Question

What happens to an efficient frontier when you add a risk-free asset to it? You can project a picture of the graph or hand draw one to explain

Answer 1

Feasible set of Portfolios

With a limited number of securities an investor can create a large number of portfolio with different proportion of securities.These consitutes feasible set of protfolio in which an investor can invest . Its also known as Portfolio opportunity set.

Each portfolio is portfolio opportunity set characterised by an expected returns & risk . Investors are not interested in all portfolio in opportunity set. Some portfolios are dominated by others either if Higher returns & same risk or Lower risk & same returns. Portfolio domintaed by others is known as inefficient portfolio.

Efficent Frontier

It can be illustrated with following graph of XY where x represents Risk & Y represents Returns . Each portfolio has its own risk and return are deplected by asingle point in the risk return space enclosed within two axis of the graph. Graph shows set of portfolio created with given securities and its shape is concave because it consist of securities that are lessthan perfectly correlated to each other.

While comparing each portfolis as follws ;

1 . E & F - E is selected

2. C & E - E is selected (High returns and low risk)

3. C & A - C is selected (Lower risk) C is lowest risky portfolio compared to all others , C in this graph represent Global Minimum Variance Portfolio

4 . A & B - (same risk) B is selected because it has High returns .

ie , Portfolio lying on North west boundary of shaped area is more efficent than others. This boundary of shaped area is called as Efficent Frontier because it contain more efficent portfolio set.

Introducing Risk free assets to Efficent Frontier

Linear combinations of the risk free asset and a risky portfolio have risk-return characteristics such that they fall on a straight line:

The expected return of the combination portfolio is a weighted average of the two assets’ individual returns. And because the covariance of the risk free asset and any risky asset is zero (and the variance of the risk-free asset is zero), the formula for the variance of the portfolio is simply the weight of the risk asset squared times the variance of the risky asset. So, the std deviation is the weight of the risk times the std. deviation of the risky.

These two things mean that that combinations of the risk-free asset and any one risky portfolio will fit on a straight line on the std deviation/expected return graph. So, if you combined the risk-free asset with the portfolio on the risky asset-only efficient frontier that made the resulting line just tangent to the the risky-asset-only efficient frontier, a potfolio on the the resulting line would always give a superior risk-return tradeoff to any other portfolio with the same standard deviation.

In other words, this line would become the “new” efficient frontie

Rf is a risk free asset . It has a guaranteed expected return with no risk . In other words , its expected returns is certain .

Each green point is a portfolio created by combining our previous risky assets ( A , B , & C) and the risk - free asset (Rf). Again, there is a limit to the effect of diversification illustrated by the red and black lines .The least risky portfolio is now constituted only of the risk - free asset.The efficient frontier is now a straight line because the risk - free asset has a risk of 0 . It includes the risk free asset and the risky securities.

There is one portfolio that belongs to the both efficient frontiers (with and without the risk- free asset). It is called theTangency portfolio and contains only risk assets. In fact , it is the only portfolio on the red line to contain only risky assets.