should be the premium earned by stocks over long-term
bonds.
3.Arithmeticandgeometricaverages.Thefinalstickingpoint
whenit comestoestimating historicalpremiumsrelates to
how the average returns on stocks, Treasury bonds, and
Treasurybillsarecomputed. Thearithmeticaverage return
measuresthe simple meanof the series ofannual returns,
whereas the geometric average looks at the compounded
return.
13 Conventionalwisdomarguesfortheuseofthearithmetic
average.Infact,ifannualreturnsareuncorrelatedovertime
andourobjectivewastoestimatetheriskpremiumforthe
nextyear,thearithmeticaverageisthebestunbiasedestimate
of the premium. In reality, however, there are strong
arguments that can be made for the use of geometric averages.
First,empiricalstudiesseemtoindicatethatreturnsonstocks
are negatively correlated over time.
14 Consequently,thearithmetic averagereturn islikely to
overstate thepremium. Second,while assetpricing models
maybesingleperiodmodels,theuseofthesemodelstoget
expectedreturnsoverlongperiods(suchasfiveortenyears)
suggeststhatthesingleperiodmaybe muchlongerthana
year.Inthiscontext,thecaseforgeometricaveragepremiums
becomes even stronger.
In summary,therisk premium estimatesvary across users
because ofdifferences in time periods used, the choice of
Treasurybillsorbondsastherisk-freerate,andtheuseof
arithmeticasopposed togeometric averages.Theeffect of
thesechoicesissummarizedinTable2.2,whichusesreturns
from 1928 to 2004.
15