UTS16657 Chap.7 The Efficient Frontier and Optimal Weights
The Efficient Frontier and Optimal Weights
Vary the weights of a two-asset portfolio continuously and the resulting risk and return pairs trace a curve. Given the number of listed securities it is effectively impossible to enumerate every feasible portfolio, but it has been proven mathematically that for a large number of securities the set of feasible portfolios is enveloped by a parabola.
Everything achievable lies inside or on that boundary, and the boundary is what portfolio construction is looking for.
Two definitions that sound alike are not. The minimum variance set is the subset of feasible portfolios offering the lowest risk for each level of return, and its left-most point, where risk is as low as it can be made, is the global minimum variance portfolio.
The efficient portfolio is the part of that set which offers the highest return available at each level of risk, effectively the top half starting at the minimum variance point, and the line mapping it out is the efficient frontier.
Every point on the lower arm is dominated: read straight up from it and another portfolio offers the same risk with a higher return, so no rational investor holds there.
The frontier does not name one portfolio.
Different investors target different points on it according to a minimum required return, a maximum allowable risk or constraints limiting exposure to any single asset, and with enough computing power the weights satisfying all of those can be found by iteration.
The curve bows left because the assets are imperfectly correlated, so the visible distance between the curve and the straight chord joining the two assets is the diversification benefit, drawn.
One result is genuinely counter-intuitive and worth stating in a sentence: with a negative correlation the minimum variance portfolio is less risky than either asset held alone, so adding a slice of the volatile asset to the safe one reduces risk while raising return.
Finally, weights are not chosen on the frontier alone: the price to net asset value ratio, sustainability screening and the macroeconomic landscape all enter the decision.
What this chapter covers
- 01
The feasible set and the parabola that envelopes it
- 02
The minimum variance set and the global minimum variance portfolio
- 03
The efficient portfolio and the frontier that maps it
- 04
Why no rational investor holds on the lower arm
- 05
Building the set across weights from 0 to 1 at 0.05 intervals
- 06
The closed-form minimum variance weight, and what the grid approximates
- 07
Price to net asset value, sustainability screening and the macro overlay
Finding the exact minimum variance portfolio, and checking it against the grid
- +1Covariance first: −0.18 × 0.1420 × 0.0680 = −0.0017381.
- +1Numerator of the minimum variance weight: σb2 − σab = 0.004624 − (−0.0017381) = 0.0063621.
- +1Denominator: σa2 + σb2 − 2σab = 0.020164 + 0.004624 + 0.0034762 = 0.0282642.
- +1Weight: 0.0063621 ÷ 0.0282642 = 0.2251, so about 22.51% in the volatile asset and 77.49% in the defensive one, giving a return of 5.661% and a risk of 5.6497%.
- +1Explain the grid. At a weight of 0.20 the risk is 5.6655% and at 0.25 it is 5.6652%, a difference of three ten-thousandths of a point, because the true minimum sits between them. Report the grid minimum, state that the exact minimum lies between the two adjacent weights, and you have answered the question and shown you know what the grid approximates.
Key terms
- Feasible set
- Every risk and return combination attainable from a given group of assets by varying the weights. For a large number of securities it is enveloped by a parabola.
- Minimum variance set
- The subset of feasible portfolios offering the lowest risk at each level of return. Its left-most point is the global minimum variance portfolio.
- Minimum variance point
- The weight combination producing the lowest attainable portfolio risk. Below it, risk rises while return falls, so it marks where the efficient frontier begins.
- Dominated portfolio
- A portfolio for which another exists with the same risk and a higher return. Every point on the lower arm of the minimum variance set is dominated and none is rationally held.
- Price to net asset value
- The traded price of a share or unit divided by the net asset value behind it. Above one the security trades at a premium and below one at a discount.
The Efficient Frontier and Optimal Weights FAQ
Why does the efficient frontier start at the minimum variance point rather than at the safest asset?
Because everything below that point is dominated. Take any portfolio on the lower arm and read straight up: there is another mix with exactly the same risk and a higher return, so nobody rational would hold the lower one. In the worked sweep a portfolio at 15% in the growth asset carries 5.7891% risk for a 5.4905% return, while one at 30% carries almost the same risk at 5.7884% and returns 5.831%.
Same risk, a third of a point more return, purely from sitting on the right arm.
Can a portfolio really be less risky than both of the assets in it?
Yes, whenever the correlation is negative, and it is the most counter-intuitive result in this part of the subject. In the worked case the defensive asset alone carries 6.80% risk while a mix of 22.51% growth and 77.49% defensive carries 5.6497% and returns more.
The mechanism is the cross term in the variance formula: for small weights in the volatile asset the reduction it produces outweighs the addition from that asset's own variance, so total risk falls before it starts to rise.
Is the frontier a forecast?
No, and saying so is worth a mark. Everything on it is conditional on estimates of two standard deviations and a correlation, all measured from history and none of them stable. A correlation estimated over a decade that included a pandemic is not the correlation of the next decade, and correlations between risky assets tend to rise toward one exactly when diversification is most wanted.
Present the frontier as the best available reading of the past rather than as a prediction.
What else decides the weights, if not the frontier?
The subject names three considerations the frontier cannot see. The price to net asset value ratio compares the traded price with the assets behind it, and a persistent discount is the market disagreeing with the valuer. Environmental, social and governance factors are increasingly integrated to surface the strengths, weaknesses, opportunities and threats bearing on performance, with rating providers scoring each listed trust.
And the macroeconomic landscape, along with industry and sector analysis, feeds the weighting decision through the top-down route.
Exam move
Build the twenty-one row sweep in a spreadsheet once, then plot column five against column three and look at the shape until the turning point is obvious by eye. That picture is what Question 2 of the individual report asks you to produce, and doing it once makes the written part far easier because you will have seen the dominated arm rather than read about it.
Keep the closed-form weight in your notes as a check on the grid rather than as a replacement for it.
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