Question

In our discussion of efficient portfolios, we saw the we can directly find the optimal portfolio of risky assets (at least for an investor whose entire preference is defined by mean and variance) by maximizing the so-called Sharpe ratio. Explain why this optimization procedure works – that is, why is a maximum Sharpe ratio considered optimal. (Hints: Think about what a combination of the risk-free asset and any risky portfolio selected from the efficient frontier would look like. In other words, how would the set of combinations of the risky and the risk-free plot visually in our risk-return space? Consider the fact that a risk-free asset, of course, has zero variance and, thus, has zero-covariance with any other asset or portfolio.)

Discuss about that with approximately 400 words.

Answer 1

Efficient frontier consists of investment portfolios that furnish the highest expected return for a specific amount of risk.

Returns are relied on the investment combinations that make up the portfolio.

The standard deviation is synonymous with risk. Lower covariance between portfolio securities gives lower portfolio standard deviation, thus lower risk and lower return. Portfolio optimization of the return versus risk regime should place a portfolio along the efficient frontier line. Optimal portfolios on the efficient frontier tend to have a higher degree of diversification.

The benefit of diversification is the curvature of the efficient frontier. It signifies how diversification improves the portfolio's risk / reward profile. It also reveals the diminishing marginal return to risk, i.e the relationship is non-linear. Thus, for an additional return there is requirement of adding less risk to a portfolio.It can be calculated using Sharpe ratio which is additional return per 1 unit of risk. The formula for Sharpe ratio is (Rp - Rf)/?p

Rp = Return on portfolio

Rf = Risk free interest rate

?p = Standard deviation of portfolio

Limitations:

The efficient frontier have many assumptions that may not represent reality. For example, one of the assumptions is that asset returns follow a normal distribution. In reality, securities may experience returns that are more than 3 standard deviations away from the mean. Consequently, asset returns are said to have a leptokurtic distribution or fat-tailed distribution.

Additionally, assumptions which are different from reality are:

1) investors are rational and avoid risk when possible;

2) not enough investors are present to influence market prices;

3) and investors have unlimited access to borrowing and lending money at the risk-free interest rate.

However, it is proven that market includes irrational investors and risk-seeking investors, there are large market participants that can influence market prices, and not all investors have unlimited access to borrowing and lending money at risk free interest rate