What it means
The idea starts from a simple observation. Every investment has an expected return, which is the mean, and a spread of possible outcomes around it, which is measured by variance or its square root, standard deviation.
Investors like higher means and dislike higher variance. The surprising insight is that combining assets can reduce risk without giving up all the return.
If two assets do not move perfectly together, a loss on one is partly offset by a gain on the other, so the portfolio's variance is lower than the weighted average of the parts. This benefit is called diversification.
Mean-variance analysis uses three inputs for each asset: expected return, standard deviation and the correlation with every other asset. Correlation is a number between -1 and 1 that shows how closely two assets move together.
With those inputs, an analyst can plot every possible mix and find those that offer the best return for each level of risk, known as the efficient frontier. The output is practical.
A finance team choosing how to split a pension scheme or a cash reserve between shares, bonds and property can see how different mixes behave. The investor then picks a point on the frontier that suits their appetite for risk.
The method has well-known limits. It relies on estimates of future returns and correlations, which are uncertain, and small errors in the inputs can change the recommended mix a lot.
It also treats upside and downside swings as equally bad and assumes returns follow a neat bell-shaped pattern, which real markets do not always do.
In practice
Real-world examples.
Example
A charity with $10,000,000 of reserves wants to split the money between a share fund and a bond fund. The finance committee uses mean-variance analysis to compare mixes and chooses a 50/50 split with a modest risk level. The result is documented in the investment policy.
Example
A founder who has sold her company holds most of her wealth in one technology share. Her adviser shows that adding bonds and property reduces total variance with little loss of expected return. She agrees to sell part of the holding and diversify.
Example
A corporate treasury team has to decide how much of a $50,000,000 pension surplus to place in growth assets. An analyst plots the efficient frontier and shows the board which mixes fall below it. The board rejects those mixes because they take on risk without being paid for it.
Formula
Calculation
Portfolio expected return = w1 x R1 + w2 x R2
Portfolio variance = w1^2 x s1^2 + w2^2 x s2^2 + 2 x w1 x w2 x correlation x s1 x s2
Here w is the weight, R is expected return and s is standard deviation.
Suppose Asset 1 is shares with a weight of 60%, an expected return of 8% and a standard deviation of 20%. Asset 2 is bonds with a weight of 40%, a return of 4% and a standard deviation of 10%. The correlation is 0.25. The expected return is 0.6 x 8 + 0.4 x 4 = 4.8 + 1.6 = 6.4%. The variance is 0.36 x 0.04 + 0.16 x 0.01 + 2 x 0.6 x 0.4 x 0.25 x 0.2 x 0.1 = 0.0144 + 0.0016 + 0.0024 = 0.0184. The standard deviation is the square root of 0.0184, about 13.56%, which is lower than the 60/40 weighted average of 16%.Case study
Seen in the real world.
Alder Point Endowment is an illustrative, fictional fund supporting a small college. It held 90% in shares and 10% in bonds, with expected return of 8.0% and a standard deviation of about 18%.
The investment committee ran a mean-variance analysis and found that moving to 70% shares and 30% bonds cut the standard deviation to about 14% while lowering the expected return only to 7.2%. The trade looked attractive because the endowment had to fund scholarships every year and could not afford deep losses.
The committee adopted the new mix and agreed to review the inputs annually. In this illustrative case the main benefit was not a higher return but a steadier one.
Watch out
Common mistakes.
- Treating the estimates as facts, when expected returns and correlations are uncertain and small changes in them can alter the answer a lot.
- Assuming that diversification removes all risk, when it only reduces the part that is specific to individual assets.
- Forgetting that correlations tend to rise in a crisis, so the diversification benefit can shrink when it is needed most.
Questions
People also ask.
What is the efficient frontier?
It is the set of portfolios that offer the highest expected return for each level of risk, so any portfolio below it is taking risk that is not rewarded.
Why use variance as the measure of risk?
It is mathematically convenient and captures how widely returns spread, though it counts upside swings as risk as well.
Who developed this approach?
Harry Markowitz introduced it in the 1950s, and it earned him a Nobel prize in economics.
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