109 Sept 2003 Q2(ii)

[lucky999]

The examiners answer for this says:

"At the point of tangency, the portfolio is a diversified one without risk-free
assets. To the left of the point of tangency, the portfolios will have a mix of
diversified assets and risk-free assets. To the right of the point of tangency,
the portfolios will consist of more than 100% diversified assets, as the investor
would have borrowed at risk-free rate and invested in diversified assets."


BUT in Chapter 6, page 5 of the CMP notes, it says
"All rational investors will hold a combination of the risk-free asset and the portfolio of risky assets at the point of tangency".

So how come the examiner answer does not have the risk-free asset at the point of tangency?

 

[morrisja]

Is there context for that quote from the CMP? I always thought that the tangent portfolio was 100% invested in risky assets.. otherwise you're not at the point of tangency, you're somewhere along the CAL.

 

[Graham Aylott]

With a risk-free asset, the efficient frontier is a straight line in E-sigma space from the risk-free rate of return to a tangency point of the efficient frontier with risky assets only. As this tangency point is by definition part of the opportunity set of risky assets only, it therefore consists only of risky assets.

However, all points along the straight line efficient frontier correspond to a different combination of the risk-free asset and the combination of risky assets at the tangency point. For example, 1/2 way between the risk-free asset and the tangency point corresponds to a portfolio half invested in the risk-free asset and half invested in the combination of risky assets at the tangency point.

The investor will choose the point (the optimal portfolio) along this straight line efficient frontier at which his or her expected utility is maximised. In theory, this will be where one or his or her indifference curves is tangential to the efficient frontier and the exact point will determine the mix of the risk-free asset and the risky assets. However, as all points along the straight line efficient frontier correspond to some combination of the risk-free asset and the risky assets at the tangency point (of the straight line to the risky assets only efficient frontier), the combination of risky assets at the optimal portfolio must itself be some combination of the risk-free asset and the combination of risky assets at that tangency point .