You invest in a country named Paf. You consider explaining the difference in returns by two common factors, with a linear factor model as given in Equation (6.9). Your candidate for factors are movements in interest rates and changes in the popularity of the president of Paf, as measured by polls. The various values of the factor returns and of the returns on four stocks (A, B, C, and D) for the ten past periods are given below:
$$\begin{array} { r c c r r r r }
\hline \text { Periad } & \text { Interet Rate } & \text { Papularity } & { \mathbf { A } } & { \text { B } } & { \mathbf { C } } & { \mathbf { D } } \
\hline 1 & - 3 \% & 2 \% & - 2.57 & - 4.43 & - 2.16 & - 5.02 \
2 & 2 \% / 6 & 7 \% & 1.90 & 2.53 & 2.01 & 3.81 \
3 & 5 \% & 6 \% & 5.46 & 7.42 & 4.10 & 9.17 \
4 & - 2 \% & 4 \% & - 1.62 & - 4.70 & - 0.59 & - 3.24 \
5 & 4 \% & - 3 \% & 4.51 & 6.43 & 3.01 & 8.79 \
6 & 2 \% & 4 \% & 2.16 & 1.56 & 1.36 & 4.66 \
7 & - 5 \% & - 7 \% & - 5.54 & - 4.73 & - 4.08 & - 8.60 \
8 & - 6 \% & 0 \% & - 5.83 & - 8.12 & - 5.41 & - 11.47 \
9 & 3 \% & 2 \% & 3.04 & 4.71 & 2.42 & 6.43 \
10 & 1 \% & - 2 \% & 1.18 & 2.44 & 0.00 & 2.50 \
\hline
\end{array}$$ You will try to assess whether the two factors have an influence on stock returns. To do so:
a. Estimate the factor exposures for each of the four assets. (You can do a times-series regression for each asset against the two factors.)
b. Are the factors «priced», that is, is there a relation between mean returns on the assets and their factor exposures? (You can do a cross-sectional regression between the mean return on the asset and its exposures, that is, you have one observation for each asset.)

Question

You invest in a country named Paf. You consider explaining the difference in returns by two common factors, with a linear factor model as given in Equation (6.9). Your candidate for factors are movements in interest rates and changes in the popularity of the president of Paf, as measured by polls. The various values of the factor returns and of the returns on four stocks (A, B, C, and D) for the ten past periods are given below:
 $$\begin{array} { r c c r r r r } 
\hline \text { Periad } & \text { Interet Rate } & \text { Papularity } &  { \mathbf { A } } & { \text { B } } & { \mathbf { C } } & { \mathbf { D } } \
\hline 1 & - 3 \% & 2 \% & - 2.57 & - 4.43 & - 2.16 & - 5.02 \
2 & 2 \% / 6 & 7 \% & 1.90 & 2.53 & 2.01 & 3.81 \
3 & 5 \% & 6 \% & 5.46 & 7.42 & 4.10 & 9.17 \
4 & - 2 \% & 4 \% & - 1.62 & - 4.70 & - 0.59 & - 3.24 \
5 & 4 \% & - 3 \% & 4.51 & 6.43 & 3.01 & 8.79 \
6 & 2 \% & 4 \% & 2.16 & 1.56 & 1.36 & 4.66 \
7 & - 5 \% & - 7 \% & - 5.54 & - 4.73 & - 4.08 & - 8.60 \
8 & - 6 \% & 0 \% & - 5.83 & - 8.12 & - 5.41 & - 11.47 \
9 & 3 \% & 2 \% & 3.04 & 4.71 & 2.42 & 6.43 \
10 & 1 \% & - 2 \% & 1.18 & 2.44 & 0.00 & 2.50 \
\hline
\end{array}$$ You will try to assess whether the two factors have an influence on stock returns. To do so:
a. Estimate the factor exposures for each of the four assets. (You can do a times-series regression for each asset against the two factors.)
b. Are the factors «priced», that is, is there a relation between mean returns on the assets and their factor exposures? (You can do a cross-sectional regression between the mean return on the asset and its exposures, that is, you have one observation for each asset.)

Quizplus · Accepted Answer

The Answer is a. We estimate the factor exposures for each stock by running a multiple regression between the stock returns and the two factors' returns:

R _ { i } = \alpha _ { i } + \beta _ { 1 i } f _ { 1 } + \beta _ { 2 i } f _ { 2 } + \varepsilon _ { i }

where:

R _ { i }

is the return on stock i

f _ { 1 }

is the factor "movements in interest rate"

f _ { 2 }

is the factor "change in popularity"

\alpha _ { i }

is the intercept of stock i

\beta _ { 1 i }

is the exposure of stock i to factor 1

\beta _ { 2 i }

is the exposure of stock i to factor 2

\begin{array} { l c c c c } & \text { Stack A } & \text { Stack B } & \text { Stack C } & \text { Stack D } \\hline \text { Mean Stack Return } & 0.27 \% & 0.31 \% & 0.07 \% & 0.70 \% \\text { Constant } & 0.14 & 0.46 & - 0.10 & 0.60 \( 5 t a n d a r d ~ e r r o r ) & ( 0.11 ) & ( 0.28 ) & ( 0.16 ) & ( 0.14 ) \\text { Beta 1 } & 1.02 & 1.48 & 0.77 & 1.94 \( 5 t a n d a r d \text { error) } & ( 0.03 ) & ( 0.08 ) & ( 0.05 ) & ( 0.04 ) \\text { Beta 2 } & 0.02 & - 0.23 & 0.07 & - 0.08 \( 5 t a n d a r d ~ e r r o r ) & ( 0.03 ) & ( 0.07 ) & ( 0.04 ) & ( 0.04 )\end{array}

The first line gives the mean return of each stock over the ten periods. The next lines give the estimated coefficients of the regressions and their standard errors. It appears that the interest
rate factor has a strong influence on stock returns, and the estimates are highly significant (low standard errors). The exposures to the popularity factor are smaller and much less significant statistically. All R-squares are quite large (above 90%).
b. The second question asked is whether the risk factors are priced. In other words, is there a link between the risk exposures of an asset and its return? If a risk factor is priced, assets with different risk exposures should have different mean returns. This is a cross-sectional conclusion that can be tested by running a regression between the mean return on each stock and their risk exposures. Here we have four observations (four different stocks) and two factors, the cross-sectional regression can be written as:

\bar { R } _ { i } = \gamma _ { 0 } + \gamma _ { 1 } \beta _ { 1 i } +\gamma _ { 2 } \beta _ { 2 i } + \xi _ { i }

\begin{array}{c}\begin{array}{l}Y_{0} \\text { (standard error) } \Y_{1} \\text { (standard error) } \Y_{2} \\text { (standard error) } \end{array}\begin{array}{lll} -0.41\(0.11)\0.60\(0.09)\0.73\ (0.35)\end{array}\end{array}

The two factors are priced. Their slopes are statistically significant, especially for the first factor. The higher the risk exposure to each factor, the higher the mean stock return. However, you should note that the first factor has much more importance in stock pricing. This is because the risk exposures to the first factor vary between 0.77 and 1.94, while the risk exposures to the second factor only vary between -0.23 and 0.07.

You Invest in a Country Named Paf You Will Try to Assess Whether the Two Factors Have