1.3 Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility
1.3.4 Covariance and Correlation
1.3.4.2 Properties of Covariance and Correlation A covariance has the following properties
• If the covariance is positive, then one asset tends to have high (low) returns whenever the other one also has high (low) returns. Usually, this is the case for stocks of companies from the same industry.
39Esch et al. (2005, p. 42).
40Hull (2009, p. 284). We assume252trading days per year.
• If the covariance is negative, then one asset tends to have high (low) returns when the other one has low (high) returns. For example, high oil prices negatively affect airlines. Hence, the covariance between the returns of airline stocks and oil prices is negative.
• If the covariance is zero, then there is no linear relation between the asset returns:
knowledge of the return of one asset will not lead to any knowledge about the return of the other asset.
A correlation has the following properties41:
• The correlation between two assets lies between1and1:
11;21: (1.50)
• If the correlation is positive, then one asset tends to have high (low) returns whenever the other one also has high (low) returns. If the correlation is1, then the relationship between the asset returns is positively linear.
• If the correlation is negative, then one asset tends to have high (low) returns whenever the other one has low (high) returns. If the correlation is1, then the relationship between the asset returns is negatively linear.
• If the correlation is 0, then there is no linear relationship between the asset returns.
• The greater the absolute value of the correlation, the stronger the association between the asset returns.
To illustrate the last property of this list, let us first look at Table 1.10.
It lists different absolute values for the correlation coefficient together with the corresponding strength of the association between the asset returns. The degree of association is very high for0:80and above, and very low for values below0:20.
Next, we will visualize various correlations by using ascatter plot. A scatter plot42 is a type of mathematical diagram using Cartesian coordinates to display
Table 1.10 Absolute value of correlation coefficient and strength of corresponding association
Absolute Strength of association
value of correlation between asset returns
0.80–1.00 Very strong association
0.60–0.79 Strong association
0.40–0.59 Moderate association
0.20–0.39 Weak association
0.00–0.19 Little if any association
Source: Lhabitant (2004, p. 129)
41DeFusco et al. (2004, pp. 207–208).
42Also called scatter chart, scattergram, scatter diagram or scatter graph.
Fig. 1.14 Scatter plots for different correlations. Each graph illustrates returns of assets1and2 during some time periodŒ0; T with100subintervals in which the returns are measured.Source:
Own, for illustrative purposes only
values for two variables for a set of data. The data are displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis.
The scatter plots in Figs.1.14,1.15,1.16, and1.17illustrate correlation graphi- cally. They show examples with two different assets1and2during some time period Œ0; T with100subintervals where returns
r11; r12; : : : ; r1100 and
r21; r22; : : : ; r2100 are measured. The points
Fig. 1.15 Scatter plots for different correlations. Each graph illustrates the returns of assets1and 2during some time periodŒ0; T with100subintervals in which the returns are measured.Source:
Own, for illustrative purposes only
.r11; r21/; .r12; r22/; : : : ; .r1100; r2100/
are plotted on the graphs which represent the absolute returns of these assets in the respective subintervals, for example,.r150; r250/represents the returns of assets1and 2in the50th subinterval.
These scatter plots show the typical pictures for different correlations.
• For correlation1(1), the points lie exactly on a line with positive (negative) slope.
• For˙0:95, the points are still close to a line.
• For˙0:90and˙0:80, the graphs show a strong relationship between the asset returns.
• The relationship is weaker for˙0:60and˙0:40.
• For˙0:20, you can hardly spot any relationship at first sight, and for0:00, there is no relationship between the asset returns.
Fig. 1.16 Scatter plots for different correlations. Each graph illustrates the returns of assets1and 2during some time periodŒ0; T with100subintervals in which the returns are measured.Source:
Own, for illustrative purposes only
In Fig.1.18, we see the scatter plot of the monthly returns of US Airways (LCC) against Delta (DAL) for the period January2008–June2010. Given the monthly returns
rDAL1 ; rDAL2 ; : : : ; rDAL30
of Delta and
rLCC1 ; rLCC2 ; : : : ; rLCC30
of US Airways, the graph plots the points
.rDAL1 ; rLCC1 /; .rDAL2 ; rLCC2 /; : : : ; .rDAL30 ; rLCC30 /:
Fig. 1.17 Scatter plots for different correlations. Each graph illustrates the returns of assets1and 2during some time periodŒ0; T with100subintervals in which the returns are measured.Source:
Own, for illustrative purposes only
The correlation, which is calculated below in Eq. (1.66), is 0:789, i.e., the relationship between the monthly returns of the respective airline stocks is strong.
This is graphically supported by the plot in Fig.1.18.
Figure 1.19 shows the scatter plot of monthly returns of US Airways (LCC) against oil for the period January2008–June2010. Given the monthly returns
rLCC1 ; rLCC2 ; : : : ; rLCC30
of US Airways and
rOil1 ; rOil2 ; : : : ; rOil30
of oil, the graph plots the points
.rOil1 ; rLCC1 /; .rOil2 ; rLCC2 /; : : : ; .rOil30; rLCC30 /:
Fig. 1.18 Scatter plot of monthly returns of US Airways (LCC) against Delta (DAL) for the period January 2008–June 2010.Source:
Yahoo! Finance
20 40 60 80 100
−20
−40
−60
20 40
−20
−40
rkLCC(in %)
rkDAL(in %) Correlation =0.789
Fig. 1.19 Scatter plot of monthly returns of US Airways (LCC) against oil (Crushing, OK Crude Oil Futures Contract) for the period January 2008–June 2010.Sources: Yahoo!
Finance and U.S. Energy Information Administration
The correlation, which is calculated below in Eq. (1.68), is 0:435. The plot shows that the monthly returns of US Airways tend to be negatively affected by oil.
We will now extend our business case with additional data displayed in Table 1.11. Please note, that we will refer to Table 1.11 again in Chap.2 when we continue with this business case in the context of regression analysis.
Business Case (cont.)
Let us determine the covariances and correlations between the monthly returns of the airline stocks Delta (DAL) & US Airways (LCC), and the monthly returns on crude oil. The calculation is based on the monthly data from the period January 2008–June 2010 as shown in Table 1.11. To calculate the different correlations, we have to start with the covariances using Eq. (1.45). This requires to calculate the arithmetic mean monthly returns first:
(continued)
Table 1.11 Time series of end-of-month stock values of Delta (NYSE:DAL), US Airways (NYSE:LCC), both paying no dividends, and oil (Crushing, OK Crude Oil Futures Contract, price per barrel) in the period January 2008–June 2010, together with the respective monthly returns
Time point VDALtk rDALk VLCCtk rLCCk VOiltk rOilk
k tk Month (in USD) (in %) (in USD) (in %) (in USD) (in %)
0 0 Dec 2007 14:89 14:71 95:98
1 1/12 Jan 2008 16:82 12.96 13:84 5.91 91:75 4.41
2 2/12 Feb 2008 13:35 20.63 12:40 10.40 101:84 11.00
3 3/12 Mar 2008 8:60 35.58 8:91 28.15 101:58 0.26
4 4/12 Apr 2008 8:51 1.05 8:59 3.59 113:46 11.70
5 5/12 May 2008 6:15 27.73 3:96 53.90 127:35 12.24
6 6/12 Jun 2008 5:70 7.32 2:50 36.87 140:00 9.93
7 7/12 Jul 2008 7:54 32.28 5:06 102.40 124:08 11.37
8 8/12 Aug 2008 8:13 7.82 8:49 67.79 115:46 6.95
9 9/12 Sep 2008 7:45 8.36 6:03 28.98 100:64 12.84
10 10/12 Oct 2008 10:98 47.38 10:14 68.16 67:81 32.62
11 11/12 Nov 2008 8:81 19.76 5:96 41.22 54:43 19.73
12 1 Dec 2008 11:46 30.08 7:73 29.70 44:60 18.06
13 13/12 Jan 2009 6:90 39.79 5:67 26.65 41:68 6.55
14 14/12 Feb 2009 5:03 27.10 2:85 49.74 44:76 7.39
15 15/12 Mar 2009 5:63 11.93 2:53 11.23 49:66 10.95
16 16/12 Apr 2009 6:17 9.59 3:79 49.80 51:12 2.94
17 17/12 May 2009 5:81 5.83 2:58 31.93 66:31 29.71
18 18/12 Jun 2009 5:79 0.34 2:43 5.81 69:89 5.40
19 19/12 Jul 2009 6:93 19.69 2:93 20.58 69:45 0.63
20 20/12 Aug 2009 7:22 4.18 3:40 16.04 69:96 0.73
21 21/12 Sep 2009 8:96 24.10 4:70 38.24 70:61 0.93
22 22/12 Oct 2009 7:14 20.31 3:06 34.89 77:00 9.05
23 23/12 Nov 2009 8:19 14.71 3:69 20.59 77:28 0.36
24 2 Dec 2009 11:38 38.95 4:84 31.17 79:36 2.69
25 25/12 Jan 2010 12:23 7.47 5:31 9.71 72:89 8.15
26 26/12 Feb 2010 12:92 5.64 7:33 38.04 79:66 9.29
27 27/12 Mar 2010 14:59 12.93 7:35 0.27 83:76 5.15
28 28/12 Apr 2010 12:08 17.20 7:07 3.81 86:15 2.85
29 29/12 May 2010 13:58 12.42 8:83 24.89 73:97 14.14
30 30/12 Jun 2010 11:75 13.48 8:61 2.49 75:63 2.24
Sources: Yahoo! Finance (for DAL and LCC) and U.S. Energy Information Administration (for oil)
rDAL D 1 30
X30 kD1
rDALk
D 1
30.12:96%C.20:63%/C: : :C.13:48%//
D 1:59%: (1.51)
rLCC D 1 30
X30 kD1
rLCCk
D 1
30.5:91%C.10:40%/C: : :C.2:49%//
D 4:73%: (1.52)
rOil D 1 30
X30 kD1
rOilk
D 1
30.4:41%C11:00%C: : :C2:24%/
D 0:04%: (1.53)
Using Eq. (1.45), we calculate the covariances between the monthly returns:
DALmonthly;LCC D 1 29
X30 kD1
.rDALk rDAL/.rLCCk rLCC/
D 1
29Œ.12:96%1:59%/.5:91%4:73%/C: : : C.13:48%1:59%/.2:49%4:73%/
D 0:065368: (1.54)
DALmonthly;Oil D 1 29
X30 kD1
.rDALk rDAL/.rOilk rOil/
D 1
29Œ.12:96%1:59%/.4:41%.0:04%//C: : : C.13:48%1:59%/.2:24%.0:04%/
D 0:010083: (1.55)
(continued)
LCCmonthly;Oil D 1 29
X30 kD1
.rLCCk rLCC/.rOilk rOil/
D 1
29Œ.5:91%4:73%/.4:41%.0:04%//C: : : C.2:49%4:73%/.2:24%.0:04%/
D 0:019952: (1.56)
Now, we annualize the covariances using Eq. (1.48):
DALp:a:;LCC D 12DALmonthly;LCCD120:065368D0:7844; (1.57) DALp:a:;Oil D 12DALmonthly;OilD12.0:010083/D 0:1210; (1.58) LCCp:a:;Oil D 12LCCmonthly;Oil D12.0:019952/D 0:2394: (1.59) The next step is to calculate the monthly volatilities using Eq. (1.22):
DALmonthly D vu ut1
29 X30 kD1
.rDALk rDAL/2
D r1
29Œ.12:96%1:59%/2C: : :C.13:48%1:59%/2
D 0:21914: (1.60)
LCCmonthly D vu ut1
29 X30 kD1
.rLCCk rLCC/2
D r1
29Œ.5:91%4:73%/2C: : :C.2:49%4:73%/2
D 0:37816: (1.61)
(continued)
OilmonthlyD vu ut1
29 X30 kD1
.rLCCk rLCC/2
D r1
29Œ.4:41%.0:04%//2C: : :C.2:24%.0:04%//2
D 0:12133: (1.62)
Using Eq. (1.24), the annualized monthly volatilities are:
DALp:a: D p
12DALmonthlyD p
120:21914 D 0:7591; (1.63) LCCp:a: D p
12LCCmonthlyD p
120:37816 D 1:3100; (1.64) Oilp:a: D p
12OilmonthlyD p
120:12133 D 0:4203: (1.65) We use Eq. (1.43) to calculate the correlations:
p:a:DAL;LCC D DAL;LCC
DALLCC D 0:7844
0:75911:3100 D 0:789; (1.66) DAL;Oilp:a: D DAL;Oil
DALOil
D 0:1210
0:75910:4203 D 0:379; (1.67) p:a:LCC;Oil D LCC;Oil
LCCOil D 0:2394
1:31000:4203 D 0:435: (1.68) The correlations indicate that the stock prices of both airlines are strongly correlated and have a tendency to move together, whereas their correlations with the oil price are negative, i.e., their stock prices and the oil price tend to move in opposite directions.
Having demonstrated several applications of our formulas of covariance and correlation in the business case, we now turn to our hypothetical examples in order to show how these values can be easily calculated using Excelrformulas.
Example 7
Table1.12shows the consecutive monthly relative returns of two assetsAand B. In this example we assume that the two assets are two portfolios available to investors as retail funds. The goal is to calculate the covariance and correlation of these two funds over a given time period of18consecutive months.
Using again the column and row notation from MicrosoftrExcelr, the first column is labeledA(month), the secondB (monthly performance of fundA) and the thirdC (monthly portfolio of fundB). Applying Excelrfunctions, the
Table 1.12 Example 7: Covariance and correlation calculation
A B C
Monthly performance Monthly performance
1 Month of fund A of fund B
2 07/2012 6.10 % 7.71 %
3 08/2012 5.50 % 6.38 %
4 09/2012 4.70 % 4.83 %
5 10/2012 5.00 % 7.70 %
6 11/2012 5.10 % 4.90 %
7 12/2012 6.70 % 5.32 %
8 01/2013 6.03 % 6.48 %
9 02/2013 3.23 % 3.95 %
10 03/2013 5.12 % 3.80 %
11 04/2013 5.21 % 5.20 %
12 05/2013 4.10 % 3.00 %
13 06/2013 4.50 % 2.01 %
14 07/2013 1.75 % 3.54 %
15 08/2013 3.71 % 4.90 %
16 09/2013 4.20 % 1.62 %
17 10/2013 4.26 % 6.05 %
18 11/2013 4.00 % 4.10 %
19 12/2013 5.10 % 6.20 %
20 A;BmonthlyD 0.00224
21 A;BD 0.9261
Source: Own, for illustrative purposes only
covariance and the correlation of fundsAandB over this time period can be calculated as:
• Covariance of the monthly percentage fund returnsA;Bmonthlyin cellC 20:
0:00224 D COVARIANCE:S.B2WB19; C 2WC19/
• Correlation of the monthly percentage fund returnsA;Bmonthlyin cellC 21: 0:9261 D CORREL.B2WB19; C 2WC19/
This concept will be needed in the next section where beta is introduced as a measure of the systematic risk of a portfolio (or single security) versus an index.
End of Example 7