1.3 Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility
1.3.5 Beta
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
Fig. 1.20 Systematic and unsystematic risk in a portfolio in relation to the number of portfolio constituents.Source: Own, for illustrative purposes only
above, the critical component when constructing this portfolio is the covariance of all its securities. This idea is illustrated in Fig.1.20.
Figure1.20distinguishes betweensystematicand unsystematic risk. Systematic risk is the inherent risk in the market which cannot be diversified away, hence it issystematic. The second risk component of the portfolio volatility is unsystematic risk which is the unique risk of the portfolio. As this figure shows, the effect of diversification reduces unsystematic risk. Systematic risk is calledbeta(ˇ).
Figure1.21illustrates the idea of beta. In this diagram, the horizontal axis shows the250daily relative returns (not all data points are shown) of a benchmark for a portfolio. The corresponding250daily returns of the portfolio are displayed on the vertical axis. The daily returns of the benchmark and the portfolio are denoted withrBmdaily andrPfdaily, respectively. This diagram also applies if the portfolio only comprises one single security. In such a situation, beta is the security’s beta versus the index. Now, a simple linear regression is run (for example, using Microsoftr Excelr) which yields a linear regression line shown in Fig.1.21. The regression output shows two parameters which determine the regression line as a function ofrBmdaily:
• The slope of the regression line which is the systematic risk measureˇPfdaily.
• The valuePfdailywhere the regression line intersects with the vertical axis.
A detailed introduction to linear regression will be presented in Sect.2.2in order to explain the tests of the capital asset pricing model (CAPM). However, regression is not needed in the calculation of beta as we will see in the following definition.
Definition: Beta
ˇPf measures the interaction of a portfolio with an index as a benchmark, i.e., how the portfolio return changes depending on the returns of the benchmark.
To measure this interaction mathematically, we again split time intervalŒ0; T (continued)
Fig. 1.21 Description of the regression’s beta. See also Fig.2.3on page106. ParametersPfdaily
andˇPfdailyare the regression parameters (real values).Source: Own, for illustrative purposes only
intoN equidistant subintervals. These are usually days or months. Let further rPf1; rPf2 ; : : : ; rPfN
be the subperiod percentage returns of the portfolio and rBm1 ; rBm2 ; : : : ; rBmN
the subperiod percentage returns of the benchmark. Depending on the chosen subperiods we get the respective beta: daily returns as input lead to a daily beta, monthly returns lead to a monthly beta. Using the monthly covariance Pf;Bm2;monthlyof portfolio and benchmark as well as the monthly varianceBm2;monthly
of the benchmark, the monthly beta can be calculated as43 ˇPfmonthlyD Pfmonthly;Bm
Bm2;monthly
(1.69) withN representing the number of months inŒ0; T . Formula (1.69) can be easily modified for daily returns as input. Then,N is the number of days in Œ0; T :
ˇdailyPf D Pfdaily;Bm
Bm2;daily
(1.70) Note that the use of daily data is more common in practice.
43Esch et al. (2005, p. 91).
1.3.5.1 Interpretation
By using Fig.1.21 and our notation from above, it is easy to interpret beta. The regression equation for the regression line in Fig.1.21is
rPfdailyDPfdailyCˇdailyPf rBmdaily: (1.71) To be precise: In statistics, the left termrPfdailyis called estimator. If we interpret this term as a scalar parameter we would have to include an error term on the right hand side of the equation.
Equation (1.71) leads to the interpretation of beta as a sensitivity measure. If the benchmark value changes byrBmdaily in 1 day, then the portfolio value on that day changes by
ˇdailyPf rBmdaily:
For example, if the portfolio beta for daily data is0:8and the benchmark return on a certain day is2:4%, then the expected percentage return of the portfolio on that day is 0:8.2:4/ D 1:92, i.e., the portfolio is less sensitive than the benchmark.
Obviously, a beta below one is advantageous in falling markets and a beta of above one is advantageous in rising markets. A beta of one indicates that the portfolio return is in line with the benchmark return. The choice of beta, based on the portfolio manager’s expectations on the future development of the benchmark, is often referred to asmarket timing. However, smart market timing is a very difficult task.
Example 8
Table1.13shows the monthly percentage returns of fund A and its benchmark.
The goal is to calculate the portfolio beta using Eq. (1.70).
A beta of 0:9345 as calculated in Table 1.13 means that the monthly percentage returns of the portfolio are highly correlated with the respective benchmark returns. Simply speaking, a1% return of the benchmark will lead (on average) to a0:9345% return of the portfolio.
As usual, the first column is columnA (month). In column B and C the monthly performance of the portfolio and the benchmark are displayed. Using Excelr, the calculation of beta is easy if the covariance and the correlation are calculated up-front as done in the previous section:
• Covariance of the monthly portfolio and benchmark returnsPfmonthly;Bm in cell C 21:
0:00226 D COVARIANCE:S.B2WB19; C 2WC19/
• Monthly benchmark volatilityBmmonthlyin cellC 23: 4:91% D STDEV:S.C 2WC19/
Table 1.13 Example 8: Calculation of the portfolio beta
A B C
Monthly benchmark
1 Month Monthly portfolio performance
2 07/2012 6.10 % 6.01 %
3 08/2012 5.50 % 5.45 %
4 09/2012 4.70 % 4.63 %
5 10/2012 5.00 % 6.99 %
6 11/2012 5.10 % 4.16 %
7 12/2012 6.70 % 7.07 %
8 01/2013 6.03 % 5.97 %
9 02/2013 3.23 % 2.95 %
10 03/2013 5.12 % 4.66 %
11 04/2013 5.21 % 4.91 %
12 05/2013 4.10 % 4.01 %
13 06/2013 4.50 % 3.87 %
14 07/2013 1.75 % 2.95 %
15 08/2013 3.71 % 4.52 %
16 09/2013 4.20 % 3.93 %
17 10/2013 4.26 % 4.99 %
18 11/2013 4.00 % 3.84 %
19 12/2013 5.10 % 4.99 %
21 Pf;BmmonthlyD 0.00226
23 BmmonthlyD 4.91 %
25 ˇPfmonthlyD 0.9345
Source: Own, for illustrative purposes only
• Monthly portfolio betaˇPfmonthlyvs. benchmark in cellC 25:
0:9345DCOVARIANCE:S.B2WB19; C 2WC19/=STDEV:S.C 2WC19/^2 End of Example 8
1.3.5.2 Note
ˇis often understood as a sort of correlation. But although the formulas look the same,ˇ can have values that are less than1and greater than1. For example, a beta of two states a higher sensitivity to the market, i.e., on average the portfolio earns a return twice as high as the market return.
In the long run, the beta of a portfolio or single security versus a comparable index is positive. However, there may be periods with negative beta. A prominent example is Volkswagen in 2008, when it was engaged in take-over and merger discussions with Porsche. Although the German stock index DAX showed primarily negative returns in 2008, the Volkswagen stock increased, resulting in a negative beta over the time period October 23, 2007–October 24, 2008, see Fig.1.22.
However, if the time period only changes slightly like in Fig.1.23, the stock beta is not negative any more.
Fig. 1.22 Example of a negative stock beta over the time period October 23, 2007–October 24, 2008.
Source: Bloomberg (Tickers:
VOW for Volkswagen and DAX for DAX Index)
Fig. 1.23 Example of a positive stock beta over the time period November 1, 2007–October 7, 2008.
Source: Bloomberg (Tickers:
VOW for Volkswagen and DAX for DAX Index)
1.3.5.3 Conclusion
Beta is an important risk measure for assessing the performance of a portfolio versus the market represented by an index. The value of beta depends on the chosen subperiod length, for example, daily or monthly return data. As seen in the Volkswagen example, the value of beta is very sensitive to the chosen historical time period. However, if a security reacts differently in up and down markets, this cannot be captured by beta. For this, we would need to look at a more specialized concept of beta, the bull and bear market beta.