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Risk and Return Analysis. Sam Chung. Statistical Terms: Introduction. Here is a brief summary of the most common statistical terms and how they relate to the performance evaluation of traditional and alternative portfolios. Mean Value Standard Deviation Variance

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statistical terms introduction
Statistical Terms: Introduction
  • Here is a brief summary of the most common statistical terms and how they relate to the performance evaluation of traditional and alternative portfolios.
  • Mean Value
  • Standard Deviation
  • Variance
  • Correlation Coefficient
  • Normal Distribution
mean value
Mean Value

By mean one often refers to the simple average of a number of observations. This value is more correctly denoted as arithmetic mean, to distinguish it from the geometric mean. In order, the formulas below show the arithmetic and the geometric mean, respectively:

The arithmetic mean is more related to sums of values while the geometric one by its nature has more to do with products. Geometric average is a more accurate measure of past performance. On the other hand, arithmetic average is an unbiased estimate of the next period’s rate of return. Unless returns are constant, arithmetic average is higher than the geometric average. Also, holding the arithmetic average constant, the higher the volatility of returns the lower the geometric average.


The variance measures the fluctuation of the observations around their mean. The larger the value of variance, the greater the fluctuation. The population variance is given by:

Where µis the population mean and n represents the size of the population. The sample variance is given by:

Where X is the sample mean and n is the number of observations in the sample. In most applications, the sample variance is calculated rather than the population variance because calculation of the latter is possible only when every value in the population is known (i.e. the population variance has no practical application for empirical analyses). Dividing by (n-1) instead of by n (which may seem more logical) is done to make the sample variance correspond to . Division by n can be proven to produce an s-value that underestimates the population variance. Since our analyses are based on sample data with no complete understanding of the population, the sample variance from now on will be referred to as simply variance.

standard deviation
Standard Deviation

The standard deviation also measures the variability of observations around the mean. It is defined as the square root of the variance. The standard deviation will as a consequence have the same unit as the observation and is in a way easier to interpret. In financial terms, variability measured as standard deviation equals risk and the notion of risk has a very central place in the financial theory. From the definition of variance above follows that the standard deviation is given by:

correlation coefficient
Correlation Coefficient

A correlation coefficient is a measure of the strength of the linear relationship between two variables. If two variables are denoted by X and Y, then the correlation coefficient r of a sample of observations is found from:

where Xi and Yi denote the coordinates of the ith observation, X-bar is the sample mean of the X values, Y-bar is the sample mean of the Y-values and n is the sample size. The sample correlation coefficient is always between –1 and 1, where a r-value of 1 denotes a perfect linear relationship and –1 the opposite. A correlation coefficient of zero indicates that the two variables X and Y are uncorrelated, meaning that knowing the change in the variable X does not help in predicting the value of Y, X and Y are in other words independent of each other. Correlation measure is central in determining how beneficial it would be to add a particular investment to an existing portfolio of assets. For reasons to be presented in the section covering the financial background, adding an asset with favorable risk/return characteristics to a portfolio is generally most effective if the assets correlation with the existing portfolio is low (or even negative).

risk and return
Risk and Return

Risk and return will be very central terms in our analysis and it is essential that the reader clearly understands the meaning of each term and how assets with different payout structures can be compared. General utility theory suggests that the average investor is risk averse. Given the same expected return of two assets with different risks, he would prefer the one with less risk. (This assumption may not be perfectly true for all individuals in all situations, but for the investor community as a whole it is probably true). For an asset with uncertain cash flows and payoffs, which are normally distributed, the mean of the distribution will be the expected return while the standard deviation forms some kind of “risk”. Choosing the “less risky” asset therefore comes down to choosing the asset with the lowest standard deviation in its payout distribution. An investor could also approach the problem from the other direction, choosing among assets with the same risk and then choose the asset with the highest expected return.

portfolio theory
Portfolio Theory

When many assets are held together, assets decreasing in value can often be offset by other assets increasing in value, thus the decreasing risk The total variance of the portfolio is therefore almost always lower than a simple weighted average of the individual variances. If the number of assets is large enough, the total variance does in fact stem more from the covariances than from the variances of the assets. It is in other words more important how the assets tend to move together than how much each individual asset fluctuates in value.  

portfolio theory9
Portfolio Theory

The total risk (variance) of a portfolio of investments can be computed as:

where the xiyi are the weights of each investment, the ρ’s are the correlations between two investments, while the ’s are the standard deviations of each investment. The risk of additional investments is only relevant to the degree they correlate with the investments already in the portfolio. If n is large, the second term dominates the first one, making the individual variances completely unimportant.

efficient frontier
Efficient Frontier

As shown in the last section, the total risk of a portfolio is obtained through weighting and adding the risks of the individual investments in the portfolio. The same is true for the expected return of a portfolio, only this relationship is linear as opposed to the risk relationship displayed above for the variance of the portfolio.

This implies that it in most cases, as long as the investments are not perfectly correlated, it will be possible to obtain more favorable risk/return combinations by holding several investments at the same time. For each level of risk, there exists a maximum return obtainable through choosing wisely among the investments. If the level of risk is varied and that maximum return is calculated for each level, the result can be plotted in a diagram and that graph will have the form of an arc. The arc is called the efficient frontier, as it shows the most efficient allocation of funds possible. The efficient frontier is very central in the analysis of how hedge funds perform and what value they add to an existing portfolio.  

sharpe ratio
Sharpe Ratio

It is now time to discuss how investors should go about choosing their investments in a world where the simple idea that more return and less risk is good does not help in deciding between assets. Most often, the choice is between two (or more) assets, where one asset has both higher risk and higher return than the other. In that case, one would think that it is the individual preferences of the investor that decide which asset to choose. In a way, that is true, but it does on the other hand imply ranking assets despite their having different returns and risks.

One way of ranking investments taking into account both risk and return is by using the Sharpe ratio. This ratio essentially divides the return by the risk, after first subtracting the risk-free rate of return from the return, since any asset with a lower return should never be chosen. The higher the ratio the more favorable the risk/return characteristics of the investment. The Sharpe ratio is computed as:

Where R-bar is the mean rate of return of the asset and R-f is the risk-free rate of return. This measure can be taken to show return obtained per unit of risk. Apart from being very intuitive (dividing return above the risk-free rate by the risk of the asset), the Sharpe ratio does have some theoretic merit.

capm and sharpe alternatives
CAPM and Sharpe Alternatives

As stated above, one of the disadvantages of the Sharpe measure is for instance that it is not well suited to rank investments which are to be included in a portfolio since covariances matter at least as much as the standalone risk of the assets. The Sharpe ratio will be the predominant measure in the analyses, but a description of other measures is due.

Another risk measure suggested in the literature is the Treynor measure. This measure is not as straightforward and intuitive as the Sharpe ratio and requires an understanding of CAPM, the Capital Asset Pricing Model. The model was put forth in the 1960s and answers what the expected return should be for an asset of certain risk.

The model states that the return of an asset should equal the risk-free rate added together with some “risk premium” multiplied by the asset’s sensitivity to market movements, called “beta”. The risk premium is defined as the return of a “market portfolio” minus the return of the risk-free asset, where the market portfolio in theory is a portfolio consisting of all risky assets in the market. The beta is defined as:

capm and sharpe alternatives16
CAPM and Sharpe Alternatives

The beta of the market portfolio is thus, per definition, equal to one, while the betas of assets are more volatile than the market, i.e. IT stocks etc., can be far higher. CAPM suggests that in an efficient market, every investor is compensated exactly for the amount of risk he takes on, the higher beta his portfolio has, the higher his expected return is. With the basics of CAPM having been outlined, the Treynor measure can now be defined:

The Treynor measure is strikingly similar to the Sharpe ratio, only this time the excess return of the asset is related to the beta and not the standard deviation of the asset. The beta measures only the asset’s sensitivity to the market’s movements, while the standard deviation tells the full story of the asset’s volatility.

As a consequence, the Treynor measure addresses one of the drawbacks mentioned earlier regarding the Sharpe ratio; the Treynor measure works well when adding assets to a portfolio as the betas of the assets in a sense measure the covariances that become so important when the number of assets in the portfolio grows large.

capm and sharpe alternatives17
CAPM and Sharpe Alternatives

CAPM is however not a generally accepted model and the debate regarding it has raged ever since its inception almost 40 years ago. Finding the “market portfolio” is a difficult task, as it is supposed to include all risky assets in their relative proportion, of which only a fraction are traded and quoted with up-to-date prices. Proxies may of course be used but it is not clear what the scope of the proxy should be, whether the portfolio can be taken as domestic-only for a US-based investor or how much one would depart from the real market portfolio if the S&P 500 or DJIA were used as proxies.

The Treynor measure and all other measures (i.e. the Jensen differential performance index among others) based on betas, rely on a correctly defined market portfolio, but such a portfolio does obviously not exist. With CAPM evidence being inconclusive and the market portfolio non-existent, no beta-based performance measures will be used in the analyses. The reader should however be aware of the alternatives and that they in essence all try to measure the same thing, whether the asset is generating returns in line with its risk or not.

What is VAR of an asset portfolio?
  • Maximum expected loss on the portfolio over a particular horizon (eg. 1day, 1 week, 1 month, etc.) under normal market conditions at a prespecified confidence level, eg:
  • Daily VAR on a $10,000,000 fixed income plusequity portfolio: $100,000 at a 95% confidence level
  • Interpretation: Under normal market conditions, we can expect to lose no more than $100,000 (19 out of 20 times) over the next 24 hours

Mathematical definition of portfolio VAR

For normal portfolio return distributions

MVp = Market value of asset portfolio

a = Lower quantile of standard normal distribution

sp = standard deviation of daily portfolioreturns

t = time horizon period for forecasting