Portfolio and Portfolio Selection Model

Portfolio:

All types of assets that people diversify. So, the important issue in portfolio is the portion of each type of asset. How to choose (or allocate) the combination of riskiness and rate of return will depend upon each person’s attitude toward riskiness or uncertainty.

Portfolio Selection Model:

Two types of assets: safe vs. risky assets

rf : rate of return of safe (or risk-free) asset
rk : rate of return of risky asset.

We can simply imagine that rf < rk .

To express the level of riskiness we are able to adopt the standard deviation of the rate of return (σ).

Therefore let’s assume that rk has the standard deviation of σk .

Suppose a person has $1 worth of wealth and s/he wants to have a as a type of risky asset and (1− a) as safe asset. Based on this information we can calculate the rate of return of this portfolio (rp) and level of riskiness (σp) as follows:

rp = ark + (1− a)rf  ........(ii)

σp = aσk      ...............(iii)

Finally, the question is to get the specific value of a in this portfolio selection model. Using utility function with the insertion of two factors, rp and σp , we can set up:

U = U(rp , σp)  .........(iv)

If this person is risk averse, s/he thinks that σ p will be a “bad.”

From (ii) we are able to get:

rp = rf + a(rk − rf)  ..........(v)

Plugging a = σpk into (v), we can get the final expression:

1803_portfolio.jpg

1781_portfolio.jpg

A: a = 0. B: a = 1

(rk −rf )/σk: price of risk

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