What you will learn
- Explain why diversification reduces risk
- See how correlation drives the benefit of combining assets
- Understand what portfolio variance measures
- Recognize the limits of diversification
Now the statistics you built starts paying off. This lesson is where covariance, correlation, and variance come together into one of the key ideas in investing: diversification. It is often called the only free lunch in finance, because done right it reduces risk without necessarily reducing expected return. Understanding why requires everything you learned about how assets move together.
Diversification (The Plain Bagel)
A clear, grounded explanation of why spreading money across assets lowers risk. Focus on how assets that do not move together smooth the combined result.
Why not putting all your eggs in one basket works
The intuition is old and simple: do not put all your eggs in one basket. If you hold a single stock, your whole outcome rides on that one company. If you hold many different stocks, a disaster at any one of them dents only a small part of your portfolio. But the statistical reason it works is more precise than the proverb. When you combine assets that do not move in perfect lockstep, their ups and downs partially cancel out. When one zigs, another zags, and the combined result is smoother than any single piece. The key quantity controlling this is the correlation between the assets.
Correlation is the engine
Recall correlation runs from negative one to positive one. Combining two assets with a correlation below positive one always reduces risk compared to holding either alone, and the lower the correlation, the greater the benefit. If two assets were perfectly correlated at positive one, combining them would give no risk reduction at all, because they move together anyway. At the other extreme, two assets with a correlation of negative one could in principle be combined to cancel risk almost entirely, since when one falls the other rises. Real assets sit in between, but the lesson holds: the less two assets move together, the more diversification helps. This is exactly why covariance and correlation mattered so much earlier in the unit.
Portfolio variance
Portfolio variance is the formal measure of a portfolio's total risk, and here is the key feature. It does not depend only on the variances of the individual assets. It also depends heavily on the covariances, or correlations, between every pair of assets in the portfolio. This is the mathematical heart of diversification. Because the covariance terms can be small or even negative, the variance of a diversified portfolio can be substantially lower than the average variance of its individual holdings. In other words, the whole can be less risky than the sum of its parts, and it is the pairwise co-movements that make this possible. For a two-asset portfolio, this is captured by a formula you can actually compute, shown below.
The two-asset portfolio variance formula
- w1, w2 = the weights (fraction in each asset)
- σ1², σ2² = the two variances
- Cov(1,2) = covariance between the assets
The first two terms are just the individual risks scaled by how much you hold. The third term is the diversification term, and because a covariance can be small, zero, or negative, it is what pulls the portfolio's total risk down. Since Cov(1,2) also equals the correlation times σ1 times σ2, a lower correlation shrinks that third term and lowers the portfolio variance.
Two 20 percent assets combine into less risk
Two assets each have a standard deviation of 20 percent, meaning a variance of 0.04. You split your money equally, so w1 = w2 = 0.5. Their covariance is 0 (correlation 0). What is the portfolio's standard deviation?
- Scale each variance by the squared weight. 0.5² times 0.04 is 0.25 times 0.04, which is 0.01. The second asset gives another 0.01.
- Add the covariance term. 2 times 0.5 times 0.5 times 0 is 0, because the covariance is zero.
- Add them for the portfolio variance. 0.01 plus 0.01 plus 0 is 0.02.
- Take the square root for the standard deviation. The square root of 0.02 is about 0.141, or 14.1 percent.
Why it matters: Two assets that were each 20 percent risky combine into a portfolio only about 14.1 percent risky, purely because their covariance was zero. That drop from 20 to 14.1 is the diversification benefit made concrete.
Compute a portfolio variance
Two assets each have a variance of 0.04 and equal weights of 0.5. This time their covariance is 0.02. Using w1²σ1² + w2²σ2² + 2·w1·w2·Cov, what is the portfolio variance?
Two stocks, less combined risk
You hold two stocks, each fairly volatile on its own. They operate in unrelated industries, so their returns have a low correlation. What happens to the risk of holding both equally versus holding just one?
- Consider each alone. Each stock swings a lot, so holding just one exposes you to that full swing.
- Combine them. Because their returns are only weakly correlated, on many days one rises while the other falls, partially canceling each other.
- Read the result. The combined portfolio's ups and downs are smoother, so its variance is lower than the average of the two individual variances.
Why it matters: The risk reduction came entirely from the low correlation between them. Same expected return, less risk, is the free lunch.
Which pair diversifies better?
You will add one more stock to a portfolio that currently holds one airline stock. Which candidate gives the most diversification benefit?
The low-correlation stock from an unrelated industry diversifies best. Diversification benefit grows as the correlation between holdings falls.Diversification and risk (Khan Academy)
Connects the risk-reward tradeoff to spreading investments. Reinforces why combining imperfectly correlated assets lowers risk.
The limits of diversification
Diversification is powerful but not unlimited, and this sets up the next lesson. Spreading across many assets can eliminate the risk specific to individual companies, the risk that one firm has a scandal or a bad quarter. But it cannot eliminate the risk that affects the entire market at once, like a recession or a financial crisis, because in those events almost everything falls together and correlations spike toward positive one. This leftover, undiversifiable risk is called systematic or market risk, and it is the reason diversification reduces risk but can never drive it to zero. That split between diversifiable and undiversifiable risk is the subject of an upcoming lesson.
Diversification is the only free lunch in finance, but the kitchen closes in a crisis, when correlations move toward one and almost everything falls together.
Match the correlation to its effect
Explain the free lunch
In your own words, explain why combining assets with low correlation reduces risk without necessarily reducing expected return.
Write an answer before comparing it with the model response.
Model answer
When assets have low correlation, they do not move together, so on many occasions one rises while another falls and their swings partially cancel out. That makes the combined portfolio's returns smoother, which means lower variance and lower risk. The expected return of the portfolio is just the weighted average of the assets' expected returns, so it does not have to fall when I diversify. Getting the same expected return with less risk is why diversification is called a free lunch. It works because the covariances between assets can be low or negative, pulling the portfolio's total variance below the average of the individual variances.
You have seen that risk splits into a part you can diversify away and a part you cannot. The next lesson turns diversification into a full framework for choosing the best possible portfolio: modern portfolio theory.