Finance

# Efficient Frontier

Efficient Frontier

Efficient frontier or Portfolio frontier, a part of modern portfolio theory comprises efficient parts of the risk-return spectrum. It occupies investment portfolios that offer the highest expected return for a specific risk level. It is a set of optimal portfolios that are expected to give higher returns for a minimum level of return. How Does an Efficient Frontier Work?

Expected returns of a portfolio and standard deviations of returns are plotted to represent efficient frontier. Y-axis contains expected returns of a portfolio and the X-axis represents the standard deviation of returns, which measures risk. A portfolio is plotted on a graph on the basis of its returns and standard deviation. The portfolio is compared to the efficient frontier wherein the portfolio on right represents a high level of risk and the portfolio below the frontier represents low risk.

Example of an Efficient Frontier Portfolio:

Assuming that there are 2 assets B1 & B2 in a particular portfolio. Calculate the risks and returns for assets whose returns and standard deviation are as follows:

 Particulars B1 B2 Expected Returns 15% 25% Standard Deviation 20% 30% Correlation Coefficient 0.05

The weights of the assets, i.e., portfolio possibilities of investing in such assets is given below:

 Portfolio Weights in % B1 B2 1 0 100 2 25 75 3 50 50

Using the formulae, we calculate:

Expected Returns= (Weight of B1 * Return of B1) + (Weight of B2 * Return of B2)

Portfolio 1: ER= (0*0.15) + (100*0.25)

= (0 + 25)

= 25

Portfolio 2: ER = (25 * 0.15) + (75 * 0.25)

= (3.75 + 18.75)

= 22.5

Portfolio 3: ER = (50 * 0.15) + (50 * 0.25)

= (7.5 + 12.5)

= 20

Portfolio Risk = √ [(Weight of A12 * Standard Deviation of A12) + (Weight of A22 * Standard Deviation of A22) + (2 X Correlation Coefficient * Standard Deviation of A1 * Standard Deviation of A2)]

Portfolio 1:

Risk = √ [ (02 * 0.22) + (12 * 0.32) + (2* 0.05 * 0* 1)]

= √ [(0*0.04) + (1*0.09) + (0)]

= √ [(0) + (0.09) + (0)]

= √ (0.09)

= 0.3 = 30%

Portfolio 2:

Risk = √ [ (0.252 * 0.22) + (0.752 * 0.32) + (2* 0.05 * 0.25 * 0.75)]

= √ [(0.0625*0.04) + (0.5625*0.09) + (0.006)]

= √ [(0.002) + (0.050) + (0.019)]

= √ (0.071)

= 0.27 = 27%

Portfolio 3:

Risk = √ [ (0.52 * 0.22) + (0.52 * 0.32) + (2* 0.05 * 0.5 * 0.5)]

= √ [(0.25*0.04) + (0.25*0.09) + (0.025)]

= √ [(0.01) + (0.022) + (0.025)]

= √ (0.057)

= 0.24 = 24%

 Portfolio Risk Return 1 30 25 2 27 22.5 3 24 20

On the basis of this table, we plot the portfolio where X-axis represents risk and Y-axis represents returns and what we get is an efficient frontier.

Assumptions involved in the Efficient Frontier:

1. All the investors have a common goal of avoiding risk because they do risk aversion and maximize returns.
2. It assumes that all investors are vigilant and have rational knowledge about the stock movements.
3. Not many investors affect the market price.
4. There is no limit on investors borrowing power and therefore trade at the risk-free interest rate.
5. The markets are efficient and all the assets follow a normal distribution.
6. Information is quickly absorbed by the market and the actions are accordingly planned.
7. The investors totally depend on expected returns and standard deviation.

Benefits & Drawbacks:

Benefits:

1. It highlights the significance of diversification.
2. Investors are enabled with the choice of various portfolio combinations having high returns and low risk.
3. It displays all the dominant portfolios in the risk-return space.

Drawbacks:

1. The assumption that all investors have rational knowledge may not always be true.
2. It works only where diversification is involved.
3. The assumption that there is no limit on borrowing is faulty.
4. The real costs are not taken into consideration and are thus ignored.
5. All the assets may not follow normal distribution every time.

Bottom Line:

Efficient frontier displays the best combination of assets that give higher returns on a given level of risk. When the portfolios fall on the line of frontier, they are efficient and the ones outside the line are not optimal. The location of the frontier totally depends on the investor’s degree of risk tolerance.

Author: Urvi Surti