FINM7008 Chap.6 Two Risky Assets and the Optimal Risky Portfolio
Two Risky Assets and the Optimal Risky Portfolio
Correlation decides the shape
With two risky funds, the portfolio mean is a weighted average but the variance includes a covariance term.
Perfect positive correlation gives a straight line, perfect negative correlation a broken line through zero risk, and anything in between a curve that bulges left.
Tangency and the complete portfolio
The chapter finds the minimum-variance mix, the tangency portfolio with the highest Sharpe ratio, and the investor's complete portfolio of bills and both funds.
The lecture's timed question
The Week 3 lecture sets an assessment-style problem in which a second, perfectly negatively correlated asset is added to a one-asset portfolio while risk is held fixed.
Questions like it test whether you keep the absolute value in the standard deviation and read the constraint correctly, and the chapter practises one with new numbers.
What this chapter covers
- 01
Portfolio mean and variance
- 02
Correlation between minus one and one
- 03
Perfect positive correlation
- 04
Perfect negative correlation and zero risk
- 05
The minimum-variance portfolio
- 06
Tangency weights in excess returns
- 07
From P to the complete portfolio
Worked example · free
The minimum-variance mix of a bond fund and a stock fund
- 1Covariance: 0.2 × 0.24 × 0.10 = 0.0048.
- 1Bond weight: (0.0576 − 0.0048) / (0.0576 + 0.01 − 2 × 0.0048) = 0.0528 / 0.058 = 0.910.
- 1Mean 0.910 × 6% + 0.090 × 14% = 6.72%; risk 9.76%, below the bond fund's own 10%.
Key terms
- Correlation Coefficient
- A number between minus one and one measuring how closely two returns move together.
- Covariance
- The expected product of two returns' deviations from their means, equal to correlation times both standard deviations.
- Tangency Portfolio
- The risky portfolio whose capital allocation line from the risk-free rate is steepest.
- Opportunity Set
- All the mean and standard deviation pairs attainable by mixing the available assets.
- Perfect Hedge
- A mix of perfectly negatively correlated assets weighted so that portfolio risk is zero.
- Minimum-Variance Portfolio
- The mix of the two funds with the lowest possible standard deviation.
Two Risky Assets and the Optimal Risky Portfolio FAQ
Why does a lower correlation reduce portfolio risk?
Portfolio variance includes twice the weights times the covariance, and the covariance equals correlation times the two standard deviations. A lower correlation shrinks that term, so the same holdings produce less variation in the combined return.
Does the optimal risky portfolio depend on risk aversion?
No. Investors with the same inputs hold the same tangency portfolio, because it maximises the Sharpe ratio whatever their preferences. Risk aversion only decides how much of their wealth goes into it and how much stays in bills.
Why use excess returns in the tangency formula?
The tangency portfolio maximises the slope of a line from the risk-free rate, and that slope is the excess return per unit of risk. Writing the weights in excess returns over the risk-free rate builds the slope directly into the formula.
Can an asset with a lower return and higher risk be worth holding?
Yes, if its correlation with the rest of the portfolio is low enough. A small position can then reduce total risk. With a correlation of one it would be dominated, and nobody would hold it at that price.
What does a correlation of minus one allow?
It allows the two assets' risks to offset completely. Weighting each asset by the other's standard deviation produces a portfolio with zero standard deviation, which in equilibrium must earn the risk-free rate, since a riskless portfolio cannot offer more than a riskless bill.
How do I find the complete portfolio from the tangency portfolio?
Compute y* from the tangency portfolio's risk premium and variance and the investor's A, then multiply each fund's weight inside the tangency portfolio by y*. Whatever is left, one minus y*, goes into bills.
Why does the top-down process build the risky portfolio first?
Because the best risky portfolio does not depend on the investor. Once it is found from the opportunity set and the risk-free rate, each investor's risk aversion only sets how much to hold, so the same technical step serves everyone.
Exam move
Draw the three correlation cases from memory, then work one tangency problem fully: covariance, weights, mean, standard deviation and Sharpe ratio. Finish by scaling P into a complete portfolio for a given A. Keep a one-line check for each answer: the weights sum to one and the tangency Sharpe ratio beats both funds on their own.
Rehearse the timed two-asset question with a stopwatch, since the lecture sets thirty minutes for its own version and the method has several steps.
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