$\overline{r}^{+}_{.95}$ is the mean of the highest 5% of returns,
$\overline{r}^{-}_{.05}$ is the mean of the smallest 5% of returns,
$\overline{r}^{+}_{.5}$ is the mean of the top half of returns,
$\overline{r}^{-}_{.5}$ is the mean of the bottom half of returns; and $r_{x}$
is the observation corresponding to the $x$%ile of the return data. S.R.
(ann.) is the Sharpe ratio of the monthly excess returns multiplied by
$\sqrt{12}$. VWI and EWI are the value-weighted and equally-weighted
portfolios of sample stocks. The out-of-sample period contains 528 monthly
returns– January 1975 through December 2018.

Panel A: Benchmarks
---
Metric | Resampled |  | Resampled
| Out of sample |  | Out of sample
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | VWI |  | EWI
$\cal{CE}$2 | 82.5 | 86.7 | 91.0 |  | 100.6 | 102.7 | 104.8
$\cal{CE}$5 | 51.6 | 56.2 | 60.8 |  | 57.1 | 59.3 | 61.6
$\cal{CE}$9 | 4.4 | 9.8 | 15.2 |  | -13.4 | -10.3 | -7.2
$E(r)$ | 1.02 | 1.06 | 1.10 |  | 1.27 | 1.29 | 1.31
$E(r)-r_{f}$ | 0.65 | 0.69 | 0.73 |  | 0.90 | 0.92 | 0.94
$\sigma(r)$ | 4.27 | 4.31 | 4.37 |  | 5.04 | 5.07 | 5.09
Median | 1.21 | 1.33 | 1.44 |  | 1.43 | 1.52 | 1.62
IQ Range | 5.04 | 5.23 | 5.42 |  | 5.90 | 6.05 | 6.19
Minimum | -23.30 | -22.27 | -21.25 |  | -26.49 | -25.91 | -25.32
Skewness | -0.09 | -0.06 | -0.04 |  | -0.06 | -0.04 | -0.03
Kurtosis | 0.31 | 0.36 | 0.41 |  | 0.36 | 0.38 | 0.40
S.R. (ann.) | 0.52 | 0.55 | 0.59 |  | 0.61 | 0.63 | 0.64

Table 3 -Continued
Sampling distributions of out-of-sample certainty equivalent returns,
portfolio moments and properties

Panel B: $\gamma^{*}=3:\,\gamma$ 2 loss function’s perspective
---
Metric | Resampled |  | Resampled
| Out of sample |  | Out of sample
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio : | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$
$\cal{CE}$2 | 257.4 | 301.6 | 351.7 |  | -10,000.0 | -1,139.4 | 278.9
$E(r)$ | 4.34 | 5.11 | 6.00 |  | 3.58 | 4.22 | 4.95
$\sigma(r)$ | 12.40 | 14.11 | 16.10 |  | 11.57 | 13.06 | 14.71
Median | 4.26 | 5.18 | 6.19 |  | 3.46 | 4.25 | 5.10
IQ Range | 14.37 | 16.71 | 19.35 |  | 12.45 | 14.35 | 16.45
Minimum $r$ | -68.33 | -53.56 | -43.16 |  | -115.01 | -79.65 | -47.53
Skewness | -0.05 | -0.01 | 0.04 |  | -0.05 | 0.00 | 0.04
Kurtosis | 0.28 | 0.41 | 0.55 |  | 0.41 | 0.56 | 0.71
S.R. (ann.) | 1.07 | 1.17 | 1.27 |  | 0.92 | 1.02 | 1.12
Neg.Wts. | -599.2 | -515.4 | -441.4 |  | -557.1 | -485.8 | -419.9
Portfolio : | M, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, S, $\overline{r}$, $\sigma_{\epsilon}$
$\cal{CE}$2 | 169.7 | 208.0 | 248.2 |  | 238.5 | 279.2 | 320.6
$E(r)$ | 3.11 | 3.76 | 4.48 |  | 4.19 | 4.86 | 5.62
$\sigma(r)$ | 10.99 | 12.64 | 14.52 |  | 12.33 | 13.95 | 15.74
Median | 2.53 | 3.23 | 4.02 |  | 4.16 | 4.97 | 5.85
IQ Range | 12.04 | 14.07 | 16.37 |  | 13.78 | 15.75 | 17.89
Minimum $r$ | -67.95 | -54.11 | -42.98 |  | -68.29 | -54.10 | -44.23
Skewness | 0.00 | 0.04 | 0.08 |  | -0.05 | -0.01 | 0.04
Kurtosis | 0.43 | 0.57 | 0.72 |  | 0.39 | 0.52 | 0.65
S.R. (ann.) | 0.83 | 0.93 | 1.03 |  | 1.03 | 1.12 | 1.21
Neg.Wts. | -479.5 | -400.1 | -329.0 |  | -569.7 | -493.7 | -423.4
Portfolio: | M, S, $\beta$, $\sigma_{\epsilon}$ |  | M, S, $\beta$, $\overline{r}$
$\cal{CE}$2 | 114.5 |  182.2 | 230.2 |  | 167.4 |  199.6 | 233.3
$E(r)$ | 3.30 | 3.78 | 4.33 |  | 3.28 | 3.79 | 4.36
$\sigma(r)$ | 11.38 | 12.56 | 13.86 |  | 11.62 | 12.89 | 14.24
Median | 3.06 | 3.73 | 4.48 |  | 3.23 | 3.92 | 4.65
IQ Range | 12.04 | 13.62 | 15.43 |  | 12.81 | 14.49 | 16.26
Minimum $r$ | -86.13 | -67.20 | -51.26 |  | -62.46 | -51.41 | -43.02
Skewness | -0.04 | 0.00 | 0.04 |  | -0.05 | -0.01 | 0.03
Kurtosis | 0.50 | 0.63 | 0.76 |  | 0.38 | 0.51 | 0.65
S.R. (ann.) | 0.86 | 0.94 | 1.03 |  | 0.84 | 0.92 | 1.01
Neg.Wts. | -422.0 | -366.2 | -317.4 |  | -506.0 | -443.2 | -387.6

Table 3 -Continued
Sampling distributions of out-of-sample certainty equivalent returns,
portfolio moments and properties

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | M, V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, V, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$
$\cal{CE}$2 | 247.4 | 271.3 | 350.9 |  | 167.6 | 200.2 | 294.9
$E(r)$ | 4.31 | 5.09 | 6.00 |  | 3.49 | 4.20 | 5.02
$\sigma(r)$ | 12.29 | 14.10 | 16.21 |  | 10.99 | 12.81 | 14.92
Median | 4.23 | 5.12 | 6.14 |  | 3.16 | 3.95 | 4.82
IQ Range | 13.74 | 16.16 | 18.89 |  | 12.38 | 14.65 | 17.30
Minimum $r$ | -75.30 | -52.75 | -40.13 |  | -84.54 | -55.33 | -38.39
Skewness | -0.05 | 0.00 | 0.04 |  | -0.02 | 0.02 | 0.06
Kurtosis | 0.34 | 0.48 | 0.62 |  | 0.28 | 0.43 | 0.58
S.R. (ann.) | 1.06 | 1.16 | 1.26 |  | 0.93 | 1.03 | 1.14
Neg.Wts. | -583.7 | -496.9 | -420.5 |  | -512.5 | -431.1 | -358.7
Panel C: $\gamma^{*}=10:\,\gamma$ 5 loss function’s perspective
Portfolio: | M, S, $\overline{r}$, $\sigma_{\epsilon}$ |  | S, $\overline{r}$, $\sigma_{\epsilon}$
$\cal{CE}$5 | 132.7 | 151.9 | 170.2 |  | 56.5 | 116.2 | 146.5
$E(r)$ | 2.20 | 2.44 | 2.71 |  | 1.98 | 2.18 | 2.41
$\sigma(r)$ | 5.33 | 5.82 | 6.36 |  | 5.21 | 5.62 | 6.08
Median | 2.32 | 2.64 | 2.97 |  | 2.02 | 2.30 | 2.59
IQ Range | 5.57 | 6.22 | 6.95 |  | 5.05 | 5.61 | 6.22
Minimum $r$ | -37.75 | -27.35 | -20.12 |  | -53.51 | -40.88 | -29.22
Skewness | -0.07 | -0.03 | 0.00 |  | -0.06 | -0.02 | 0.02
Kurtosis | 0.52 | 0.64 | 0.77 |  | 0.58 | 0.71 | 0.85
S.R. (ann.) | 1.15 | 1.24 | 1.32 |  | 1.03 | 1.12 | 1.21
Neg.Wts. | -168.0 | -144.3 | -122.8 |  | -152.8 | -131.0 | -111.4
Portfolio: | M, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, S, $\sigma_{\epsilon}$
$\cal{CE}$5 | 92.3 | 107.6 | 121.7 |  | 110.5 | 123.4 | 135.9
$E(r)$ | 1.62 | 1.79 | 1.98 |  | 1.90 | 2.05 | 2.21
$\sigma(r)$ | 4.81 | 5.19 | 5.62 |  | 5.06 | 5.37 | 5.72
Median | 1.48 | 1.71 | 1.95 |  | 1.96 | 2.20 | 2.43
IQ Range | 4.86 | 5.34 | 5.86 |  | 5.08 | 5.53 | 6.02
Minimum $r$ | -36.51 | -26.55 | -18.26 |  | -35.13 | -28.02 | -22.33
Skewness | -0.02 | 0.02 | 0.05 |  | -0.06 | -0.03 | 0.01
Kurtosis | 0.55 | 0.67 | 0.80 |  | 0.58 | 0.68 | 0.79
S.R. (ann.) | 0.87 | 0.95 | 1.03 |  | 1.02 | 1.09 | 1.16
Neg.Wts. | -127.1 | -107.9 | -91.1 |  | -120.6 | -107.7 | -96.1

Table 3 -Continued
Sampling distributions of out-of-sample certainty equivalent returns,
portfolio moments and properties

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| 2.5%ile |  Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | M, S, $\overline{r}$ |  | M, V, S, $\overline{r}$, $\sigma_{\epsilon}$
$\cal{CE}$5 | 58.4 | 65.6 | 73.2 |  | 121.8 | 146.7 | 166.7
$E(r)$ | 1.31 | 1.42 | 1.53 |  | 2.20 | 2.44 | 2.70
$\sigma(r)$ | 5.09 | 5.31 | 5.54 |  | 5.42 | 5.92 | 6.46
Median | 1.39 | 1.58 | 1.77 |  | 2.30 | 2.61 | 2.94
IQ Range | 5.40 | 5.71 | 6.03 |  | 5.60 | 6.25 | 6.94
Minimum $r$ | -25.38 | -23.72 | -22.12 |  | -42.87 | -31.24 | -22.05
Skewness | -0.06 | -0.03 | 0.00 |  | -0.07 | -0.03 | 0.01
Kurtosis | 0.51 | 0.58 | 0.66 |  | 0.54 | 0.66 | 0.78
S.R. (ann.) | 0.63 | 0.68 | 0.74 |  | 1.13 | 1.22 | 1.31
Neg.Wts. | -65.4 | -52.6 | -41.3 |  | -163.4 | -140.9 | -120.0
Portfolio: | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | $\sigma_{\epsilon}$
$\cal{CE}$5 | 129.8 | 150.6 | 171.1 |  | 66.2 | 74.2 | 81.9
$E(r)$ | 2.16 | 2.42 | 2.72 |  | 1.10 | 1.16 | 1.23
$\sigma(r)$ | 5.29 | 5.80 | 6.37 |  | 3.83 | 3.95 | 4.07
Median | 2.32 | 2.62 | 2.95 |  | 1.05 | 1.20 | 1.34
IQ Range | 5.51 | 6.23 | 7.04 |  | 4.00 | 4.23 | 4.47
Minimum $r$ | -37.16 | -26.76 | -19.73 |  | -29.39 | -24.22 | -19.34
Skewness | -0.08 | -0.03 | 0.01 |  | -0.04 | -0.01 | 0.03
Kurtosis | 0.49 | 0.62 | 0.74 |  | 0.49 | 0.57 | 0.66
S.R. (ann.) | 1.13 | 1.23 | 1.33 |  | 0.63 | 0.70 | 0.76
Neg.Wts. | -173.3 | -148.1 | -125.6 |  | -51.4 | -47.1 | -43.1

Table 3 -Continued
Sampling distributions of out-of-sample certainty equivalent returns,
portfolio moments and properties

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| 2.5%ile |  Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Panel D: $\gamma^{*}=16$: $\gamma$ 9 loss function’s perspective
Portfolio: | M, S, $\sigma_{\epsilon}$ |  | S $\sigma_{\epsilon}$
$\cal{CE}$9 | 44.2 | 62.2 | 77.4 |  | 25.2 | 47.3 | 62.9
$E(r)$ | 1.64 | 1.76 | 1.89 |  | 1.33 | 1.42 | 1.51
$\sigma(r)$ | 4.33 | 4.54 | 4.77 |  | 3.96 | 4.12 | 4.29
Median | 1.70 | 1.89 | 2.08 |  | 1.27 | 1.41 | 1.56
IQ Range | 4.33 | 4.66 | 5.02 |  | 3.80 | 4.07 | 4.35
Minimum $r$ | -27.55 | -22.29 | -18.44 |  | -30.61 | -24.83 | -19.47
Skewness | -0.06 | -0.03 | 0.01 |  | -0.03 | 0.00 | 0.04
Kurtosis | 0.58 | 0.68 | 0.78 |  | 0.67 | 0.77 | 0.87
S.R. (ann.) | 0.99 | 1.07 | 1.14 |  | 0.81 | 0.88 | 0.95
Neg.Wts. | -85.6 | -77.0 | -69.0 |  | -59.8 | -53.5 | -47.4
Portfolio: | M, $\sigma_{\epsilon}$ |  | M, S
$\cal{CE}$9 | 32.6 | 45.0 | 56.1 |  | -4.4 | 2.8 | 9.8
$E(r)$ | 1.25 | 1.33 | 1.41 |  | 1.00 | 1.06 | 1.12
$\sigma(r)$ | 3.96 | 4.09 | 4.22 |  | 4.40 | 4.47 | 4.55
Median | 1.24 | 1.40 | 1.56 |  | 1.03 | 1.18 | 1.33
IQ Range | 4.19 | 4.44 | 4.70 |  | 4.90 | 5.14 | 5.37
Minimum $r$ | -24.48 | -20.30 | -16.73 |  | -23.86 | -22.59 | -21.30
Skewness | -0.05 | -0.02 | 0.02 |  | -0.06 | -0.03 | 0.00
Kurtosis | 0.47 | 0.56 | 0.65 |  | 0.33 | 0.39 | 0.44
S.R. (ann.) | 0.75 | 0.81 | 0.88 |  | 0.49 | 0.53 | 0.58
Neg.Wts. | -62.0 | -57.4 | -53.1 |  | -35.4 | -30.3 | -25.2
Portfolio: | M, $V_{-f}$, S, $\sigma_{\epsilon}$ |  | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$
$\cal{CE}$9 | 36.5 | 57.4 | 74.3 |  | 13.1 | 61.3 | 89.7
$E(r)$ | 1.65 | 1.77 | 1.90 |  | 1.74 | 1.92 | 2.13
$\sigma(r)$ | 4.44 | 4.66 | 4.89 |  | 4.41 | 4.72 | 5.07
Median | 1.72 | 1.91 | 2.11 |  | 1.94 | 2.16 | 2.39
IQ Range | 4.45 | 4.80 | 5.16 |  | 4.41 | 4.84 | 5.33
Minimum $r$ | -29.11 | -23.54 | -19.08 |  | -35.73 | -27.59 | -20.14
Skewness | -0.07 | -0.03 | 0.00 |  | -0.09 | -0.05 | -0.01
Kurtosis | 0.56 | 0.67 | 0.77 |  | 0.56 | 0.68 | 0.81
S.R. (ann.) | 0.97 | 1.05 | 1.12 |  | 1.05 | 1.15 | 1.25
Neg.Wts. | -86.0 | -77.7 | -70.1 |  | -109.0 | -92.8 | -78.8

Table 4
Sampling distribution of $\theta$ coefficients on characteristics

Feasible portfolio rules are obtained by fixing $\gamma^{*}$, the set of
characteristics, and the updating protocol, and then maximizing the following
using $T$ months of sample data:
$\max_{\theta}\sum\limits_{t=0}^{T-1}\frac{(1+r_{p,t+1})^{1-\gamma^{*}}}{1-\gamma^{*}}\left(\frac{1}{T}\right)$
by allowing portfolio weights to depend on observable stock characteristics:
$r_{p,t+1}=\sum\limits_{i=1}^{N_{t}}\left(\overline{\omega}_{i,t}+\frac{1}{N_{t}}\theta^{{}^{\prime}}x_{i,t}\right)\cdot
r_{i,t+1}\,,$ where: $x_{i,t}$ is the $K$-vector of characteristics on firm
$i$, measurable at time $t$; $\overline{\omega}_{i,t}$ is the weight of stock
$i$ in the (value-weighted) market portfolio at time $t$; and $N_{t}$ is the
number of stocks in the sample at time $t$. The $\theta$-vector is updated
annually, (and applied to the next 12 months of out-of-sample returns, to
construct 528 monthly out-of-sample returns). This generates 44 (annual)
$\theta$-vectors. This table reports small sample properties of the mean of
these 44 vectors for 30 different feasible portfolio rules: characteristic set
and curvature of loss function ($\gamma^{*}$) under the updating protocol
(from 10,000 pseudosamples). Characteristic abbreviations are: M - momentum;
$V$ \- book-to-market ratio (all stocks); $V_{-f}$ \- book-to-market ratio
exempting financial stocks; S - log size; $\beta$ \- market model regression
coefficient estimated over previous 60 months; $r_{t-12}$ \- last year’s same-
month return; $\overline{r}$ \- average same-month return from previous five
years; $\sigma_{\epsilon}$ \- residual standard deviation from the market
model regression estimated over the previous 60 months.

Panel A: $\gamma^{*}=3$. Emboldened portfolio optimal for $\gamma$ 2 investor.
---
| Resampled |  | Resampled
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Characteristics: | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$
M | 3.65 | 5.31 | 7.04 |  |  |  |
S | -14.09 | -10.62 | -7.84 |  | -13.21 | -9.88 | -7.09
$\beta$ | -1.64 | 3.59 | 7.52 |  | -5.05 | 0.43 | 4.45
$\overline{r}$ | 6.26 | 9.20 | 12.13 |  | 7.66 | 11.06 | 14.64
$\sigma_{\epsilon}$ | -15.95 | -11.88 | -6.80 |  | -11.39 | -7.77 | -3.33
Characteristics: | M, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, S, $\overline{r}$, $\sigma_{\epsilon}$
M | 3.16 | 4.80 | 6.43 |  | 2.88 | 4.85 | 6.70
S |  |  |  |  | -14.31 | -10.82 | -7.96
$\beta$ | 2.89 | 5.77 | 8.80 |  |  |  |
$\overline{r}$ | 5.81 | 8.32 | 11.87 |  | 6.24 | 9.20 | 12.05
$\sigma_{\epsilon}$ | -13.06 | -9.50 | -5.88 |  | -11.03 | -8.17 | -5.83
Characteristics: | M, S, $\beta$, $\sigma_{\epsilon}$ |  | M, S, $\beta$, $\overline{r}$
M | 3.95 | 6.32 | 8.30 |  | 1.79 | 3.74 | 5.41
S | -12.59 | -9.02 | -6.17 |  | -13.15 | -8.82 | -5.76
$\beta$ | -1.58 | 3.43 | 6.98 |  | -9.12 | -6.17 | -3.67
$\overline{r}$ |  |  |  |  | 6.39 | 8.88 | 11.54
$\sigma_{\epsilon}$ | -14.26 | -10.53 | -6.20 |  |  |  |

Table 4 -Continued
Sampling distribution of $\theta$ coefficients on characteristics

| Resampled |  | Resampled
---|---|---|---
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Characteristics: | M, V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, V, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$
M | 4.79 | 6.25 | 7.92 |  | 5.71 | 6.78 | 7.89
V / $V_{-f}$ | 1.23 | 3.92 | 6.81 |  | 5.93 | 7.60 | 9.37
S | -12.46 | -8.59 | -5.68 |  |  |  |
$\beta$ | -2.24 | 3.41 | 7.43 |  | 2.13 | 4.57 | 7.20
$\overline{r}$ | 6.14 | 9.12 | 12.13 |  | 6.14 | 8.12 | 10.25
$\sigma_{\epsilon}$ | -14.39 | -10.27 | -4.87 |  | -10.38 | -7.23 | -4.30
Panel B: $\gamma^{*}=10$. Emboldened portfolio optimal for $\gamma$ 5
investor.
Characteristics: | M, S, $\overline{r}$, $\sigma_{\epsilon}$ |  | S, $\overline{r}$, $\sigma_{\epsilon}$
M | 1.17 | 1.77 | 2.47 |  |  |  |
S | -5.21 | -3.99 | -2.75 |  | -4.52 | -3.53 | -2.43
$\overline{r}$ | 1.76 | 2.73 | 3.77 |  | 1.75 | 3.21 | 4.56
$\sigma_{\epsilon}$ | -5.12 | -3.87 | -2.67 |  | -4.54 | -3.34 | -2.01
Characteristics: | M, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, S, $\sigma_{\epsilon}$
M | 0.96 | 1.44 | 1.94 |  | 1.07 | 1.96 | 2.82
S |  |  |  |  | -4.79 | -3.58 | -2.31
$\overline{r}$ | 1.54 | 2.47 | 3.65 |  |  |  |
$\sigma_{\epsilon}$ | -2.71 | -2.04 | -1.49 |  | -4.74 | -3.56 | -2.33
Characteristics: | M, S, $\overline{r}$ |  | M, V, S, $\overline{r}$, $\sigma_{\epsilon}$
M | 0.41 | 0.85 | 1.90 |  | 1.27 | 1.97 | 2.60
V |  |  |  |  | -0.10 | 0.86 | 1.95
S | -0.06 | 0.39 | 0.81 |  | -4.79 | -3.54 | -2.49
$\overline{r}$ | 0.56 | 1.43 | 2.66 |  | 1.74 | 2.71 | 3.76
$\sigma_{\epsilon}$ |  |  |  |  | -4.72 | -3.56 | -2.51
Characteristics: | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | $\sigma_{\epsilon}$
M | 1.27 | 1.79 | 2.45 |  |  |  |
S | -5.31 | -4.04 | -2.75 |  |  |  |
$\beta$ | -1.51 | 0.09 | 1.53 |  |  |  |
$\overline{r}$ | 1.80 | 2.76 | 3.78 |  |  |  |
$\sigma_{\epsilon}$ | -5.37 | -4.03 | -2.70 |  | -2.09 | -1.52 | -1.03
Panel C: $\gamma^{*}=16$. Emboldened portfolio optimal for $\gamma$ 9
investor.
Characteristics: | M, S, $\sigma_{\epsilon}$ |  | S, $\sigma_{\epsilon}$
M | 0.68 | 1.26 | 1.96 |  |  |  |
S | -3.44 | -2.46 | -1.38 |  | -2.57 | -1.92 | -1.23
$\sigma_{\epsilon}$ | -3.91 | -2.92 | -1.89 |  | -3.13 | -2.30 | -1.50

Table 4 -Continued
Sampling distribution of $\theta$ coefficients on characteristics

| Resampled |  | Resampled
---|---|---|---
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Characteristics: | M, $\sigma_{\epsilon}$ |  | M, S
M | 0.61 | 0.98 | 1.42 |  | -0.06 | 0.35 | 1.36
S |  |  |  |  | 0.09 | 0.78 | 1.18
$\sigma_{\epsilon}$ | -2.27 | -1.70 | -1.24 |  |  |  |
Characteristics: | M, $V_{-f}$, S, $\sigma_{\epsilon}$ |  | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$
M | 1.36 | 1.58 | 1.80 |  | 0.76 | 1.11 | 1.67
V | 0.73 | 1.16 | 1.60 |  |  |  |
S | -2.35 | -1.91 | -1.47 |  | -3.85 | -2.84 | -1.65
$\beta$ |  |  |  |  | -1.71 | -0.58 | 0.58
$\overline{r}$ |  |  |  |  | 0.96 | 1.66 | 2.42
$\sigma_{\epsilon}$ | -2.79 | -2.50 | -2.22 |  | -3.64 | -2.53 | -1.36

Panel D: $\theta$ coefficient standard-deviation / correlation matrices
---
M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon};\,\gamma^{*}=3$
Optimal portfolio for $\gamma$ 2 loss function
Characteristic: | M | S | $\beta$ | $\overline{r}$ | $\sigma_{\epsilon}$
M | 0.95 | -0.60 | -0.34 | 0.01 | -0.05
S |  | 1.72 | 0.60 | -0.02 | -0.15
$\beta$ |  |  | 2.50 | 0.22 | -0.85
$\overline{r}$ |  |  |  | 1.45 | -0.32
$\sigma_{\epsilon}$ |  |  |  |  | 2.28
M, S, $\overline{r}$, $\sigma_{\epsilon};\,\gamma^{*}=10$
---
Optimal portfolio for $\gamma$ 5 loss function
Characteristic: | M | S | $\overline{r}$ | $\sigma_{\epsilon}$
M | 0.37 | -0.75 | 0.09 | -0.73
S |  | 0.71 | -0.26 | 0.92
$\overline{r}$ |  |  | 0.51 | -0.31
$\sigma_{\epsilon}$ |  |  |  | 0.80
M, S, $\sigma_{\epsilon};\,\gamma^{*}=16$
---
Optimal portfolio for $\gamma$ 9 loss function
Characteristic: | M |  S | $\sigma_{\epsilon}$
M | 0.40 | -0.87 | -0.80
S |  | 0.64 | 0.91
$\sigma_{\epsilon}$ |  |  | 0.71

Table 5
Decomposition of portfolio expected return and variance based on Fama-French-
Carhart Factors

Panel A reports results from the linear regression:
$r_{p,t}-r_{f,t}=\alpha+\beta_{M}\cdot(r_{M,t}-r_{f,t})+\beta_{S}\cdot
r_{S,t}+\beta_{H}\cdot r_{H,t}+\beta_{m}\cdot r_{m,t}+\epsilon_{t}$. Where
$r_{p,t}$ is the portfolio return in month $t$, $r_{f,t}$ is the return on
one-month Treasury bill in month $t$, $r_{M,t}$ is the return on the value-
weighted market in month $t$, $r_{S,t}$ is the return on the SMB factor
portfolio in month $t$, $r_{H,t}$ is the return on the HML factor portfolio in
month $t$, and $r_{m,t}$ is the return on the momentum factor portfolio in
month $t$. This decomposes the portfolio variance into 10 (additive)
components. We report only covariance terms greater than 5% in absolute value
(so the reported components will not sum to unity). The volatility-correlation
matrix in Panel B reports the monthly factor standard deviations along the
diagonal and correlations between the factors, off-diagonal. Panel A $\alpha$
values and Panel B factor means are reported in basis points per month.

Panel A. Portfolio factor exposures
---
Metric | Resampled |  | Resampled
| Out of sample |  | Out of sample
| regression coefficient |  | variance component (%)
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | Value-weighted portfolio of all sample stocks
$\alpha$ / Orthog. | -2.00 | 2.65 | 7.17 |  | 1.40 | 1.70 | 2.04
MKT | 0.98 | 0.99 | 1.00 |  | 101.32 | 102.40 | 103.41
SMB | -0.11 | -0.09 | -0.07 |  | 0.23 | 0.41 | 0.62
HML | 0.02 | 0.04 | 0.07 |  | 0.02 | 0.10 | 0.20
MOM | -0.02 | 0.00 | 0.01 |  | 0.00 | 0.01 | 0.04
Cov(MKT , SMB) |  |  |  |  | -4.08 | -3.30 | -2.49
Cov(MKT , HML) |  |  |  |  | -2.22 | -1.46 | -0.71
Portfolio: | Equally-weighted portfolio of all sample stocks
$\alpha$ / Orthog. | 6.23 | 8.49 | 10.76 |  | 2.75 | 2.95 | 3.15
MKT | 1.02 | 1.03 | 1.03 |  | 79.13 | 79.74 | 80.34
SMB | 0.47 | 0.48 | 0.49 |  | 7.86 | 8.21 | 8.58
HML | 0.25 | 0.26 | 0.27 |  | 2.05 | 2.22 | 2.39
MOM | -0.09 | -0.08 | -0.07 |  | 0.37 | 0.46 | 0.56
Cov(MKT , SMB) |  |  |  |  | 12.79 | 13.04 | 13.29
Cov(MKT , HML) |  |  |  |  | -6.69 | -6.43 | -6.18
Portfolio: | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 201.29 | 269.39 | 347.78 |  | 61.31 | 68.63 | 75.47
MKT | 0.30 | 0.52 | 0.75 |  | 0.87 | 2.85 | 5.69
SMB | 0.59 | 1.02 | 1.45 |  | 1.49 | 5.05 | 9.87
HML | 1.61 | 1.99 | 2.39 |  | 11.92 | 16.60 | 21.36
MOM | 1.18 | 1.46 | 1.75 |  | 13.60 | 20.39 | 27.93
Cov(SMB , HML) |  |  |  |  | -5.47 | -3.61 | -1.87
Cov(HML , MOM) |  |  |  |  | -9.86 | -8.11 | -6.35

Table 5 -Continued
Decomposition of portfolio expected return and variance based on Fama-French-
Carhart Factors

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| regression coefficient |  | variance component
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 248.66 | 311.30 | 382.04 |  | 75.94 | 81.15 | 86.13
MKT | 0.23 | 0.43 | 0.63 |  | 0.58 | 2.23 | 4.63
SMB | 0.11 | 0.47 | 0.83 |  | 0.06 | 1.38 | 3.68
HML | 1.49 | 1.79 | 2.11 |  | 11.49 | 15.70 | 20.07
MOM | -0.53 | -0.32 | -0.12 |  | 0.16 | 1.23 | 2.83
Cov(MKT , HML) |  |  |  |  | -4.04 | -2.76 | -1.50
Portfolio: | M, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 124.60 | 182.27 | 248.00 |  | 50.07 | 57.72 | 65.06
MKT | 1.18 | 1.35 | 1.52 |  | 17.16 | 22.36 | 28.06
SMB | -1.52 | -1.19 | -0.88 |  | 4.76 | 8.11 | 11.82
HML | -0.11 | 0.12 | 0.37 |  | 0.00 | 0.14 | 0.63
MOM | 1.24 | 1.49 | 1.74 |  | 18.80 | 26.38 | 34.65
Cov(MKT , SMB) |  |  |  |  | -8.27 | -6.79 | -5.29
Cov(MKT , MOM) |  |  |  |  | -6.24 | -5.25 | -4.29
Portfolio: | M, S, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 194.41 | 256.16 | 328.01 |  | 59.49 | 66.62 | 73.30
MKT | 0.11 | 0.28 | 0.44 |  | 0.12 | 0.87 | 2.15
SMB | 0.57 | 0.97 | 1.36 |  | 1.45 | 4.62 | 8.89
HML | 1.67 | 2.02 | 2.41 |  | 13.14 | 17.53 | 22.12
MOM | 1.30 | 1.55 | 1.80 |  | 16.89 | 23.40 | 30.50
Cov(SMB , HML) |  |  |  |  | -5.29 | -3.55 | -1.90
Cov(HML , MOM) |  |  |  |  | -10.59 | -8.94 | -7.31
Portfolio: | M, S, $\beta$, $\sigma_{\epsilon}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 80.16 | 118.27 | 162.00 |  | 40.88 | 46.15 | 52.06
MKT | 0.16 | 0.34 | 0.52 |  | 0.31 | 1.57 | 3.51
SMB | 0.92 | 1.23 | 1.55 |  | 5.22 | 8.83 | 12.97
HML | 1.88 | 2.18 | 2.49 |  | 20.35 | 25.14 | 29.75
MOM | 1.60 | 1.80 | 2.01 |  | 31.12 | 38.56 | 46.16
Cov(SMB , HML) |  |  |  |  | -7.49 | -5.96 | -4.43
Cov(HML , MOM) |  |  |  |  | -15.05 | -13.77 | -12.35
Portfolio: | M, S, $\beta$, $\overline{r}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 133.58 | 182.53 | 237.68 |  | 61.91 | 68.70 | 75.21
MKT | -0.03 | 0.14 | 0.30 |  | 0.00 | 0.32 | 1.18
SMB | 0.45 | 0.81 | 1.15 |  | 1.10 | 3.77 | 7.34
HML | 1.08 | 1.37 | 1.66 |  | 6.17 | 9.43 | 13.02
MOM | 1.32 | 1.53 | 1.74 |  | 20.29 | 26.73 | 33.55
Cov(HML , MOM) |  |  |  |  | -8.44 | -7.00 | -5.55

Table 5 -Continued
Decomposition of portfolio expected return and variance based on Fama-French-
Carhart Factors

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| regression coefficient |  | variance component
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | M, V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 186.86 | 255.51 | 334.57 |  | 59.41 | 66.50 | 73.23
MKT | 0.51 | 0.76 | 1.02 |  | 2.52 | 5.87 | 10.33
SMB | 0.38 | 0.85 | 1.31 |  | 0.62 | 3.63 | 8.09
HML | 2.11 | 2.59 | 3.11 |  | 20.28 | 28.32 | 36.89
MOM | 0.85 | 1.16 | 1.47 |  | 7.05 | 12.97 | 19.76
Cov(MKT , HML) |  |  |  |  | -8.38 | -6.12 | -4.02
Cov(HML , MOM) |  |  |  |  | -10.53 | -8.40 | -6.21
Portfolio: | M, V, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=3$
$\alpha$ / Orthog. | 125.79 | 187.91 | 259.35 |  | 58.45 | 64.19 | 69.79
MKT | 1.25 | 1.44 | 1.64 |  | 18.42 | 24.99 | 31.97
SMB | -1.04 | -0.65 | -0.30 |  | 0.56 | 2.51 | 5.31
HML | 1.54 | 2.02 | 2.54 |  | 13.17 | 20.86 | 29.27
MOM | 0.61 | 0.89 | 1.18 |  | 4.17 | 9.43 | 15.99
Cov(MKT , HML) |  |  |  |  | -13.17 | -10.94 | -8.68
Cov(HML , MOM) |  |  |  |  | -8.21 | -6.09 | -4.06
Portfolio: | M, S, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 80.40 | 101.90 | 126.70 |  | 53.73 | 58.33 | 63.02
MKT | 0.50 | 0.56 | 0.62 |  | 13.26 | 18.51 | 24.66
SMB | -0.14 | -0.01 | 0.11 |  | 0.00 | 0.12 | 0.61
HML | 0.97 | 1.09 | 1.22 |  | 24.99 | 29.59 | 34.04
MOM | 0.51 | 0.59 | 0.68 |  | 14.83 | 19.55 | 24.60
Cov(MKT , HML) |  |  |  |  | -12.99 | -11.27 | -9.58
Cov(HML , MOM) |  |  |  |  | -12.13 | -10.64 | -9.16
Portfolio: | S, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 98.89 | 118.35 | 140.22 |  | 55.29 | 59.97 | 64.77
MKT | 0.58 | 0.63 | 0.69 |  | 19.26 | 25.00 | 31.14
SMB | -0.25 | -0.13 | -0.02 |  | 0.01 | 0.64 | 1.83
HML | 0.85 | 0.96 | 1.07 |  | 20.29 | 24.20 | 28.24
MOM | -0.14 | -0.08 | -0.02 |  | 0.03 | 0.43 | 1.03
Cov(MKT , HML) |  |  |  |  | -13.28 | -11.85 | -10.40

Table 5 -Continued
Decomposition of portfolio expected return and variance based on Fama-French-
Carhart Factors

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| regression coefficient |  | variance component
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | M, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 43.95 | 60.46 | 78.48 |  | 27.75 | 32.85 | 38.36
MKT | 0.80 | 0.84 | 0.88 |  | 43.95 | 51.44 | 59.04
SMB | -0.94 | -0.84 | -0.76 |  | 20.95 | 24.19 | 27.49
HML | 0.28 | 0.34 | 0.42 |  | 2.38 | 3.71 | 5.20
MOM | 0.50 | 0.57 | 0.65 |  | 17.99 | 23.01 | 28.23
Cov(MKT , SMB) |  |  |  |  | -19.42 | -17.95 | -16.37
Cov(MKT , HML) |  |  |  |  | -8.05 | -6.65 | -5.26
Cov(MKT , MOM) |  |  |  |  | -8.38 | -7.47 | -6.50
Portfolio: | M, S, $\sigma_{\epsilon}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 47.18 | 60.10 | 74.05 |  | 44.90 | 48.00 | 51.02
MKT | 0.45 | 0.50 | 0.56 |  | 12.82 | 17.42 | 22.67
SMB | -0.03 | 0.07 | 0.17 |  | 0.00 | 0.24 | 0.92
HML | 1.06 | 1.16 | 1.26 |  | 35.23 | 38.90 | 42.44
MOM | 0.57 | 0.64 | 0.71 |  | 22.27 | 26.97 | 31.90
Cov(MKT , HML) |  |  |  |  | -14.15 | -12.54 | -10.91
Cov(HML , MOM) |  |  |  |  | -15.39 | -14.35 | -13.27
Portfolio: | M, S, $\overline{r}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 10.51 | 20.80 | 32.04 |  | 9.29 | 12.17 | 15.46
MKT | 1.09 | 1.12 | 1.15 |  | 82.81 | 87.19 | 91.25
SMB | -0.30 | -0.23 | -0.15 |  | 0.74 | 1.69 | 2.91
HML | -0.20 | -0.15 | -0.10 |  | 0.28 | 0.64 | 1.09
MOM | 0.23 | 0.29 | 0.35 |  | 3.64 | 5.61 | 7.83
Cov(MKT , SMB) |  |  |  |  | -8.11 | -6.10 | -4.08
Portfolio: | M, V, S, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 78.26 | 99.65 | 123.90 |  | 49.34 | 54.85 | 60.28
MKT | 0.53 | 0.60 | 0.67 |  | 14.76 | 20.55 | 27.19
SMB | -0.21 | -0.07 | 0.07 |  | 0.00 | 0.25 | 1.10
HML | 1.08 | 1.23 | 1.38 |  | 29.64 | 36.19 | 42.64
MOM | 0.45 | 0.54 | 0.63 |  | 11.22 | 15.99 | 21.05
Cov(MKT , HML) |  |  |  |  | -15.54 | -13.14 | -10.86
Cov(HML , MOM) |  |  |  |  | -12.18 | -10.62 | -8.98

Table 5 -Continued
Decomposition of portfolio expected return and variance based on Fama-French-
Carhart Factors

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| regression coefficient |  | variance component
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 77.68 | 100.42 | 126.87 |  | 53.86 | 58.39 | 63.05
MKT | 0.49 | 0.56 | 0.63 |  | 13.41 | 18.51 | 24.45
SMB | -0.14 | -0.01 | 0.13 |  | 0.00 | 0.12 | 0.62
HML | 0.98 | 1.10 | 1.23 |  | 25.57 | 30.43 | 35.11
MOM | 0.48 | 0.57 | 0.66 |  | 13.68 | 18.53 | 23.86
Cov(MKT , HML) |  |  |  |  | -13.14 | -11.42 | -9.74
Cov(HML , MOM) |  |  |  |  | -12.07 | -10.50 | -8.88
Portfolio: |  $\sigma_{\epsilon}$; $\gamma^{*}=10$
$\alpha$ / Orthog. | 22.43 | 29.94 | 37.44 |  | 19.07 | 21.27 | 23.55
MKT | 0.73 | 0.76 | 0.79 |  | 67.04 | 72.54 | 77.92
SMB | -0.70 | -0.66 | -0.62 |  | 22.77 | 25.45 | 28.18
HML | 0.39 | 0.42 | 0.46 |  | 8.05 | 9.59 | 11.28
MOM | -0.02 | 0.01 | 0.05 |  | 0.00 | 0.06 | 0.26
Cov(MKT , SMB) |  |  |  |  | -22.66 | -21.87 | -21.04
Cov(MKT , HML) |  |  |  |  | -13.65 | -12.74 | -11.82
Portfolio: | M, S, $\sigma_{\epsilon}$; $\gamma^{*}=16$
$\alpha$ / Orthog. | 40.30 | 50.84 | 61.98 |  | 42.92 | 46.24 | 49.41
MKT | 0.54 | 0.58 | 0.63 |  | 26.01 | 32.44 | 39.66
SMB | -0.27 | -0.18 | -0.08 |  | 0.30 | 1.49 | 3.31
HML | 0.87 | 0.96 | 1.04 |  | 33.16 | 37.38 | 41.34
MOM | 0.36 | 0.42 | 0.48 |  | 12.08 | 16.26 | 20.60
Cov(MKT , HML) |  |  |  |  | -18.04 | -16.79 | -15.42
Cov(HML , MOM) |  |  |  |  | -12.32 | -10.91 | -9.32
Portfolio: | S, $\sigma_{\epsilon}$; $\gamma^{*}=16$
$\alpha$ / Orthog. | 38.40 | 46.83 | 55.80 |  | 35.30 | 39.15 | 42.78
MKT | 0.59 | 0.63 | 0.67 |  | 38.55 | 46.13 | 54.18
SMB | -0.32 | -0.24 | -0.16 |  | 1.43 | 3.21 | 5.54
HML | 0.71 | 0.78 | 0.86 |  | 25.59 | 30.06 | 34.42
MOM | -0.07 | -0.03 | 0.00 |  | 0.00 | 0.17 | 0.58
Cov(MKT , SMB) |  |  |  |  | -8.58 | -6.14 | -3.91
Cov(MKT , HML) |  |  |  |  | -18.84 | -17.96 | -17.02

Table 5 -Continued
Decomposition of portfolio expected return and variance based on Fama-French-
Carhart Factors

Metric | Resampled |  | Resampled
---|---|---|---
| Out of sample |  | Out of sample
| regression coefficient |  | variance component
| 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
Portfolio: | M, $\sigma_{\epsilon}$; $\gamma^{*}=16$
$\alpha$ / Orthog. | 15.90 | 23.43 | 31.27 |  | 20.31 | 22.48 | 24.69
MKT | 0.74 | 0.76 | 0.79 |  | 63.67 | 68.81 | 73.86
SMB | -0.72 | -0.67 | -0.63 |  | 22.36 | 24.97 | 27.54
HML | 0.40 | 0.45 | 0.49 |  | 8.37 | 10.09 | 11.92
MOM | 0.33 | 0.38 | 0.43 |  | 12.80 | 16.68 | 20.83
Cov(MKT , SMB) |  |  |  |  | -22.03 | -21.10 | -20.08
Cov(MKT , HML) |  |  |  |  | -13.73 | -12.72 | -11.65
Portfolio: | M, S; $\gamma^{*}=16$
$\alpha$ / Orthog. | -4.88 | 0.74 | 6.48 |  | 4.26 | 5.17 | 6.21
MKT | 1.01 | 1.02 | 1.04 |  | 100.60 | 102.16 | 103.56
SMB | -0.43 | -0.38 | -0.32 |  | 4.76 | 6.44 | 8.25
HML | -0.06 | -0.03 | 0.01 |  | 0.00 | 0.04 | 0.15
MOM | 0.09 | 0.14 | 0.18 |  | 0.70 | 1.78 | 3.11
Cov(MKT , SMB) |  |  |  |  | -14.76 | -13.04 | -11.25
Portfolio: | M, $V_{-f}$, S, $\sigma_{\epsilon}$; $\gamma^{*}=16$
$\alpha$ / Orthog. | 39.84 | 50.16 | 61.17 |  | 36.12 | 40.02 | 44.01
MKT | 0.58 | 0.63 | 0.68 |  | 29.48 | 36.34 | 43.75
SMB | -0.32 | -0.23 | -0.13 |  | 0.77 | 2.30 | 4.40
HML | 0.98 | 1.07 | 1.17 |  | 40.05 | 44.93 | 49.67
MOM | 0.30 | 0.36 | 0.42 |  | 8.04 | 11.71 | 15.69
Cov(MKT , HML) |  |  |  |  | -21.10 | -19.49 | -17.75
Cov(HML , MOM) |  |  |  |  | -11.79 | -10.14 | -8.36
Portfolio: | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$; $\gamma^{*}=16$
$\alpha$ / Orthog. | 53.46 | 69.72 | 87.92 |  | 48.43 | 52.29 | 56.22
MKT | 0.52 | 0.57 | 0.62 |  | 22.93 | 28.86 | 35.32
SMB | -0.29 | -0.19 | -0.10 |  | 0.39 | 1.67 | 3.50
HML | 0.84 | 0.94 | 1.03 |  | 28.55 | 33.03 | 37.50
MOM | 0.34 | 0.41 | 0.47 |  | 10.21 | 14.06 | 18.06
Cov(MKT , HML) |  |  |  |  | -16.39 | -14.89 | -13.35
Cov(HML , MOM) |  |  |  |  | -10.97 | -9.53 | -8.04
Panel B. Factor volatility / correlation matrix |
|  |  |  |  |  |  | Mean
|  | MKT | SMB | HML | MOM |  | return
| MKT | 4.41 | 0.25 | -0.24 | -0.11 |  | 67.49
| SMB |  | 3.01 | -0.20 | 0.06 |  | 23.25
| HML |  |  | 2.88 | -0.22 |  | 28.85
| MOM |  |  |  | 4.32 |  | 59.62