Optimal Portfolios

Risk-free assets are typically those issued by a government and considered to have zero risk. When we combine a risk-free asset with a portfolio of risky assets, we create a capital allocation line that we can represent graphically on the efficient frontier curve. The capital allocation line connects the optimal risky portfolio with the risk-free asset.

The Two-Fund Separation Theorem

The two-fund separation theorem states that all investors regardless of taste, risk preference and initial wealth will hold a combination of two portfolios or funds: a risk-free asset and an optimal portfolio of risky assets. This allows us to break the portfolio construction problem into two distinct steps: an investment decision and a financing decision. Firstly, the optimal risky asset portfolio using the risk, return and correlation characteristics of the underlying assets dictates the investment decision. And secondly, considering an investor’s risk preference, a determination is made on the allocation to the risk-free asset.  Plotting graphically the risk-free asset with the risky portfolio creates the capital allocation line (CAL).

Investor Preferences

A highly risk-averse investor may choose to invest only in the risk-free asset while a less risk-averse investor may have a small portion of their wealth invested in the risk-free asset and a large portion invested in the risky portfolio. Investors with a high-risk tolerance may, in fact, choose to borrow from the risk-free asset to invest in the risky portfolio in order to invest more than 100% of their assets creating a leveraged portfolio.

Utility and Indifference Curves

Utility is a measure of relative satisfaction that an investor derives from different portfolios. We can generate a mathematical function to represent this utility that is a function of the portfolio expected return, the portfolio variance and a measure of risk aversion.

U = E(r) – ½Aσ2

where

U = utility

E(r) = portfolio expected return

A = risk aversion coefficient

σ2 = portfolio variance

In determining the risk aversion (A), we measure the marginal reward an investor needs in order to take on more risk. A risk averse investor will need a high margin reward for taking on more risk. The utility equation shows the following:

  • Utility can be positive or negative – it is unbounded.
  • High returns add to utility.
  • High variance reduces utility.
  • Utility does not measure satisfaction but can be used to rank portfolios.

The risk aversion coefficient, A, is positive for risk-averse investors (any increase in risk reduces utility), it is 0 for risk-neutral investors (changes in risk do not affect utility) and negative for risk-seeking investors (additional risk increases utility).

An indifference curve plots the combination of risk and return that an investor would accept for a given level of utility. For risk-averse investors, indifference curves run “northeast” as an investor must be compensated with higher returns for increasing risk and has the greatest slope. An investor that is more risk-seeking has an indifference curve that is much flatter as their demand for increased returns as risk increases is much less acute.

We can overlay an investor’s indifference curve with the capital allocation line to determine the investor’s optimal portfolio.

Question

Using the utility function U = E(r) – ½Aσ2 and assuming A = -4, which best describes the investor’s attitude to risk?

A. The investor is risk neutral.

B. The investor is risk-averse.

C. The investor is risk-seeking.

Solution

The correct answer is C.

A negative risk aversion coefficient (A = -4) means the investor receives a higher utility (more satisfaction) for taking on more portfolio risk. A risk-averse investor would have a risk aversion coefficient greater than 0 and a risk neutral investor would have a risk aversion coefficient equal to 0.

 

Reading 41 LOS 41h:

Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line

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