# Asset Allocation, Performance Measurement and Downside Risk

## Zusammenfassung

Investors should not and in fact do not hold a single asset, they hold groups or portfolios of assets. An important aspect in portfolio theory is that the risk of a portfolio is more complex than the risk of its components. It depends on how much the assets represented in the portfolio move together, that is, on the correlation between the single assets. In portfolio theory, there are several definitions of risk: First of all, the Capital Asset Pricing Model (CAPM) relies on the beta factor of an asset relative to the market as a measure for the assets risk. On the other hand, also downside risk can be used in order to determine a portfolios risk. The kind of risk in question is market risk, which is the risk of losses arising from adverse movements in market prices or rates. Market risk can be subdivided into interest rate risk, equity price risk, exchange rate risk and commodity price risk.

For many investment decisions, there is a minimum return that has to be reached in order to meet different criteria. Returns above this minimum acceptable return ensure that these goals are reached and thus are not considered risky. Standard deviation captures the risk associated with achieving the mean, while downside risk assumes that only those returns that fall below the minimal acceptable return incur risk. One has to distinguish between good and bad volatility. Good volatility is dispersion above the minimal acceptable return, the farther above the minimal acceptable return, the better it is.

One way of measuring downside risk is to consider the shortfall probability or chances of falling below the minimal acceptable return. Another possibility is measuring downside variance, i.e. variance of the returns falling below the minimal acceptable return.

As a consequence, downside variance is very sensitive to the estimate of the mean of the return function, while standard deviation does not suffer from this problem. Thus the calculation of downside deviation is more difficult than the calculation of standard deviation.

The quality of the calculation also depends on the choice of differencing interval of the time series. The calculation of downside risk assumes that financial time series follow either a normal or lognormal distribution.

Finally, there is no universal risk measure for the many broad categories of risk. For example, standard deviation captures the risk of not achieving the mean, beta captures the risk of investing […]

## Leseprobe

## Inhaltsverzeichnis

1 Introduction

2 Asset Allocation in a Downside Risk Framework

2.1 Expected Return

2.2 Variance and Standard Deviation

2.3 The Benefits of International Diversification

2.4 The Investment Process

2.4.1 Portfolio Selection

2.4.2 Asset Allocation Based on Alternative Risk Measures

2.4.2.1 Downside Risk Measures

2.4.2.2 Downside Risk Optimization

3 Estimation of Correlation and Volatility

3.1 Correlation

3.1.1 Computation of Correlation

3.1.2 Properties of Correlation

3.1.3 Forecasting Correlation

3.1.3.1 Simple Moving Averages

3.1.3.2 Exponentially Weighted Moving Average (EWMA)

3.1.3.3 Factor Models

3.1.4 The Influence of Correlation on Portfolio Weights

3.1.5 Autocorrelation

3.2 Volatility

3.2.1 Calculation of Volatility

3.2.2 Properties of Volatility

3.2.3 Forecasting Volatility

3.2.3.1 Simple Moving Average (SMA)

3.2.3.2 Exponentially Weighted Moving Average (EWMA)

3.2.3.3 ARCH(p)

3.2.3.4 GARCH (p,q)

3.2.3.5 Exponential General Autoregressive Conditional Heteroscedasticity

(EGARCH)

3.2.3.6 Multivariate Density Estimation (MDE)

3.3 The Link between Correlation and Volatility

3.4 The Influence of Volatility on Portfolio Weights

4 Performance Measurement

4.1 Traditional Performance Measures

4.1.1 Jensen Index

4.1.2 Treynor Index

4.1.3 Sharpe Ratio

4.2 Limitations of Traditional Performance Measures

4.3 Traditional Performance Measures Using Downside Risk Measures

5 Value at Risk

5.1 Definition of Value at Risk

5.2 Value at Risk as a Performance Evaluation Tool

5.4 Value at Risk as a Long Term Risk Measure

5.5 Limitations of Value at Risk

5.5.1 Aggregation Problems in Value at Risk

5.5.2 Tail-Fatness

5.5.3 Estimation Error in Value at Risk

5.5.4 Model Performance

6 New Risk Management Tools

6.1 Extreme Value Theory

6.2 Expected Shortfall

6.3 Maximum Loss

6.4 Factors at Risk

7 Empirical Part

7.1 Data Description

7.2 Autocorrelation

7.3 Estimation of Correlation and Volatility

7.4 Asset Allocation

7.5 Performance Measurement

7.5.1 Performance Measurement based on the Jensen and Treynor Indices

7.5.2 Performance Measurement based on the Sharpe Ratio

7.5.3 Performance Measurement Based on Value at Risk

7.5.4 Performance Measurement Based on Maximum Loss

8 Summary and Conclusion

A Appendix

A.1 Approaches to Measure Value at Risk

A.1.1 Delta-Normal Approach

A.1.2 Delta-Gamma Approach

A.1.3 Full Valuation

A.1.3.1 Historical Simulation

A.1.3.2 Structured Monte Carlo Simulation

A.1.4 Stress Testing

References

List of Abbreviations

List of Tables

## 1 Introduction

Investors should not and in fact do not hold a single asset, they hold groups or portfolios of assets. An important aspect in portfolio theory is that the risk of a portfolio is more complex than the risk of its components. It depends on how much the assets represented in the portfolio move together, that is, on the correlation between the single assets. In portfolio theory, there are several definitions of risk: First of all, the Capital Asset Pricing Model (CAPM) relies on the beta factor of an asset relative to the market as a measure for the asset’s risk. On the other hand, also downside risk can be used in order to determine a portfolio’s risk. The kind of risk in question is market risk, which is the risk of losses arising from adverse movements in market prices or rates. Market risk can be subdivided into interest rate risk, equity price risk, exchange rate risk and commodity price risk.

For many investment decisions, there is a minimum return that has to be reached in order to meet different criteria. Returns above this minimum acceptable return ensure that these goals are reached and thus are not considered risky. Standard deviation captures the risk associated with achieving the mean, while downside risk assumes that only those returns that fall below the minimal acceptable return incur risk. One has to distinguish between good and bad volatility. Good volatility is dispersion above the minimal acceptable return, the farther above the minimal acceptable return, the better it is.

One way of measuring downside risk is to consider the shortfall probability or chances of falling below the minimal acceptable return. Another possibility is measuring downside variance, i.e. variance of the returns falling below the minimal acceptable return.

As a consequence, downside variance is very sensitive to the estimate of the mean of the return function, while standard deviation does not suffer from this problem. Thus the calculation of downside deviation is more difficult than the calculation of standard deviation.

The quality of the calculation also depends on the choice of differencing interval of the time series. The calculation of downside risk assumes that financial time series follow either a normal or lognormal distribution.

Finally, there is no universal risk measure for the many broad categories of risk. For example, standard deviation captures the risk of not achieving the mean, beta captures the risk of investing in the assets available in the market, and downside deviation captures the risk of not achieving the minimal acceptable return necessary to accomplish some goal. They all provide useful information, but none of them provides all the information necessary to manage risk in every situation.

### 2 Asset Allocation in a Downside Risk Framework

### 2.1 Expected Return

The expected rate of return to a portfolio is simply the weighted average of the expected rates of return of the securities that are included in the portfolio. The weights are equal to the fractions of the investor’s wealth that is invested in each of the securities. For a portfolio with M securities, the expected return is equal to

This calculation method is necessary due to the fact that all portfolio optimization models do not directly indicate the amount that should be invested in each asset. Instead, they yield the optimal proportions for each security in the optimal portfolio.

### 2.2 Variance and Standard Deviation

In order to compute the variance of a portfolio of securities, the covariance matrix for the securities represented in the portfolio is required. For a portfolio consisting of three securities, the covariance matrix is given as follows:

Abbildung in dieser Leseprobe nicht enthalten

For the three-security portfolio, variance is calculated as

Abbildung in dieser Leseprobe nicht enthalten

More generally, the variance for a portfolio consisting of M securities is calculated as

Abbildung in dieser Leseprobe nicht enthalten

or, in matrix notation,

Abbildung in dieser Leseprobe nicht enthalten

where x is the vector of portfolio weights an dis the variance-covariance matrix of the assets represented in the portfolio.

The number of variances and covariances that need to be estimated increases with the number of securities available for investment. For a portfolio consisting of M securities, M variances have to be estimated, while the number of covariances that have to calculated is equal to components of the variance-covariance matrix, M are variances, while the rest of the matrix is symmetric, which implies that only half of the remaining covariances have to be estimated.

To compute the standard deviation of the portfolio, only the square root of the portfolio variance needs to be taken.

### 2.3 The Benefits of International Diversification

More and more investors rely on cross-border investments. One of the reasons for this phenomenon is the risk reduction that can be reached by investing internationally. This in turn is due to the fact that the correlations between international markets are usually lower than the correlations between markets in the same country, thus increasing the effect of diversification. An investor who diversifies internationally has to consider not only the return level, but also the risk reduction arising from adding foreign investments into his portfolio. The expected gain from international diversification can therefore be written as

Abbildung in dieser Leseprobe nicht enthalten^{[1]}

where internationally diversified portfolio including the world portfolio and the risk-free asset at time t, is the excess return on a portfolio consisting exclusively of domestic investments at time t, is the excess return on the world portfolio at time t and is the market price of risk per unit of risk. This can be explained as follows: If the investor is not diversified, his risk exposure will be equal to the volatility of the domestic market. For the same level of volatility, the world portfolio of stocks, combined with the risk-free investment, has an expected excess return which is equal to the variance of the domestic market times the market price of risk per unit of risk. The term in brackets can be interpreted as a measure of the nonsystematic risk in the domestic market, for which investors are not compensated.

Empirical results suggest that there was a positive effect of market liberalization on the correlation of international returns, which is partly responsible for changes in the expected gains from diversification.

Moreover, markets tend to move together especially in bear phases, i.e. the highest correlations can be found during pronounced market declines. This phenomenon has serious implications for international portfolio diversification. When poor performance is spread across markets, investors have only limited opportunities to shield their portfolios, which in turn leads to a decrease in the expected gain from international portfolio diversification. However, this should not be an argument against diversifying internationally, as the expected gains from international diversification are still positive, though small. A second factor that can be observed is that after bear market phases, the country specific volatility tends to increase more than the systematic risk. This in turn implies that the gains from international diversification increase considerably when the market starts to recover. As can be seen from the above discussion, the gains from international diversification are not constant over time, but they react to changes in the economic environment.

When investments into foreign equities and bonds are considered, the investor is faced with two types of risk and return: First, the risk and return generated from price changes of the foreign securities and second, the risk and return arising from fluctuations in the value of the foreign currency. The volatility of the exchange rates makes short-term commitments highly risky. The currency factor has an impact both on the return and the risk of the internationally diversified portfolio. Exchange rate gains improve the return of the foreign investment, while exchange rate risk accounts for a part of the total risk of the foreign investment. Usually the impact of currency risk on total risk is more pronounced in bond than in equity markets. Exchange rate risk is usually smaller than the systematic risk of the respective stock market, but it is often higher than the volatility of the corresponding bond market, which means that foreign investors can, in the short run, lose more on currency than on bond price movements.

A major drawback on international diversification is the fact that correlations tend to move together in bear market phases, thus eliminating a part or all of the diversification gain.

Generally, we can say that spreading investments over a number of foreign markets increases the chance to yield an improved return-risk ratio than investing exclusively in the domestic market.

### 2.4 The Investment Process

### 2.4.1 Portfolio Selection

Following classical portfolio theory, the decision about the composition of the portfolio depends on expected return and risk. The main rule is that investors should maximize the discounted value of future returns.^{[2]} The investment process is subdivided into three steps:

First of all, the investor needs to form his expectation about risk and return of the securities. This step is a particularly important part of the investment process. The forecasts can be obtained from a quantitative model or a qualitative model, or a combination of both. Naive models include historical averages, equal means and risk-adjusted equal means. The historical average approach assumes that excess returns will equal their historical averages. The problem with this approach is that historical means provide very poor forecasts of future returns. When applying the equal means approach, the investor assumes equal means for returns across all countries for each asset class. The problem here is that equal expected excess returns do not compensate investors in an appropriate way for the different levels of risk in assets of different countries. A similar approach is the use of risk-adjusted equal means as a predictor of future returns. Here the investor assumes that bonds and equities have the same expected excess return per unit of risk, where risk is measured by the volatility of asset returns. This approach has the disadvantage that it does not take the correlations of the asset returns into account. As an alternative, the investor can use the set of expected returns that would clear the market if all investors had identical views. Alternatively, the expected return of an equity or bond market can be modeled using fundamental variables, such as dividend yields, short and long interest rates, a January dummy variable, the spreads between the interest rates in the markets that are compared and lagged market returns.^{[3]} For foreign markets, also the foreign exchange rate should be included in the forecast.^{[4]} For the regression analysis, either the raw values of the variables can be used, or the variables can be set on an equal footing by standardizing them by subtracting the mean and dividing by the sample standard deviation. As these are linear combinations of the raw variables, this has no effect on the explanatory power of the model, but it makes it easier to detect the magnitude of the influence of the single variables on returns.

In a second step, the investor is concerned with the computation of the portfolio weights. This step is particularly complex when the investors considers investing internationally. In this case, the investor has to decide how much to allocate to each market, to each currency and finally to each individual asset available in the market. The portfolio weights derived according to mean-variance optimization are extremely sensitive to the estimate of the expected return in a market. This effect can be handled through a Bayesian refinement of the raw expected returns.^{[5]} The estimation is based on an unconditional or consensus expected return. The consensus return can be derived based on the benchmark portfolio. The consensus expected excess return depends on the beta of the individual asset class with respect to the benchmark portfolio:

Abbildung in dieser Leseprobe nicht enthalten

where is the consensus expected return to the benchmark.

Then the refined expected return conditional on the information available to the investor is derived. The forecast for the expected return is then given by

Abbildung in dieser Leseprobe nicht enthalten

where is the refined expected return conditional on the forecast This refined expected return differs from the unconditional expected return only if the forecast differs from its expectations. The extent to which the forecast differs from the consensus expectation depends on the correlation between the expected return and the forecast. This correlation coefficient is also referred to as the information coefficient (IC) of the forecast. The score of the forecast is defined as the standardized value of , that is

Abbildung in dieser Leseprobe nicht enthalten

Then the refined forecast simplifies to

Abbildung in dieser Leseprobe nicht enthalten

As a last step, the performance of the investment has to be assessed, which will be discussed in later sections.

For fixed probability beliefs about the expected risk and return, the investor has a choice of various combinations of return and risk depending on his choice of the portfolio weights. A rational investor prefers allocation with minimum risk for a given level of return and the maximum return for a given level of risk.

The basic model on which the investor bases his decisions was derived by Harry Markowitz. The investment decision process is given by the maximization of the investor’s utility subject to the following constraints.

Abbildung in dieser Leseprobe nicht enthalten

A portfolio that meets the two last requirements is also referred to as a feasible, obtainable or legitimate portfolio.^{[6]} A portfolio is called inefficient if there exists another obtainable combination which exhibits either a higher return at the same standard deviation or a lower standard deviation for a given level of return. A combination of return and risk is referred to as infeasible if no portfolio can meet its constraints.

The above portfolio optimization model can be expanded by imposing a maximum share that can be invested into each of the securities. This adds another constraint to the model, i.e.

Abbildung in dieser Leseprobe nicht enthalten

where are given constants that represent the maximum share of the investor’s wealth that can be invested into the individual securities. This modification reduces the feasible set of portfolios for the investor if the upper bounds for the individual securities are smaller than one.

Alternatively, a portfolio optimization model which allows for both short and long positions in all assets, i.e. which has as the sole constraint of the portfolio weights, is defined as Black’s model.

This model implies that portfolios including a large long position in one asset and a large short position in the other one are feasible solutions. For large amounts, this solution is not possible for every investor, while for smaller short positions, even if they are permitted, a collateral is required. The required collateral can be included in the optimization model. The constraints on the portfolio weights change to

Abbildung in dieser Leseprobe nicht enthalten

where A represents the total wealth of the investor and and denote long positions and short positions in the individual assets respectively. The percentage of the value of the short positions which is required as a collateral is given by. The first condition implies that the sum of the total long positions in the portfolio are not allowed to exceed the investor’s total wealth, while the second condition requires that the sum of the total short positions times the percentage that is required as a collateral does not exceed the investor’s total equity minus the value of the positions which cannot be used as a collateral.

The models of Markowitz and Black can be extended by allowing an investment in the risk-free asset. This model is then referred to as Tobin’s model. Funds can be invested at the risk-free rate of interest, while short selling of the risk-free asset is possible by borrowing at a cost that is equal to the risk-free rate of return. The efficient frontier in the Tobin model is a straight line which connects all portfolios that are linear combinations of the risk-free investment and the so-called tangency portfolio.^{[7]} The tangency portfolio is located at the point where the efficient frontier is exactly tangent to the efficiency curve of the Markowitz model.

The investor is confronted with a trade-off between expected return and risk. The higher the level of return the investors wants to reach, the higher is the risk he has to bear. On the other hand, an investor that is prepared to bear a higher level of risk can expect a higher return from his portfolio.

The total risk of a portfolio of securities depends on the correlation between the individual securities. A correlation coefficient that is smaller than 1 allows the investor to reduce risk, while for perfect positive correlation this is possible only with permission of short sales.

There are two main problems of quantitative models of asset allocation: First, expected returns are very difficult to estimate. Most investors are informed about returns for only a limited number of markets, while the standard optimization model requires them to provide expected returns for all assets and currencies. Thus investors need to rely on a set of auxiliary assumptions, such as the use of historical returns as predictors of future returns. The second problem is closely related to the first one: The asset weights and currency positions of standard asset allocation models are extremely sensitive to the return assumptions on which they are based. Equilibrium risk premia can help providing a neutral reference point for expected returns.^{[8]}

### 2.4.2 Asset Allocation Based on Alternative Risk Measures

The portfolio selection process mentioned above concentrates on the standard deviation of the portfolio return as a measure of risk. Other approaches rely on asymmetric measures of risk in order to determine the optimal portfolio. They are of interest because investors usually distinguish between upside potential and downside risk, while standard deviation implies that upside and downside risks are viewed with equal distaste. Asymmetric measures of risk are measures of risk that focus on a portion of the return distribution rather than the spread or dispersion of the overall distribution. All downside-risk measures are asymmetric, because they isolate return deviations in the left tail of the distribution that fall below the target rate.

##### 2.4.2.1 Downside Risk Measures

An example for a downside-risk measure is semivariance, which focuses on squared return deviations below the mean of the distribution. In a more general definition, all return dispersions lying below the target level of return are considered. Semivariance is considered the most theoretically robust measure of risk. It captures the notion of downside risk and is an appropriate definition of investment risk because investors are more concerned about losses relative to a certain benchmark level. An important feature of this measure is that it does not increase with growing upside potential, as standard deviation does.

Another concept of downside risk is target shortfall, which captures the severity of not achieving some minimum target or benchmark return. It represents the expected deviation of returns falling below the target rate.

Downside-risk optimization is thus referred to as an alternative formulation of the investor’s decision problem using a downside measure of risk instead of the variance or standard deviation of asset returns.

All downside-risk measures involve the tail of the relevant distribution of returns below a specified threshold level. As in the calculation only the left-hand tail of the return distribution is used, they are also referred to as „lower partial moments“.

The lower partial moment for an empirical distribution of portfolio returns is given by

Abbildung in dieser Leseprobe nicht enthalten

where pp is the probability that return Rp occurs. The type of moment is given by n, which captures the investor’s preferences by fixing the way in which the return dispersion below the target is characterized. For n = 1, LPM1 becomes the expected deviation of returns below the target, or the target shortfall. For n = 2, LPM2 is a probability weighting of squared deviations, thus equal to variance. The only difference to variance is that the deviations are determined with respect to the target rate. A special case arises when n is set equal to 0. Then, LPM0 is simply the probability of a loss. In this case, the investor does not care about the size of a gain or loss. His preferences are characterized by the fact that a shortfall, i.e. a return that falls below the threshold return, does not create any utility for him, while a return than exceeds the threshold level adds positively to his total utility, regardless of the size of the gain. If the utility for the investor from a return above the target return is equal to 1, the investor’s expected utility will be equal to the probability of exceeding the threshold return.^{[9]} For symmetric distributions, LPM2 is equal to one half of variance.

##### 2.4.2.2 Downside Risk Optimization

The downside risk measures mentioned above can also be used in the context of portfolio optimization. As in every portfolio choice problem, the investor selects a portfolio of assets so as to minimize risk for a given level of return. The optimization problem is similar to the optimization under the mean-variance approach.

Abbildung in dieser Leseprobe nicht enthalten

Total risk is captured in the optimization problem, but only in the mean of the return distribution and not in its risk.

If the optimization is based on historical data, pp can be replaced by Abbildung in dieser Leseprobe nicht enthalten , where T is equal to the number of observations.

Downside risk optimization yields substantially different results compared to mean-variance optimization in case of large skewness in asset returns.

Another tool for downside risk optimization is offered by the shortfall line concept.^{[10]} It gives the portfolio manager an indication about the amount that has to be invested into risky assets in order to meet specified requirements, such as a limitation of the maximum possible shortfall probability. As for the efficient frontier, expected return is plotted against the standard deviation of different portfolios. The shortfall line is the connection between the risk-free rate of interest and the mean returns on all portfolios with different risk levels that meet the shortfall constraint, i.e. the maximum shortfall probability for a given threshold return. For a maximum shortfall probability p and a threshold return, all return-risk combinations that plot above the shortfall line exhibit at least a probability of (1-p) of reaching a return that exceeds the threshold return. Contrarily, any portfolio that has at least a probability of p of falling below the threshold return will plot below the shortfall line. The maximum holding of risky assets that is consistent with this shortfall constraint is located at the intersection of the shortfall line and the efficient frontier.

This allocation is exposed to changes in a number of factors, such as volatility, risk premium estimation and the choice of the threshold return level. Generally, the lower volatility, the steeper is the efficient frontier and the higher the maximum holding of risky assets in order to reach the shortfall goal. The same phenomenon is true for the risk premium estimate. The higher the risk premium, the steeper is the efficient frontier and the higher the maximum holding of risky assets. As the shortfall line always emanates from the return threshold, an increase in the return threshold leads to a parallel shift of the shortfall line, i.e. the higher the return threshold, the lower the maximum holding of risky assets. Similarly, a decrease in the allowed shortfall probability leads to a steepening of the shortfall line. Thus, the lower the allowed shortfall probability, the lower the maximum holding of risky assets. When varying the time horizon of the investment, the lower volatility for longer investment horizons allows for more aggressive allocations with a higher share of risky assets than in the short run.

Alternatively, the optimization using the shortfall probability as a risk measure can be done by deriving the optimal portfolio weights from the weight of the respective asset in the minimum variance portfolio:^{[11]}

Abbildung in dieser Leseprobe nicht enthalten

where is equal to the return on a zero beta portfolio and is the deterministic target return level specified by the investor. The optimal share that should be invested in an individual asset is thus obtained by correcting the share of the asset in the minimum variance portfolio. The correction factor is equal to the ratio of excess returns over the target return. This correction increases the risk minimizing share. The allocation thus depends crucially on the target return level specified by the investor.

Another variation of this downside risk optimization problem is the tracking error approach^{[12]}. In this case, the portfolio manager is not given a deterministic threshold return, but a stochastic target, which means that instead of a fixed target return the portfolio manager is given a certain index as a benchmark and a value, by which the portfolio return is allowed to differ from the benchmark return. The tolerated difference can be interpreted as the allowed shortfall. Using the shortfall probability concept, the utility function of the investor is characterized as follows: Every return that exceeds the benchmark return corrected for, the investor’s utility increases, while any return that falls below the benchmark return corrected for does not give the investor any additional utility. If the additional utility generated from a positive excess return is equal to 1, the expected utility of the investor is equal to the probability of exceeding the target. Again, similar to the derivation for the deterministic target, the optimal weight for an asset can be derived from its weight in the minimum tracking error (MTE) portfolio:

Abbildung in dieser Leseprobe nicht enthalten

An alternative to the these asset allocation approaches is the asset allocation based on Value at Risk. In this case, the shortfall constraint is formulated as that the return on the portfolio will not fall below the target return with a sufficiently high probability.^{[13]} Formally, the asset allocation problem is given by

Abbildung in dieser Leseprobe nicht enthalten

where is the maximum probability of falling below the threshold return. It can be specified according to the risk aversion of the investor.

Under this approach, the result is very sensitive to the assumptions made about the distribution of the underlying return series. The normal distribution is subject to measurement errors when extreme returns occur. This is also referred to as tail-fatness, which means that extreme returns occur more often than is implied by the normal probability distribution. This problem is better solved by the Student-t distribution, which allows for fat tails according to the number of degrees of freedom. The smaller the number of degrees of freedom, the fatter the tails of the distribution. As increases towards infinity, the Student-t distribution becomes identical with the normal distribution. The use of the Student-t distribution instead of the normal distribution leads to a more aggressive allocation for higher shortfall probabilities. This is due to two effects. First, the use of the Student-t distribution leads to a more prudent allocation, because the probability of extreme negative returns is adequately taken into account. Second, the precision of the distribution increases, which leads to more certainty about the spread of the outcome, thus allowing for a more aggressive strategy. The latter effect dominates in case of higher shortfall probabilities, such as 5 percent, thus leading to the more aggressive asset allocation. On the other hand, for lower shortfall probabilities such as 1 percent, the first of the two effects dominates and leads to a more prudent asset allocation than under the assumption of a normal distribution of the underlying returns.

## 3 Estimation of Correlation and Volatility

### 3.1 Correlation

#### 3.1.1 Computation of Correlation

In financial theory, the notion of correlation is central, as important models such as the Capital Asset Pricing Model (CAPM) rely on the correlation coefficient as a measure of dependence between different financial instruments. Correlation is a measure of linear dependence, which is frequently used in risk management and portfolio optimization.

Correlation can be calculated in two steps: First, the covariance between the two time series has to be computed. In a second step, the resulting value is standardized to the correlation coefficient by dividing it by the product of the standard deviations of the two return series.

Covariance is computed as

Abbildung in dieser Leseprobe nicht enthalten

It is the weighted sum of the product of the deviations from the expected returns of two assets. The correlation coefficient can be computed by standardizing the covariance over the product of standard deviations of the two assets:

Abbildung in dieser Leseprobe nicht enthalten

By definition, the correlation coefficient always lies between -1 and 1. A positive correlation coefficient indicates that the there is a positive dependence between the two time series, while the opposite is true for negative correlations.

#### 3.1.2 Properties of Correlation

The future correlation structure of returns is an important component of asset allocation. The usual measure of correlation represents the average co-movement in both up and down markets. Separate correlation estimates in different return environments permit detection of whether correlation increases or decreases in down markets. Increased correlation in down markets reduces the benefit of portfolio diversification. In other words, the gains from portfolio diversification will be weakest in periods when these benefits are most needed. To detect differences in correlations in up and down markets, we can calculate a correlation coefficient for months with below-average return (negative semicorrelation) and for months with above-average performance (positive semicorrelation). In a more refined partitioning, the returns are classified so that one-third of the normal distributions falls into each bear, calm or bull market category.^{[14]} The observed correlation profiles are more u-shaped than predicted by the bivariate normal model. The lower tail of the empirically observed correlation profiles, that is the correlation in bear markets, is steeper than predicted by the bivariate normal model. The shape of the observed correlation profiles can be examined by comparing the differences between correlations in each of the three categories of return to the differences expected from the bivariate normal model. The difference between bear market and calm market correlation measures the curvature of the distribution at the left tail, while the difference between bull market and calm market correlations measures the curvature of the distribution at the right tail. Finally, the difference between bear market and bull market correlations gives indications about asymmetry in the correlation profiles.

The implication for portfolio management arising from this phenomenon is that portfolios formed on the basis of average correlations, which implicitly assume symmetry, might exhibit a worse performance in down markets than those formed on the basis of expected correlations. An internationally diversified portfolio becomes riskier in bear markets because of higher volatility and greater correlation among the national markets. The higher correlation amplifies higher portfolio variance through the covariance. This in turn decreases the benefits from international diversification in times when these are most needed. The benefits from international asset allocation are likely to be overestimated for markets with higher-than-normal correlations during bear markets. Because actual portfolio performance is lower than predicted performance in bear markets, countries which have higher-than-normal correlations during bear markets should receive lower portfolio weights than suggested by mean-variance optimization.

Correlations exhibit a strong interaction with the business cycle. The international cross-correlations are highest when both economies are contracting, while they are lowest when both economies show an upward trend.^{[15]}

Correlation is not constant over time, so it is difficult to identify a stable trend. They are most volatile for shorter observation periods and more stable for longer intervals, such as three or five years. This is due to the fact that longer moving averages are used, which makes the curve smoother. Generally, countries with an open economy have higher correlations with other markets, as they are open for foreign ownership of firms and foreign investments. This phenomenon can be observed for Great Britain after the deregulation and opening of the economy initiated by the Thatcher administration, and for European Union countries due to the progressive economic and monetary integration.

Among the shortcomings of correlation as a dependence measure there is the fact that the variances of both variables have to be finite, which is problematic in the case of heavy-tailed distributions.^{[16]}

#### 3.1.3 Forecasting Correlation

##### 3.1.3.1 Simple Moving Averages

This method is based on moving averages using a fixed window length, e.g. one month or one quarter. As in the estimation of volatility, the main problem of this model is that it places the same weight on all observations in the sample, even though more recent observations contain more decision-relevant information than older ones. Moreover, dropping observations from the window sometimes has severe effects on the measured correlations.

##### 3.1.3.2 Exponentially Weighted Moving Average (EWMA)

The Exponentially Weighted Moving Average which is used by JPMorgan’s RiskMetrics in order to forecast volatility can also be used as a forecast tool for covariances and correlations. The first step in the estimation process is the estimation of the covariance between the two time series. The covariance between the returns of two assets can be written as

Abbildung in dieser Leseprobe nicht enthalten

or alternatively as

Abbildung in dieser Leseprobe nicht enthalten

where is set to be 0,94 for daily data sets and 0,97 for monthly data sets. The number of daily returns that are used to generate these results is equal to 550.

In order to obtain the correlation coefficient between the two time series, the following formula has to be used:

Abbildung in dieser Leseprobe nicht enthalten

In order to scale covariance values to higher time intervals, it is necessary to multiply the value derived on a daily basis by the factor , where T is equal to the number of trading days during the desired time interval. The correlation forecast is then equal to the forecast on a daily basis, because the ratio of the scaling parameters used for volatility and covariance is equal to 1:

Abbildung in dieser Leseprobe nicht enthalten

##### 3.1.3.3 Factor Models

A correlation forecast can be based on a number of instrumental variables^{[17]}, such as lagged correlation, the lagged returns in both markets, dividend yields as well as the term structure of interest rates in both countries, i.e. the difference between long-term government bond yield and short-term bill yield. This last variable is important in order to capture business cycle effects, which are clearly reflected by the term structure of interest rates in the two countries. The lag length depends on the forecasting horizon, i.e. for a five-year forecasting horizon, the lag is 60 months.

For correlations between currencies, the dividend yield as an explaining variable can be replaced by the short-term interest rate differential.

Because of the strong interaction between correlation and volatility, volatility forecasts can be used to forecast changes in international correlation.

#### 3.1.4 The Influence of Correlation on Portfolio Weights

Correlation is one of the most important input factors that are necessary for portfolio optimization. The weights for the single assets depend heavily on the estimation of the correlation coefficient between the assets represented in the portfolio, which in turn implies that correlation also affects the return generated from the portfolio of securities. The following three tables consider three possible degrees of correlation: 1, zero and -1. If the correlation coefficient is equal to 1, this means that the assets are perfectly positively correlated. In this case, the investor cannot take advantage of the diversification effect, because adverse movements of one of the assets imply that also the other assets will exhibit an unfavorable development for the investor. If the assets are uncorrelated, risk can be reduced by investing into both assets rather than only in one of them. The diversification effect is most advantageous when the assets are perfectly negatively correlated. This implies that the investor can eliminate risk completely, the mimimum variance portfolio in this case has zero variance. In reality, perfectly correlated assets cannot be found, however, strong correlation helps the investor reduce the risk of his portfolio.

In a two-asset portfolio or a portfolio consisting of two sub-portfolios^{[18]}, the weight of asset A in the minimum variance portfolio (MVP) is given by^{[19]}

Abbildung in dieser Leseprobe nicht enthalten

This value depends crucially on the estimate for the correlation coefficient . To quantify this dependence, we need to analyze the first derivative of with respect to , i.e. According to the differentiation rule for quotients,

Abbildung in dieser Leseprobe nicht enthalten

For , the first derivative is

Abbildung in dieser Leseprobe nicht enthalten

In order to determine the sign of the derivative, a separate analysis of the numerator and the denominator is necessary. The numerator is positive, if

Abbildung in dieser Leseprobe nicht enthalten

that is, if

Abbildung in dieser Leseprobe nicht enthalten

Analogously, the numerator is negative if

Abbildung in dieser Leseprobe nicht enthalten

The analysis of the denominator is more complex. According to the degree of correlation, the individual terms of the denominator behave as follows:

Abbildung in dieser Leseprobe nicht enthalten

Table 1: Sign of Denominator

The numerator is always positive for negative and zero correlations, while it has its minimum of zero and intersect with the horizontal axis at. However, this value is greater than 1 for all , so it does not need to be considered any further. For all other positive correlations, the denominator is positive.

Abbildung in dieser Leseprobe nicht enthalten

So we have to distinguish between the following cases:

Abbildung in dieser Leseprobe nicht enthalten

Table 2: Sign of First Derivative of Asset Weight with Respect to Correlation

The analogous result holds for the share of the second asset, xB.

An underestimation of the correlation coefficient leads to an underestimation of the weight of the less volatile asset in the portfolio and to an overestimation of the weight of the more volatile asset in the minimum variance portfolio. This implies that the portfolio chosen bears a higher risk than then true minimum variance portfolio.

Vice versa, an overestimation of the correlation coefficient leads to an overestimation of the weight of the less volatile asset in the portfolio and to an underestimation of the weight of the more volatile asset in the minimum variance portfolio.

In down markets, correlation tends to increase, and for high correlations, the sensitivity of asset weights to differences in the correlation coefficient is particularly high and positive, as the graphs show. For this reason, the estimation of correlation and its accuracy are especially important in down markets.

Abbildung in dieser Leseprobe nicht enthalten

Table 3: Weight of for

Abbildung in dieser Leseprobe nicht enthalten

Table 4: Weight for for

Abbildung in dieser Leseprobe nicht enthalten

Table 5: Risk and Return with Correlation = -1

Abbildung in dieser Leseprobe nicht enthalten

Table 6: Risk and Return with Correlation = 0

Abbildung in dieser Leseprobe nicht enthalten

Table 7: Risk and Return for Correlation = 1

#### 3.1.5 Autocorrelation

Autocorrelation measures the dependence between a variable and its past realizations, the autocorrelation coefficient measures the correlation of returns across time. Stationary time series fluctuate around their means. They are also referred to as mean-reverting since, regardless of the fluctuations’ amplitudes, the series reverts to its mean. Non-stationary time series have a drift term that grows with time. This drift term is represented by the term.

The kth order autocorrelation coefficient is defined as^{[20]}

Abbildung in dieser Leseprobe nicht enthalten

where k is equal to the time lag that is used in the calculation. For a given sample of returns, the sample autocorrelation coefficient can be calculated as

Abbildung in dieser Leseprobe nicht enthalten

where denotes the sample mean. If a large amount of historical data is available, a 95 percent confidence band for each autocorrelation coefficient can be calculated as . If a time series is not autocorrelated, then estimates of will not be significantly different from zero.

A time series can be tested for autocorrelation in two different ways. First, plotting the autocorrelation coefficient against the time lag k gives a general idea about the autocorrelation behavior of the time series. The more values are situated within the confidence band, the more likely it is that the time series follows a random walk. A more formal way of detecting autocorrelation in a time series is the Box-Ljung test statistic. It is defined as

Abbildung in dieser Leseprobe nicht enthalten

Under the null hypothesis that a time series is not autocorrelated, BL(p) is distributed chi-squared with p degrees of freedom. p denotes the number of autocorrelations used to estimate the statistic. Values for the BL(p) statistic that are higher than the appropriate quantile of the chi-squared distribution indicate that there is a significant degree of autocorrelation.

### 3.2 Volatility

#### 3.2.1 Calculation of Volatility

Volatility is defined as the standard deviation of historical returns. It is equal to the square root of the variance of return, which in turn is the weighted sum of squared deviations of returns from the average return:

Abbildung in dieser Leseprobe nicht enthalten

This measure, which gives information about the variability of an asset’s returns, can be interpreted as follows:^{[21]} In about two out of three cases, the returns lie within one standard deviation from the mean, that is, in the interval . In 95 out of 100 cases, that is, with a probability of 95 percent, the returns lie within two standard deviations from the mean, i.e. in the interval.

#### 3.2.2 Properties of Volatility

One of the main characteristics of volatility is volatility clustering. It gives indications about the predictability of volatility. If large changes in financial markets tend to be followed by more large changes in either direction, then volatility must be predictably high after such large movements. However, the fact that volatility is predictable does not imply that the size or direction of market moves can be predicted.

Volatility is an important risk measure that is relevant not only for the pricing of derivative instruments, but also for the determination of optimal portfolio weights.

Volatility is not constant over time. In statistics, changing variances or volatilities are often denoted by the term heteroscedasticity.

The properties of volatility vary from market to market. The foreign currency market is an example for a market that exhibits different features from other time series, such as equity or bond markets. This special behavior has several reasons: One of these is that foreign exchange markets are highly liquid, the transaction costs are low and trading is possible around the clock. In foreign exchange markets, it is also possible to capture the effects of trading on the basis of private or public information. The effects of trading on the basis of public information can be observed rather easily, because foreign exchange rates are very sensitive to announcements of macroeconomic data, e.g. inflation, economic growth or the unemployment rate. In the United States, these announcements are usually made on Thursdays and Fridays between 7:15 a.m. and 8:30 a.m. Central Time.^{[22]} Another reason for the concentration of volatility on Friday morning is the reduced market liquidity in the time interval between the closing of the European markets and the opening of the United States market. Within this time band, a significant increase in volatility can be observed. Also trading on the basis of private information becomes observable in foreign exchange markets. In the US dollar market, the most important privately informed trader is the New York Federal Reserve Bank. They usually intervene via open market operations around 10:35 a.m. and 11:15 a.m. during weekdays.

For foreign currencies, the ratio of week-day and week-end volatilities is significantly lower than in equity markets. This is due to the fact that foreign exchange markets are active 24 hours a day and that there are relatively few macroeconomic news during week-ends. The quotations for European currencies against the US dollar are most volatile when the US markets are active, while cross rates between European currencies exhibit the highest volatility during European trading hours. This can be explained by the fact that European currencies react more to European economic news disclosures. Also, volatility estimates can differ according to the time at which the prices are measured. Differences might occur when using daily high or low prices instead of instead or end-of-day data or vice versa.

**[...]**

^{[1]} cf. De Santis/Gerard (1997) p. 1905

^{[2]} cf. Markowitz (1952), p. 77

^{[3]} cf. Kahn/Roulet/Tajbakhsh (1996) p. 24

^{[4]} cf. Kahn/Roulet/Tajbakhsh (1996) p. 25

^{[5]} cf. Kahn/Roulet/Tajbakhsh (1996) p. 28

^{[6]} cf. Markowitz (1959) p.4

^{[7]} cf. Alexander/Francis (1986) p. 61

^{[8]} cf. Black/Litterman (1992), p. 29

^{[9]} cf. Reichling (1996) p. 32

^{[10]} cf. Leibowitz/Kogelman (1991) p. 19f.

^{[11]} cf. Reichling (1996) p. 35

^{[12]} cf. Reichling (1996) p. 40f.

^{[13]} cf. Lucas/Klaassen (1998) p. 72

^{[14]} cf. Butler/Joaquin (2000) p. 1

^{[15]} cf. Erb/Harvey/Viskanta (1994) p. 34

^{[16]} cf. Embrechts/McNeil/Straumann (1999) p. 7

^{[17]} cf. Erb/Harvey/Viskanta (1994) p. 39

^{[18]} This simplification can be justified by the two-fund theorem, which says that the efficient frontier can be described by combinations of only two efficient portfolios.

^{[19]} cf. Fischer (1996) p.46

^{[20]} cf. RiskMetrics (1996) p. 56

^{[21]} cf. Auckenthaler (1991) p. 65

^{[22]} cf. Harvey/Huang (1992) p. 14

## Details

- Seiten
- Erscheinungsform
- Originalausgabe
- Jahr
- 2001
- ISBN (eBook)
- 9783832432218
- ISBN (Paperback)
- 9783838632216
- DOI
- 10.3239/9783832432218
- Dateigröße
- 1 MB
- Sprache
- Englisch
- Institution / Hochschule
- Universität Wien – unbekannt, Betriebswirtschaftslehre
- Erscheinungsdatum
- 2001 (März)
- Note
- 1,0
- Schlagworte
- value-at-risk korrelationen extreme value theory portfoliomanagement risikomessung