Question

Compare and contrast different portfolio performance measurements. Analyze and explain the following metrics:

- Annualized Return (APY)

- Standard Deviation

- Beta - Sharpe Ratio

- Alpha or Jensen Ratio

- Treynor Ratio

Answer 1

Hi,

ANNUALIZED RETURN : An annualized total return is the geometric average amount of money earned by an investment each year over a given time period. It is calculated as a geometric average to show what an investor would earn over a period of time if the annual return was compounded. An annualized total return provides only a snapshot of an investment's performance and does not give investors any indication of its volatility.

The generalized formula to calculate annualized return needs only two variables: the returns for a given period of time and the time the investment was held. The formula is:

For example, take the annual returns of Mutual Fund A Returns = 3%, 7%, 5%, 12% and 1% for 5 years. An analyst substitutes each of the "r" variables with the appropriate return, and "n" with the number of years the investment was held. In this case, five. The annualized return of Mutual Fund A is calculated as:

Annualized Return = ((1 + 3%) x (1 + 7%) x (1 + 5%) x (1 + 12%) x (1 + 1%)) ^{(1 / 5)} -1 = 130.9% ^(0.20) -1 = 105.55% - 1 = 5.53%.

STANDARD DEVIATION : Standard deviation is a measure of the dispersion of a set of data from its mean. It is calculated as the square root of variance by determining the variation between each data point relative to the mean. If the data points are further from the mean, there is higher deviation within the data set.

In finance, standard deviation is a statistical measurement; when applied to the annual rate of return of an investment, it sheds light on the historical volatility of that investment. The greater the standard deviation of a security, the greater the variance between each price and the mean, indicating a larger price range. For example, a volatile stock has a high standard deviation, while the deviation of a stable blue-chip stock is usually rather low.

The formula for standard deviation uses three variables. The first variable is to be the value of each point within the data set, traditionally listed as x, with a sub-number denoting each additional variable (x, x1, x2, x3, etc.). The mean, or average, of the data points is applied to the value of the variable M, and the number of data points involved is assigned to the variable n. To determine the mean value, the values of the data points must be added together, and that total is then divided by the number of data points that were included.

For example, if the data points were 5, 7, 3 and 7, the total would be 22. That total of 22 would then be divided by the number of data points, in this case four, resulting in a mean of 5.5. This leads to the following determinations: M = 5.5 and n = 4.

The variance is determined by subtracting the value of the mean from each data point, resulting in -0.5, 1.5, -2.5 and 1.5. Each of those values are then squared, resulting in 0.25, 2.25, 6.25 and 2.25. The square values are then added together, resulting in a total of 11, which is then divided by the value of n-1, which is 3 in this instance, resulting in a variance approximately of 3.67.

The square root of the variance is then calculated, resulting in the standard deviation of approximately 1.915.

BETA - SHARPE RATIO :

Beta, also known as the "beta coefficient," is a measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole. Beta is calculated using regressionanalysis, and you can think of it as the tendency of an investment's return to respond to swings in the market. By definition, the market has a beta of 1.0. Individual security and portfolio values are measured according to how they deviate from the market.

A beta of 1.0 indicates that the investment's price will move in lock-step with the market. A beta of less than 1.0 indicates that the investment will be less volatile than the market, and, correspondingly, a beta of more than 1.0 indicates that the investment's price will be more volatile than the market. For example, if a fund portfolio's beta is 1.2, it's theoretically 20% more volatile than the market.

Conservative investors looking to preserve capital should focus on securities and fund portfolios with low betas, whereas those investors willing to take on more risk in search of higher returns should look for high beta investments.

Developed by Nobel laureate economist William Sharpe, this ratio measures risk-adjusted performance. It is calculated by subtracting the risk-free rate of return (U.S. Treasury Bond) from the rate of return for an investment and dividing the result by the investment's standard deviation of its return.

The Sharpe ratio tells investors whether an investment's returns are due to smart investment decisions or the result of excess risk. This measurement is very useful because although one portfolio or security can reap higher returns than its peers, it is only a good investment if those higher returns do not come with too much additional risk. The greater an investment's Sharpe ratio, the better its risk-adjusted performance.

ALPHA - JENSEN RATIO :  Jensen's Alpha, or just "Alpha", is used to measure the risk-adjusted performance of a security or portfolio in relation to the expected market return (which is based on the capital asset pricing model (CAPM).The higher the alpha, the more a portfolio has earned above the level predicted.

The measure was first used by Michael Jensen in 1968 and was originally designed to evaluate fund managers, i.e. to gauge if it was possible for them to consistently outperform the markets. Jenson's results, however, suggested that this is rarely the case.Jensen's Alpha is also known as "Jensen's Performance Index" and "Jensen's Measure".

Jensen's Alpha is important to investors because they need to look not only at the total return of a security or portfolio, but also at the amount of risk involved in achieving that return.Usually, investors will aim to achieve a high return with a minimum amount of risk. So if, for example, two portfolios yielded identical returns, but one involved lower risk, the one with lower risk would rationally be the more attractive option.

Jensen's Alpha can help determine if the average return generated is acceptable based on the amount of risk involved. If the return is higher than that predicted by the CAPM, the security or portfolio is said to have a positive alpha (or an abnormal return). Investors are always looking for opportunities where a positive alpha is involved.

TREYNOR RATIO :  

The Treynor ratio, also known as the reward-to-volatility ratio, is a metric for returns that exceed those that might have been gained on a risk-less investment, per each unit of market risk. The Treynor ratio, developed by Jack Treynor, is calculated as follows:

(Average Return of a Portfolio – Average Return of the Risk-Free Rate)/Beta of the Portfolio

In essence, the Treynor ratio is a risk-adjusted measurement of a return, based on systematic risk. It is a metric of efficiency that makes use of the relationship that exists between risk and annualized risk-adjusted return. The ratio attempts to measure how successful an investment is in providing investors compensation, with consideration for the investment’s inherent level of risk. The Treynor ratio is reliant upon beta – that is, the sensitivity of an investment to movements in the market – to judge risk. The Treynor ratio is based on the premise that risk inherent to the entire market (as represented by beta) must be penalized, because diversification will not remove it.

When the value of the Treynor ratio is high, it is an indication that an investor has generated high returns on each of the market risks he has taken. The Treynor ratio allows for an understanding of how each investment within a portfolio is performing. It also gives the investor an idea of how efficiently capital is being used.

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