evestment-investment-statistics-a-reference-guide-september-2012.pdf

www.evestment.com | 1

Table of Contents

Introduction ................................ ................................ ................................ .............................. 4

I. Using Statistics to Understand Return Characteristics ................................ ......................... 5

Using Standard Deviation to Predict Possible Return Ranges ................................ .............. 6

Assessing Skewness and Kurtos is in the Return Distribution ................................ ............... 8

Predicting Returns using Monte Carlo Simulation ................................ ............................... 10

II. R isk Statistics and Risk -adjusted Statistics ................................ ................................ ....... 12

Standard Deviation ................................ ................................ ................................ .............. 12

Sharpe Ratio ................................ ................................ ................................ ........................ 14

Sortino Ratio ................................ ................................ ................................ ........................ 15 Omega Ratio ................................ ................................ ................................ ........................ 16

Drawdown Analysis ................................ ................................ ................................ ............. 21

Calmar Ratio ................................ ................................ ................................ ........................ 23 Sterling Ratio ................................ ................................ ................................ ....................... 23

Comparing Risk Statistics and Risk -adjusted Statistics ................................ ...................... 23

III. Correlation and Regression Analysis ................................ ................................ ................ 24

The Correlation Coefficient (R) ................................ ................................ ............................ 24

Alpha and Beta ................................ ................................ ................................ ..................... 23

The Coeffic ient of Determination (R 2) ................................ ................................ ................ 23 Benchmark Ratios ................................ ................................ ................................ ................ 24

IV. Peer Group Analysis ................................ ................................ ................................ ........... 26

Top Quartile Performance ................................ ................................ ................................ ... 26 Bottom Quartile Performance ................................ ................................ .............................. 27

Manager Search Criteria ................................ ................................ ................................ ...... 27

V. Composite Returns: Portfolio Construction, Optimization, Simulation .............................. 31

Portfolio Construction ................................ ................................ ................................ ......... 31

Optimization ................................ ................................ ................................ ........................ 36

Simulation ................................ ................................ ................................ ............................ 38

VI. Fat Tail Analysis, Risk Budgeting, Factor Analysis & Stress Testing ................................ . 40

Fat Tail Analysis ................................ ................................ ................................ ................... 40

VaR (Value at Risk) ................................ ................................ ................................ .......... 40 - ................... 40

ETL (Expected Tail Loss) ................................ ................................ .......................... 41

ETR (Expected Tail Return) ................................ ................................ ....................... 41

STARR Performance ................................ ................................ ................................ .. 41

Rachev Ratio ................................ ................................ ................................ ............. 41

Marginal Contribution to Risk (MCTR) / Marginal Contribution to Expected Tail Loss

(MCETL) ................................ ................................ ................................ ............................ 41

Percentage Contribution to Risk (PCTR) / Percentage Contribution to Expected Tail

Loss (PCETL) ................................ ................................ ................................ .................... 41


Skew ................................ ................................ ................................ .......................... 41

Excess Kurtosis ................................ ................................ ................................ ......... 41

Implied Return ................................ ................................ ................................ .......... 42

Risk Budgeting ................................ ................................ ................................ ..................... 43

Factor Analysis & Factor Contribution to Risk ................................ ................................ ..... 45

Stress Testing ................................ ................................ ................................ ...................... 47

Conclusion ................................ ................................ ................................ ............................... 48

Appendix I: Key Investment Statistics ................................ ................................ ................... 49

I. Absolute Ret urn Measures ................................ ................................ ............................... 50 1. Monthly Return (Arithmetic Mean): ................................ ................................ ............. 50

2. Average Monthly Gain (Gain Mean): ................................ ................................ ............ 50

3. Average Monthly Loss (Loss Mean): ................................ ................................ ........... 50

4. Compound Mo nthly Return (Geometric): ................................ ................................ .... 51

II. Absolute Risk -adjusted Return Measures ................................ ................................ ...... 52

1. Sharpe Ratio: ................................ ................................ ................................ ............... 52 2. Calmar Ratio: A return/risk ratio. ................................ ................................ ............... 52

3. Sterling Ra tio: ................................ ................................ ................................ ............. 53

4. Sortino Ratio: ................................ ................................ ................................ .............. 53

5. Omega: ................................ ................................ ................................ ........................ 53

III. Absolute Risk Measures ................................ ................................ ................................ 54

1. Monthly Standard Devi ation: ................................ ................................ ...................... 54 2. Gain Standard Deviation: ................................ ................................ ............................ 54

3. Loss Standard Deviation: ................................ ................................ ............................ 55

4. Downside Deviation: ................................ ................................ ................................ ... 55

5. Semi Deviation: ................................ ................................ ................................ ........... 56

6. Skewness: ................................ ................................ ................................ ................... 56

7. Kurtosis: ................................ ................................ ................................ ...................... 57

8. Maximum Drawdown: ................................ ................................ ................................ . 58

9. Gain/Loss Ratio: ................................ ................................ ................................ .......... 58

IV. Relative Return Measures ................................ ................................ .............................. 59

1. Up Capture Ratio: ................................ ................................ ................................ ........ 59

2. Down Capture Ratio: ................................ ................................ ................................ ... 59

3. Up Number Ratio: ................................ ................................ ................................ ........ 60 4. Down Number Ratio: ................................ ................................ ................................ ... 61

5. Up Percentage Ratio (Proficiency Ratio): ................................ ................................ ... 61

6. Down Percentage Ratio (Proficiency Ratio): ................................ .............................. 62 V. Rel ative Risk -adjusted Return Measures ................................ ................................ ....... 63

1. Annualized Alpha: ................................ ................................ ................................ ....... 63

2. Treynor Ratio: ................................ ................................ ................................ ............. 63 3. Jensen Alpha: ................................ ................................ ................................ .............. 64

4. Information Ratio: ................................ ................................ ................................ ...... 64

VI. Relative Risk Measure ................................ ................................ ................................ ... 65 1. Beta: ................................ ................................ ................................ ............................ 65

VII.Tail Risk Measures ................................ ................................ ................................ ......... 66

1. Value at Risk (Parametric V aR): ................................ ................................ ................. 66


2. Modified Value at Risk: ................................ ................................ ............................... 66

3. Expected Tail Loss (ETL): ................................ ................................ ............................ 66 4. Modifed Expected Ta il Loss (ETL): ................................ ................................ .............. 66

5. Jarque -Bera: ................................ ................................ ................................ ................ 67

6. STARR (Stable Tail Adjusted Return Ratio): ................................ ............................... 67

7. Rachev Ratio: ................................ ................................ ................................ .............. 67

About eVestment ................................ ................................ ................................ ..................... 68


Introduction

The purpose of this Guide is to assist you in gaining a better understanding of how to derive meaningful

conclusions from investment statistics.

A glossary of the key investment terms used in this Guide is provided in Appendix I at the end of the

document.

Learning Objectives

1. Explain how to use statistics to predict future investment returns.

2. Interpret the different investment risk statistics and risk -adjusted statistics.

3. Explain the concepts of correlation and regression analysis for investment analysis.

4. Describe the key characteristics of peer group analysis and its usefulness in the search process.


I. Using Statistics to Understand Return Characteristics

Investment statistics can be used in two ways:

To compare the performance histories of multiple investment managers

To try to predict a range of future returns for an investment.

When using statistics to predic

average annual return of the S&P 500 Index, over the 36 -year period from January 1975 to June

2011, was 11.84%. However, as Figure 1 highlights, if we assess the same data using only 1- or 3-year

rolling returns, they range between 61% and -43% on a 1 -year rolling basis, and between 33% and -

16% on a 3 -year rolling basis.


Similarly, if we assess a 5-year rolling period, the returns range between -3% and 20%. As Figure 2

illustrates, only when we lengthen the period to 10 years do we begin to see a true reversion to the

mean, or a narrowing of the spread of actual returns close to the long-term average or mean return. This

means that if an investment has a 20% return one year, per - haps the best prediction for its return the

- ch to

draw conclusions. Therefore, investors should not rely exclusively on statistics that only cover 1-, 3- or

even 5-year periods, since they may not be significant or meaningful over the long term.

B. Understanding Investment Return Characteristics

In this section, we review some of the methods and statistics used to predict investment returns,

including standard deviation, skewness and kurtosis, and Monte Carlo simulation.

Using Standard Deviation to Predict Possible Return Ranges

Can we use historical returns to predict future investment returns? As you can see with Figure 2, despite

all of our carefully analyzed averages of historical returns, the S&P 500 Index still experienced one of its

worst returns ever in 2008.

To help us predict future returns, we can generate a range of probabilities for the expected returns using

standard deviation as a mathematical measure of predictability, rather than using historical averages.


Standard deviation enables us to generate a probable range of expected returns. To demonstrate this, we

can assess the returns of the S&P 500 Index and develop a normal, bell-shaped distribution of returns for

the Index. Figure 3 illustrates the distribution of monthly returns for the S&P 500 Inde x.

From Figure 3, we see that the mean monthly return for the S&P 500 Index is 1.04% for the period

January 1975 to June 2011.

there is a 50% chance that the return will exceed 1. 04%, and a 50% chance it will not achieve this

return. As Figure 4 illustrates, there is a 75% chance that the next monthly return will be greater than -

1.66%, according to the shaded area under the curve. While some might find this information beneficial,

there are significant problems with relying too heavily upon standard deviation as a predictive statistic.

Perhaps the biggest problem is that very few investments display a normal distribution.


Assessing Skewness and Kurtosis in the Return Dist ribution

When returns fall outside of a normal distribution, the distribution exhibits skewness or kurtosis.

moment of the return distribution, with the me an and the variance being the first and second moments,

respectively. (Variance is a statistic that is closely related to standard deviation; both measure the

ments or

Skewness: Skewness measures the degree of asymmetry of a distribution around its mean.

Positive skewness indicates a distribution with an asymmetric tail extending toward more positive

values. Negative skewness indicates a distribution with an asymmetric tail extending toward more

negative values.

Kurtosis: Kurtosis measures the degree to which a distribution is more or less peaked than a

normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis

indicates a relatively flat distribution. A normal distribution has a kurtosis of 3. Therefore, an

investment characterized by high kurtosis will have

the extreme negative and positive ends of the distribution curve. A distribution of returns

exhibiting high kurtosis tends to overestimate the probability of achieving the mean return.

Figure 5 illustrates both the skewness and kurtosis in the return distribution for the S&P 500 Index from

Figure 4. The skewness is negative, which tells us that the returns are negatively biased. Because

kurtosis measures the steepness of the curve, we can tell that there is a steep curve by reviewing the

kurtosis number. A kurtosis less than zero indicate a relatively flat distri bution.


Skewness and kurtosis are important because few investment returns are normally distributed. Investors

often predict future returns based on standard deviation, but such predictions assume a normal

d kurtosis measure how its distribution differs from a normal

distribution and therefore provide an indication of the reliability of predictions based on the standard

deviation. As Figure 6 highlights, two investments with very different distribution profi les can have the

same mean and standard deviation. Therefore, it is useful to consider other methods for predicting

returns.

n to Omega, Con Keating and William Shadwick, The Finance Development

Center, 2002

Table 1 summarizes the key characteristics of a return distribution.


Predicting Returns using Monte Carlo Simulation

One method that can be used to predict returns is Monte Carlo simulation. Monte Carlo simulation is a

method of generating thousands of series representing potential outcomes of possible returns,

drawdowns, Sharpe ratios, standard deviations and other investments statistics of a specific investment

returns given its range of potential

returns. Software that uses this simulation method can assess the probability of an individual achieving a

allocation.

Monte Carlo simulation using a bootstrapping technique allows for both skewness and kurtosis to be

preserved. The bootstrapping technique involves resampling the actual data rather than assuming a

normal distribution like standard deviation does. Monte Carlo simulations randomly construct a

distribution of many possible returns for a portfolio over a specified time horizon. Thousands of possible

results are calculated, and a probability profile is constructed for the various statistics.

To see how this works, we can look at the stock market cr ash of 1987. From the period of January 1975

to August 1987, the largest drawdown for the S&P 500 Index was -16.52%, and the average return was

19.45%. Based on these numbers, few investors would have anticipated the crash of October 1987.

However, using Monte Carlo simulation, we can see that there was the possibility of a market crash even

in August 1987. Figure 7 shows the results of 10,000 Monte Carlo simulations on the S&P 500 Index.

Note that the 99th percentile indicates a possibility of a 28.83% dra wdown. This percentile indicates that,

however remote, there is the possibility of a significant drawdown, one which historical returns and

standard deviations do not predict.


Figure 8 shows the results of a Monte Carlo simulation that was run as of 6/30/2011. Each bar

represents the range of worst potential returns which have a 10% probability of occurring. As can be

seen in the chart, from 1975 to 2001 the S&P

500 Index never had a 3 year to 10 year period that fell within the range. However, as of June 2011, the

S&P 500 Index had experienced its worst performance in over a 25 year history. This example indicates

that although there may be a discrete probability that an event might oc cur, it does not specify exactly at

which time it will occur.


II. Risk Statistics and Risk -adjusted Statistics

Many investors approach manager selection and analysis with pre-conceived statistical prejudices based

statistic measures something that it does not. Others encounter difficulties trying to use a pre-defined

toolkit of investment statistics b

choose. It is important to remember, however, that investors have different notions of risk. To some, risk

is the uncertainty of achieving an expected return. To others, it is not ac hieving

a minimal acceptable return (MAR). Still others define risk as flat -out losing money. To illustrate this

Standard Deviation

Investors some

Because of the historical ties between risk and standard deviation in the world of traditional investments,

they equate high standard deviation with high risk, and the n use standard deviation as a comparative

statistic. However, in truth, standard deviation is merely a statistic that measures predictability. A high

standard deviation means that the fund is volatile, not that the fund is risky or will lose money, while a

low standard deviation means a fund is generally consistent in producing similar returns.

A fund can have extremely low standard deviation and lose money consistently, or have high standard

deviation and never experience a losing period. For example, without looking at the returns the fund in

Figure 9 exhibits a return pattern with overall consistency, which results in a low annualized standard

deviation of 3.8%.


Is the fund in Figure 9 a good investment? If we assess the same chart with returns plotted on the x -

axis, the exact opposite is true. As Figure 10 highlights, this fund, while maintaining a low standard

deviation, has a compound annual return of less than 1% (see circled area), and the fund has lost money

almost as often as it has generated profits.

Assessing funds based on standard deviation also tends to unjustly penalize funds with high upside

volatility. The fund in Figure 11 has a standard deviation of 22.5%, which is generally considered high.

However, the monthly returns are skew ed to the upside as the result of several months of 15+% returns

(see circled area).


One of the main differences between traditional return analysis and absolute return analysis is accepting

the fact that volatility is good, provided it is on the upside. Indeed, most investors should be less

concerned with upside volatility, and consider d

to achieve its return goal. For this reason, investors should acquaint themselves with downside deviation.

Downside deviation introduces the concept of minimum acceptable return (MAR) as a risk factor. If a

retirement plan has annual liabilities of not whether it has a

high or low standard deviation.

Downside deviation considers only the returns that fall below the MAR, ignoring upside volatility. As

Figure 12 illustrates, if the MAR is set at 8%, downside deviation measures the variation of returns below

this value.

So, with standard deviation out of the equation, what statistics can we use to compare funds? While fund

returns may seem useful, they do

use risk-adjusted statistics such as the Sharpe, Sortino, Sterling or Calmar ratios.

Sharpe Ratio

The Sharpe ratio is the best-known risk-adjusted statistic. You calculate an investmen

taking the average period return, subtracting the risk -free rate, and dividing it by the standard deviation

for the period.

This calculation generates a number we can use to compare investments. Note that for meaningful

comparisons, all comparative investment statistics must be calculated over the same time period.


deviation of 8%, while Fund B has a return of 20% and standard devia tion of 16%. If the risk -free rate is

4%, Fund A has a Sharpe ratio of 0.75, and Fund B has a Sharpe ratio of 1.0.

Comparing the Sharpe ratios, Fund B would have been the better investment because:

If we invested $1,000,000 in Fund A, with a 10% return, we would have $1,100,000 after 1 year.

If we invested $500,000 in Fund B, with a 20% return, and $500,000 in a bank account with a

4% return, we would have $1,120,000 after 1 year. This is called de -leveraging.

According to Sharpe, a higher standard deviation is not bad, provided it is accompanied by a

proportionally higher return. Note that there is no such thing as a good or bad absolute number for a

comparative investment statistic, only its relative relationship to other peers. Also note that in real w orld

investing, investors do not, in fact, de -leverage.

Sortino Ratio

Since upside volatility will decrease the Sharpe ratio of some investments, the Sortino ratio can be used

as an alternative. The Sortino ratio is similar to the Sharpe ratio; however it uses downside deviation

instead of standard deviation in the denominator of the formula, as well as substituting a minimum

acceptable return for the risk free rate. In other words, the Sortino ratio equals the return minus the

MAR, divided by the downside deviation.

Table 2 highlights the difference between the Sharpe and Sortino ratios using the S&P 500 Index and the

Barclays Aggregate Bond Index.

We can see from Table 2 that bonds have a somewhat higher Sharpe ratio than stocks (0.58 vs. 0.45).

However, if our goal is to achieve a MAR of 10%, the Sortino ratio favors stocks (0.09). For lower MARs,

the Sortino ratio favors bonds.


Omega Ratio

The omega ratio is a relative measure of the likelihood of achieving a given return, such as a minimum

acceptable return (MAR) or a target return. The higher the omega value, the greater the probability that

a given return will be met or exceeded. Omega represents a ratio of the cumulative probability of an

n level (a threshold level), to the cumulative

expected returns into two parts gains and losses, or returns above the expected rate (the upside) and

those below it (the downside). Therefore, in simple terms, consider omega as the ratio of upside returns

(good) relative to downside returns (bad).

Omega Ratio -

of return s.

Where

r is the threshold return, and

F is cumulative density function of

returns.

There are several ways to estimate the risk of not achieving a given return, but most of them assume

that returns are normally distributed. However, as stated above, investment returns are not normally

-

by the theoretical normal distribution). The omega calculation s are important as they use the actual

return distribution rather than a theoretical normal distribution. Thus the omega ratio and its components

more accurately reflect the historical experience of the investment being measured.

Since omega considers all

decisions are not static for at least two reasons:

1. As return informat ion is updated, the probability distribution will change and omega must be updated.

2.

change.

Therefore, omega allows investors to visualize the trade-off between risk and return at different threshold

levels for various investment choices. Note that when the threshold is set to the mean of the distribution,

the omega ratio is equal to 1.


Figure 13a and 13b highlights the omega ratios for two different threshold levels (a 0% return and a

10% return) for the S&P minimum

acceptable return (MAR) is 10% rather than 0%, the omega rankings among the investments will

change. For example, for a threshold return above 0%, F1 has the highest omega ratio followed by F2

and F3, but for a threshold return above 10%, F8 has the highest omega ratio followed by F9.


A key question to consider is how the Omega Ratio compares when ranking Omega Ratio relative to more

commonly used Statistics. Table 3 summarizes the return data for Fund A and a mix of asset class

benchmarks. The table is sorted on Sharpe Ratio, and the Omega Ratio has a 1% monthly return

threshold. When looking at the rankings, Bonds appear to be a good choice when looking at Sharpe

Ratio, however the Bonds have the lowest Omega Ratio at the 1% monthly return threshold. Fund A

provides the highest Sharpe and Omega ratios and has the smallest drawdown.


Because the Sharpe ratio is calculated from return data that has been averaged or annualized, the

resulting ranking of the investments do not include higher levels of information specific to the shape of

the distribution of the underlying return data. Therefore, it is reaso nable to conclude that the observed

differences in rankings are due to the higher levels of information contained in the Omega calculations. In

effect, Omega as a risk-adjusted measure provides investors with additional information to better

understand the

Figure 14 illustrates the omega ratios for five different investments (Funds A -E) for a 12-month holding

period. The two selected thresholds are a minimum acceptable return (MAR) of 2% (dashed purple line)

Figure 14 shows that there is considerable difference between the funds, as Fund A (the red line) has a

much better chance of exceeding the downside (i.e. it has a higher omega ratio at 2%).

However, there is less difference between the funds as far as earning a target return of 8%. This means

that choosing between the funds should be based more on the downside risk than on the expected

return. The 8% target is close to the crossover point of all the funds. The Omega ratio is a useful

investment tool because it can be used in a compact way to show how different investment options relate

to a target return and to a MAR.


Figure 14 used a 12-month holding period, which is appropriate for short -term expectations. However,

Figure 15 uses an investment horizon of 5 years (60-month holding period), and shows that downside

risk is less of a consideration for this period. The main decision is the target return. Once again, it is

essential to consider the specific time period when analyzing investment returns.

evestment-investment-statistics-a-reference-guide-september-2012.pdf

Documents

risk statistics

risk adjusted statistics

r isk statistics

sharpe ratio

sortino ratio

omega ratio

calmar ratio

sterling ratio