Question

(1) Discuss the SHARP RATIO and its significance to Mean Variance portfolio.

(2) Higher the coefficient of the risk aversion (RA) higher the investors expected utility or risk adjusted return. - True OR False

Answer 1

Answer of question =1 , Sharp Ratio and its significance to mean variance portfolio.

sharp ratio meaning

• The Sharpe ratio was developed by Nobel laureate William F. Sharpe
• It is used to help investors understand the return of an investment compared to its risk.
• The ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk.
• Subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking activities.
• Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.

Sharp ratio Formula :

= (return of portfolio - risk free rate)/std. Deviation of portfolio.

More details of sharp ratio :

• Usually, any Sharpe ratio greater than 1.0 is considered acceptable to good by investors. A ratio higher than 2.0 is rated as very good. A ratio of 3.0 or higher is considered excellent.

• The Sharpe ratio has become the most widely used method for calculating the risk-adjusted return.

• Modern Portfolio Theory states that adding assets to a diversified portfolio that has low correlations can decrease portfolio risk without sacrificing return.

• Adding diversification should increase the Sharpe ratio compared to similar portfolios with a lower level of diversification.

• For this to be true, investors must also accept the assumption that risk is equal to volatility which is not unreasonable but may be too narrow to be applied to all investments.

• The Sharpe ratio can be used to evaluate a portfolio’s past performance (ex-post) where actual returns are used in the formula.

• Alternatively, an investor could use expected portfolio performance and the expected risk-free rate to calculate an estimated Sharpe ratio (ex-ante).

• The Sharpe ratio can also help explain whether a portfolio's excess returns are due to smart investment decisions or a result of too much risk.

• Although one portfolio or fund can enjoy higher returns than its peers, it is only a good investment if those higher returns do not come with an excess of additional risk.

• The greater a portfolio's Sharpe ratio, the better its risk-adjusted-performance.

• If the analysis results in a negative Sharpe ratio, it either means the risk-free rate is greater than the portfolio’s return, or the portfolio's return is expected to be negative. In either case, a negative Sharpe ratio does not convey any useful meaning.

Limitations of sharp ratio :

• The Sharpe ratio uses the standard deviation of returns in the denominator as its proxy of total portfolio risk, which assumes that returns are normally distributed. A normal distribution of data is like rolling a pair of dice. We know that over many rolls, the most common result from the dice will be 7 and the least common results will be 2 and 12.

• However, returns in the financial markets are skewed away from the average because of a large number of surprising drops or spikes in prices.

• Additionally, the standard deviation assumes that price movements in either direction are equally risky.

• The Sharpe ratio can be manipulated by portfolio managers seeking to boost their apparent risk-adjusted returns history. This can be done by lengthening the measurement interval. This will result in a lower estimate of volatility.

• For example, the annualized standard deviation of daily returns is generally higher than that of weekly returns which is, in turn, higher than that of monthly returns.

• Choosing a period for the analysis with the best potential Sharpe ratio, rather than a neutral look-back period, is another way to cherry-pick the data that will distort the risk-adjusted returns.

mean variance portfolio theory meaning :

• Mean-variance analysis is one part of modernportfolio theory, which assumes that investors will make rational decisions about investments if they have complete information.
• In modern portfolio theory, an investor would choose different securities to invest in with different levels of variance and expected return.

significance of sharp ratio to mean variance portfolio

• Sharpe Ratio is a measurement of risk-adjusted return of a portfolio.

• The concept is named after William F. Sharpe of Stanford University.

• The ratio measures the return on the funds in excess of proxy for a risk-free guaranteed investment relative to the standard deviation. Generally, the 90-day Treasury bill rate is the proxy for risk-free rate.

• A portfolio with higher Sharpe ratio is superior to its peers.

• Among the many formulas that are available, mutual fund managers use this ratio to measure the risk-adjusted returns on their portfolio.

• Understanding the relationship between the Sharpe ratio and risk often comes down to measuring the standard deviation, also known as the total risk.

• The square of standard deviation is the variance, which was widely used by Nobel Laureate Harry Markowitz, the pioneer of Modern Portfolio Theory.

• Here , sharp ratio calculates the average return over and above the risk free rate of return Per unit of portfolio risk.

• Thus it describes how much excess return you are receiving for the extra volatility that you endure for holding a riskier asset.

• Sharp ratio tells us whether a portfolio's return are due to smart investment decision or a result of excess risk.

Answer of question 2 =true = Higher the coefficient of risk aversion , higher the investors expected utility or risk adjusted return , is true , due to reasons are below :

Meaning of risk aversion :

• Definition: A risk averse investor is an investor who prefers lower returns with known risks rather than higher returns with unknown risks.
• In other words, among various investments giving the same return with different level of risks, this investor always prefers the alternative with least interest.
  
• Description: A risk averse investor avoids risks. S/he stays away from high-risk investments and prefers investments which provide a sure shot return.
• Such investors like to invest in government bonds, debentures and index funds.
• risk aversion is the behavior of humans (especially consumersand investors), who, when exposed to uncertainty, attempt to lower that uncertainty.
• It is the hesitation of a person to agree to a situation with an unknown payoff rather than another situation with a more predictable payoff but possibly lower expected payoff.
• For example, a risk-averse investor might choose to put their money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value.

Meaning of Coefficient of risk aversion :

• A quantitative and practical method is the following:

• we attributed a number from 1 (lowest risk aversion) to 5 (highest risk aversion) to an investor. We then assign this number the letter A, which is called the "risk aversion coefficient".

• To get it, we use the following utility formulafootnote1: U = E(r) – 0,5 x A x σ2.

• In this formula, U represents the utility or score to give this investment in a given portfolio by comparing it to a risk-free investment, such as treasury bills.

Meaning of Risk adjusted return :

• Risk-adjusted return defines an investment's return by measuring how much risk is involved in producing that return, which is generally expressed as a number or rating.

• Risk-adjusted returns are applied to individual securities, investment funds, and portfolios.

• Some common risk measures include alpha, beta, R-squared, standard deviation and the Sharpe ratio.

• When comparing two or more potential investments, an investor should always compare the same risk measures to each different investment to get a relative performance perspective.

• risk-adjusted return is of how much return your investment has made relative to the amount of risk the investment has taken over a given period of time.

• If two or more investments have the same return over a given time period, the one that has the lowest risk will have the better risk-adjusted return.

• However, considering that different risk measurements give investors very different analytical results,

• it is important to be clear on what type of risk-adjusted return is being considered

Conclusion :

It is true that , higher the coefficient of risk aversion , higher the risk adjusted return .

A risk averse investor is an investor who prefers lower returns with known risks rather than higher returns with unknown risks.

High risk aversion means they are not willing to take more risks .they will avoids risks.so they invests in safe investment securities like govt.bonds..etc.

Higher the risk aversion Higher risk adjusted return because , If two or more investments have the same return over a given time period, the one that has the lowest risk will have the better risk-adjusted return.

In other words, among various investments giving the same return with different level of risks, this investor always prefers the alternative with least interest.

A risk averse investor avoids risks. S/he stays away from high-risk investments and prefers investments which provide a sure shot return. Such investors like to invest in government bonds, debentures and index funds.

Hope you understand both answers .comment me please.