Portfolio Management and Measurements Flashcards
Portfolio Standard Deviation
A useful measure of risk should take into account both the probabilities of various bad outcomes and their associated magnitudes. Instead of measuring the probability of a number of different possible outcomes, the measure of risk should estimate the extent to which the actual outcome is likely to diverge from the expected outcome. Standard deviation accomplishes this objective.
The clearest example arises when the probability distribution for a portfolio’s returns can be approximated by the familiar bell-shaped curve used to describe a normal distribution. This is often considered a plausible assumption for analyzing returns on diversified portfolios when the holding period being studied is relatively short. As you may recall, over time there are other moments that will influence compounded returns, those being Skew and Kurtosis.
How is the standard deviation of a portfolio calculated? For the three-security portfolio consisting of Able, Baker, and Charlie, the formula is:
σp=√W2iσ2i+W2jσ2j+2WiWjCOVij
Covariance
Covariance is a statistical measure of the relationship between two random variables. That is, it is a measure of how two random variables, such as the returns on securities i and j, “move together.” A positive value for covariance indicates that the securities’ returns tend to move in the same direction. For example, a better-than-expected return for one is likely to occur along with a better-than-expected return for the other.
COV=SDi×SDj×corr. coeff.ij
Correlation
Closely related to covariance is the statistical measure known as correlation coefficient. In fact, when it comes to diversification, the correlation coefficient is the most important statistic. Correlation coefficients always lie between -1.0 and +1.0. A value of -1.0 represents perfect negative correlation, and a value of +1.0 represents perfect positive correlation. In the real world, most financial assets have positive correlation coefficients ranging in value from .4 to .9. However, for purposes of diversification, combining assets with anything other than perfect positive (+1.0) correlation will have diversification benefits. The lower the coefficient (say .4 vs. .7) the better, and negative is much better than positive. If you could ever find perfect negatively correlated assets (in theory anyway), you could have zero risk with just two assets. Your return would be with complete certainty.
The difference between correlation coefficient and covariance is that covariance is more of a refined statistic, designed to take specific asset risk into account. Correlation coefficients are raw figures, which simply measure the degree of variation between two assets returns from one period to the next.
The correlation coefficient squared is known as the coefficient of determination in the statistical-world, but commonly known as R squared in the every-day world. The R squared is another extremely important statistic, in that it tells you the degree to which a fund or a portfolio is diversified. Technically, it tells you the degree to which a dependent variable’s variation in returns (say a stock mutual fund), are explained by the variation of returns of an independent variable (say a benchmark such as the S&P). To now think of this statistic in a managerial context is the key. For example, if I have a fund with an R squared of .92, that tells me that 92% of the variation of the fund’s returns are due to systematic forces (non-diversifiable). More importantly, it tells me 8% of the variation of the fund’s returns are due to unsystematic or diversifiable risk.
Coefficient of Variation
Coefficient of Variation is a relative measure for determining if the return is worth the risk. Under the CAPM, it is an investment statistic that determines which investment is more efficient. The formula is standard deviation divided by the expected return. For example, if you are comparing two securities:
Security A Security B Expected Return 8% 5% Standard Deviation 10% 7% Coefficient of Variation 1.25 (10/8) 1.40 (7/5)
For every unit of return (for which Security A had 8 units), Security A has 1.25 units of risk. For every unit of return (for which Security B had 5 units), Security B has 1.40 units of risk.
Without using the Coefficient of Variation formula, you might select security B because it appears less risky. When in fact, security B took more risk to achieve its return versus security A.
Risk-Adjusted Performance Measures
As we have noted previously, many investors spend the bulk of their time looking at returns, however, professional investors spend just as much or more time reviewing risk. When it comes to compounded returns over time, avoiding major losses are critical to attaining higher returns.
After the periodic returns for a portfolio during a time interval have been measured, next we need to determine the performance of these returns. This requires an estimate of the portfolio’s risk level during the time interval. Two kinds of risk can be estimated:
1) the portfolio’s market (or systematic) risk, measured by its beta, and
2) the portfolio’s total risk, measured by its standard deviation.
Although a portfolio’s return and a measure of its risk can be compared individually with those of other portfolios, it is often not clear how the portfolio performed on a risk-adjusted basis relative to these other portfolios. In terms of the return, the portfolio might perform slightly above average. In terms of standard deviation, it might turn out less risky than approximately three-quarters of the other portfolios. Overall, these results suggest that the portfolio can do better on a risk-adjusted basis than the others, but it does not give the client a clear and precise sense of how much better.
Such a sense can be conveyed by certain CAPM-based measures of portfolio performance. Each one of these measures provides an estimate of a portfolio’s risk-adjusted performance, thereby allowing the client to see how the portfolio performed relative to other portfolios and relative to the market.
The following are CAPM-based measures of portfolio performance:
Sharpe ratio
Treynor ratio
Jensen ratio
Information ratio
Sharpe Ratio
Ranking portfolios’ returns averaged over several years is oversimplified because such rankings ignore risk. Thus, what is needed is an index of portfolio performance, which is determined by both the return and the risk.
William F. Sharpe devised the reward-to-variability index of portfolio performance, denoted SHARPEp. This defines a single parameter portfolio performance index that is calculated from both the risk and return statistics.
(r - Rf )/ Sd
r - Average return of the portfolio
Rf - risk free rate
Sd - standard deviation of returns for the portfolio
Sharpe’s index of portfolio performance measures the risk premium per unit of risk. The SHARPE index considers both risk and return and yields one index number for each portfolio. These index numbers may be used to rank the desirability of heterogeneous portfolios
Treynor Ratio
Jack Treynor conceived an index of portfolio performance that is based on systematic risk, as measured by a portfolio’s beta coefficient. In fact, the only difference between the Sharpe Ratio and the Treynor Ratio, is the different measures of risk in the denominator. Both ratios use “excess returns” in their numerators as the measure of return. Again, excess return in finance is something very specific: it is the realized returns on the security or portfolio, less the risk free rate.
(r - Rf )/ B
r - Average return of the portfolio
Rf - risk free rate
B - beta of returns for the portfolio. The systematic risk of the portfolio
Treynor uses beta coefficients and average returns to derive an index number suitable for ranking the desirability of assets in Beta - E(r) space. Some analysts prefer the TREYNOR portfolio performance measure because systematic risk is more relevant than total risk in certain applications and because the TREYNOR measure can be used to compare individual assets and portfolios. The TREYNOR performance measure has the disadvantage that its values can be sensitive to the market index used to estimate the investments’ betas.
Alpha
Michael C. Jensen developed another performance measure that uses beta as the measure of (systematic) risk. Although the name of this index is the Jensen Index, it is known commonly as “Jensen’s alpha” or “alpha.” Subtracting the required return from the realized return on the security or portfolio generates alpha. This required return portion of the formula should look familiar [everything in square brackets in the formula]. It is simply the CAPM (also referred to as the SML) used to generate a security’s required return.
Said another way, Alpha takes the actual return of a portfolio and subtracts the expected CAPM return based on the investment’s Beta.
α=Rp−[Rf+(Rm−Rf)b]
Interpreting Alpha–
When alpha is negative, it means that the return on the asset was less than was required given the amount of risk. The asset is overvalued, and should be avoided.
When alpha equals zero, the portfolio’s managers were successful in obtaining the minimum return that was required, given the securities risk premium. The asset is priced in equilibrium, and purchasing would be appropriate.
When alpha is a positive number, the asset managers achieved superior performance; they actually provided return beyond what was required given the asset’s risk premium. This necessarily means that the asset is undervalued, and should be purchased.
Jensen’s alpha measures risk-adjusted returns and is useful for evaluating the performance of both portfolios and individual assets. Alpha is used to compare risk premiums over systematic risks. However, the Alpha is not quite as easy to use for rankings as the other one-parameter performance measures.
Information Ratio
Also known as an appraisal ratio, the information ratio is another widely used performance measure. It measures a portfolio’s average return in excess of a benchmark portfolio, divided by the standard deviation of those excess returns.
(r - Rb )/ Sd
r - Average return of the portfolio
Rb - the benchmark return.
Sd - standard deviation of the excess return for the portfolio
The information ratio is useful in determining the investment manager’s skill in obtaining a portfolio return that differs from the benchmark against which the investment manager’s performance is being measured. The additional unsystematic risk the manager took to obtain these risks is also considered. The Information Ratio (like Sharpe and Treynor) is a relative measure that must be compared to other information ratios in order to make informed decisions.
