Sample vs. Population

1.3 Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility

1.3.1.1 Sample vs. Population

Ideally, when doing a statistical analysis, one would like to have all data points available. For example, if the daily returns of the Dow Jones Industrial Average Index were known since its inception in 1884, then all important key figures, such as arithmetic mean, variance or standard deviation, could be exactly calculated.

The totality of data points for a random variable is calledpopulation. However, in reality, we usually only have access to asampleof this population. Furthermore, it is sometimes not meaningful to take all data points into consideration, for example, the standard deviation of the Dow Jones Industrial Average Index since its creation might be of no use since the market activity may have changed over time. Therefore, data of a recent time period is preferred and inferences are made from a sample to the population.

Let us have a look at, for example, the standard deviation. The formula for the population standard deviation of a random variableX withN observations is¹⁶

D

vu ut1

N XN iD1

.XiX/² (1.23)

with X being the arithmetic mean 1 N

XN iD1

Xi. When we observe a population and calculate the arithmetic mean, then this is the expected value of X. If you

16Compare Esch et al. (2005, p. 41).

Table 1.6 The importance

of sample vs. population A B C

Monthly portfolio Monthly benchmark

1 Month performance performance

2 07/2012 6.10 % 6.01 %

3 08/2012 5.50 % 5.45 %

4 09/2012 4.70 % 4.63 %

Source: Own, for illustrative purposes only

compare Eq. (1.23) with the sample standard deviation in Eq. (1.22), then the only difference is the factor _N¹ instead of_N1¹ .

The reason for this distinction between the factors _N¹ or _N1¹ is the so-called unbiased estimator. If the expected value of the sample standard deviation is equal to the population standard deviation, the estimator for the population standard deviation in Eq. (1.22) is unbiased. This is achieved using _N¹₁ instead of _N¹ in Eq. (1.22).

In the following, we will illustrate the difference in the formulas for population and sample estimators. Let us first look at Table1.6where you find the monthly performance of a hypothetical portfolio and its benchmark for a period of3months, i.e.,N D3.

Below you see the outcome of the different Excel^rformulas either using:P in case the three data points characterize a population and:S if they represent only a sample of a population.

• Population monthly portfolio variancePf^2;monthly: 0:003% D VAR:P .B2WB4/

• Sample monthly portfolio variancePf^2;monthly: 0:005% D VAR:S.B2WB4/

• Population monthly portfolio volatilityPf^monthly: 0:57% D STDEV:P .B2WB4/

• Sample monthly portfolio volatilityPf^monthly: 0:70% D STDEV:S.B2WB4/

Obviously, there are differences that cannot be ignored. They are evident, because we look at a population or sample that consists only of three data points. It makes a big difference if one divides by2or by3. Since, as a rule of thumb, the deviations vanish if the number of observationsN is greater than 30, the size of available data is crucial. On the other hand, the factors _N¹ or _N1¹ have no impact on the correlation as long as it is computed by using either the population formulas or the sample formulas. This is because in the formula for the correlation, see Eq. (1.47), the terms_N¹ or_N1¹ simply vanish.

Let us look at a concrete example.

Example 3

An analyst follows the stock of companyXYC Ltd.over a 3-month period. At the beginning of the first month, the stock has a value ofe100. It rises toe110 at the end of the first month, and toe122 at the end of the second month. At the end of the third month the stock is priced ate136.

In order to calculate the volatility using the formula in Eq. (1.22), we first have to calculate the percentage stock returns every month. Then we calculate the arithmetic average of these three percentage returns and plug the results into Eq. (1.22). Knowing that, in practice, only three relative returns are insufficient to get a meaningful volatility value, this example is only for illustrative purposes.

Calculation for the percentage returnsrXYC^k for each subperiodk D 1; 2and 3DN:

rXYC¹ D e110e100

e100 D10% rXYC² D e122e110

e110 D10:91% rXYC³ D e136e122

e122 D11:48%:

Calculation of the arithmetic average of the percentage returns in the three subperiods:

rXYCD rXYC¹ CrXYC² CrXYC³

N

D 10%C10:91%C11:48% D10:80%: 3

According to Eq. (1.22) the stock’s monthly volatilityXYC^monthlyis then

XYC^monthly .1.22/D

vu ut 1

N 1 X3 kD1

.rXYC^k r^XYC/²

D s 1

31Œ.10%10:80%/²C.10:91%10:80%/²C C.11:48%10:80%/²

D0:75%: End of Example 3

In practice, neither a monthly nor a daily volatility is used, rather an annual volatility. This would call for annual returns over non-overlapping consecutive years and the time series would need to be sufficiently long, for example, 20 years, in order to get a meaningful volatility figure. However, this is not practicable: Over such a

long time, the management of a fund will most likely have changed or the fund will have existed only for a much shorter time period. But to calculate an annual volatility using only a few years of returns is meaningless. This brings us to the question of how to calculate anannualizedvolatility using less data.

In order to do this we use daily or monthly return data and annualize them.

At least 1 year’s worth of data are required, i.e.,T 1year. For example, we look at a historical3-year period using either36monthly percentage returns or (roughly) 750daily percentage returns.

Let us assume we haveN 12monthly percentage returns. Using this data in Eq. (1.22) yields the monthly volatilityPf^monthly. We now can annualize this monthly volatility which yields the annualized volatilityPf^p:a: by scaling Pf^monthly with the square root of12, i.e.:

Pf^p:a: D p

12 Pf^monthly: (1.24)

Similarly, if daily percentage return data are used, the calculated daily volatility Pf^dailycan be annualized by scaling it with the square root of business days per year which is roughly252, i.e.:

Pf^p:a: D p

252 Pf^daily: (1.25)

For the interpretation of volatility we have to turn to a research field called financial engineering. Assuming that the portfolio value follows a so-called Geo- metric Brownian Motion (GBM), the percentage returns of the portfolio in the subperiods are normally distributed.¹⁷ Accordingly, we assume that for a future 1-year time period Œ0; 1 the annualized return R_i^p:a:, as a random variable, is normally distributed. The interpretation of the annualized volatility _i^p:a: of an asseti, as shown in Fig.1.7, is then as follows:

Let iDEŒR^p:a:_i be the expected annualized return and _i^p:a: its annualized volatility. Then the probability distribution for the annualized returns R_i^p:a: is described by a symmetrical bell-shaped curve which peaks atR_i^p:a:Di. In this figure, the probability of the return being between two return values is the stated area under the curve (the entire area under the graph is1). The percentages in the diagram show the area under the graph in the respective intervals, for example, the probability for the return to lie betweeni andiC_i^p:a:is34:13%. Thus, the probability of the annualized returnR_i^p:a:to be

• at most_i^p:a:off from the expected annualized returni is68:3%.

• at most2_i^p:a:off from the expected annualized returni is95:4%.

• at most3_i^p:a:off from the expected annualized returni is99:7%.

17Please note that no further mathematical analysis on this will be provided here. For further details, please see the basic introductions in Schulmerich (2010a, Chap. 2), Schulmerich (2005, Chap. 2), Schulmerich (2008a, Chaps. 1 and 2) or Schulmerich (1997, Chaps. 1 and 2). For a rigorous mathematical introduction to this topic see, for example, Neftci (2000) or Øksendal (1995).

Return density

Annual returnR^p.a.i

μi

μi−3σ_i^p.a.

μi−2σ^p.a._i

μi−σ^p.a._i μi+σ^p.a._i

μi+ 2σ^p.a._i

μi+ 3σ_i^p.a.

34.13 % 34.13 %

13.59 % 13.59 %

2.14 % 2.14 %

0.13 % 0.13 %

Fig. 1.7 Graphical illustration of annualized volatility for normally distributed returns.Source:

Reilly and Brown (1997, Appendix D, p. 1047)

Returndensity

Annualized returnR^p.a._i μi

μi−1.65·σ^p.a._i μi−1.96·σ^p.a._i μi−2.33·σ_i^p.a.

5 % 2.5 % 1 %

Fig. 1.8 Graphical illustration of the worst-case scenarios (1,2:5,5%) for normally distributed returns.Source: Reilly and Brown (1997, Appendix D, p. 1047)

Volatility can also be used to illustrate worst-case scenarios, as shown in Fig.1.8.

Typically investors look at the worst5,2:5and1% outcomes when they want to assess risk. The lines below the return axis show the left tails which make up5,2:5 and1% of the total area under the graph, corresponding to our worst-case scenarios.

The probability that the annualized returnR_i^p:a:is less than

• the expected annualized return i minus 1.65 times the annualized volatility _i^p:a:is5:0%.

• the expected annualized return i minus 1.96 times the annualized volatility _i^p:a:is2:5%.

• the expected annualized returniminus 2.33 times the annualized volatility_i^p:a:

is1:0%.

Let us now look at a practical example for calculating volatility.

Example 4

In Table 1.7the monthly and annualized volatility is calculated based on the percentage returns of a portfolio and its benchmark for18consecutive months.

The return data are the same as shown in Table1.4for Example 1. Using the column and row notation from Microsoft^rExcel^r, the first column is labeled A(month) and the secondB(monthly portfolio performance). Applying Excel^r functions, the volatility of the portfolio can easily be calculated:

• Monthly portfolio volatilityPf^monthlyin cellB20:

4:76% D STDEV:S.B2WB19/

• Annualized portfolio volatilityPf^p:a:in cellB21: 16:49% D SQRT.12/STDEV:S.B2WB19/

Table 1.7 Example 4: Volatility calculation

A B C D

Monthly portfolio Monthly benchmark

1 Month performance performance Monthly alpha

2 07/2012 6.10 % 6.01 % 0.09 %

3 08/2012 5.50 % 5.45 % 0.05 %

4 09/2012 4.70 % 4.63 % 0.07 %

5 10/2012 5.00 % 6.99 % 1.99 %

6 11/2012 5.10 % 4.16 % 0.94 %

7 12/2012 6.70 % 7.07 % 0.37 %

8 01/2013 6.03 % 5.97 % 0.06 %

9 02/2013 3.23 % 2.95 % 0.28 %

10 03/2013 5.12 % 4.66 % 0.46 %

11 04/2013 5.21 % 4.91 % 0.30 %

12 05/2013 4.10 % 4.01 % 0.09 %

13 06/2013 4.50 % 3.87 % 0.63 %

14 07/2013 1.75 % 2.95 % 4.70 %

15 08/2013 3.71 % 4.52 % 0.81 %

16 09/2013 4.20 % 3.93 % 0.27 %

17 10/2013 4.26 % 4.99 % 0.73 %

18 11/2013 4.00 % 3.84 % 0.16 %

19 12/2013 5.10 % 4.99 % 0.11 %

20 Pf^monthlyD4:76% Bm^monthlyD4:91%

21 Pf^p:a:D16:49% Bm^p:a:D17:02% Source: Own, for illustrative purposes only

• Monthly benchmark volatilityBm^monthlyin cellC 20:

4:91% D STDEV:S.C 2WC19/

• Annualized benchmark volatilityBm^p:a:in cellC 21: 17:02% D SQRT.12/STDEV:S.C 2WC19/

Using the interpretation of a normal distribution in Fig.1.7this means: With a probability of 68 %, the 1-year percentage return of our portfolio lies between ^Pf 16:49% and^Pf C16:49% with^Pf being the average of the historical yearly portfolio returns.

Using^Pf D 15:79% as calculated in Example 1 (same data series as in Example 4), there is a probability of 68 % that the 1-year percentage return of our portfolio lies in the intervalŒ0:70%; 32:28%.

End of Example 4

We will now return to our business case and expand it to include gold. For our calculations we will use Table1.8which shows the end-of-month gold prices in EUR together with the monthly absolute returns (in percentages) in the period January 2009–March 2010.

Table 1.8 Calculation of absolute returns for one ounce of gold from time series of end-of-month stock values

Time point Gold value (one ounce) Absolute return

k tk Month VGold^tk (in EUR) rGold^tk (in %)

0 0 Dec 2008 627.10

1 1/12 Jan 2009 724.18 15.48

2 2/12 Feb 2009 742.54 2.54

3 3/12 Mar 2009 693.13 6.65

4 4/12 Apr 2009 669.67 3.38

5 5/12 May 2009 692.11 3.35

6 6/12 Jun 2009 659.94 4.65

7 7/12 Jul 2009 667.30 1.12

8 8/12 Aug 2009 663.53 0.56

9 9/12 Sep 2009 688.26 3.73

10 10/12 Oct 2009 709.76 3.12

11 11/12 Nov 2009 786.08 10.75

12 1 Dec 2009 765.89 2.57

13 13/12 Jan 2010 780.06 1.85

14 14/12 Feb 2010 820.11 5.13

15 15/12 Mar 2010 824.08 0.48

Ticker: XAUEUR.Source:http://www.fxhistoricaldata.com

Business Case (cont.)

In our business case, the annualized volatility for Lufthansa stock is41:26%, whereas the annualized volatility for gold is 19:71%, implying that the Lufthansa stock fluctuated roughly twice as much as gold. These quantities, which show the greater risk of Lufthansa stock, will be calculated below [Eqs. (1.28) and (1.31)]. Let us first calculate the volatility for Lufthansa based on the monthly returns in the period January 2009–March 2010 from Table1.3. First, we start with the arithmetic mean monthly return:

r^LHA D 1 15

X15 kD1

rLHA^k

D 1

15.15:10%C.8:42%/C: : :C12:04%/

D 1:73%: (1.26)

Then, we use Eq. (1.22) to calculate the volatility (LHA^monthlyin our example):

LHA^monthly D vu ut1

14 X15 kD1

.rLHA^k r^LHA/²

D r1

14Œ.15:10%1:73%/²C: : :C.12:04%1:73%/²

D 11:91%: (1.27)

To obtain the annualized volatility, multiply the monthly volatility by the factorp

12, see Eq. (1.24):

LHA^p:a: D p

12LHA^monthly D p

1211:91% D 41:26%: (1.28) Let us now calculate the volatility for one ounce of gold based on the monthly returns in the period January 2009–March 2010 from Table 1.8.

First, we start with the arithmetic mean monthly return:

rGold D 1 15

X15 kD1

rGold^k D 1

15.15:48%C.2:54%/C: : :C0:48%/

D1:98%: (1.29)

Then, we use Eq. (1.22) to calculate the volatility (Gold^monthlyin our example):

(continued)

Gold^monthly D vu ut 1

14 X15 kD1

.rGold^k rGold/²

D r 1

14Œ.15:48%1:98%/²C: : :C.0:48%1:98%/²

D 5:69%: (1.30)

To obtain the annualized volatility, multiply the monthly volatility by the factorp

12, see Eq. (1.24):

Gold^p:a: D p

12Gold^monthly D p

125:69% D 19:71%: (1.31) To illustrate the volatility, we need to calculate the annualized return of gold. The cumulative return is [using Eq. (1.3)]

rGold^cum D VGold¹⁵⁼¹²VGold⁰

VGold⁰

D e824.08 e627.10 e627.10

D 31:41%; (1.32)

and the annualized return is [using Eq. (1.5), withT D15=12] rGold^p:a: D .1CrGold^cum/^1=T 1

D .1C31:41%/¹²⁼¹⁵1

D 24:42%: (1.33)

Based on our results the

• annualized return for Lufthansa stocks is13:9% [see Eq. (1.14)] and the annualized volatility is41:3% [Eq. (1.28)].

• annualized return for gold is 24:4% [Eq. (1.33)] and the annualized volatility is19:7% [Eq. (1.31)].

Assuming that the expected return equals the past annualized return, we can illustrate the volatilities in Figs.1.9and1.10, based on the general pictures of Figs.1.7and1.8.

(continued)

Fig. 1.9 Graphical illustration of annualized volatility for Lufthansa stock, assuming normally distributed returns.Sources: Reilly and Brown (1997, Appendix D, p. 1047), Yahoo! Finance

The graph shows that for Lufthansa stock, the probability that

• the annualized return lies between27:4and55:2% is68:3%.

• the annualized return lies between68:6and96:4% is95:4%.

For gold, the probability that

• the annualized return lies between4:7and44:1% is68:3%.

• the annualized return lies between15:0and63:8% is95:4%.

Comparing Figs.1.9and1.10we see that because of the smaller volatility the ranges of the annual returns are narrower for gold.

Let us compare the left tails: For Lufthansa stock, the expected return is

• less than54:2% with a probability of5%.

• less than66:9% with a probability of2:5%.

• less than82:2% with a probability of1%.

For gold, the expected return is

• less than8:1% with a probability of5%.

• less than14:2% with a probability of2:5%.

• less than21:5% with a probability of1%.

The lower volatility of gold has the effect that the left tail risks are very low compared to Lufthansa.

Fig. 1.10 Graphical illustration of annualized volatility for gold (in EUR), assuming normally distributed returns. Sources: Reilly and Brown (1997, Appendix D, p. 1047), http://www.

fxhistoricaldata.com