Computing Capital Asset Pricing Model with tidyquant

This note came from Matt Dancho’s webpage.

Factor models are used in many financial appplications, suhc as:

• identifying the determinants of a security’s return
• cost of capital (rate of return required by an investor to hold the share) calculations

Capital Asset Pricing Model (CAPM) is the most popular factor model. Although CAPM does not perform well in empirical testing, it’s important because it provides a starting point of many cost of capital calculations.

## # A tibble: 16,019 x 8
## # Groups:   symbol [3]
##    symbol date        open  high   low close   volume adjusted
##    <chr>  <date>     <dbl> <dbl> <dbl> <dbl>    <dbl>    <dbl>
##  1 AAPL   1990-04-30  1.40  1.42  1.39  1.41 34098400     1.14
##  2 AAPL   1990-05-01  1.42  1.43  1.41  1.42 40902400     1.15
##  3 AAPL   1990-05-02  1.42  1.43  1.40  1.42 33857600     1.15
##  4 AAPL   1990-05-03  1.42  1.44  1.42  1.43 41577200     1.16
##  5 AAPL   1990-05-04  1.43  1.46  1.40  1.43 42383600     1.16
##  6 AAPL   1990-05-07  1.42  1.49  1.42  1.48 33997600     1.21
##  7 AAPL   1990-05-08  1.46  1.5   1.46  1.49 28114800     1.21
##  8 AAPL   1990-05-09  1.49  1.5   1.47  1.50 24309600     1.22
##  9 AAPL   1990-05-10  1.49  1.49  1.45  1.48 44760800     1.20
## 10 AAPL   1990-05-11  1.48  1.53  1.46  1.52 53810400     1.24
## # … with 16,009 more rows

## # A tibble: 766 x 3
## # Groups:   symbol [3]
##    symbol date            Ra
##    <chr>  <date>       <dbl>
##  1 AAPL   1990-04-30  0     
##  2 AAPL   1990-05-31  0.0505
##  3 AAPL   1990-06-29  0.0848
##  4 AAPL   1990-07-31 -0.0615
##  5 AAPL   1990-08-31 -0.116 
##  6 AAPL   1990-09-28 -0.216 
##  7 AAPL   1990-10-31  0.0603
##  8 AAPL   1990-11-30  0.199 
##  9 AAPL   1990-12-31  0.170 
## 10 AAPL   1991-01-31  0.291 
## # … with 756 more rows

## # A tibble: 361 x 2
##    date             Rb
##    <date>        <dbl>
##  1 1990-04-30  0      
##  2 1990-05-31  0.0926 
##  3 1990-06-29  0.00719
##  4 1990-07-31 -0.0521 
##  5 1990-08-31 -0.130  
##  6 1990-09-28 -0.0963 
##  7 1990-10-31 -0.0427 
##  8 1990-11-30  0.0888 
##  9 1990-12-31  0.0409 
## 10 1991-01-31  0.108  
## # … with 351 more rows

## # A tibble: 766 x 4
## # Groups:   symbol [3]
##    symbol date            Ra       Rb
##    <chr>  <date>       <dbl>    <dbl>
##  1 AAPL   1990-04-30  0       0      
##  2 AAPL   1990-05-31  0.0505  0.0926 
##  3 AAPL   1990-06-29  0.0848  0.00719
##  4 AAPL   1990-07-31 -0.0615 -0.0521 
##  5 AAPL   1990-08-31 -0.116  -0.130  
##  6 AAPL   1990-09-28 -0.216  -0.0963 
##  7 AAPL   1990-10-31  0.0603 -0.0427 
##  8 AAPL   1990-11-30  0.199   0.0888 
##  9 AAPL   1990-12-31  0.170   0.0409 
## 10 AAPL   1991-01-31  0.291   0.108  
## # … with 756 more rows

## # A tibble: 3 x 13
## # Groups:   symbol [3]
##   symbol ActivePremium  Alpha AnnualizedAlpha  Beta `Beta-` `Beta+` Correlation
##   <chr>          <dbl>  <dbl>           <dbl> <dbl>   <dbl>   <dbl>       <dbl>
## 1 AAPL          0.0949 0.0122           0.156  1.08    1.04   1.12        0.543
## 2 GOOG          0.126  0.0115           0.147  1.02    1.06   0.889       0.557
## 3 NFLX          0.290  0.0306           0.435  1.17    1.15   0.799       0.354
## # … with 5 more variables: `Correlationp-value` <dbl>, InformationRatio <dbl>,
## #   `R-squared` <dbl>, TrackingError <dbl>, TreynorRatio <dbl>

Interpretation

• r RaRb_capm[RaRb_capm$symbol == “AAPL”, “Alpha”] is the y-intercept: it means that, in the case of an actively managed portfolio, the manager provided an incremental return of r RaRb_capm[RaRb_capm$symbol == “AAPL”, “Alpha”] per month beyond that of what is expected from its sensitivity to the benchmark. In the case of a sinlge stock, however, the y-intercept captures part of the return that is not captured by the overall market movement but rather attributed to the individual stock.
• r RaRb_capm[RaRb_capm$symbol == “AAPL”, “Beta”]is the beta: + It measures how sensitive the stock's return is to the movement of the overall market. It means that, if the market goes up by 1%, we expect the stock to go up by r RaRb_capm[RaRb_capm$symbol == "AAPL", "Beta"]%. A beta less than one is considered defensive stocks as these stocks are less affected by adverse market movements.
  • It is also used to calculate the cost of equity (rate of return required by an investor to hold the company’s share). \(cost of equity = Yintercept + beta * ERP\) The general range of Equity Risk Premeum (ERP) is 5-8%.

Q1 Import stock prices of Apple, Microsoft and Amazon for the last 20 years.

## # A tibble: 15,090 x 8
## # Groups:   symbol [3]
##    symbol date        open  high   low close    volume adjusted
##    <chr>  <date>     <dbl> <dbl> <dbl> <dbl>     <dbl>    <dbl>
##  1 AAPL   2000-04-28  4.54  4.55  4.33  4.43  62395200     3.85
##  2 AAPL   2000-05-01  4.46  4.47  4.35  4.44  56548800     3.85
##  3 AAPL   2000-05-02  4.40  4.51  4.20  4.21  59108000     3.65
##  4 AAPL   2000-05-03  4.25  4.33  3.99  4.11 122449600     3.57
##  5 AAPL   2000-05-04  4.11  4.12  3.95  3.95  99878800     3.43
##  6 AAPL   2000-05-05  3.96  4.10  3.95  4.04  71019200     3.51
##  7 AAPL   2000-05-08  4.00  4.06  3.93  3.93  46225200     3.41
##  8 AAPL   2000-05-09  3.94  3.97  3.75  3.77  81785200     3.27
##  9 AAPL   2000-05-10  3.72  3.75  3.53  3.55 133772800     3.08
## 10 AAPL   2000-05-11  3.62  3.72  3.54  3.67 124936000     3.19
## # … with 15,080 more rows

Q2 Calculate monthly returns.

## # A tibble: 723 x 3
## # Groups:   symbol [3]
##    symbol date            Ra
##    <chr>  <date>       <dbl>
##  1 AAPL   2000-04-28  0     
##  2 AAPL   2000-05-31 -0.323 
##  3 AAPL   2000-06-30  0.247 
##  4 AAPL   2000-07-31 -0.0298
##  5 AAPL   2000-08-31  0.199 
##  6 AAPL   2000-09-29 -0.577 
##  7 AAPL   2000-10-31 -0.240 
##  8 AAPL   2000-11-30 -0.157 
##  9 AAPL   2000-12-29 -0.0985
## 10 AAPL   2001-01-31  0.454 
## # … with 713 more rows

Q3 Import baseline prices and calculate its monthly return.

## # A tibble: 241 x 2
##    date            Rb
##    <date>       <dbl>
##  1 2000-04-28  0     
##  2 2000-05-31 -0.119 
##  3 2000-06-30  0.166 
##  4 2000-07-31 -0.0502
##  5 2000-08-31  0.117 
##  6 2000-09-29 -0.127 
##  7 2000-10-31 -0.0825
##  8 2000-11-30 -0.229 
##  9 2000-12-29 -0.0490
## 10 2001-01-31  0.122 
## # … with 231 more rows

Q4 Merge the two sets of monthly returns.

## # A tibble: 723 x 4
## # Groups:   symbol [3]
##    symbol date            Ra      Rb
##    <chr>  <date>       <dbl>   <dbl>
##  1 AAPL   2000-04-28  0       0     
##  2 AAPL   2000-05-31 -0.323  -0.119 
##  3 AAPL   2000-06-30  0.247   0.166 
##  4 AAPL   2000-07-31 -0.0298 -0.0502
##  5 AAPL   2000-08-31  0.199   0.117 
##  6 AAPL   2000-09-29 -0.577  -0.127 
##  7 AAPL   2000-10-31 -0.240  -0.0825
##  8 AAPL   2000-11-30 -0.157  -0.229 
##  9 AAPL   2000-12-29 -0.0985 -0.0490
## 10 AAPL   2001-01-31  0.454   0.122 
## # … with 713 more rows

Q5 Calculate performance metrics, CAPM measures.

## # A tibble: 3 x 13
## # Groups:   symbol [3]
##   symbol ActivePremium  Alpha AnnualizedAlpha  Beta `Beta-` `Beta+` Correlation
##   <chr>          <dbl>  <dbl>           <dbl> <dbl>   <dbl>   <dbl>       <dbl>
## 1 AAPL          0.197  0.0188          0.251  1.20    1.11    1.31        0.638
## 2 MSFT          0.0661 0.0069          0.0864 0.867   0.594   0.998       0.679
## 3 AMZN          0.165  0.0174          0.231  1.34    1.34    1.50        0.616
## # … with 5 more variables: `Correlationp-value` <dbl>, InformationRatio <dbl>,
## #   `R-squared` <dbl>, TrackingError <dbl>, TreynorRatio <dbl>

Q6 Interpet the alpha.

All three stocks did better than the broad market. Apple beats the market by .019% per month on average. Microsoft beats the market by .007% per month on average. Amazon beats the market by .017% per month on average.

Q7 Interpret the beta.

Amazon is the riskiest stock, providing a 33% greater return than the market, but also suffering from 33% greater losses during downturns. Apple provides a 20% greater return than the market, while also suffering from 20% greater losses during downturns. Microsoft is the least risky stock, providing 10% less returns during positive market runs, but 10% less losses during downturns.