One examination of the relationship between portfolio returns and risk is the efficient frontier, a curve that is a part of the modern portfolio theory. The curve forms from a graph plotting return and risk indicated by volatility, which is represented by standard deviation. According to the modern portfolio theory, funds lying on the curve are yielding the maximum return possible given the amount of volatility.

Notice that as standard deviation increases, so does the return. In the above chart, once expected returns of a portfolio reach a certain level, an investor must take on a large amount of volatility for a small increase in return. Obviously portfolios that have a risk/return relationship plotted far below the curve are not optimal as the investor is taking on a large amount of instability for a small return. To determine if the proposed fund has an optimal return for the amount of volatility acquired, an investor needs to do an analysis of the fund's standard deviation.

Note that the modern portfolio theory and volatility are not the only means investors use to determine and analyze risk, which may be caused by many different factors in the market. Not all investors therefore evaluate the chance of losses the same way - things like risk tolerance and investment strategy will affect how an investor views his or her exposure to risk.

Jensen's model proposes another risk adjusted performance measure. This measure was developed by Michael Jensen and is sometimes referred to as the Differential Return Method. This measure involves evaluation of the returns that the fund has generated vs. the returns actually expected out of the fund given the level of its systematic risk (ßi). The surplus between the two returns is called Alpha, which measures the performance of a fund compared with the actual returns over the period. Required return of a fund at a given level of risk (ßi) can be calculated as:

**r _{i} = r_{f} + ß_{i} (r_{M} - r_{f})**

Where, r_{M} is average market return during the given period. After calculating it, alpha can be obtained by subtracting required return from the actual return of the fund.

Higher alpha represents superior performance of the fund and vice versa. Limitation of this model is that it considers only systematic risk not the entire risk associated with the fund and an ordinary investor can not mitigate unsystematic risk, as his knowledge of market is primitive.

Alpha shows the fund’s performance relative to the benchmark and can demonstrate the value added by the fund manager. The higher the 'alpha' the better the manager.

**Alpha** is a risk-adjusted measure of the so-called "excess return" on an investment. It is a common measure of assessing an active manager's performance as it is the return in excess of a benchmark index or "risk-free" investment.

The **alpha coefficient(α _{i}) ** is a parameter in the capital asset pricing model. In fact it is the intercept of the

E(α_{i}) = r_{f}.

Therefore the alpha coefficient can be used to determine whether an investment manager has created economic value:

- α
_{i}< r_{f}: the manager has destroyed value - α
_{i}= r_{f}: the manager has neither created nor destroyed value - α
_{i}> r_{f}: the manager has created value

The difference (α_{i}- rf ) is called **Jensen's alpha.**

The concept and focus on Alpha comes from an observation increasingly made during the middle of the twentieth century, that around 75 percent of stock investment managers did not make as much money picking investments as someone who simply invested in every stock in proportion to the weight it occupied in the overall market in terms of market capitalization, or indexing. Many academics felt that this was due to the stock market being "efficient" which means that since so many people were paying attention to the stock market all the time, the prices of stocks rapidly moved to the correct price at any one moment, and that only luck made it possible for one manager to achieve better results than another, before fees or taxes were considered. A belief in efficient markets spawned the creation of market capitalization weighted index funds that seek to replicate the performance of investing in an entire market in the weights that each of the equity securities comprises in the overall market..

In fact, to many investors, this phenomenon created a new standard of performance that must be matched: an investment manager should not only avoid losing money for the client and should make a certain amount of money, but in fact should make more money than the passive strategy of investing in everything equally (since this strategy appeared to be statistically more likely to be successful than the strategy of any one investment manager).The name for the additional return above the expected return of the beta adjusted return of the market is called "Alpha".

A rational investor would not invest in an asset which does not improve the risk-return characteristics of his existing portfolio. Since a rational investor would hold the market portfolio, the asset in question will be added to the market portfolio.

Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Systematic risk, or market risk, refers to the risk common to all securities - except for selling short as noted below, systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with the risk (standard deviation) of the market portfolio.

Since a security will be purchased only if it improves the risk / return characteristics of the market portfolio, the risk of a security will be the risk it adds to the market portfolio. In this context, the volatility of the asset, and its correlation with the market portfolio, is historically observed and is therefore a given (there are several approaches to asset pricing that attempt to price assets by modelling the stochastic properties of the moments of assets' returns - these are broadly referred to as conditional asset pricing models). The (maximum) price paid for any particular asset (and hence the return it will generate) should also be determined based on its relationship with the market portfolio.

The Information Ratio measures the excess return of an investment manager divided by the amount of risk the manager takes relative to a benchmark. It is used in the analysis of performance of mutual funds etc. Specifically, the information ratio is defined as excess return divided by Tracking Error.Excess return is the amount of performance over or under a given benchmark index. Thus, excess return can be positive or negative. Tracking error is the standard deviation of the excess return. An alternative calculation of Information ratio is alpha divided by tracking error, although it is preferable to use pure excess return in the calculation.

The ratio compares the annualized returns of the Fund in question with those of a selected benchmark (e.g, 3 month Treasury Bills ). Since this ratio considers the annualized standard deviation of both series (as measures of risks inherent in owning either the fund or the benchmark), the ratio shows the risk-adjusted excess return of the Fund over the benchmark. The higher the Information Ratio, the higher the excess return of the Fund, given the amount of risk involved, and the better a Fund manager.

__The Information Ratio__ of a manager series vs. a benchmark series is the quotient of the annualized excess
return and the annualized standard deviation of excess return.

Information Ratio = (AnnRtn(r_{1}, ..., r_{n}) - AnnRtn(s_{1},
..., s,_{n})) / AnnStdDev(e_{1}, ..., e_{n})

where:

r_{1}, ..., r_{n} = manager return series

s_{1}, ..., s_{n} = benchmark return series

e_{1}, ..., e_{n} = r_{1} - s_{1}, ..., r_{n} - s_{n}

The Information ratio is similar to the Sharpe ratio, but there is a major difference. The Sharpe ratio compares the return of an asset against the return of Treasury bills, but the Information Ratio compares excess return to the most relevant equity (or debt) benchmark index.

The Information Ratio measures the consistency with which a manager beats a benchmark.

It is very important to realize that __annualized
and cumulative excess__ return are not calculated
in the naive way, by taking the annualized or
cumulative return of the excess return series.
Instead, one must take the annualized and cumulative
return of the two original series and then form
the difference between the two:

AnnExRtn = AnnRtn(r_{1}, ..., r_{n}) - AnnRtn(s_{1}, ...,
s_{n})

The __annualized standard deviation__ is the standard
deviation multiplied by the square root of the
number of periods in one year.

AnnStdDev(r_{1}, ..., r_{n}) = StdDev(r_{1}, ..., r_{n})
*

where r_{1}, ..., r_{n} is a return series, i.e., a
sequence of returns for n time periods.

Standard deviation of return measures the average deviations of a return series from its mean, and is often used as a measure of risk. A large standard deviation implies that there have been large swings in the return series of the manager.

There exists a close connection between the Information Ratio and the statistical significance of excess returns. The hypothesis that the set of relative returns is positive and statistically significant on average can be tested with the t-statistic.

The __t-Statistic__ of a manager series vs a benchmark
series is the information ratio multiplied by
the square root of the number of years.

t-Statistic = (Information Ratio) *

If a fund's beta is close to one, its information ratio times the square root of the number of observations is about equal to the t-statistic for testing the significance of positive relative returns. A statistical test for over performance is therefore also a test for a significant information ratio.

**Treynor Ratio (Reward to Variability Ratio)**

Developed by Jack Treynor, Treynor ratio is a measurement of the returns earned in excess of that which could have been earned on a riskless investment (i.e. Treasury Bill) (per each unit of market risk assumed).

Treynor Ratio is a ratio of return generated by the fund over and above risk free rate of return (generally taken to be the return on securities backed by the government, as there is no credit risk associated), during a given period and systematic risk associated with it.

Symbolically, it can be represented as:

**Treynor Ratio (T) = (R _{i} - R _{f}) / ß _{i}.**

R_{i }: Portfolio Return

R_{f} : Riskfree Return

ß_{i} : Portfolio Beta

All risk-averse investors would like to maximize this value. While a high and positive Treynor Ratio shows a superior risk-adjusted performance of a fund, a low and negative Treynor Ratio is an indication of unfavorable performance. T does not quantify the value added of active portfolio management. It is a ranking criterion only. However, it can be expected that portfolio managers, which possess private information, will have a higher T than the T of the uninformed market strategy. A ranking of portfolios based on T measure is only useful if the funds under consideration are sub funds of a broader, fully diversified portfolio. If this is not the case, portfolios with identical systematic risk, but different total risk, will be rated the same. But the portfolio with a higher total risk is less diversified and therefore has a higher unsystematic risk which is not priced in the market.

**Sharpe Ratio**

Performance of a fund is also evaluated on the basis of Sharpe Ratio, which is a ratio of returns generated by the fund over and above risk free rate of return and the total risk associated with it. According to Sharpe, it is the total risk of the fund that the investors are concerned about. So, the model evaluates funds on the basis of reward per unit of total risk. Symbolically, it can be written as:

Sharpe ratio (S_{i}) = (R_{i} - R_{f})/S_{i}

Where, S_{i} is standard deviation of the fund.

Betas are widely used to measure the volatility of a stock fund's price relative to the general market. The beta relates the volatility of a single security to the volatility of the market as a whole.

This common measure compares a mutual fund's volatility with that of a benchmark and is supposed to give some sense of how far you can expect a fund to fall when the market takes a dive, or how high it might climb if the bull is running hard. A fund with a beta greater than 1 is considered more volatile than the market; less than 1 means less volatile.

An issue with a beta of 1.5 for example, tends to move 50% more than the total market, in the same direction. An issue with a beta of 0.5 tends to move 50% less. If a stock or stock fund moved exactly as the market moved, it would have a beta of 1.0. Thus, high beta is typical of a volatile stock. Low beta is typical of a stock that moves less than the market as a whole. A stock with a negative beta moves in the direction opposite to that of the market. With a beta of -1.0 a stock has the same volatility as the market, but tends to rise when the market falls, and vice versa.

The beta coefficient is a key parameter in the capital asset pricing model (CAPM). It measures the part of the asset's statistical variance that cannot be mitigated by the diversification provided by the portfolio of many risky assets, because it is correlated with the return of the other assets that are in the portfolio.

This correlated risk, measured by Beta, is what actually creates almost all of the risk in a diversified portfolio.

The formula for the Beta of an asset is

where r_{a} measures the rate of return of the asset
and rp measures the rate of return of the portfolio
of which the asset is a part. In the CAPM formulation,
the portfolio is the market portfolio that contains
all risky assets, and so the rp terms in the formula
are replaced by r_{m}, the rate of return of the
market.

Beta is also referred to as **financial elasticity**
or correlated relative volatility, and can be
referred to as a measure of the asset's sensitivity
of the asset's returns to market returns, its
non-diversifiable risk, its systematic risk or
market risk. On a portfolio level, measuring beta
is thought to separate a manager's skill from
his or her willingness to take risk.

The beta movement should be distinguished from the actual returns of the stocks. For example, a sector may be performing well and may have good prospects, but the fact that its movement does not correlate well with the broader market index may decrease its beta. However, it should not be taken as a reflection on the overall attractiveness or the loss of it for the sector, or stock as the case may be. Beta is a measure of risk and not to be confused with the attractiveness of the investment.

**Important** to note: Beta, though a useful guide, is far from perfect, especially when used as a proxy
for "risk." The problem here, as with many risk measures, is the benchmark. The benchmark has to be a correct measure of
comparison only then will the beta hold any indicative value.

Volatility most frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon. It is often used to quantify the risk of the instrument over that time period. Volatility is typically expressed in annualized terms, and it may either be an absolute number ($5) or a fraction of the initial value (5%).

For a financial instrument, the volatility increases by the square-root of time as time increases. Conceptually, this is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases.

**Historical volatility** is the standard deviation of a financial instrument based on historical returns. This phrase is used particularly when it is wished to distinguish between the actual volatility of an instrument in the past, and the current volatility implied by the market.

Volatility of returns of a fund is measured by standard deviation which is a measure of total risk of a fund. Volatility indicates the tendency of the funds NAV (Net Asset Value) to rise and fall in a short period. It measures the extent to which the NAV fluctuates as compared to the average returns during a period.

A fund that has a consistent four year return of 3 %, for example, would have a mean , or average, of 3 %. The standard deviation for this fund would then be zero because the fund’s return in any given year does not differ from its four year mean of 3 %. On the other hand, a fund that in each of the last four years returned -5%, 17%, 2% and 30% will have a mean return of 11%.The fund will also exhibit a high standard deviation because each year the return of the fund differs from mean return. This fund is therefore more risky because it fluctuates widely between negative and positive returns within a short period.

A higher standard Deviation means that the returns of the fund have been more volatile than a fund having low standard deviation. In other words high standard deviation means high risk.

Tracking Error is performance measurement term which quantifies the extent to which mutual fund portfolio’s return is at variance with the underlying benchmark. In the case of Index Funds, this number is very important. An index fund is expected to replicate the index and therefore have a minimal tracking error. Index funds are compared and ranked on the basis of their tracking errors.

If tracking error is measured historically, it is called 'realised' or 'ex post' tracking error. If a model is used to predict tracking error, it is called 'ex ante' tracking error. The former is more useful for reporting or analysis purposes, whereas exante is generally used by portfolio managers to control risk to satisfy client guidelines.

Tracking error is mathematically the same as Active Risk, and has historically been used in the context of index portfolio or fund management, but, especially in Europe, is now typically used to describe the standard deviation of returns, either active or passive. The active return is the difference in the return of a portfolio and its benchmark. An index manager aiming to match the return of a benchmark index seeks to minimize realised tracking error, i.e., the standard deviation of returns about the benchmark. An active portfolio manager, on the other hand, aims to achieve a positive active return with a low active risk.

**Tracking Error = stdev(RETURN(portfolio) - beta * RETURN(index))**

r^{2} is a measure of the association between a fund and its benchmark. Values are between
0 and 1. 1 indicates a perfect correlation and 0 indicates no correlation. This measure is useful in determining if the
fund manager is adding value in their investment choices or acting as a closet tracker mirroring the market and making
little difference. For example, an index fund will have an R-squared with its benchmark index very close to 1,
indicating close to perfect correlation the index fund's fees and tracking error prevent the correlation from ever
equalling 1).

In statistics, the **coefficient of determination** r^{2} is the proportion
of variability in a data set that is accounted for by a statistical model. In this definition, the term "variability"
stands for variance or, equivalently, sum of squares. There are equivalent expressions for r^{2}. The version
most common in statistics texts is based on an analysis of variance decomposition as follows:

**r ^{2} = RSS/TSS = 1-(ESS/TSS)**

In the above definition,

RSS : Residual Sum of Squares

TSS : Total Sum of Squares

ESS : Explained Sum of Squares

r^{2} is the statistic that will give information about the goodness of fit of the model. It has a drawback:
r^{2} increases as we increase the number of variables in the model r^{2} will not decrease), so the
alternative technique is to look for adjusted r^{2}.

In finance, r^{2} measures how well the Capital Asset Pricing Model (CAPM) predicts the actual
performance of an investment or portfolio

The r^{2} of a fund advises investors, if the beta of a mutual
fund is measured against an appropriate benchmark. Measuring the correlation of a fund’s movements to that of an index,
R- squared describes the level of association between the fund’s volatility and market risk, or more specifically, the
degree to which a fund’s volatility is a result of the day to day fluctuations experienced by the overall market.