1.3 Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility
1.3.6 Bull and Bear Market Beta
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.
Definition: Bull and Bear Market Beta
Let us split time intervalŒ0; T into N equidistant subintervals which 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. Using the notation from above,bullˇPf andbearˇPf can mathematically be calculated in a similar way as the overall betaˇPf. However, the overall covariance and volatility have to be replaced with conditional measures.
For this letNbull be the number of subperiods inŒ0; T with a positive benchmark return andNbearthe number of subperiods inŒ0; T with a zero or negative benchmark return. Then, we obviously haveNbullCNbear DN.
Covariance bullPf;Bm between the portfolio and the benchmark in case of bull markets is then calculated using Eq. (1.45) but applying only Nbull subintervals with a positive benchmark return.
Similarly, covariancebearPf;Bm between the portfolio and the benchmark in case of bear markets is calculated using Eq. (1.45) but applying onlyNbear subintervals with a zero or negative benchmark return.
As a special case we get the bull and bear variances for bull and bear markets:
bullBm2 D bullBm;Bm and bearBm2 D bearBm;Bm: (1.72) The bull and bear market beta can be calculated as
bullˇPf D bullPf;Bm bullBm2
and bearˇPf D bearPf;Bm bearBm2
(1.73)
These definitions can be done using daily subintervals or monthly subintervals.
We then get
bullPf;Bmdaily and bullPf;Bmmonthly as the bull market covariance,
bearPfdaily;Bm and bearPfmonthly;Bm
as the bear market covariance,
bullBm2;daily and bullBm2;monthly
as the bull market variance,
bearBm2;daily and bearBm2;monthly
as the bear market variance,
bullˇPfdaily and bullˇmonthlyPf
as the bull market beta and
bearˇPfdaily and bearˇPfmonthly: as the bear market beta.
1.3.6.1 Interpretation
While ˇPf measures the sensitivity of a portfolio to its benchmark over all N subintervals in time periodŒ0; T , the bull market betabullˇPf is more specific and considers only subintervals with a positive benchmark return. On the other hand, the bear market betabearˇPf measures this sensibility relative to subintervals with a zero or negative benchmark performance. The interpretation of the bear-beta and bull-beta is, therefore, straightforward:
• If bearˇPf 1, then the portfolio return drops faster than the benchmark in a downward trending market.
• If0 bearˇPf 1, then the portfolio return drops, but less than the benchmark in a declining market.
• If bearˇPf 0, then the performance of the asset rises, if the benchmark’s performance drops.
Therefore, in a bear market, the best is to have a negative bear beta: the portfolio wins even when the benchmark loses. The interpretation of the bull market beta
bullˇPf is as follows:
• If bullˇPf 1, then the portfolio return rises faster than the benchmark in an upward trending market.
• If0bullˇPf 1, then the portfolio return rises, but less than the benchmark in a rising market.
• IfbullˇPf 0, then the performance of the portfolio drops, if the benchmark rises.
Obviously, it is advantageous to have a bull beta above1in rising markets. In combination, an ideal portfolio has a below 1 or, even better, negative bear beta and a rather high bull beta (at a minimum above 1).
Example 9
The following example analyzes a portfolio versus its benchmark over a period of 18months, i.e.,T D 1:5years andN D18months. The monthly performance data of portfolio and benchmark are provided in Table1.14 on page62which displays the same performance figures shown in Table1.13for Example 8. To calculate the bull and bear market beta, new columnsD toG were added. In these columns, an ExcelrIF-function is used to obtain the positive and negative values of the benchmark returns only.
• Positive monthly benchmark returnrBmmonthlyin cellD2: 6:01% D IF.C 2 > 0; C 2;“”)
• Negative monthly benchmark returnrBmmonthlyin cellF 2: D IF.C 2 < 0; C 2;“”)
• Monthly portfolio returnrPfmonthly, ifrBmmonthlyis positive, in cellE2:
6:10% D IF.D2 <>“”,B2,“”)
• Monthly portfolio returnrPfmonthly, ifrBmmonthlyis negative, in cellG2: D IF.F 2 <>“”,B2,“”)
• Bull market covariancebullPfmonthly;Bm for positive benchmark returns in cellD20: 0:000064 D COVARIANCE:S.D2WD19; E2WE19/
• Bear market covariance bearPf;Bmmonthly for negative benchmark returns in cell F 20:
0:000146D COVARIANCE:S.F 2WF19; G2WG19/
• Bull market variancebullBm2;monthlyfor positive benchmark returns in cellD21: 0:000066D VAR:S.D2WD19/
• Bear market variancebearBm2;monthlyfor negative benchmark returns in cellF 21: 0:00159D VAR:S.F 2WF19/
Then, the bull and bear market beta can be calculated via Excelras follows:
• Monthly portfolio bull market betabullˇmonthlyPf in cellD22: 0:9726D D20=D21
• Monthly portfolio bear market betabearˇmonthlyPf in cellF 22: 0:9188D F 20=F 21
Table1.14Example9:Calculationofbullandbearmarketbeta ABCDEFG MonthswithMonthswith MonthlyMonthlypositiveCorrespondingnegativeCorresponding portfoliobenchmarkbenchmarkportfoliobenchmarkportfolio 1Monthperformanceperformanceperformanceperformanceperformanceperformance 207/20126.10%6.01%6.01%6.10% 308/20125.50%5.45%5.45%5.50% 409/20124.70%4.63%4.63%4.70% 510/20125.00%6.99%6.99%5.00% 611/20125.10%4.16%4.16%5.10% 712/20126.70%7.07%7.07%6.70% 801/20136.03%5.97%5.97%6.03% 902/20133.23%2.95%2.95%3.23% 1003/20135.12%4.66%4.66%5.12% 1104/20135.21%4.91%4.91%5.21% 1205/20134.10%4.01%4.01%4.10% 1306/20134.50%3.87%3.87%4.50% 1407/20131.75%2.95%2.95%1.75% 1508/20133.71%4.52%4.52%3.71% 1609/20134.20%3.93%3.93%4.20% 1710/20134.26%4.99%4.99%4.26% 1811/20134.00%3.84%3.84%4.00% 1912/20135.10%4.99%4.99%5.10% 20bullmonthly Pf;BmD0.000064bearmonthly Pf;BmD0.000146 21bull2;monthly BmD0.000066bear2;monthly BmD0.000159 22bullˇmonthly PfD0.9726bearˇmonthly PfD0.9188 Source:Own,forillustrativepurposesonly
All calculation results are provided in Table 1.14. As can be seen in cell D22, the bull market beta stands at0:9726, i.e., if the benchmark increases by 1%, the portfolio, on average, increases only by0:9726%. The bear market beta of0:9191indicates that when the benchmark decreases by 1%, the fund decreases only by0:9191%. Such a portfolio would behave nicely during up- and downward movements of the market: the bull beta should be much higher than the bear beta which is here the case. Ideally, the bull beta should be higher than1and the bear beta should be close to0 or even negative, but the latter is difficult to achieve with long-only portfolios.
End of Example 9
1.3.6.2 Conclusion
The bull and bear beta are plausible concepts that build on the general beta concept.
Investors can more thoroughly analyze a portfolio’s behavior compared to the market development. Therefore, it makes sense to not only look at the overall beta but also to consider the bull and bear market beta.