\mathbf{w}^{\mathsf T}\boldsymbol\mu.$$

Variance instead depends on all co-movements:

$$\sigma_P^2=\sum_iw_i^2\sigma_i^2 +2\sum_{i<j}w_iw_j\sigma_{ij} =\mathbf{w}^{\mathsf T}\Sigma\mathbf{w}.$$

The equivalent double sum over all $i\ne j$ already counts each distinct pair twice; do not add another factor of two to it. For two assets:

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

Worked two-asset example

Take $\mu_A=10\%$, $\sigma_A=20\%$, $\mu_B=6\%$, and $\sigma_B=10\%$. With 50% in each and correlation 0.25:

$$\mu_P=0.5(0.10)+0.5(0.06)=8\%,$$ $$\sigma_P^2=0.25(0.04)+0.25(0.01)+2(0.5)(0.5)(0.005) =0.015,$$

so volatility is approximately 12.2474%, below the weighted average standalone volatility of 15%.

Holding the same weights and standalone volatilities:

Correlation	Portfolio volatility
-1	5.00%
0	11.18%
0.25	12.25%
1	15.00%

Correlation -1 does not automatically produce zero risk at arbitrary weights. In this example, weights $1/3$ in A and $2/3$ in B exactly offset the return deviations when correlation is -1.

6. What diversification can and cannot remove

For nonnegative weights summing to one,

$$\sigma_P\le\sum_iw_i\sigma_i.$$

The inequality is strict for a genuinely mixed portfolio with positive-volatility exposures that are not all perfectly positively correlated. With short positions, the corresponding general bound uses absolute weights; the simple long-only statement should not be applied unchanged.

Why covariances matter in large portfolios

For $N$ equally weighted assets with the same variance $\sigma^2$ and common pairwise correlation $\rho$:

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

For a fixed nonnegative $\rho$, as $N$ becomes large, variance approaches $\rho\sigma^2$. The standalone contribution shrinks while the common covariance contribution remains. “Covariances dominate” describes this kind of broadly spread portfolio, not every possible set of weights or covariance matrix.

If individual volatility is 30% and correlation is 0.20:

Number of equal-weight assets	Portfolio volatility
1	30.00%
10	15.87%
100	13.68%
Very large	Approaches 13.42%

Adding names has diminishing benefits when they share common exposures. Fixed negative pairwise correlation cannot be extended to arbitrarily many assets: a valid equal-correlation matrix requires $\rho\ge-1/(N-1)$.

Systematic versus idiosyncratic components

A simple one-factor model writes

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

If residuals are uncorrelated with the market and with one another, and individual positions remain small, their portfolio contribution can approach zero as diversification increases. Economy-wide market exposure remains unless separately hedged or reduced.

This result is conditional. Concentration, correlated residuals, sector exposures, and changing correlations can leave substantial residual risk. Diversification reduces company-specific exposure; it does not make every finite portfolio safe, and it does not guarantee stable correlations during crises.

7. Minimum variance and the efficient frontier

For a target expected return $\mu^*$, mean–variance optimization solves

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

Additional constraints can prohibit short sales, bound exposures, or limit turnover. Repeating this across feasible targets traces the minimum-variance boundary. Its upper branch, starting at the global minimum-variance portfolio, is the efficient frontier: portfolios below it are dominated by an attainable portfolio with better mean–variance characteristics.

The familiar bullet-shaped region describes a standard nondegenerate multi-asset setting. With only two risky assets, fully invested weights trace a curve rather than a filled region, and constraints alter the shape.

Worked example: the lowest-risk two-asset mix

For two assets, unconstrained global minimum-variance weight in A is

$$w_A^{GMV}=\frac{\sigma_B^2-\sigma_{AB}} {\sigma_A^2+\sigma_B^2-2\sigma_{AB}}.$$

Now use the earlier A and B means and volatilities but set their correlation to zero for the rest of this lesson. Then covariance is zero:

$$w_A^{GMV}=\frac{0.01}{0.04+0.01}=0.20,\qquad w_B^{GMV}=0.80.$$

Expected return is $0.2(10\%)+0.8(6\%)=6.8\%$. Variance is $0.2^2(0.04)+0.8^2(0.01)=0.008$, giving volatility 8.9443%.

Minimum risk is not the same as maximum return, maximum Sharpe ratio, or the best choice for every investor’s liabilities and preferences.

8. Add a risk-free asset: the tangency portfolio

Let $x$ be the wealth fraction invested in a risky portfolio T, with $1-x$ in a risk-free asset. Then

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

For $x\ge0$, these combinations form a straight capital allocation line. Its slope is the risky portfolio’s Sharpe ratio:

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

The tangency portfolio maximizes this slope under the assumed opportunity set. With nonsingular covariance, unconstrained risky positions, and an appropriate positive normalization:

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

The expression requires the denominator to be nonzero and the usual well-posed positive-premium setting. With long-only or other constraints, solve the constrained optimization instead of applying this formula mechanically.

Worked tangency portfolio

Continue with uncorrelated A and B and set $r_f=2\%$. The unnormalized risky allocations are

$$\frac{\mu_A-r_f}{\sigma_A^2}=\frac{0.08}{0.04}=2, \qquad \frac{\mu_B-r_f}{\sigma_B^2}=\frac{0.04}{0.01}=4.$$

Normalize to get $w_A=1/3$ and $w_B=2/3$. Then

$$\mu_T=7.3333\%,\qquad \sigma_T=\sqrt{(1/3)^2(0.04)+(2/3)^2(0.01)}\approx9.4281\%,$$ $$\operatorname{SR}_T=\frac{0.073333\ldots-0.02}{0.0942809\ldots}\approx0.5657.$$

Risky allocation	Expected return	Volatility	Sharpe ratio at 2% risk-free rate
20% A / 80% B: global minimum variance	6.80%	8.94%	0.5367
One-third A / two-thirds B: tangency	7.33%	9.43%	0.5657
50% A / 50% B	8.00%	11.18%	0.5367

The highest-return row is not the highest-Sharpe row. The lower-volatility row is not the tangency portfolio either.

Choosing the total risk level

Putting half of total wealth in T and half in the risk-free asset gives mean 4.6667% and volatility 4.7140%. Putting 150% in T, financed by borrowing 50% at the assumed same 2% rate, gives mean 10% and volatility 14.1421%.

Leverage scales losses as well as gains. Real borrowing rates, margin constraints, and funding availability can differ from the idealized lending opportunity.

9. Two-fund separation is a conditional theorem

Within the standard mean–variance model, investors facing the same expected returns, covariance matrix, investable assets, and common risk-free borrowing/lending rate choose the same tangency risky portfolio. Their risk aversion changes the split between that portfolio and the risk-free asset.

One convenient objective is

$$\max_x\ \mu_C-\frac{\gamma}{2}\sigma_C^2,$$

where $\gamma>0$ is risk aversion. Differentiating gives

$$x^*=\frac{\mu_T-r_f}{\gamma\sigma_T^2}.$$

For the example, $\gamma=6$ gives $x^*=1$: all wealth in T. With $\gamma=12$, $x^*=0.5$. With $\gamma=4$, $x^*=1.5$, assuming borrowing is allowed at the same risk-free rate.

This does not establish that every rational real-world investor should hold identical assets. Heterogeneous forecasts, taxes, constraints, liabilities, labor income, horizons, and preferences beyond mean and variance can change the optimal risky allocation. Mean–variance preferences are an assumption, often motivated by quadratic utility or suitable return-distribution conditions.

A capital allocation line becomes the capital market line when its tangency portfolio is identified with the market portfolio under additional equilibrium assumptions. Optimization alone does not establish that identification.

10. Applying the framework responsibly

A practical analysis should specify the horizon and objective, build consistent total-return observations, estimate means and a valid covariance matrix, define constraints, and test sensitivity. Re-estimate after accounting for costs and check performance on information not used to fit the allocation.

Expected-return errors can make optimized weights unstable, especially when assets are highly correlated. Regularization, shrinkage, exposure limits, and simple diversified benchmarks can help assess robustness. A high backtested Sharpe ratio is not proof of a high future Sharpe ratio.

Volatility is only one lens. Drawdowns, liquidity demands, leverage, inflation exposure, and asymmetric losses can matter even when two portfolios have the same expected return and variance.

11. Practice questions

1. An asset returns 15% or -5%, each with probability one-half. Find mean, variance, and standard deviation.
2. A holding earns +25% followed by -20%. Find its arithmetic average and annual geometric return.
3. Asset A has volatility 30%, asset B 20%, and correlation 0.4. Find covariance and the volatility of a 50/50 portfolio.
4. Two uncorrelated assets have means 12% and 7%, and volatilities 20% and 10%. Find the global minimum-variance weights and their portfolio mean and volatility.
5. With 25 equal-weight assets, each at 20% volatility and common correlation 0.1, find portfolio volatility and its large-$N$ limit.
6. A risky portfolio has expected return 9%, volatility 15%, and risk-free rate 3%. Find Sharpe ratio, and the mean and volatility of a 60% risky / 40% risk-free mix.
7. Using the tangency example in Section 8, how much total wealth goes into A, B, and the risk-free asset when $x=0.5$?
8. Why can two investors rationally choose different risky portfolios despite two-fund separation? Does a January pattern prove an exploitable inefficiency?

12. Answers and reasoning

1. Mean is 5%. Deviations are +10 and -10 percentage points, so variance is 0.01 and standard deviation is 10%.
2. Arithmetic average is 2.5%. Compound wealth multiplier is $1.25(0.80)=1$, so annual geometric return is zero.
3. Covariance is $0.4(0.30)(0.20)=0.024$. Portfolio variance is $0.25(0.09)+0.25(0.04)+0.5(0.024)=0.0445$; volatility is approximately 21.0950%.
4. Minimum-variance weights are 20% and 80%. Mean is $0.2(12\%)+0.8(7\%)=8\%$, and volatility is $\sqrt{0.008}\approx8.9443\%$.
5. Variance is $0.04[0.1+0.9/25]=0.00544$, so volatility is approximately 7.3756%. Its limit is $0.20\sqrt{0.1}\approx6.3246\%$.
6. Sharpe ratio is $(0.09-0.03)/0.15=0.4$. Mixed expected return is $0.03+0.6(0.06)=6.6\%$, and volatility is $0.6(15\%)=9\%$.
7. A receives $0.5(1/3)=16.6667\%$, B receives $0.5(2/3)=33.3333\%$, and the risk-free asset receives 50% of total wealth.
8. Different beliefs, constraints, liabilities, or preferences violate the common-opportunity-set assumptions. A calendar pattern must survive an appropriate risk benchmark, costs, data-bias checks, and out-of-sample testing before supporting an inefficiency claim.

Further reading

• Andrew W. Lo, MIT OpenCourseWare, Risk and Return: statistical foundations and historical patterns.
• Andrew W. Lo, MIT OpenCourseWare, Portfolio Theory: the lecture sequence on efficient portfolios and tangency.
• MIT OpenCourseWare, Risk Analytics and Portfolio Theory slides: the course’s Lectures 13–14 source material.
• Kenneth R. French, Data Library and Momentum Factor Construction: historical research portfolios and their definitions.

Feynman study lab

Explain it simply

Expected return describes an average over possible outcomes, while risk describes uncertainty around it.

Concrete example. An investment returning +20% or −10% with equal probability has expected return 5%.

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:

Does a 5% expected return mean an investor will receive 5% in either outcome?

Check your explanation after you try

No. The two realized outcomes are +20% and −10%. Expected returns also differ from compounded long-run growth.

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.