UTS16657 Chap.5 Portfolio Return, Risk and Diversification
Portfolio Return, Risk and Diversification
Combining assets introduces a third element beyond the individual risks and returns, which the subject calls the portfolio effect. The revelation of modern portfolio theory, developed by Harry Markowitz, is that in combining investments the returns are averaged but the risks are not. A portfolio return is the weighted average of the component returns and must therefore land between them.
A portfolio risk is not the weighted average of the component standard deviations, and the gap between the two is the economic content of this chapter.
Measuring that gap requires a statistic for how two series move together.
Covariance describes the relationship: positive means both tend to be above their means at the same time, negative means one is above when the other is below, and zero means no reliable relationship. Covariance shows that a relationship exists and never why.
Because its magnitude depends on the units of the two series it is unreadable on its own, so it is standardised into a correlation by dividing by the product of the standard deviations, which confines the result to the range between minus one and plus one.
Portfolio variance is then a bookkeeping rule.
Each asset contributes its own variance weighted by the square of its weight, and each distinct pair contributes twice its covariance weighted by the product of the two weights. Two assets give one pair, three give three, five give ten, which is why the calculation is laid out as a covariance matrix rather than written as an equation.
All of the diversification benefit lives in the cross terms, because the own-variance terms are always positive.
A negative correlation makes a cross term subtract outright; a positive but imperfect one still helps, because the term is smaller than it would be at perfect correlation.
How much diversification is needed depends on what you believe about markets, and the phrase that does the real work is unrelated industries.
Five listed property trusts share a single set of drivers, so their correlations stay high and the cross terms stay large, which is why holding count and diversification are not the same thing.
What this chapter covers
- 01
The portfolio return as a plain weighted average
- 02
Why the same trick fails on risk
- 03
Covariance, its sign, and what it does not tell you
- 04
Correlation as covariance standardised into a readable range
- 05
The two-asset and three-asset variance formulas
- 06
The covariance matrix and why it scales where an equation does not
- 07
Unrelated industries, and why holding count is not diversification
A two-asset portfolio, and the diversification benefit stated as a number
- +1Expected return: 0.65 × 7.42 + 0.35 × 5.15 = 4.8230 + 1.8025 = 6.6255%, which lands between the two asset returns as a weighted average always must.
- +1Covariance, because the variance formula needs it and the question gives a correlation: −0.18 × 0.1420 × 0.0680 = −0.0017381.
- +1Own-variance terms: 0.652(0.1420)2 = 0.0085192 and 0.352(0.0680)2 = 0.0005664. Both are necessarily positive.
- +1Cross term: 2 × 0.65 × 0.35 × (−0.0017381) = −0.0007908. Total variance 0.0082948, so the risk is √0.0082948 = 9.1076%.
- +1Compare with the naive blend: 0.65 × 14.20 + 0.35 × 6.80 = 11.61%, so holding the assets together rather than averaging their risks saves 2.5024 percentage points at no cost in return.
Key terms
- Portfolio effect
- What happens to risk and return when more than one investment is held together. Returns average and risks do not, and the difference is what diversification captures.
- Covariance matrix
- A grid whose diagonal cells hold each asset's own variance and whose off-diagonal cells hold the covariance of each pair. Weighting every cell and summing them gives the portfolio variance.
- Diversification benefit
- The amount by which a portfolio's standard deviation falls short of the weighted average of its holdings' standard deviations, measured in percentage points.
- Efficient market hypothesis
- The proposition that share prices already reflect all available information, so a stock picker cannot reliably beat the market. It is what separates passive managers, who accept it, from active managers, who dispute it.
- Weighted average return
- The portfolio return, formed by multiplying each holding's return by its weight and summing. Because it is linear in the weights it always lies between the highest and lowest component return.
Portfolio Return, Risk and Diversification FAQ
Why does a portfolio's risk fall below the average of its components' risks?
Because the variance formula carries cross terms that the simple average does not. Alongside each asset's own variance, weighted by the square of its weight, sits a term for every pair equal to twice the product of the two weights and their covariance. Whenever the assets are less than perfectly correlated that term is smaller than it would be at perfect correlation, and when the correlation is negative it subtracts outright.
Only at a correlation of exactly plus one does the whole expression collapse to the weighted average and diversification deliver nothing.
A client holds twenty different Australian listed property trusts. Are they diversified?
Holding count is not diversification; correlation is. Twenty trusts in one country and one sector face the same interest rate cycle, the same credit conditions, the same construction pipeline and the same domestic demand, so their pairwise correlations are high and the cross terms stay large and positive. The measured risk falls only slightly below the weighted average.
Adding assets whose drivers genuinely differ, such as fixed income, offshore equities or a commodity exposure, does far more for the cross terms than another trust would.
Do I have to know the maths, or can I just use the spreadsheet functions?
Both, for different reasons. The subject is explicit that you will use spreadsheet functions rather than building formulas by hand, and a report submitted with correlations typed in as constants fails the stated requirement that every formula works.
But the final exam is closed book and its short answers ask you to explain why risk diversifies, which you cannot do without knowing what the cross term is and why its sign matters. Know the structure so you can interpret the output, and use the functions so the workbook stays live and auditable.
Exam move
Build the covariance matrix by hand for three assets once, then rebuild it in a spreadsheet with absolute references on the covariance block, and check that the two agree to the seventh decimal. That single exercise is the technical core of the individual report.
For the exam, what is worth rehearsing is not the arithmetic but the sentence: returns average, risks do not, the difference lives in the cross terms, and the cross terms carry the covariance and therefore its sign.
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