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One of the factors to consider when selecting the optimal portfolio for a particular investor is degree of risk aversion. This level of aversion to risk can be characterized by defining the investor's indifference curve. This curve consists of the family of risk/return pairs defining the trade-off between the expected return and the risk. It establishes the increment in return that a particular investor will require in order to make an increment in risk worthwhile. Typical risk aversion coefficients range between 2.0 and 4.0, with the higher number representing lesser tolerance to risk. The equation used to represent risk aversion in Financial Toolbox™ software is

U = E(r) - 0.005*A*sig^2

where:

`U` is the utility value.

`E(r)` is the expected return.

`A` is the index of investor's aversion.

`sig` is the standard deviation.

This example computes the optimal risky portfolio on the efficient
frontier based upon the risk-free rate, the borrowing rate, and the
investor's degree of risk aversion. You do this with the function `portalloc`.

First generate the efficient frontier data using either `portopt` or `frontcon`.
This example uses `portopt` and the same asset data
from the previous example.

ExpReturn = [0.1 0.2 0.15];

ExpCovariance = [ 0.005 -0.010 0.004; -0.010 0.040 -0.002; 0.004 -0.002 0.023];

This time consider 20 different points along the efficient frontier.

NumPorts = 20; [PortRisk, PortReturn, PortWts] = portopt(ExpReturn,... ExpCovariance, NumPorts);

As with `frontcon`, calling `portopt` while specifying output arguments
returns the corresponding vectors and arrays representing the risk,
return, and weights for each of the portfolios along the efficient
frontier. Use them as the first three input arguments to the function `portalloc`.

Now find the optimal risky portfolio and the optimal allocation of funds between the risky portfolio and the risk-free asset, using these values for the risk-free rate, borrowing rate and investor's degree of risk aversion.

RisklessRate = 0.08 BorrowRate = 0.12 RiskAversion = 3

Calling `portalloc` without
specifying any output arguments gives a graph displaying the critical
points.

portalloc (PortRisk, PortReturn, PortWts, RisklessRate,... BorrowRate, RiskAversion);

Calling `portalloc` while
specifying the output arguments returns the variance (`RiskyRisk`),
the expected return (`RiskyReturn`), and the weights
(`RiskyWts`) allocated to the optimal risky portfolio.
It also returns the fraction (`RiskyFraction`) of
the complete portfolio allocated to the risky portfolio, and the variance
(`OverallRisk`) and expected return (`OverallReturn`)
of the optimal overall portfolio. The overall portfolio combines investments
in the risk-free asset and in the risky portfolio. The actual proportion
assigned to each of these two investments is determined by the degree
of risk aversion characterizing the investor.

[RiskyRisk, RiskyReturn, RiskyWts,RiskyFraction, OverallRisk,... OverallReturn] = portalloc (PortRisk, PortReturn, PortWts,... RisklessRate, BorrowRate, RiskAversion) RiskyRisk = 0.1288 RiskyReturn = 0.1791 RiskyWts = 0.0057 0.5879 0.4064 RiskyFraction = 1.1869 OverallRisk = 0.1529 OverallReturn = 0.1902

The value of `RiskyFraction` exceeds 1 (100%),
implying that the risk tolerance specified allows borrowing money
to invest in the risky portfolio, and that no money will be invested
in the risk-free asset. This borrowed capital is added to the original
capital available for investment. In this example the customer will
tolerate borrowing 18.69% of the original capital amount.

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