Modern Portfolio Theory – Part Two

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We continue our journey into the wonderland of alpha, taking up with leverage and the Efficient Frontier.  We documented last time that a mix of a risky portfolio and the risk-free rate (Rf) yields a linear Capital Asset Line (CAL) within risk-return space. When the mix is varied to decrease the amount of Rf, the riskiness and expected return of the portfolio increase.  The mirror image occurs as we increase the relative percentage of Rf in the portfolio mix.  We can even borrow at the risk-free rate to purchase additional risky assets for our portfolio – one form of a practice known as leverage.

We mentioned risk-return space, but what exactly is it? It is a two-dimensional graph upon which the risk and return of individual portfolios is plotted.  Each portfolio has its own unique mix of risky assets, so risk-return space contains the universe of different portfolios.  Now, out of all of these portfolios, some are winners and some are dogs.  The very best portfolios provide the maximum expected return possible for a given level of risk (as measured by standard deviation).  If you were to plot these best portfolios, you would have a bullet-shaped line in risk-return space known as the Efficient Frontier (EF).  [For you set theorists out there, the EF is the intersection of two sets of portfolios: the Minimum Variance and the Maximum Return portfolios].

The shape of the EF is a parabola, which is convex rather than a straight line.  This means that as you take on additional risk, your incremental rate of additional expected return decreases above a certain point. And what point marks the inflection from increasing incremental returns to decreasing ones?  In MPT, it’s the Market Portfolio, which by definition has a beta of one.

The Sharpe Ratio is a formula that calculates the return/risk ratio:

Sharpe Ratio = (Ra – Rf) / σ where Ra = return on an asset, Rf is the risk-free rate of return, and σ is the standard deviation of returns of an asset (i.e. its riskiness).

The maximum Sharpe Ratio on the Efficient Frontier is the Market Portfolio. It is the portfolio on the Capital Asset Line that is tangent to the EF. This is what it looks like:

Efficient Frontier

Some important facts about the diagram:

  1. Any portfolio below the Efficient Frontier line is suboptimal. It has too much risk for the given reward (or alternatively, too little reward for a given level of risk).
  2. There are no attainable risky portfolios above the EF without leverage.
  3. You can think of individual points in the diagram as portfolios of different assets in different proportions
  4. The risk-free rate intersects the y-axis where the standard deviation equals zero (i.e. zero risk).
  5. The tangent line drawn from the risk-free expected return to the market portfolio is the Capital Asset Line, which we discussed in our previous blog. The point of tangency, the Market Portfolio, has the highest possible Sharpe Ratio available.

The discussion implies that you can’t get better risk-adjusted returns than holding the market portfolio, but that is not quite right.  Leverage can have a profound effect on your portfolio’s risk/reward characteristics.

You could deleverage your portfolio by adding risk-free asset to your risky portfolio.  Your new portfolio would now lie on the CAL extending from the Rf to your old portfolio.  Your new portfolio would have less risk (and less reward) than your old portfolio, but, it would still be above the EF line.  You have created a super-efficient portfolio just by adding cash.

Alternatively, you could leverage your portfolio by borrowing at the risk-free rate and use the proceeds to buy more of your old portfolio.  Your new portfolio with be riskier and have a higher potential reward.  It is also above the EF line.

There is a special version of the Capital Asset Line.  It is the line that ranges from risk-free rate, touches the Efficient Frontier at the Market Portfolio, and extends upward beyond the tangent point.  It is known as the Capital Market Line, and we’ll explore it in our next installment.

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Copyright 2011 Eric Bank, Freelance Writer

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