5.3Lesson 5.3 · Module 5 — Risk and Return

Covariance, Correlation, and Diversification

How assets combine into portfolios, what covariance and correlation measure, and why diversification reduces risk.

Portfolio return = weighted average
Portfolio volatility is NOT a weighted average
Correlation determines diversification benefit
Two risky assets can form a safer portfolio
Diversification reduces idiosyncratic but not systematic risk

Start

Learning objectives

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

1Compute portfolio weights and portfolio returns as weighted averages.
2Explain why portfolio volatility is not a simple weighted average of asset volatilities.
3Define covariance and correlation and interpret their signs and magnitudes.
4Compute two-asset portfolio volatility using the variance formula.
5Show how lower correlation increases the diversification benefit.
6Distinguish diversifiable (idiosyncratic) risk from systematic risk.

5.3.1Section 1

Portfolio weights

A portfolio is a collection of assets. Each asset's weight is the fraction of total wealth invested in it. Weights always sum to 1 (or 100%).

Portfolio weight

w_{i} = \frac{value _{i}}{total value}

Each weight is the dollar amount in asset i divided by the total portfolio value. The weights across all assets sum to 1.

$value_{i}$: Dollar amount invested in asset i.
$total value$: Sum of all asset values.

w_{A} = \frac{60 , 000}{100 , 000} = 0.60, w_{B} = \frac{40 , 000}{100 , 000} = 0.40

$w_{A}$ + $w_{B}$ = 60% + 40% = 100%

5.3.2Section 2

Portfolio return

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

The portfolio return is a weighted average of the asset returns, using the portfolio weights.

$w_{A}, w_{B}$: Weights on assets A and B.
$R_{A}, R_{B}$: Returns on assets A and B.

R_{P} = 0.60 (10%) + 0.40 (5%) = 6% + 2% = 8%

Portfolio return = 8%

The same weighted-average logic applies to expected returns: E[ $R_{P}$ ] = $w_{A}$ E[ $R_{A}$ ] + $w_{B}$ E[ $R_{B}$ ]. Returns combine linearly.

Try itPortfolio return worksheet

Compute a weighted-average return

You hold w_A = 60% in stock A (return 12%) and w_B = 40% in asset B (return 4%). What is the portfolio return?

Contribution of stock A (w_A × R_A)

Attempt 1 of 3

Contribution of asset B (w_B × R_B)

Attempt 1 of 3

Portfolio return R_P

Attempt 1 of 3

5.3.3Section 3

Why portfolio risk is different

Expected return is a weighted average. But portfolio volatility is not. Here is the key two-state example. Stock A does well in expansions and poorly in recessions; asset B (think bonds) does the opposite.

State	Stock A	Asset B	50/50 portfolio
Expansion	+30%	−2%	0.5×30% + 0.5×(−2%) = 14%
Recession	−10%	+18%	0.5×(−10%) + 0.5×18% = 4%

Stock A alone swings between −10% and +30% — a 40-point range. The 50/50 portfolio swings only between 4% and 14% — a 10-point range. The two assets partially offset each other, smoothing the portfolio's outcomes. That offset is the essence of diversification, and it is driven by how the assets co-move.

5.3.4Section 4

Covariance

Cov (R_{A}, R_{B}) = E [(R_{A} - E [R_{A}]) (R_{B} - E [R_{B}])]

Covariance measures how two assets move together. It averages the product of each asset's deviation from its own mean.

$R_{A} - E [R_{A}]$: Surprise in asset A's return.
$R_{B} - E [R_{B}]$: Surprise in asset B's return.

A positive covariance means the assets tend to move in the same direction; negative means opposite directions; near zero means little linear co-movement. But covariance depends on units, making it hard to interpret on its own — which motivates correlation.

5.3.5Section 5

Correlation

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

Correlation standardizes covariance by dividing by the product of the two standard deviations, pinning it to the range [−1, 1].

$σ_{A}, σ_{B}$: Standard deviations of assets A and B.
$ρ_{A, B}$: Correlation, always between −1 and +1.

ρ = +1 means perfect positive co-movement; ρ = −1 means perfect opposite movement; ρ = 0 means no linear relationship. The lower the correlation, the more diversification two assets provide.

Try itCorrelation explorer

See how correlation shapes the scatter

Each plot shows returns of asset A (x-axis) against asset B (y-axis). Click a correlation level to see how the cloud of points changes.

Strong positive — assets move mostly together.

As ρ moves from +1 toward −1, the cloud rotates from a tight upward line to a loose disk to a downward line. The tighter the cloud hugs a line, the stronger the linear co-movement — and the smaller the diversification benefit.

5.3.6Section 6

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 terms: each asset's weighted variance, plus a co-movement term. The third term is where diversification lives.

$w_{A}^{2} σ_{A}^{2}$: Weighted variance of asset A.
$w_{B}^{2} σ_{B}^{2}$: Weighted variance of asset B.
$2 w_{A} w_{B} ρ_{A, B} σ_{A} σ_{B}$: Co-movement (covariance) term — the diversification channel.

When ρ is less than 1, the third term is smaller than it would be for perfectly correlated assets, so portfolio variance falls below the weighted-average variance. That reduction is the diversification benefit.

5.3.7Section 7

Correlation comparison

Hold the weights at 50/50 and the volatilities at σ_A = σ_B = 20%. Only the correlation changes. Watch what happens to portfolio volatility — while expected return stays fixed.

Try itCorrelation comparison

w_A = w_B = 50%, σ_A = σ_B = 20%

Compute σ_P for each correlation

Expected return is fixed — only correlation changes. Compute the portfolio volatility for each case using the two-asset variance formula. Each field reveals its answer after attempts.

Portfolio volatility σ_P when ρ = 1

Attempt 1 of 3

Portfolio volatility σ_P when ρ = 0

Attempt 1 of 3

Portfolio volatility σ_P when ρ = −0.5

Attempt 1 of 3

Portfolio volatility σ_P when ρ = −1

Attempt 1 of 3

5.3.8Section 8

A realistic portfolio example

Now a realistic case: w_A = 60%, w_B = 40%, E[R_A] = 8%, E[R_B] = 12%, σ_A = 12%, σ_B = 18%, and ρ = 0.25. The portfolio's expected return is a weighted average, but its volatility must come from the variance formula.

Expected return

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

0.60 (8%) + 0.40 (12%) = 4.8% + 4.8% = 9.6%

E[ $R_{P}$ ] = 9.6%

Portfolio volatility

σ_{P} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ σ_{A} σ_{B}

σ_{P}^{2} = 0.005184 + 0.005184 + 0.002592 \approx 0.01296

σ_P ≈ 11.38%

σ_P ≈ 11.38% is below both σ_A (12%) and σ_B (18%). The portfolio is safer than either asset held alone — the payoff of imperfect correlation.

Try itTwo-asset portfolio worksheet

w_A = 60%, w_B = 40%, σ_A = 12%, σ_B = 18%, ρ = 0.25

Build the portfolio variance step by step

Enter expected returns as percents (e.g. 9.6) and the variance terms as decimals (e.g. 0.0052). Work each piece, then assemble the total.

Expected return E[R_P] (percent)

w_A² σ_A² (decimal)

w_B² σ_B² (decimal)

2 w_A w_B ρ σ_A σ_B (decimal)

Total portfolio variance σ_P² (decimal)

Portfolio standard deviation σ_P (percent)

5.3.9Section 9

Diversification is not just the number of stocks

Owning many stocks does not automatically mean you are diversified. What matters is whether those stocks are exposed to distinct risk sources.

Not diversified

Twenty semiconductor companies all share the same demand cycle, the same supply chain, and the same technology risk. When chip demand falls, they fall together.

Also not diversified

Twenty regional banks all depend on the same interest-rate environment and the same local credit conditions. A rate shock hits all of them at once.

Diversification comes from combining assets whose risks are genuinely different — not from stacking many tickers that share the same exposures.

5.3.10Section 10

Diversifiable versus systematic risk

Definition · Diversifiable (idiosyncratic) risk

Risk specific to a single company: product failure, fraud, a factory accident, a lawsuit. Because it is company-specific, holding many unrelated assets drives this risk toward zero.

Definition · Systematic (market) risk

Risk that affects all assets: recession, inflation, rate shocks, financial crises. Diversification cannot eliminate it because no asset is immune.

Diversification reduces company-specific risk, but it cannot eliminate market-wide risk. A portfolio of fifty stocks still falls when the whole market crashes. The risk that remains after full diversification is the systematic risk — and it is the only risk that earns a premium.

5.3.11Section 11

Limitations of correlation

Historical and estimated. Correlation is measured from past data and is itself an estimate, not a fixed constant.
Can change over time. Correlations shift as industries, economies, and market structures evolve.
Linear only. Correlation captures straight-line co-movement. It can miss nonlinear dependencies (e.g. assets that crash together but rise independently).
Rises in crises. Assets that looked uncorrelated in calm periods often move together in panics — exactly when diversification is needed most.
Low correlation is not enough. A low-correlation asset with poor expected return or high standalone risk can still be a bad addition to a portfolio.

5.3.QACommon questions

Diversification Q&A

Try itConcept check

Pulling it together

A portfolio has w_A = 60%, w_B = 40%, E[R_A] = 8%, E[R_B] = 12%, σ_A = 12%, σ_B = 18%, ρ = 0.25. Compute the expected return and the volatility.

Portfolio expected return E[R_P]

Attempt 1 of 3

Portfolio volatility σ_P

Attempt 1 of 3

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
$60K in A and $40K in B, total $100K. What is $w_{A}$ ?
02
$w_{A}$ = 60%, $R_{A}$ = 10%, $w_{B}$ = 40%, $R_{B}$ = 5%. What is the portfolio return?
03
ρ = 1, equal weights, σ_A = σ_B = 20%. What is σ_P?
04
ρ = −1, equal weights, σ_A = σ_B = 20%. What is σ_P?
05
$w_{A}$ = 60%, $w_{B}$ = 40%, σ_A = 12%, σ_B = 18%, ρ = 0.25. What is σ_P (approximately)?
06
Why is portfolio volatility not a simple weighted average of asset volatilities?

Lesson summary

1Portfolio return is a weighted average of asset returns.
2Portfolio volatility is NOT a weighted average — it depends on correlation.
3Covariance measures how two assets co-move. Correlation standardizes it to [−1,1].
4Lower correlation creates more diversification benefit.
5Two risky assets can form a less risky portfolio when ρ < 1.
6Diversification reduces idiosyncratic risk but not systematic risk.
7Diversification depends on distinct risk sources, not just the number of stocks.

Replay this lesson Continue to Systematic Risk and Beta