When Does Idiosyncratic Risk
Really Matter?
Tony Ruan
Xiamen Univ.
Qian Sun
Fudan Univ.
Yexiao Xu
UT Dallas
December 11, 2010
2010 NTU International Conference on Finance
1
What Is Idiosyncratic Risk?

Consider a factor model (e.g., the CAPM or APT) for
stock returns
rit  rft  1i r1t   2i r2t  3i r3t   4i r4t   it
where r1t , r2 t , r3t , and r4 t are systematic factors,
and  it is idiosyncratic risk with mean 0 and
variance  it2 .
In this setup, only systematic risk is priced and idiosyncratic
risk is not priced due to a full level of diversification. (Sharp
(1964), Lintner (1965), Mossin (1966), Black (1972), Ross
2
(1976)…).
Could Idiosyncratic Risk Be Priced?

Idiosyncratic risk may matter.

Divergence of opinion and short-sales
constraints. (negatively, Miller (1977))
 Imperfect
information. (positively, Merton
(1987))
 Difference between effective and published
supplies of securities. (positively, Malkiel
and Xu (2002))
 Loss aversion (positively, Barberis and
Huang (2001)).
3
The Recent Empirical Debate
and Research Question


Issues related to times-series empirical evidence
 Idiosyncratic risk increased over the past decades
(Campbell, Lettau, Malkiel, and Xu (2001), known as
CLMX (2001)).
 Idiosyncratic risk positively predicts future expected
market premium (Goyal and Santa-Clara (2003), known
as GS (2003)).
 GS’s results are NOT robust to subsamples, alternative
measures, and subperiods (Bali, Cakici, Yan, and Zhang
(2005) and Wei and Zhang (2005)).
What could explain the seemingly ambiguous results?
4
Our Idea

In a market with frictions, investors in general hold
under-diversified portfolios (Goetzmann and Kumar,
2008) so that investors should be exposed to
idiosyncratic risk to the extent it is not diversified away.

Hence, researchers have used aggregate idiosyncratic
risk measures that are too noisy to deliver consistent
and robust results in a time-series predictive regression.
We use a simple method to reduce the noise effect in
the time-series test.
5
Merton (1987) Revisited
For individual stock k
Rk  R f   Var ( Rm )  k  k
Shadow cost of under-diversification
k   (1/ qk  1) wk  k2 , where  k2 is stock k's idiosyncratic risk.
Examples:
Undiversified
1. q1  w1  0.1 (a small stock), q2  w2  0.9 (a large stock); idiosyncratic risk
1   (0.9) 12 , 2   (0.1) 22 .
2. q3  0.1, w3  0.01, 3   (0.09) 32 .
At the market level:
Rm  R f   Var ( Rm )  m
Aggregate measure of
undiversified idiosyncratic risk
n
m   wk k
k 1
But only aggregate measures of
idiosyncratic risk is observed.
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A Simple Econometric Method
Suppose the true model is
rt 1  vt   t 1 .
But only noisy measures of vt are observed
x1  a1v  b1 s1
x2  a2 v  b2 s2 ,
To reduce the noise effect, our idea is to include both
noisy measures in the predictive regression.
where v is the signal; s1 and s2 are measurement errors with a
correlation coefficient  . v, s1 , s 2 , and  have mean zeros and
unit variances. a1 (a2 ) and b1 (b2 ) are non-negative real numbers.
a1
a2
Define 1 
and  2  . We assume 1 > 2 .
b1
b2
1 and  2 are interpreted as signal-to-noise ratios.
7
Simple Simulations
High
Correlation
B/W Noises
Low
8
Specifications for Time-Series Tests
Rmt    1,dual x1,t 1   2,dual x2,t 1   t ,
(1)
Rmt    1,dual x1,t 1   2,dual x2,t 1  Z t'1   t ,
(2)
where R mt is the market excess return and Z t is a
vector of control variables, which may include
lagged market return, market risk, and some financial
macro variables.
Our hypothesis predicts
1. 1,dual  0,
2.  2,dual < 0 if  is high relative to the difference b/w 1 and  2 ;
3. Increased R 2 .
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How to Construct Different Measures
of Aggregate Idiosyncratic Risk?


The premise is that small stocks have larger
priced idiosyncratic risk components than large
stocks.
Using different weighting schemes (i.e., equalweighted and value-weighted)
to construct n-stock portfolios
 to aggregate portfolio idiosyncratic risks

10
A Simple Test Using CRSP VW
Index Excess Return
Signal measure - Equal-weighted idiosyncratic risk;
Noise measure - Value-weighted idiosyncratic risk;
11
Main Results (Using CRSP VW Index
Excess
Return)
Signal
Noise
12
Sharpe Ratios of Trading Strategies
Based on Different Forecasts
13
Profitability of Trading Strategies
Based on Different Forecasts
P-value
14
Conclusion

Our results suggest that in a time-series
predictive regression idiosyncratic risk matters
when investors are under-diversified.
 when researchers take into account the noise effect
resulting from under-diversification in testing for the
effect of idiosyncratic risk.

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