# IFM7 Chapter 3

Document Sample

```					101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                                   Model
Ch 03 Tool Kit                                                                                           5/29/2002

Chapter 3. Tool Kit for Risk and Return: Portfolio Theory and Asset Pricing Models

PROBABILITY DISTRIBUTIONS

The probability distribution is a listing of all possible outcomes and the corresponding probability.

Probability of
Occurrence                              Rate of Return Distribution
E                 F                G                  H
0.10                  10%               6%               14%                4%
0.20                  10%               8%               12%                6%
0.40                  10%              10%               10%                8%
0.20                  10%              12%               8%                 15%
0.10                  10%              14%               6%                 22%
1.00

EXPECTED RATE OF RETURN AND STANDARD DEVIATION

The expected rate of return is the rate of return that is expected to be realized from an investment. It is
determined as the weighted average of the probability distribution of returns.

To calculate the standard deviation, there are a few steps. First find the differences of all the possible returns
from the expected return. Second, square that difference. Third, multiply the squared number by the
probability of its occurrence. Fourth, find the sum of all the weighted squares. And lastly, take the square
root of that number.

Calculation of expected return and standard deviation for E
Expected rate of return for E               Standard deviation for E
Probability of                                         Deviation from    Squared
Occurrence          Rate of Return        Product         r hat         deviation      Sq Dev * Prob.

10%                    10%               1.00%            0.000%            0.00%               0.00%
20%                    10%               2.00%            0.000%            0.00%               0.00%
40%                    10%               4.00%            0.000%            0.00%               0.00%
20%                    10%               2.00%            0.000%            0.00%               0.00%
10%                    10%               1.00%            0.000%            0.00%               0.00%
100%                                                                           Sum:             0.00%
Expected                                            Std. Dev. =
Rate of Return, r                                      Square root of
hat =                     10%                          sum =                    0.00%

If the probabilities are fairly simple, then a short-cut method is to use the excel functions for AVERAGE and
ay       "weigh
STDEVP, but to "trick" them by entering arguments more than once, in a way that "weights" them like
ten probability.
the probabilities. For example, for stock E we would enter 6% once, since it has only a one innten probability
lity.
We would enter 8% twice, since it has a two in ten probability. We would enter 10% 4 times, since it has
a four in ten probability. We can do the same thing with the standard deviation function. Note that we use
STDEVP and not STDEV, since we are measuring the standard deviation for the entire population and not
for a sample. We call this the "indirect" method.

Indirect method                    r hat =        10%                                       s=          0.00%

Calculation of expected return and standard deviation for F

Michael C. Ehrhardt                                            Page 1                                                   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                          Model
Expected rate of return for F                 Standard deviation for F
Probability of                                        Deviation from     Squared
Occurrence         Rate of Return      Product           r hat          deviation      Sq Dev * Prob.

10%                   6%             0.60%             -4%             0.16%           0.02%
20%                   8%             1.60%             -2%             0.04%           0.01%
40%                  10%             4.00%             0%              0.00%           0.00%
20%                  12%             2.40%             2%              0.04%           0.01%
10%                  14%             1.40%             4%              0.16%           0.02%
100%                                                                      Sum:         0.05%
Expected                                         Std. Dev. =
Rate of Return, r                                   Square root of
hat =                   10%                         sum =                2.19%
Indirect method                 r hat =       10%                                     s=           2.19%

Calculation of expected return and standard deviation for G
Expected rate of return for G                Standard deviation for G
Probability of                                         Deviation from     Squared
Occurrence          Rate of Return       Product          r hat         deviation      Sq Dev * Prob.

10%                  14%             1.40%             4%              0.16%           0.02%
20%                  12%             2.40%             2%              0.04%           0.01%
40%                  10%             4.00%             0%              0.00%           0.00%
20%                   8%             1.60%             -2%             0.04%           0.01%
10%                   6%             0.60%             -4%             0.16%           0.02%
100%                                                                      Sum:         0.05%
Expected                                         Std. Dev. =
Rate of Return, r                                   Square root of
hat =                   10%                         sum =                2.19%
Indirect method                 r hat =      10.00%                                   s=       2.19%

Calculation of expected return and standard deviation for H
Expected rate of return for H                Standard deviation for H
Probability of                                         Deviation from     Squared
Occurrence          Rate of Return       Product          r hat         deviation      Sq Dev * Prob.

10%                   4%             0.40%             -6%             0.36%           0.04%
20%                   6%             1.20%             -4%             0.16%           0.03%
40%                   8%             3.20%             -2%             0.04%           0.02%
20%                  15%             3.00%             5%              0.25%           0.05%
10%                  22%             2.20%             12%             1.44%           0.14%
100%                                                                      Sum:         0.28%
Expected                                         Std. Dev. =
Rate of Return, r                                   Square root of
hat =                  10.00%                       sum =                5.27%
Indirect method                 r hat =      10.00%                                   s=       5.27%

Michael C. Ehrhardt                                        Page 2                                              9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                               Model
COVARIANCE

The covariance is a measure that combines the variance of a stock's return with the tendency of those returns
to move up or down at the same time another stock moves up or down.

To calculate the covariance, there are a few steps. First find the differences of all the possible returns from
the expected return; do this for both stocks. Second, multiply the differences of both stocks. Third, multiply
the previous product by the probability of its occurrence. Fourth, find the some of all the weighted products.
The result is the covariance.

Calculation of covariance between F and G

Probability of       Deviation of F     Deviation of G     Product of       Product *
Occurrence           from r hat         from r hat        deviations         Prob.

10%                   -4%                4%             -0.1600%          -0.02%
20%                   -2%                2%             -0.0400%          -0.01%
40%                    0%                0%              0.0000%          0.00%
20%                    2%                -2%            -0.0400%          -0.01%
10%                    4%                -4%            -0.1600%          -0.02%
100%
Covariance =
sum =                   -0.048%

Calculation of covariance between F and H

Probability of       Deviation of F     Deviation of H     Product of       Product *
Occurrence           from r hat         from r hat        deviations         Prob.

10%                   -4%               -6%             0.2400%           0.02%
20%                   -2%               -4%             0.0800%           0.02%
40%                    0%               -2%             0.0000%           0.00%
20%                    2%                5%             0.1000%           0.02%
10%                    4%               12%             0.4800%           0.05%
100%
Covariance =
sum =                    0.108%

Calculation of covariance between F and E

Probability of       Deviation of F     Deviation of E     Product of       Product *
Occurrence           from r hat         from r hat        deviations         Prob.

10%                   -4%                0%             0.0000%           0.00%
20%                   -2%                0%             0.0000%           0.00%
40%                    0%                0%             0.0000%           0.00%
20%                    2%                0%             0.0000%           0.00%
10%                    4%                0%             0.0000%           0.00%
100%
Covariance =
sum =                    0.000%

Michael C. Ehrhardt                                            Page 3                                               9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                                 Model

CORRELATION COEFFICIENT
Like covariance, the correlation coefficient also measures the tendency of two stocks to move together, but it is
standardized and it is always in the range of -1 to +1. The correlation coefficient is equal to the covariance
divided by the product of the standard deviations.

Calculation of the correlation between F and G

rFG = Covariance FG                ÷             SigmaF * SigmaG
=            -0.048%           ÷                2.19%           2.19%
=            -0.048%           ÷                0.048%
rFG =                 -1.0

Calculation of the correlation between F and H

rFH = Covariance FH                ÷             SigmaF * SigmaH
=            0.108%            ÷                2.19%           5.27%
=            0.108%            ÷                0.116%
rFH =              0.935

PORTFOLIO RISK AND RETURN: THE TWO-ASSET CASE

Suppose there are two assets, A and B. X is the percent of the portfolio invested in asset A.
Since the total percents invested in the asset must add up to 1, (1-X) is the percent of the
portfolio invested in asset B.

The expected return on the portfolio is the weighted average of the expected returns on
asset A and asset B.

^           ^                    ^
r   p   =X r    A   + (1 - X ) r B

The standard deviation of the portfolio, sp, is not a weighted average. It is:

sp = X2 s 2 + (1 - X) 2 sB + 2X(1 - X) rAB s A sB
A
2

ATTAINABLE PORTFOLIOS: THE TWO ASSET-CASE

Asset A              Asset B
Expected return, r hat          5%                    8%
Standard deviation, s           4%                   10%

Using the equations above, we can find the expected return and standard deviation of a
portfolio with different percents invested in each asset.

Michael C. Ehrhardt                                             Page 4                                                9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                  Model
Correlation =                            1
Proportion of
Proportion of                             Portfolio in
Portfolio in Security A                       Security B
(Value of X)                            (Value of 1-X)           rp                sp
1.00                   0.00              5.00%              4.0%
0.90                   0.10              5.30%              4.6%
0.80                   0.20              5.60%              5.2%
0.70                   0.30              5.90%              5.8%
0.60                   0.40              6.20%              6.4%
0.50                   0.50              6.50%              7.0%
0.40                   0.60              6.80%              7.6%
0.30                   0.70              7.10%              8.2%
0.20                   0.80              7.40%              8.8%
0.10                   0.90              7.70%              9.4%
0.00                   1.00              8.00%             10.0%

rAB = +1: Attainable Set of Risk/Return
Combinations
10%
Expected return

5%

0%
0%              5%          10%           15%
Risk, sp

Correlation =                            0

Proportion of
Proportion of                             Portfolio in
Portfolio in Security A                       Security B
(Value of X)                            (Value of 1-X)           rp                sp
1.00                   0.00              5.00%              4.0%
0.90                   0.10              5.30%              3.7%
0.80                   0.20              5.60%              3.8%
0.70                   0.30              5.90%              4.1%
0.60                   0.40              6.20%              4.7%
0.50                   0.50              6.50%              5.4%
0.40                   0.60              6.80%              6.2%
0.30                   0.70              7.10%              7.1%
0.20                   0.80              7.40%              8.0%
0.10                   0.90              7.70%              9.0%
0.00                   1.00              8.00%             10.0%

Michael C. Ehrhardt                                                                  Page 5            9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                            Model

rAB = 0: Attainable Set of Risk/Return
Combinations
10%

Expected return    5%

0%
0%         5%          10%          15%

Risk, sp

Correlation =                                  -1

Proportion of
Proportion of                                   Portfolio in
Portfolio in Security A                             Security B
(Value of X)                                  (Value of 1-X)              rp                 sp
1.00                              0.00                  5.00%               4.0%
0.90                              0.10                  5.30%               2.6%
0.80                              0.20                  5.60%               1.2%
0.70                              0.30                  5.90%               0.2%
0.60                              0.40                  6.20%               1.6%
0.50                              0.50                  6.50%               3.0%
0.40                              0.60                  6.80%               4.4%
0.30                              0.70                  7.10%               5.8%
0.20                              0.80                  7.40%               7.2%
0.10                              0.90                  7.70%               8.6%
0.00                              1.00                  8.00%              10.0%

rAB = -1: Attainable Set of Risk/Return
Combinations
10%
Expected return

5%

0%
0%         5%          10%               15%
Risk, sp

Michael C. Ehrhardt                                                                           Page 6             9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model
ATTAINABLE AND EFFICIENT PORTFOLIOS: MANY ASSETS

OPTIMAL PORTFOLIOS

Michael C. Ehrhardt                        Page 7    9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                 Model
EFFICIENT SET WITH A RISK-FREE ASSET

OPTIMAL PORTFOLIO WITH A RISK-FREE ASSET

Michael C. Ehrhardt                          Page 8   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                        Model

CALCULATING BETAS

Electric using its ticker symbol, GE. We also downloaded data for the S&P 500 (^SPX)
which contains most actively traded stocks, and the Fidelity Magellan mutual fund
(FMAGX). We computed returns, as shown in Chapter 2. We also obtained the monthly
rates on Treasury bills from the Federal Reserve, http://www.federalreserv.gov/.

rRF, Risk-free
RM, Market                     rp, Fidelity  rate (Monthly                  Excess stock
Return (S&P     rS, General   Magellan Fund return on T-    Excess market     return
Date              500 Index)   Electric Return    Return           bill)     return (rM-rRF)   (rs-rRF)
Mar-02             3.7%           -2.9%           3.4%          0.15%            3.5%        -3.0%
Feb-02            -2.1%            4.1%          -1.7%          0.14%           -2.2%         4.0%
Jan-02            -1.6%           -7.3%          -3.1%          0.14%           -1.7%        -7.4%
Dec-01             0.8%            4.6%           0.7%          0.14%            0.6%         4.4%
Nov-01             7.5%            5.7%           7.5%          0.16%            7.4%         5.6%
Oct-01             1.8%           -2.1%           2.4%          0.18%            1.6%        -2.3%
Sep-01            -8.2%           -8.6%          -8.1%          0.22%           -8.4%        -8.9%
Aug-01            -6.4%           -6.0%          -6.5%          0.28%           -6.7%        -6.2%
Jul-01           -1.1%          -10.9%          -1.5%          0.29%           -1.4%       -11.2%
Jun-01            -2.5%            0.0%          -2.5%          0.29%           -2.8%        -0.3%
May-01             0.5%            1.0%           0.9%          0.30%            0.2%         0.7%
Apr-01             7.7%           15.9%           8.8%          0.32%            7.4%        15.6%
Mar-01            -6.4%           -9.7%          -6.4%          0.37%           -6.8%       -10.0%
Feb-01            -9.2%            1.1%          -9.4%          0.41%           -9.6%         0.7%
Jan-01             3.5%           -4.1%           3.3%          0.43%            3.0%        -4.5%
Dec-00             0.4%           -3.0%           0.7%          0.48%           -0.1%        -3.4%
Nov-00            -8.0%           -9.6%          -8.5%          0.51%           -8.5%       -10.1%
Oct-00            -0.5%           -5.2%          -1.8%          0.51%           -1.0%        -5.7%
Sep-00            -5.3%           -1.2%          -5.4%          0.50%           -5.8%        -1.7%
Aug-00             6.1%           13.4%           6.4%          0.51%            5.6%        12.9%
Jul-00           -1.6%           -2.5%          -1.2%          0.50%           -2.1%        -3.0%
Jun-00             2.4%            0.7%           3.9%          0.47%            1.9%         0.2%
May-00            -2.2%            0.4%          -3.4%          0.48%           -2.7%        -0.1%
Apr-00            -3.1%            1.0%          -4.3%          0.47%           -3.6%         0.6%
Mar-00             9.7%           17.6%           9.0%          0.47%            9.2%        17.1%
Feb-00            -2.0%           -1.2%           1.0%          0.46%           -2.5%        -1.7%
Jan-00            -5.1%          -13.4%          -4.8%          0.44%           -5.5%       -13.9%
Dec-99             5.8%           18.9%           6.8%          0.43%            5.4%        18.5%
Nov-99             1.9%           -4.0%           2.1%          0.42%            1.5%        -4.4%
Oct-99             6.3%           14.3%           6.0%          0.41%            5.8%        13.9%
Sep-99            -2.9%            5.6%          -1.5%          0.39%           -3.2%         5.2%
Aug-99            -0.6%            3.1%          -1.2%          0.39%           -1.0%         2.7%
Jul-99           -3.2%           -3.5%          -3.4%          0.38%           -3.6%        -3.9%
Jun-99             5.4%           11.1%           6.5%          0.38%            5.1%        10.8%
May-99            -2.5%           -3.5%          -3.0%          0.38%           -2.9%        -3.9%
Apr-99             3.8%           -4.8%           2.4%          0.36%            3.4%        -5.1%
Mar-99             3.9%           10.3%           5.4%          0.37%            3.5%         9.9%
Feb-99            -3.2%           -4.3%          -3.2%          0.37%           -3.6%        -4.7%
Jan-99             4.1%            2.8%           5.6%          0.36%            3.7%         2.4%
Dec-98             5.6%           12.9%           6.4%          0.37%            5.3%        12.5%
Nov-98             5.9%            3.3%           7.8%          0.37%            5.5%         2.9%
Oct-98             8.0%           10.0%           7.7%          0.33%            7.7%         9.7%
Sep-98             6.2%           -0.5%           6.1%          0.38%            5.9%        -0.9%
Aug-98           -14.6%          -10.6%         -15.5%          0.41%          -15.0%       -11.0%
Jul-98           -1.2%           -1.6%          -0.7%          0.41%           -1.6%        -2.0%
Jun-98             3.9%            9.0%           4.3%          0.42%            3.5%         8.6%

Michael C. Ehrhardt                                      Page 9                                             9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                                                     Model
May-98                 -1.9%                  -2.1%                   -2.1%            0.42%              -2.3%           -2.5%
Apr-98                 0.9%                  -1.1%                    1.2%            0.41%               0.5%           -1.6%
Mar-98
Average (Annual)                         2.6%               10.8%                    4.2%             4.4%              -1.8%            6.3%
Standard deviation
(Annual)                             17.9%                  27.2%                   18.9%             0.4%             18.0%            27.2%
Correlation with market.                                      0.72                    0.99            -0.07              1.00             0.72
Slope                                                         1.10                    1.04             0.00              1.00             1.10
R-square                                                      0.52                    0.97             0.01              1.00             0.53

Using the AVERAGE function and the STDEV function, we found the average historical
returns and standard deviations. (We converted these from monthly figures to annual
figures. Notice that you must multiply the monthly standard deviation by the square root
of 12, and not 12, to convert it to an annual basis.) These are shown in the rows above.
We also use the CORREL function to find the correlation of the market with the other assets

Using the function Wizard for SLOPE, we found the slope of the regression line, which is
the beta coefficient. We also use the function Wizard and the RSQ function to find the
R-Squared of the regression.

Using the Chart Wizard, we plotted the Wal-Mart returns on the y-axis and the market
returns on the x-axis. We also used the menu Chart > Options to add a trend line, and to
display the regression equation and R2 on the chart. The chart is shown below. We also
used the regression feature to get more detailed data. These results are also shown below.

GE Analysis
The beta coefficient is about 1.10, as shown by the slope coefficient in the regression
equation on the chart. The beta coefficient has a t statistic of 7.07, and there is virtually a
zero chance of getting this if the true beta coefficient is equal to zero. Therefore, this is a
statistically significant coefficient. However, the confidence interval ranges from 0.78 to
1.41, which is very wide. The R2 of about 0.51 indicates that 51% of the variance of the
stock return can be explained by the market. The rest of the stock's variance is due to
factors other than the market. This is consistent with the wide scatter of points in the
graph.

Historic
GE Regression Results               (See columns J-N)
Realized                                                    Beta
Returns                                                               Coefficient              1.10
y = 1.0956x + 0.0066                   on GE, kS
R² = 0.521                                                                                                t statistic            7.07
30%
Probability of t stat.           0.0%
Lower 95% confidence interval              0.78
Upper 95% confidence interval              1.41
20%

Intercept
Coefficient              0.01
10%
t statistic            0.83
Probability of t stat.          41.0%
0%
Lower 95% confidence interval             -0.01
Historic Realized Returns              Upper 95% confidence interval              0.02
-30%      -20%       -10%          0%       10%          20%         30%
on the Market, kM

-10%

-20%

Michael C. Ehrhardt                                                   Page 10                                                            9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                                        Model

Magellan Analysis

The beta coefficient is about 1.04, as shown by the slope coefficient in the regression
equation on the chart. The beta coefficient has a t statistic of 42.22, and there is virtually a
zero chance of getting this if the true beta coefficient is equal to zero. Therefore, this is a
statistically significant coefficient. The confidence interval ranges from 0.99 to 1.09, which
is very small compared to the confidence interval for a single stock. The R2 of about 0.97
indicates that 97% of the variance of the portfolio return can be explained by the market.
This is consistent with the very narrow scatter of points in the graph. The estimate of the
intercept is equal to 0.00, and has a t statistic with a probability of 33.8%. Since this is
greater than 5%, we would say that the coefficient is not statistically significant-- in other
words, the true intercept might well be equal to zero.

Magellan Regression Results        (See columns J-N)
y = 1.0424x + 0.0012                Historic
Realized                                           Beta
R² = 0.9748
Returns                                                       Coefficient             1.04
on Magellan,
t statistic          42.22
rP
20%                                                            Probability of t stat.          0.0%
Lower 95% confidence interval               0.99
Upper 95% confidence interval               1.09

10%
Intercept
Coefficient            0.00
t statistic          0.97
Probability of t stat.        33.8%
Lower 95% confidence interval            0.00
0%
Upper 95% confidence interval            0.00
-30%     -20%    -10%          0%      10%       20%       30%
Historic Realized Returns
on the Market, rM

-10%

-20%

Michael C. Ehrhardt                                                      Page 11                                            9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                                                 Model

The Market Model vs. CAPM

We have been regressing the stock (or portfolio) returns against the market returns.
However, CAPM actually states that we should regress the excess stock returns (the stock
return minus the short-term risk free rate) against the excess market returns (the market
return minus the short-term risk free rate). We show the graph for such a regression
below. Notice that it is virtually identical to the market model regression we used earlier
for GE. Since it usually doesn't change the results whether we use the market model to
estimate beta instead of the CAPM model, we usually use the market model.

Market Model Regression Results (See columns J-N)
Beta
Excess Returns
on GE, rS-rRF                                                     Coefficient           1.09
y = 1.092x + 0.0069
t statistic         7.06
R² = 0.5201              30%
Probability of t stat.        0.0%
Lower 95% confidence interval            0.78
Upper 95% confidence interval            1.40
20%
Intercept
Coefficient         0.01
10%
t statistic       0.88
Probability of t stat.     38.6%
Lower 95% confidence interval        -0.01
Upper 95% confidence interval         0.02
0%
Excess Returns
-30%     -20%      -10%            0%      10%    20%        30%
on the Market, rM-rRF

-10%

-20%

Michael C. Ehrhardt                                            Page 12                                               9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 13   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 14   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 15   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 16   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 17   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 18   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 19   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 20   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 21   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                               Model

(See columns J-N)                  SUMMARY OUTPUT

1.10          Regression Statistics
Multiple R 0.721822
R Square 0.521027
0.510615
0.054954
Standard Error
Observations       48

ANOVA
df         SS      MS         F   Significance F
Regression          1 0.151114 0.151114   50.0389 7.06E-09
Residual           46 0.138917 0.00302
Total              47 0.290031

Standard Error t Stat
Coefficients                    P-value Lower 95%Upper 95%Lower 95.0%
Intercept   0.006601 0.007939 0.831399 0.410043 -0.00938 0.022581 -0.00938
X Variable 11.095569 0.154877 7.073818 7.06E-09 0.783819 1.407319 0.783819

Michael C. Ehrhardt                                 Page 22                                         9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                               Model

(See columns J-N)                  SUMMARY OUTPUT

1.04          Regression Statistics
Multiple R 0.987342
R Square 0.974843
0.974296
0.008761
Standard Error
Observations       48

ANOVA
df         SS      MS        F   Significance F
Regression          1 0.136809 0.136809 1782.536 1.94E-38
Residual           46 0.00353 7.67E-05
Total              47 0.140339

Standard Error t Stat
Coefficients                    P-value Lower 95%Upper 95%Lower 95.0%
Intercept   0.001225 0.001266 0.967895 0.338159 -0.00132 0.003773 -0.00132
X Variable 11.042423 0.02469 42.22009 1.94E-38 0.992724 1.092121 0.992724

Michael C. Ehrhardt                                 Page 23                                         9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                                                               Model

(See columns J-N)                  SUMMARY OUTPUT

Regression Statistics
Multiple R 0.721193
R Square 0.520119
0.509686
0.05497
Standard Error
Observations       48

ANOVA
df         SS      MS        F   Significance F
Regression          1 0.150651 0.150651 49.85703 7.38E-09
Residual           46 0.138997 0.003022
Total              47 0.289648

Standard Error t Stat
Coefficients                    P-value Lower 95%Upper 95%Lower 95.0%
Intercept   0.006949 0.007938 0.875482 0.38586 -0.00903 0.022927 -0.00903
X Variable 11.092047 0.15466 7.060951 7.38E-09 0.780733 1.403361 0.780733

Michael C. Ehrhardt                                 Page 24                                         9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 25   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 26   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 27   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 28   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 29   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 30   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 31   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 32   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                Model

Michael C. Ehrhardt                        Page 33   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                      Model

Upper 95.0%
0.022581
1.407319

Michael C. Ehrhardt                              Page 34   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                      Model

Upper 95.0%
0.003773
1.092121

Michael C. Ehrhardt                              Page 35   9/14/2012
101d1986-1e34-416e-82aa-3ebccba4f3a9.xls                      Model

Upper 95.0%
0.022927
1.403361

Michael C. Ehrhardt                              Page 36   9/14/2012

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 4 posted: 9/15/2012 language: Unknown pages: 36
How are you planning on using Docstoc?