What you will learn
- Plot any portfolio using its expected return and volatility
- Explain why portfolio return is a weighted average but portfolio risk is not
- Use the two-asset volatility formula with correlation
- See how lower correlation lowers portfolio risk
In the last lesson you learned that a portfolio's return is a simple weighted average of its holdings. If risk worked the same way, this unit would be short and boring. But it does not, and that single fact is the reason portfolio theory exists. This lesson shows why a portfolio's risk is almost always less than the weighted average of its parts, which is the math behind the free lunch.
The two axes, one more time
From Unit 5, every investment lives at a point defined by two numbers. Reward is the expected return, the average outcome you anticipate. Risk is the volatility, the standard deviation of returns, which measures how far outcomes tend to stray from that average. A whole portfolio is also just a point in this return-versus-risk space, and the job of portfolio management is to move that point in a good direction: more return, less risk, or both.
Return: the easy half
As you saw, a portfolio's expected return is the weighted average of its holdings' expected returns. Put 60 percent in something expected to return 10 percent and 40 percent in something expected to return 5 percent, and the portfolio expects 0.6 times 10 plus 0.4 times 5, which is 8 percent. Nothing surprising happens. The portfolio return always lands neatly between the returns of its parts.
Risk: the surprising half
Here is the part that makes everything work. A portfolio's volatility is almost never the weighted average of its holdings' volatilities. It is usually less, and sometimes far less. The reason is correlation, from Unit 5. When two assets are not perfectly correlated, they do not hit their highs and lows at the same time, so their swings partially cancel when combined. The combined portfolio is therefore steadier than a simple average of the pieces would suggest. That gap, between the weighted-average volatility and the actual lower portfolio volatility, is the diversification benefit measured exactly.
Expected return and standard deviation of a portfolio (Spoon Feed Me)
Shows how a portfolio's expected return and its standard deviation are computed. Notice the risk calculation is not a simple weighted average, which is exactly this lesson's point.
Key terms
- Portfolio volatility
- The standard deviation of the whole portfolio's returns, its total risk.
- Weighted-average volatility
- What you would naively get by averaging the assets' volatilities by weight. Portfolio risk is usually below this.
- Correlation (ρ)
- How closely two assets move together, from -1 to +1. Lower correlation means more diversification benefit.
The two-asset risk formula
- w1, w2 = the weights
- σ1, σ2 = the two volatilities
- ρ = the correlation between them
This is exactly the portfolio variance formula from Unit 5, with the covariance written in its correlation form, since covariance equals ρ·σ1·σ2. The portfolio volatility is the square root of that variance. The whole diversification story lives in that last term: a smaller ρ shrinks it, which shrinks the portfolio's risk.
Diversification with positive correlation
Two assets each have a volatility of 10 percent (so σ = 0.10). You hold them equally (w1 = w2 = 0.5). Their correlation is ρ = 0.5. What is the portfolio's volatility, and how does it compare to the weighted-average volatility of 10 percent?
- First two terms. 0.5² times 0.10² is 0.25 times 0.01, which is 0.0025. The second asset gives another 0.0025.
- The correlation term. 2 times 0.5 times 0.5 times 0.5 times 0.10 times 0.10 is 0.0025.
- Add for the variance. 0.0025 plus 0.0025 plus 0.0025 is 0.0075.
- Square root for volatility. The square root of 0.0075 is about 0.0866, or 8.66 percent.
Why it matters: Even with a positive correlation of 0.5, combining the two assets cut risk from 10 percent to 8.66 percent with no loss of expected return. That reduction is diversification, and it happens for any correlation below 1.
Portfolio variance with zero correlation
Take the same two assets (each σ = 0.10, equal weights 0.5), but now their correlation is ρ = 0. Using w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2, what is the portfolio variance?
A portfolio's return is a plain average of its parts. Its risk is not, and that difference is the only free lunch in finance.
Systematic vs unsystematic risk (Edspira)
A portfolio's total risk splits into market-wide risk and company-specific risk. This decomposition, which diversification acts on, runs through the rest of the unit.
The one case with no benefit
There is exactly one situation where portfolio risk does equal the weighted average of the parts: when the assets are perfectly correlated, with ρ equal to 1. Then they move in perfect lockstep, nothing cancels, and combining them buys you nothing. In the example above, setting ρ to 1 would push the variance to 0.01 and the volatility right back to 10 percent, the weighted average. The takeaway: diversification pays off to the extent that your assets are less than perfectly correlated, and it pays nothing when they move together.
Predict the risk
You combine two assets with equal volatility and equal weights. Which correlation gives the LOWEST portfolio risk?
The lower the correlation, the more the assets' swings cancel, so ρ = -1 gives the lowest portfolio risk. Diversification benefit grows as correlation falls.Explain the free lunch mathematically
In your own words, explain why a portfolio's risk is usually lower than the weighted average of its holdings' risks, referring to correlation.
Write an answer before comparing it with the model response.
Model answer
The portfolio return is a straight weighted average, but the risk formula has an extra term that depends on the correlation between the assets. Unless the assets are perfectly correlated, they do not reach their highs and lows at the same time, so when one is down another is often up and their swings partially cancel. That cancellation shows up as a smaller correlation term in the variance formula, pulling the portfolio's volatility below the weighted average of the individual volatilities. The lower the correlation, the more cancellation and the bigger the risk reduction, all without lowering the expected return, which is why it is called a free lunch.
You now understand the central fact of portfolio theory: combining imperfectly correlated assets lowers risk for free. The next lesson digs deeper into diversification itself: how correlation and the number of holdings drive it, and the floor it can never cross.