FINM1001 Chap.7 Portfolio Theory
Portfolio Theory
Let covariance determine the diversification gain
Portfolio expected return is a weighted average of component expected returns. Portfolio risk is not a weighted average in general because asset returns may move together or offset one another.
Variance therefore includes the standalone variance contributions and covariance terms linking each pair of assets.
For two assets, each standalone variance is multiplied by the square of its portfolio weight. The covariance term contains both weights, both standard deviations and correlation, multiplied by two.
This decomposition makes each role visible: allocation comes from weights, standalone spread from standard deviations and co-movement from correlation.
Perfect positive correlation removes the usual diversification reduction for positive weights, giving portfolio standard deviation equal to the weighted average of component standard deviations.
Lower correlation reduces the covariance contribution and can lower portfolio risk without changing expected return.
Perfect negative correlation can permit a zero-risk combination in the simplified two-asset setting when weights offset the movements exactly.
Do not put correlation in the return line
Holding weights and component expected returns fixed, changing correlation affects variance but not expected portfolio return. This separation is a useful error check.
If lower correlation raises the calculated expected return, the formulas have been mixed. If correlation is omitted from risk, the calculation cannot represent diversification.
Diversification depends on economic exposures as well as the number of holdings. Many securities driven by one industry or common factor can remain highly correlated.
Combining assets with different drivers can reduce firm-specific variation more effectively, though correlations themselves may rise during stress.
Minimum-variance analysis chooses weights that minimise portfolio variance under the stated inputs. A lower-risk portfolio is not automatically preferred without considering expected return and investor objectives.
Ratios such as excess return per unit of volatility can support comparison, but their interpretation remains conditional on the risk-free rate and estimated moments.
What this chapter covers
- 01
Portfolio weights
- 02
Expected portfolio return
- 03
Covariance
- 04
Correlation
- 05
Diversification
Worked example · free
Measure diversification in two assets
- 1Calculate weighted expected return.
- 1Calculate standalone variance contributions.
- 1Add the covariance term.
- 1Take the square root and compare with weighted volatility.
Key terms
- Portfolio weight
- The proportion of total portfolio value allocated to an asset, used in both expected-return and variance calculations.
- Covariance
- A measure of joint return movement whose sign and magnitude determine whether two assets reinforce or offset portfolio variation.
- Perfect correlation
- A correlation of positive or negative one, representing exact linear co-movement or exact opposing movement in the simplified model.
- Diversification benefit
- The reduction in portfolio risk produced when component returns are not perfectly positively correlated under the chosen weights.
Portfolio Theory FAQ
Why is portfolio return a weighted average?
Expected values combine linearly, so each asset contributes its expected return in proportion to the capital allocated through its portfolio weight, provided the weights describe the full investment.
Why is portfolio risk not usually a weighted average?
Joint movement matters. Covariance terms can reduce or reinforce total variance depending on correlation, so standalone volatilities alone are insufficient for measuring the risk of the combined position.
What happens when correlation falls?
With weights and component moments fixed, the covariance term falls and portfolio variance weakly falls, while expected portfolio return remains unchanged because correlation does not enter its weighted-average equation.
Does owning many assets guarantee diversification?
No. Assets exposed to the same industry or factor can remain highly correlated. Effective diversification depends on distinct return drivers, not only the count of holdings.
Exam move
Practise the return and variance equations on separate lines, then run correlation counter-cases at positive one, zero and a negative value while holding every other input fixed. Stress the relationship as well as the assets. Recompute the same weights under a higher correlation and identify how much diversification disappears. Then change weights while holding correlation fixed to separate allocation from co-movement.
A recommendation should name the investor objective, report expected return and standard deviation together, and describe the input most likely to undermine the result. If short selling is prohibited, confirm every weight remains within the permitted range. If a risk-free asset is introduced, distinguish the allocation among risky assets from the later decision about how much total wealth enters the risky portfolio.
These qualifications turn a variance exercise into a defensible portfolio decision instead of a collection of formula substitutions. Keep the covariance term visibly separate until the last line and inspect its sign before taking the square root. Then compare portfolio variance with both component variances; an unexpected result is a prompt to revisit weights, percentage units or correlation bounds.
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