6.2Lesson 6.2 · Module 6 — Portfolio Theory

Portfolio Risk, Covariance, and Correlation

Why portfolio volatility is not a weighted average. Expand the variance, build the weighted covariance matrix, and explore the opportunity curve.

Variance expansion shows cross-products
Weighted covariance matrix = raw matrix × weights
Factor of 2 from symmetric covariance
Imperfect correlation lowers volatility
Two-asset opportunity curve

Start

Learning objectives

By the end of this lesson, you should be able to:

1Explain why portfolio standard deviation is not a weighted average of individual standard deviations.
2Expand the two-asset portfolio variance and identify the cross-product covariance terms.
3Build a weighted covariance matrix and sum its entries to obtain portfolio variance.
4Apply the two-asset variance formula and interpret the factor of 2.
5Compute portfolio volatility for GM and Motorola using historical estimates.
6Show how correlation affects achievable portfolio volatility, including the opportunity curve.

Notation guide

σ_{P}

portfolio standard deviation (volatility)

σ_{P}^{2}

portfolio variance

Σ

covariance matrix (uppercase sigma)

\sum

summation operator — add a sequence of terms

Lowercase $σ$ is a standard deviation; uppercase $Σ$ is the covariance matrix. They are not interchangeable.

6.2.1Act I · Why weighted-average volatility fails

Same volatility, different portfolio risk

Before any formula, see the problem. Two assets, each with a volatility of 10%. Depending on how they move, a 50/50 portfolio can be very safe or just as risky as either asset alone.

Case A — move together

Each asset alone has $σ = 10%$ . They rise and fall together, so the 50/50 mix swings just as much — portfolio SD stays near 10%.

Case B — offset

Same individual volatilities, but when A rises B tends to fall. Gains and losses partially cancel, so portfolio SD can fall well below 10%.

Same individual volatilities in both cases — different portfolio risk. Portfolio risk depends on each asset's own volatility and how they move together. A weighted average of the individual $σ$ cannot capture that.

6.2.2Act II · Where the covariance terms come from

Expanding the two-asset variance

The cross-product covariance terms are not an accident. They appear because both assets sit inside the same squared expression. Walk through the expansion one step at a time.

Variance expansion

Step 1 · Start from portfolio return

R_{P} = w_{A} R_{A} + w_{B} R_{B}

We begin with the two-asset portfolio return and ask what happens when we measure its spread around the mean.

Two-asset portfolio variance

σ_{P}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ_{A, B} σ_{A} σ_{B}

Portfolio variance has three parts: A's weighted variance, B's weighted variance, and a cross term from their covariance (written here with correlation ρ).

$w_{A}, w_{B}$: Portfolio weights.
$σ_{A}, σ_{B}$: Individual standard deviations.
$ρ_{A, B}$: Correlation between A and B.

The factor of 2 appears because covariance shows up in both the A-B cell and the B-A cell of the weighted matrix. Since they are equal, they combine into 2 × covariance. Then σ_P = √(σ_P²).

Correlation

ρ_{A, B} = \frac{Cov ( R _{A} , R _{B} )}{σ _{A} σ _{B}}

Correlation is covariance scaled into the range [−1, 1]. Rearranging: Cov( $R_{A}$ , $R_{B}$ ) = ρ σ_A σ_B.

$ρ_{A, B}$: Correlation, always between −1 and 1.
$Cov$: Covariance between A and B.

Correlation measures only linear co-movement. It can change — often rising during market stress. Zero correlation is not the same as independence.

6.2.3Act III · Calculate and explore

Raw matrix vs weighted matrix

The expansion produces four terms that fit naturally into a 2×2 matrix. The raw covariance matrix describes the assets. The weighted matrix describes contributions to a specific portfolio.

Raw matrix

Contains relationships between assets. Do not add these directly.

MOT

σ_{A}^{2}

Cov(A,B)

MOT

Cov(B,A)

σ_{B}^{2}

Σ — asset relationships

× w_i w_j

apply weights

Weighted matrix

Contains contributions to this portfolio. Sum every cell to get σ²_P.

MOT

w²ₐ σ²ₐ

wₐw_b Cov

MOT

w_b wₐ Cov

w²_b σ²_b

wᵀΣw — portfolio contributions

Portfolio variance

σ_{P}^{2}

= the sum of all four weighted cells. Add them, then take the square root to recover

σ_{P}

Definition · Portfolio standard deviation (σ_P)

The typical size of portfolio return swings, in percentage points. It is the square root of portfolio variance.

6.2.4Worked example · GM & Motorola

A 75/25 portfolio, step by step

Historical monthly instructional estimates, 1946–2001. These are teaching numbers, not current estimates.

Asset	Mean return	Std dev σ
GM	1.08%	6.23%
Motorola	1.75%	9.73%

Correlation over this period was $ρ = 0.37$ . Build a portfolio of 75% GM and 25% Motorola.

Expected portfolio return

E [R_{P}] = w_{A} E [R_{A}] + w_{B} E [R_{B}]

E [R_{P}] = 0.75 \times 1.08% + 0.25 \times 1.75% = 0.81% + 0.4375%

E[ $R_{P}$ ] ≈ 1.25%

Now the weighted covariance matrix entries (units: percentage-points squared). Combine the two off-diagonal cells, then take the square root only after the variance sum is complete.

Weighted cell	Calculation	Value
GM diagonal	0.75² × 6.23²	21.83
GM × MOT	0.75 × 0.25 × 0.37 × 6.23 × 9.73	4.21
MOT × GM	0.25 × 0.75 × 0.37 × 9.73 × 6.23	4.21
MOT diagonal	0.25² × 9.73²	5.92
Two off-diagonals combine → 2 × 4.21		8.42
Sum = σ²_P (21.83 + 8.42 + 5.92)		36.16

Final outputs

σ_{P} = 36.16 \approx 6.01%

$E [R_{P}] \approx 1.25%$ , $σ_{P} \approx 6.01%$ .

Interpretation

The portfolio volatility of 6.01% sits below Motorola's 9.73% and close to GM's 6.23%. The partial offset from imperfect correlation pulled risk down.

6.2.5Worksheet · Build the weighted matrix

Fill in the weighted covariance matrix

Using GM (σ = 6.23%, E[R] = 1.08%) and Motorola (σ = 9.73%, E[R] = 1.75%), with ρ = 0.37, at a 75/25 mix. Each cell receives row weight × column weight. Work in percentage-points squared.

Weights

GM weight

Motorola weight

Diagonal cells (w² × σ²)

GM diagonal (0.75² × 6.23²)

Motorola diagonal (0.25² × 9.73²)

Off-diagonal cell (wₐ × w_b × ρ × σₐ × σ_b)

Each off-diagonal cell

Sum and outputs

σ²_P (21.83 + 2×4.21 + 5.92)

σ_P = √(σ²_P)

E[R_P] (weighted average)

6.2.6Worksheet · Correlation comparison

How low can correlation drive σ_P?

Two assets, equal weights (50/50), each with σ = 20%. The only thing changing is the correlation. Compute σ_P for each case with the two-asset formula.

σ_P at each correlation

σ_P when ρ = 1

σ_P when ρ = 0

σ_P when ρ = −0.5

σ_P when ρ = −1

6.2.7Exploratory · Opportunity curve

The two-asset opportunity curve

Every possible GM/Motorola mix plots as a point on a risk-return graph. Drag the Motorola weight to move along the curve; change the correlation to reshape it.

Historical monthly instructional estimates, GM and Motorola, 1946–2001 — not current estimates.

Motorola weight: 25%Correlation ρ: 0.37

GM weight

75%

MOT weight

25%

E[R_P]

1.25%

σ_P

6.01%

At ρ = 0.37, the curve bends left — imperfect correlation creates volatility reductions. The current mix has σ_P = 6.01%, below Motorola's 9.73%.

6.2.8Caution · Correlation can change

Diversification estimates are uncertain

The diversification benefit depends on the correlation you assume. But correlations are not fixed.

Regime	ρ	Curve position
Normal	0.37	Bends left — real diversification
Stress	0.80	Shifts right — benefit shrinks

When correlations rise during stress, assets that seemed to diversify begin moving together — exactly when you need the protection most. This does not mean correlations always go to 1, but the direction matters.

6.2.9Final check · Decimals to percentage

A 60/40 portfolio, ρ = 0.25

w_A = 60%, w_B = 40%, σ_A = 12%, σ_B = 18%, ρ = 0.25. Work in decimals (e.g. σ_A = 0.12) so the variance is a clean decimal, then convert σ_P back to a percentage at the end.

Diagonal cells (decimals)

A diagonal (w_A² × σ_A²)

B diagonal (w_B² × σ_B²)

Variance and volatility

Each off-diagonal (wₐ×w_b×ρ×σₐ×σ_b)

Portfolio variance (sum of all 4 cells)

σ_P = √(variance), in percent

6.2.10Optional · Common questions

Questions you may still have

Optional reminders and extensions. The core argument above stays visible; these expand on it.

Try itMastery check

Pass with 4 of 6 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
Why is portfolio SD not a weighted average of individual SDs?
02
In the two-asset formula, why is there a factor of 2 in front of the covariance term?
03
GM/Motorola at 75/25 weights. σ_P ≈ ?
04
ρ = 1, equal weights, σ_A = σ_B = 20%. σ_P?
05
ρ = −1, equal weights, σ_A = σ_B = 20%. σ_P?
06
60/40, σ_A = 12%, σ_B = 18%, ρ = 0.25. σ_P ≈ ?

Lesson summary

1Portfolio variance expands the weighted return expression, producing cross-product covariance terms.
2The weighted covariance matrix applies row and column weights to each cell.
3Portfolio variance = sum of all weighted matrix entries.
4The factor of 2 arises from two symmetric covariance cells combining.
5Imperfect correlation (ρ < 1) can produce portfolio SD below any individual asset SD.
6The two-asset opportunity curve shows achievable risk-return combinations.
7Correlations can change, especially during stress — diversification estimates are uncertain.

Replay this lesson Continue to Diversification Across Many Assets