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LOS h. describe the selection of an optimal portfolio, given an investor's utility (or risk aversion) and the capital allocation line.
Optimal portfolio.
The efficient frontier only considers the investments in risky assets. However, investors may choose to invest in a risk free asset, which is assumed to have an expected return commensurate with an asset that has no standard deviation (i.e. zero variance) around the expected return. That is, a risk-free asset's expected return is entirely certain, and is known as the risk-free rate of return (RFR). Therefore, a risk-free asset lies on the vertical axis of a portfolio graph.

When a risk-free asset is combined with a risky portfolio, a graph of possible portfolio risks-return combinations becomes a straight line between the two assets.

Assume the proportion of the portfolio the investor places in the tangency portfolio P is wP:

  • The expected rate of return for the new portfolio is the weighted average of the two returns: E(R) = (1 - wP) Rf + wP E(RT)
  • The standard deviation of the new portfolio is the linear proportion of the standard deviation of the risky asset portfolio P: σportfolio = wP σP
The introduction of a risk-free asset changes the efficient frontier into a straight line. This straight efficient frontier line is called the Capital Market Line (CML) for all investors, and Capital Allocation Line (CAL) for one investor.

  • Investors at point rf have 100% of their funds invested in the risk-free asset.
  • Investors at point P have 100% of their funds invested in portfolio P.
  • Between rf and P investors hold both the risk-free asset and portfolio P. This means investors are lending some of their funds (buying the risk-free asset).
  • To the right of P, investors hold more than 100% of portfolio P. This means they are borrowing funds to buy more of portfolio P. This represents a levered position.
Investors will choose the highest CAL, i.e. the CAL tangent to the efficient frontier. This portfolio is the solution to the optimization problem of maximizing the slope of the CAL.

Now, the line rf-P dominates all portfolios on the original efficient frontier. Thus, this straight line becomes the new efficient frontier.

Separation Theorem

Investors make different financing decisions based on their risk preferences. The separation of the investment decision from the financing decision is called the separation theorem. Now the portfolio choice problem can be broken down into two tasks:

  • Choosing P, a technical matter (can be done by the broker)
  • Deciding on the proportion to be invested in P and in the riskless asset.

Optimal Investor Portfolio

We can combine the efficient frontier and/or capital allocation line with indifference curves. The optimal portfolio is the portfolio that gives the investor the greatest possible utility.

  • Two investors will select the same portfolio from the efficient set only if their utility curves are identical.
  • Utility curves to the right represent less risk-averse investors; utility curves to the left represent more risk-averse investors.
This is portfolio selection without a risk-free asset:

The optimal portfolio for each investor is the highest indifference curve that is tangent to the efficient frontier.

This is portfolio selection with a risk-free asset:

The optimal portfolio for each investor is the highest indifference curve that is tangent to the capital allocation line.
Practice Question 1

For a portfolio to be efficient, it must exhibit:

A. the highest risk for a given level of expected return
B. the highest expected return for a given level of risk
C. the lowest risk for all levels of expected return
D. the highest expected return for all levels of risk

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Practice Question 2

Which of the following statement(s) is (are) true regarding the selection of a portfolio from those that lie on the Capital Allocation Line?

I. Less risk-averse investors will invest more in the risk-free security and less in the optimal risky portfolio than more risk-averse investors.
II. More risk-averse investors will invest less in the optimal risky portfolio and more in the risk-free security than less risk-averse investors.
III. Investors choose the portfolio that maximizes their expected utility.

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Practice Question 3

Given an optimal risky portfolio with expected return of 14% and standard deviation of 22% and a risk free rate of 6%, what is the slope of the best feasible CAL?

A. 0.64.
B. 0.33.
C. 0.36.

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Practice Question 4

Among the available portfolios, the selection of each investor's optimal portfolio depends on

A. Each investor's risk preference.
B. Market portfolio.
C. Risk free asset.

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Practice Question 5

The optimal portfolio includes all of the following characteristics except which of the following?

A. The optimal portfolio lies at the point of tangency between the efficient frontier and the highest possible utility curve.
B. The optimal portfolio may be different for each investor, but for a given investor it has the highest utility.
C. Utility curves may be different for individual investors even though the optimal portfolio for each investor is presumed to be the same.

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Practice Question 6

An investor maximizes her utility by

A. finding the highest indifference curve under the risk-return tradeoff.
B. choosing a point that is tangent to the efficient frontier and is on the highest feasible indifference curve.
C. choosing a point that is parallel to the efficient frontier and is on the highest feasible utility curve.

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