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Financial EconometricsChapter 5: Predicting Asset Returns

In Choi

Sogang University

In Choi (Sogang University) Chapter 5: Predicting Asset Returns 1 / 30

References

Cochrane, J.H. (2001) Asset Pricing, Princeton University Press.

Cochrane, J.H. (2006) The Dog That Did Not Bark: A Defense ofReturn Predictability, mimeo.

Campbell, J.Y., A.W. Lo and A.C. MacKinlay (1997) The Economicsof Financial Markets, Princeton University Press.


Conventional views on the predictability of asset returns

Stock returns are close to unpredictable and nearly identicallydistributed.Prices are random walk represented by

ln (Pt ) = ln (Pt�1) + ut ; ut � iid�0, σ2

�or log returns follow iid process.



Bond returns are nearly unpredictable.If longterm bond yields are higher than shortterm yields, it meansthat shortterm interest rates are expected to rise in the future.Therefore, owning either longterm or shortterm bond will providethe same return in the future.



Foreign exchange returns are not predictable.Holding foreign or domestic bonds yield the same rate of return.

Stock market volatility (varaince) does not change over time. (returnsare iid process)

Professional managers do not outperform simple indices and passiveportfolios once one corrects for risk.

These views mean that asset markets are informationally e¢ cient.


New, emerging views on the predictability of asset markets

Variables including the dividend/price ratio and term premium canpredict substantial amounts of stock return variates. But daily, weeklyand monthly stock returns are still close to unpredictable.

Bond returns are predictable.A steeply upward sloping yield curve means that expected returns onlongterm bonds are higher than on shortterm bonds for the nextyear.


New, emerging views on the predictability of asset markets

Foreign exchange returns are predictable.See, e.g., Mark, N.C. (1995) Exchange Rates and Fundamentals:Evidence on Long-horizon Predictability, American Economic Review.

Stock market volatility changes through time.

Some funds seem to outperform simple indices even after controllingfor risks.

Still, markets are believed to be reasonably e¢ cient.


Longhorizon stock return regressions

ModelRt+k [k ]� Rt+k ,f = α+ β (Dt/Pt ) + ut

Dt : dividend,Pt : price,Rt ,f : riskfree return.

If β > 0, high dividend/price ratio predicts high return.



Some empirical results for the long-horizon stock return regression arereported in Cochrane (2001, p.390).

Horizon k (years) β̂ s.e. R2

1 5.3 (2.0) 0.152 10 (3.1) 0.233 15 (4.0) 0.375 33 (5.8) 0.60

Sample: 19471996. Value weighted NYSE-treasury bill was regressed onthe percent value weights dividend/price ratio.



Other variables such as the term spread between longandshortterm bonds, the default spread (corporate less T-bill yield), theTbill rate and the earnings/dividend ratio, the book/market ratiohave been used with some success. See Cochrane (2001, p.391).



Longhorizon return forecastsHorizon k (years) cay d � p d � e rrel R2

1 6.7 0.181 0.14 0.08 0.041 -4.5 0.101 5.4 0.07 -0.05 -3.8 0.236 12.4 0.166 0.95 0.68 0.396 -5.10 0.036 5.9 0.89 0.65 1.36 0.42



Return : log real excess returns on the S&P composite index�r rt � r rf ,t

�cay : consumption to wealth ratiod : log of the sum of the past four quarters of dividendse : log of a single quarters earnings per sharep : log of the index

rrel : T-bill rate minus its 12 month backward moving averaged � p : dividend yieldd � e : payout ratiosample : 1952:IV1998:III

original source : Lettau and Ludvigson, Journal of Finance, 2001



At oneyear horizon, cay and rrel are more important. At sixyear horizon,d � p and d � e becomes more important.

Why does longhorizon regression provide higher coe¢ cient estimate?Consider the predictive regression

rt+1 = axt + εt+1;

xt+1 = ρxt + δt+1.



Then

rt+1 + rt+2 = a (1+ ρ) xt + aδt+1 + εt+1 + εt+2;

rt+1 + rt+2 + rt+3 = a�1+ ρ+ ρ2

�xt + aρδt+1 + aδt+2 + εt+1 + εt+2 + εt+3;

...

When ρ is close to 1, the coe¢ cient of xt increases with horizon.


Explaining high stock price

When prices are high relative to dividends,1 dividends are expected to rise in the future (Dt ")2 returns are expected to become low in the future (Pt #)3 price/dividend ratio grows explosively

Virtually all variation in price/dividend ratios has reected varyingexpected excess returns.



Price can be expressed in terms of future dividends and returns.Consider the identity

1 = (1+ Rt+1)�1 (1+ Rt+1) = (1+ Rt+1)

�1 Pt+1 +Dt+1Pt

and hence

PtDt

= (1+ Rt+1)�1 Pt+1 +Dt+1

Pt� PtDt

= (1+ Rt+1)�1�1+

Pt+1Dt+1

�Dt+1Dt

.



Taking logs of

PtDt= (1+ Rt+1)

�1�1+

Pt+1Dt+1

�Dt+1Dt

,

we have

pt � dt = �rt+1 + ∆dt+1 + ln�1+ ept+1�dt+1

�,

where rt+1 = ln(Pt+1+Dt+1

Pt) � pt+1 � pt . The Taylor expansion of

ln�1+ ept+1�dt+1

�around P/D = exp(p � d) gives1

ln�1+ ept+1�dt+1

�� ln

�1+

PD

�+

P/D1+ P/D

[pt+1 � dt+1 � (p � d)] .

1 ln(1+ ex ) � ln(1+ exo ) + exo1+exo (x � xo ).In Choi (Sogang University) Chapter 5: Predicting Asset Returns 17 / 30


Using this, we obtain

pt � dt � �rt+1 + ∆dt+1 + k + ρ (pt+1 � dt+1) .

Iterate this relation forward, then

pt � dt � const+∞

∑j=1

ρj�1 (∆dt+j � rt+j ) .

Taking conditional expectations, we obtain an ex ante relation

pt � dt � const+ Et∞

∑j=1

ρj�1 (∆dt+j � rt+j ) .

This shows that a high price/dividend ratio must be followed by highdividend growth or low returns.



Assume E (∆dt+j � rt+j ) = 0. Then,

Var (pt � dt ) = E [pt � dt � E (pt � dt )]2

= E [pt � dt � E (pt � dt )]"

∞

∑j=1

ρj�1 (∆dt+j � rt+j )#

= Cov

pt � dt ,

∞

∑j=1

ρj�1∆dt+j

!

�Cov pt � dt ,

∞

∑j=1

ρj�1rt+j

!,

positive variance of the price/dividend ratio implies that the dividendgrowth and/or the return should be correlated with the ratio. Almostall variation in price/dividend ratio is due to changing return forecasts.



Almost all variation in price/dividend ratio is due to changing returnforecasts as shown in the following table (from Cochrane, 2001, p.398).

Variance decomposition of value weighted NYSE price/dividend ratioDividends Returns

Real -34 138s.e. 10 32Nominal 30 85s.e. 41 19

Entries in the second and fourth rows are the estimates of100� Cov

�pt � dt , ∑∞j=1 ρj�1∆dt+j

�/Var (pt � dt ) and

�100� Cov�pt � dt , ∑∞j=1 ρj�1rt+j

�/Var (pt � dt ) , respectively.



Again from Cochrane (2001, p.390), regressing Dt+k/Dt on Dt/Pt yieldsthe following results.

Horizon k (years) β̂ s.e. R2

1 2.0 1.1 0.062 2.5 2.1 0.063 2.4 2.1 0.065 4.7 2.4 0.12

These results shows that the relation between the dividend growth andprice/dividend ratio is weak.


Evidence of predictability

See Tables 2.4 and 2.8 of Campbell, Lo and Mackinlay (1997).

1 Strong evidence of positive autocorrelation at lag one for indexreturns.

2 Weak negative autocorrelations are observed for individual securities.3 Autocorrelations across stocks are quite strong.


Mean reversion

Longrun regressionFama and French (1988, JPE) estimated the model

rt+k [k ] = a+ bk rt [k ] + εt+k .

Negative and signicant b coe¢ cients were found: a string of goodpast returns forecasts bad future returns.


Mean reversion

Variance ratioIf rt are uncorrelated,

Var (rt+k [k ]) = Var (rt+1 + rt+2 + � � �+ rt+k )= kVar (rt+1) .

The variance ratio statistic is dened by

Vk =Var (rt+k [k ])kVar (rt+1)

.

This should be close to one if rt are uncorrelated.If the variance ratio is less than one, it implies that stocks are safer forlong-run investorswho can tolerate ups and downs of the market.Porterba and Summers (1988, JFE) found variance ratios below one,suggesting that the returns are correlated and that negativecorrelations exist.


Mean reversion: Empirics

From Cochrane (2001, p.412),

Mean reversion using logs, 1926-1996 (log value-weighted NYSE return -long T-bill return used)Horizon k (years)1 2 3 5 7 10

σ(k�period excess return)pk

19.8 20.6 19.7 18.2 16.5 16.3

bk 0.08 -0.15 -0.22 -0.04 0.24 0.08


Mean reversion: Empirics

Evidence of mean reversion at periods 2, 3 and 5. But it disappearsat longer horizons.

Variance ratio at year 10 is (16.3/19.8)2 = 0.68 (< 1).


Relative mean reversion in international stock markets

Balvers, Wu and Gilliland (2000) Mean Reversion across NationalStock Markets and Parametric Contrarian Investment Strategies.Journal of Finance.



pit : log of the total return index of the stock market in country i atthe end of period t.

Evolution of pit is described by a mean-reverting process

pi ,t+1 � pi ,t = ai + λ(p�i ,t+1 � pit ) + εi ,t+1.

p�it : an unobserved fundamental value of the index.



Paramter λ is the speed of mean reversion and is assumed to be thesame across countries.

If 0 < λ < 1, deviations of pit from its fundamental or trend valuep�i ,t+1 will be reversed over time.

But if λ = 0, the log price follows a unit root process so that there isno mean reversion.



Assumep�it = p

�r ,t + zi + ηi ,t

where p�r ,t is a reference countrys fundamental value of the index andzi a constant.

Then,ri ,t+1 � rr ,t+1 = αi � λ(pi ,t � pr ,t ) +ωi ,t+1,

where ri ,t+1 � pi ,t+1 � pi ,t .No mean reversion (i.e., λ = 0) corresponds to the presence of a unitroot in pi ,t � pr ,t .Balvers, Wu and Gilliland report evidence of mean reversion in relativestock index prices using stock index data of 18 nations during theperiod 1969-1996.


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