Portfolio Management and Measurements

Question 1

Modern Portfolio Theory

Answer 1

According to Markowitz, the investor should view the rate of return associated with any one of these portfolios as a random variable; such variables can be “described” by their moments, two of which are:

Expected (or mean) value - measure of the potential reward associated with any portfolio
Standard deviation - measure of the risk associated with any portfolio

Question 2

Measures of Return

Answer 2

Holding Period (Terminal Value - Initial Value) / Initial Value. This method’s major weakness is it fails to take the time value of money into account.

Dollar-Weighted Return (Internal Rate of Return) Breaks up the holding period so that the market value of the account after a change will be compounded by the amount of time it was earning the interest. It is the best way to measure an individual investor’s results.

Time-Weighted Return Calculates the return for the amount prior to a change caused by deposit or withdrawal. The individual returns are added together. It is more accurate than annualized returns.

Annualized Returns Either add the returns of the quarters together, or add 1 to each quarterly return, then multiply the four figures, and finally subtract 1 from the resulting product. This could be misleading because it does not consider how long each dollar was in the investment.

Question 3

Security Expected Return

Answer 3

This procedure involves calculating the expected return on a portfolio as the weighted average of the expected returns on its component securities. The relative market values of the securities in the portfolio are used as weights. In symbols, the general rule for calculating the expected return on a portfolio consisting of N securities is:

rp= X1r1 + X2r2 + ⋯+ XNrN
where:

rp = the expected return of the portfolio

XI = the proportion of the portfolio’s initial value invested in security I

rI = the expected return of security I

N = the number of securities in the portfolio

Question 4

Portfolio Standard Deviation

Answer 4

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.

How is the standard deviation of a portfolio calculated?

σp= Sqrt (W2iσ2i+W2jσ2j+2WiWjCOVij)

Question 5

Covariance

Answer 5

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.

The formula sheet for the CFP® certification examination has the following formula for determining covariance:

COV=SDi × SDj × corr. coeff.ij

Question 6

Correlation

Answer 6

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.

Question 7

Correlation vs covariance

Answer 7

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.

Question 8

R squared

Answer 8

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.

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.

Question 9

Negative corelated assets

Answer 9

Negatively correlated assets are NOT “necessary” to reduce risk (low positives are great).

Question 10

Example

Answer 10

Example
Security A Security B
Expected Return 8% 14%
Standard Deviation 12% 30%
Portfolio Weight 40% 60%

Correlation Coef. = 0.22
Question:

What is the portfolio’s return?

What is the portfolio’s standard deviation?

Answer:

The portfolio weighted return is (40% × 8%) + (60% × 14%) = 11.60%

The portfolio standard deviation is as follows:

Since it is needed as input, calculate covariance between security A and security B first.

12 × 30 × .22 = 79.20

[(SD2i×W2i)+(SD2j×W2j)+2(Wi)(Wj)(COVij)]12
[(12 × 12 × .40 × .40) + (30 x 30 × .60 × .60) + (2 x .40 × .60 × 79.20)] ½

[ 23.04 + 324.00 + 38.02 ]½

[385.06]½

= 19.62 (This is a percent.)

Question 11

Coefficient of Variation

Answer 11

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.

The higher the number the higher the risk we are carrying per unit of return.

Question 12

Risk of a portfolio

Answer 12

Two kinds of risk can be estimated:

the portfolio’s market (or systematic) risk, measured by its beta, and
the portfolio’s total risk, measured by its standard deviation.

Question 13

Risk-Adjusted Performance Measures

Answer 13

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

Question 14

Sharpe Ratio

Answer 14

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.

Question 15

Treynor Ratio

Answer 15

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.

Question 16

Alpha

Answer 16

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.

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.

Question 17

Information Ratio

Answer 17

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.

Question 18

Comparison of Risk Adjusted Return Measures

Answer 18

Jensen (alpha) and Treynor both use beta as the measure of risk, and beta only measures systematic risk. If the asset being measured is not fully diversified, or is not part of a greater diversified situation, then the Sharpe Ratio and the Information Ratio are the only appropriate performance indices to use.

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 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.

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.

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.

Question 19

Short comparison of the risk measure

Answer 19

Alpha - compares return earned above market risk.

Treynor Ratio - compares risk premium compared to risk of the market.

Sharpe ratio - compares risk premium compared to risk of the portfolio.

The higher the index number the better.

Question 20

Appropriate Benchmarks

Answer 20

The essential idea behind performance evaluation is to compare the returns obtained by the investment manager through active management with the returns that could have been obtained for the client if one or more appropriate alternative portfolios had been chosen for investment. Such comparison portfolios are often referred to as benchmark portfolios like S&P 500 or Russell 2000 etc.

Question 21

Probability Analysis

Answer 21

Probability analysis looks at the actual performance of a portfolio in comparison to its expected outcomes. When selecting securities for a portfolio, certain assumptions are made as to the expected return. These assumptions are accompanied by the probability of attaining the expected return. Probability analysis uses standard deviation, variance, coefficient of determination, beta and alpha to determine the risk adjusted return of the portfolio.

When looking back at the actual return over a period, one can compare how a portfolio did in comparison to the expected outcome of the period.