FF Factors

Descriptive analysis

We have Fama-French pricing factor data from 2000-01 to 2024-06. We want to have a look at its descriptive statistics.

data <- read_csv("data/FF_3Factors_US_monthly.csv")
get_stat <- function(x, q_list=c(0.025, 0.5, 0.975)){
    ## Return extended summary statistics with quantiles and sd.
    #    @q_list: vector of quantiles to calculate;
    x <- na.omit(x)
    c(summary(x), quantile(x, q_list), "sd"=sd(x) )
}

Descriptive statistics

apply(data[,-1], 2, get_stat)

##             Mkt-RF         SMB         HML        RF
## Min.    -17.230000 -17.2000000 -13.8300000 0.0000000
## 1st Qu.  -2.020000  -1.8475000  -1.6925000 0.0100000
## Median    1.175000   0.0850000  -0.0500000 0.0900000
## Mean      0.577551   0.1342517   0.1777211 0.1418707
## 3rd Qu.   3.307500   1.8400000   1.7775000 0.2275000
## Max.     13.650000  21.3600000  12.8800000 0.5600000
## 2.5%     -9.424250  -5.0090000  -7.4225000 0.0000000
## 50%       1.175000   0.0850000  -0.0500000 0.0900000
## 97.5%     8.697000   6.0207500   7.8932500 0.4767500
## sd        4.609053   3.2149253   3.4991826 0.1596741

Histogram of SMB

ggplot(data) +
    geom_histogram(aes(x=SMB, y=..density..), 
                   fill="#BDBCBC", 
                   color="black", 
                   binwidth = 2, 
                   boundary=0) +
    labs(x="Small minus big, SMB")

Histogram of HML

ggplot(data) +
    geom_histogram(aes(x=HML, y=..density..), 
                   fill="#BDBCBC", 
                   color="black", 
                   binwidth = 2, 
                   boundary=0) +
    labs(x="High minus low, HML")

Risk Free Rate (RFR)

Before 2013 NIBOR: Norges Bank
After 2013 Norwegian Overnight Weighted Average rate: Norske Finansielle Referenser AS (NoRe)
Titlon: 3 months Norwegian Government Bills, from the bondindex table, CloseYield field.

Financial service companies publish surveys about financial data. For instance, PwC Norge.

Here is their annual report about RFF: https://www.pwc.no/no/publikasjoner/risikopremien.html

AAPL case study

library(quantmod)
aapl <- getSymbols("AAPL", 
           src = 'yahoo', 
           from = "2014-08-01", 
           to = "2024-09-17",   
           auto.assign = FALSE
           )
aapl

##            AAPL.Open AAPL.High AAPL.Low AAPL.Close AAPL.Volume AAPL.Adjusted
## 2014-08-01   23.7250   24.1550  23.7025    24.0325   194044000      21.20968
## 2014-08-04   24.0925   24.1450  23.7925    23.8975   159832000      21.09054
## 2014-08-05   23.8400   23.9200  23.5900    23.7800   223732000      20.98684
## 2014-08-06   23.6875   23.8700  23.6775    23.7400   154232000      20.95154
## 2014-08-07   23.7325   23.9875  23.5250    23.6200   186844000      20.94932
## 2014-08-08   23.5650   23.7050  23.3200    23.6850   167460000      21.00697
## 2014-08-11   23.8175   24.0200  23.7100    23.9975   146340000      21.28414
## 2014-08-12   24.0100   24.2200  23.9025    23.9925   135180000      21.27971
## 2014-08-13   24.0375   24.3100  24.0100    24.3100   127664000      21.56130
## 2014-08-14   24.3325   24.3925  24.2000    24.3750   112464000      21.61895
##        ...                                                                  
## 2024-09-03  228.5500  229.0000 221.1700   222.7700    50190600     222.52518
## 2024-09-04  221.6600  221.7800 217.4800   220.8500    43840200     220.60730
## 2024-09-05  221.6300  225.4800 221.5200   222.3800    36615400     222.13560
## 2024-09-06  223.9500  225.2400 219.7700   220.8200    48423000     220.57733
## 2024-09-09  220.8200  221.2700 216.7100   220.9100    67180000     220.66722
## 2024-09-10  218.9200  221.4800 216.7300   220.1100    51591000     219.86810
## 2024-09-11  221.4600  223.0900 217.8900   222.6600    44587100     222.41530
## 2024-09-12  222.5000  223.5500 219.8200   222.7700    37498200     222.52518
## 2024-09-13  223.5800  224.0400 221.9100   222.5000    36766600     222.25548
## 2024-09-16  216.5400  217.2200 213.9200   216.3200    59357400     216.08228

Candlestick Plot

chartSeries(aapl, subset="2024-09", theme=chartTheme('white',up.col='green',dn.col='red'))

Calculate monthly return, \(r_t = \frac{P_t-P_{t-1}}{P_{t-1}} = \frac{P_t}{P_{t-1}}-1\).

You may either use the function quantmod::monthlyReturn() or calculate manually.

aapl <- aapl %>% 
    apply.monthly(last)
aapl$return <- monthlyReturn(aapl$AAPL.Adjusted, type='arithmetic')
aapl <- head(aapl, -1)
aapl

##            AAPL.Open AAPL.High AAPL.Low AAPL.Close AAPL.Volume AAPL.Adjusted
## 2014-08-29   25.7150   25.7250  25.5500    25.6250   178380000      22.72762
## 2014-09-30   25.2025   25.3850  25.1325    25.1875   221056400      22.33959
## 2014-10-31   27.0025   27.0100  26.8025    27.0000   178557200      23.94715
## 2014-11-28   29.8175   29.8500  29.5125    29.7325    99257600      26.48504
## 2014-12-31   28.2050   28.2825  27.5525    27.5950   165613600      24.58101
## 2015-01-30   29.6000   30.0000  29.2125    29.2900   334982000      26.09087
## 2015-02-27   32.5000   32.6425  32.0600    32.1150   248059200      28.72022
## 2015-03-31   31.5225   31.6225  31.0900    31.1075   168362400      27.81922
## 2015-04-30   32.1600   32.1600  31.1450    31.2875   332781600      27.98019
## 2015-05-29   32.8075   32.8625  32.4750    32.5700   203538000      29.24879
##        ...                                                                  
## 2023-11-30  189.8400  190.3200 188.1900   189.9500    48794400     189.02396
## 2023-12-29  193.9000  194.4000 191.7300   192.5300    42628800     191.59138
## 2024-01-31  187.0400  187.1000 184.3500   184.4000    55467800     183.50102
## 2024-02-29  181.2700  182.5700 179.5300   180.7500   136682600     180.09834
## 2024-03-28  171.7500  172.2300 170.5100   171.4800    65672700     170.86176
## 2024-04-30  173.3300  174.9900 170.0000   170.3300    65934800     169.71590
## 2024-05-31  191.4400  192.5700 189.9100   192.2500    75158300     191.81670
## 2024-06-28  215.7700  216.0700 210.3000   210.6200    82542700     210.14528
## 2024-07-31  221.4400  223.8200 220.6300   222.0800    50036300     221.57947
## 2024-08-30  230.1900  230.4000 227.4800   229.0000    52990800     228.74834
##                  return
## 2014-08-29  0.000000000
## 2014-09-30 -0.017073105
## 2014-10-31  0.071960286
## 2014-11-28  0.105978772
## 2014-12-31 -0.071890915
## 2015-01-30  0.061424124
## 2015-02-27  0.100776504
## 2015-03-31 -0.031371661
## 2015-04-30  0.005786313
## 2015-05-29  0.045339376
##        ...             
## 2023-11-30  0.113780167
## 2023-12-29  0.013582557
## 2024-01-31 -0.042227173
## 2024-02-29 -0.018543109
## 2024-03-28 -0.051286355
## 2024-04-30 -0.006706350
## 2024-05-31  0.130222330
## 2024-06-28  0.095552593
## 2024-07-31  0.054410877
## 2024-08-30  0.032353490

# calculate manually
with(aapl, AAPL.Adjusted/lag(AAPL.Adjusted)-1)

##            AAPL.Adjusted
## 2014-08-29            NA
## 2014-09-30  -0.017073105
## 2014-10-31   0.071960286
## 2014-11-28   0.105978772
## 2014-12-31  -0.071890915
## 2015-01-30   0.061424124
## 2015-02-27   0.100776504
## 2015-03-31  -0.031371661
## 2015-04-30   0.005786313
## 2015-05-29   0.045339376
##        ...              
## 2023-11-30   0.113780167
## 2023-12-29   0.013582557
## 2024-01-31  -0.042227173
## 2024-02-29  -0.018543109
## 2024-03-28  -0.051286355
## 2024-04-30  -0.006706350
## 2024-05-31   0.130222330
## 2024-06-28   0.095552593
## 2024-07-31   0.054410877
## 2024-08-30   0.032353490

Merge equity data with FF factors.

reg_data <- aapl %>% 
    as_tibble() %>% 
    add_column(Date=index(aapl), .before = 1)
reg_data <- reg_data %>% 
    mutate(year=year(Date),
           mon=month(Date))
data <- data %>% mutate_at(vars(-Date), ~./100)
data <- data %>% 
    mutate(year=year(Date),
           mon=month(Date))
reg_data <- reg_data %>% 
    left_join(data[,-1], by=c("year","mon"))
reg_data

## # A tibble: 121 × 14
##    Date       AAPL.Open AAPL.High AAPL.Low AAPL.Close AAPL.Volume AAPL.Adjusted
##    <date>         <dbl>     <dbl>    <dbl>      <dbl>       <dbl>         <dbl>
##  1 2014-08-29      25.7      25.7     25.5       25.6   178380000          22.7
##  2 2014-09-30      25.2      25.4     25.1       25.2   221056400          22.3
##  3 2014-10-31      27.0      27.0     26.8       27     178557200          23.9
##  4 2014-11-28      29.8      29.9     29.5       29.7    99257600          26.5
##  5 2014-12-31      28.2      28.3     27.6       27.6   165613600          24.6
##  6 2015-01-30      29.6      30       29.2       29.3   334982000          26.1
##  7 2015-02-27      32.5      32.6     32.1       32.1   248059200          28.7
##  8 2015-03-31      31.5      31.6     31.1       31.1   168362400          27.8
##  9 2015-04-30      32.2      32.2     31.1       31.3   332781600          28.0
## 10 2015-05-29      32.8      32.9     32.5       32.6   203538000          29.2
## # ℹ 111 more rows
## # ℹ 7 more variables: return <dbl>, year <dbl>, mon <dbl>, `Mkt-RF` <dbl>,
## #   SMB <dbl>, HML <dbl>, RF <dbl>

Calculate excess return.

# calculate excess return
reg_data <- reg_data %>% 
    mutate(eRi = return-RF) %>% 
    rename(rmrf=`Mkt-RF`)
reg_data  %>% 
    select(-year,-mon) %>% 
    knitr::kable(floating.environment="sidewaystable", digits = 5, escape=F) %>%
    kable_styling(bootstrap_options = c("striped", "hover"), full_width = F, latex_options="scale_down") %>% 
    scroll_box(width = "100%", height = "500px")

Date	AAPL.Open	AAPL.High	AAPL.Low	AAPL.Close	AAPL.Volume	AAPL.Adjusted	return	rmrf	SMB	HML	RF	eRi
2014-08-29	25.7150	25.7250	25.5500	25.6250	178380000	22.72762	0.00000	0.0424	0.0040	-0.0045	0.0000	0.00000
2014-09-30	25.2025	25.3850	25.1325	25.1875	221056400	22.33959	-0.01707	-0.0197	-0.0371	-0.0134	0.0000	-0.01707
2014-10-31	27.0025	27.0100	26.8025	27.0000	178557200	23.94715	0.07196	0.0252	0.0421	-0.0181	0.0000	0.07196
2014-11-28	29.8175	29.8500	29.5125	29.7325	99257600	26.48504	0.10598	0.0255	-0.0206	-0.0309	0.0000	0.10598
2014-12-31	28.2050	28.2825	27.5525	27.5950	165613600	24.58101	-0.07189	-0.0006	0.0249	0.0227	0.0000	-0.07189
2015-01-30	29.6000	30.0000	29.2125	29.2900	334982000	26.09087	0.06142	-0.0311	-0.0056	-0.0358	0.0000	0.06142
2015-02-27	32.5000	32.6425	32.0600	32.1150	248059200	28.72022	0.10078	0.0613	0.0063	-0.0186	0.0000	0.10078
2015-03-31	31.5225	31.6225	31.0900	31.1075	168362400	27.81922	-0.03137	-0.0112	0.0304	-0.0037	0.0000	-0.03137
2015-04-30	32.1600	32.1600	31.1450	31.2875	332781600	27.98019	0.00579	0.0059	-0.0306	0.0182	0.0000	0.00579
2015-05-29	32.8075	32.8625	32.4750	32.5700	203538000	29.24879	0.04534	0.0136	0.0093	-0.0114	0.0000	0.04534
2015-06-30	31.3925	31.5300	31.2150	31.3575	177482800	28.15993	-0.03723	-0.0153	0.0291	-0.0079	0.0000	-0.03723
2015-07-31	30.6500	30.6600	30.2275	30.3250	171540000	27.23272	-0.03293	0.0154	-0.0417	-0.0413	0.0000	-0.03293
2015-08-31	28.0075	28.6325	28.0000	28.1900	224917200	25.43001	-0.06620	-0.0604	0.0033	0.0277	0.0000	-0.06620
2015-09-30	27.5425	27.8850	27.1825	27.5750	265892000	24.87522	-0.02182	-0.0307	-0.0262	0.0056	0.0000	-0.02182
2015-10-30	30.2475	30.3050	29.8625	29.8750	197461200	26.95004	0.08341	0.0775	-0.0188	-0.0046	0.0000	0.08341
2015-11-30	29.4975	29.8525	29.4375	29.5750	156721200	26.79361	-0.00580	0.0056	0.0360	-0.0042	0.0000	-0.00580
2015-12-31	26.7525	26.7575	26.2050	26.3150	163649200	23.84020	-0.11023	-0.0217	-0.0284	-0.0261	0.0001	-0.11033
2016-01-29	23.6975	24.3350	23.5875	24.3350	257666000	22.04641	-0.07524	-0.0577	-0.0342	0.0209	0.0001	-0.07534
2016-02-29	24.2150	24.5575	24.1625	24.1725	140865200	22.01802	-0.00129	-0.0007	0.0073	-0.0057	0.0002	-0.00149
2016-03-31	27.4300	27.4750	27.2200	27.2475	103553600	24.81894	0.12721	0.0696	0.0082	0.0119	0.0002	0.12701
2016-04-29	23.4975	23.6800	23.1275	23.4350	274126000	21.34626	-0.13992	0.0091	0.0076	0.0328	0.0001	-0.14002
2016-05-31	24.9000	25.1000	24.7050	24.9650	169228800	22.87833	0.07177	0.0178	-0.0017	-0.0166	0.0001	0.07167
2016-06-30	23.6100	23.9425	23.5750	23.9000	143345600	21.90235	-0.04266	-0.0005	0.0061	-0.0148	0.0002	-0.04286
2016-07-29	26.0475	26.1375	25.9200	26.0525	110934800	23.87494	0.09006	0.0395	0.0249	-0.0132	0.0002	0.08986
2016-08-31	26.4150	26.6425	26.4100	26.5250	118649600	24.43963	0.02365	0.0049	0.0115	0.0318	0.0002	0.02345
2016-09-30	28.1150	28.3425	27.9500	28.2625	145516400	26.04053	0.06550	0.0025	0.0212	-0.0124	0.0002	0.06530
2016-10-31	28.4125	28.5575	28.3000	28.3850	105677600	26.15340	0.00433	-0.0202	-0.0440	0.0409	0.0002	0.00413
2016-11-30	27.9000	28.0500	27.5675	27.6300	144649200	25.58846	-0.02160	0.0486	0.0571	0.0821	0.0001	-0.02170
2016-12-30	29.1625	29.3000	28.8575	28.9550	122345200	26.81556	0.04796	0.0182	0.0010	0.0353	0.0003	0.04766
2017-01-31	30.2875	30.3475	30.1550	30.3375	196804000	28.09590	0.04775	0.0194	-0.0119	-0.0275	0.0004	0.04735
2017-02-28	34.2700	34.3600	34.1750	34.2475	93931600	31.85452	0.13378	0.0357	-0.0205	-0.0167	0.0004	0.13338
2017-03-31	35.9300	36.0675	35.7525	35.9150	78646800	33.40550	0.04869	0.0017	0.0114	-0.0335	0.0003	0.04839
2017-04-28	36.0225	36.0750	35.8175	35.9125	83441600	33.40317	-0.00007	0.0109	0.0073	-0.0213	0.0005	-0.00057
2017-05-31	38.4925	38.5425	38.0950	38.1900	97804800	35.66814	0.06781	0.0106	-0.0257	-0.0378	0.0006	0.06721
2017-06-30	36.1125	36.2400	35.9450	36.0050	92096400	33.62743	-0.05721	0.0078	0.0225	0.0148	0.0006	-0.05781
2017-07-31	37.4750	37.5825	37.0325	37.1825	79383600	34.72719	0.03270	0.0187	-0.0151	-0.0031	0.0007	0.03200
2017-08-31	40.9100	41.1300	40.8700	41.0000	107140400	38.44297	0.10700	0.0016	-0.0167	-0.0210	0.0009	0.10610
2017-09-29	38.3025	38.5325	38.0000	38.5300	105199200	36.12701	-0.06024	0.0251	0.0447	0.0313	0.0009	-0.06114
2017-10-31	41.9750	42.4125	41.7350	42.2600	144187200	39.62438	0.09681	0.0225	-0.0194	0.0020	0.0009	0.09591
2017-11-30	42.6075	43.0350	42.1100	42.9625	166108800	40.42789	0.02028	0.0312	-0.0055	-0.0003	0.0008	0.01948
2017-12-29	42.6300	42.6475	42.3050	42.3075	103999600	39.81153	-0.01525	0.0106	-0.0132	0.0006	0.0009	-0.01615
2018-01-31	41.7175	42.1100	41.6250	41.8575	129915600	39.38808	-0.01064	0.0557	-0.0312	-0.0129	0.0012	-0.01184
2018-02-28	44.8150	45.1550	44.5125	44.5300	151128400	42.07376	0.06819	-0.0365	0.0026	-0.0104	0.0011	0.06709
2018-03-29	41.9525	42.9375	41.7250	41.9450	153594000	39.63134	-0.05805	-0.0235	0.0406	-0.0021	0.0011	-0.05915
2018-04-30	40.5325	41.8150	40.4600	41.3150	169709600	39.03609	-0.01502	0.0028	0.0113	0.0054	0.0014	-0.01642
2018-05-31	46.8050	47.0575	46.5350	46.7175	109931200	44.31080	0.13512	0.0265	0.0526	-0.0322	0.0014	0.13372
2018-06-29	46.5725	46.7975	45.7275	46.2775	90950800	43.89347	-0.00942	0.0048	0.0113	-0.0233	0.0014	-0.01082
2018-07-31	47.5750	48.0350	47.3350	47.5725	157492000	45.12177	0.02798	0.0319	-0.0222	0.0045	0.0016	0.02638
2018-08-31	56.6275	57.2175	56.5000	56.9075	173360400	54.16517	0.20042	0.0344	0.0115	-0.0400	0.0016	0.19882
2018-09-28	56.1975	56.4600	56.0050	56.4350	91717600	53.71543	-0.00830	0.0006	-0.0229	-0.0170	0.0015	-0.00980
2018-10-31	54.2200	55.1125	54.1550	54.7150	153435600	52.07832	-0.03048	-0.0768	-0.0474	0.0341	0.0019	-0.03238
2018-11-30	45.0725	45.0825	44.2575	44.6450	158126000	42.64185	-0.18120	0.0169	-0.0068	0.0028	0.0018	-0.18300
2018-12-31	39.6325	39.8400	39.1200	39.4350	140014000	37.66563	-0.11670	-0.0957	-0.0237	-0.0188	0.0020	-0.11870
2019-01-31	41.5275	42.2500	41.1400	41.6100	162958400	39.74304	0.05515	0.0840	0.0289	-0.0044	0.0021	0.05305
2019-02-28	43.5800	43.7275	43.2300	43.2875	112861600	41.52258	0.04478	0.0340	0.0207	-0.0270	0.0018	0.04298
2019-03-29	47.4575	47.5200	47.1350	47.4875	94256000	45.55134	0.09703	0.0110	-0.0306	-0.0413	0.0019	0.09513
2019-04-30	50.7650	50.8500	49.7775	50.1675	186139600	48.12208	0.05644	0.0397	-0.0172	0.0215	0.0021	0.05434
2019-05-31	44.0575	44.4975	43.7475	43.7675	108174400	42.14468	-0.12421	-0.0694	-0.0131	-0.0237	0.0021	-0.12631
2019-06-28	49.6700	49.8750	49.2625	49.4800	124442400	47.64538	0.13052	0.0693	0.0028	-0.0070	0.0018	0.12872
2019-07-31	54.1050	55.3425	52.8250	53.2600	277125600	51.28523	0.07639	0.0119	-0.0193	0.0047	0.0019	0.07449
2019-08-30	52.5400	52.6125	51.8000	52.1850	84573600	50.44101	-0.01646	-0.0258	-0.0239	-0.0479	0.0016	-0.01806
2019-09-30	55.2250	56.1450	55.1975	55.9925	103909600	54.12127	0.07296	0.0143	-0.0096	0.0675	0.0018	0.07116
2019-10-31	61.8100	62.2925	59.3150	62.1900	139162000	60.11165	0.11068	0.0206	0.0030	-0.0190	0.0016	0.10908
2019-11-29	66.6500	67.0000	66.4750	66.8125	46617600	64.77354	0.07755	0.0388	0.0079	-0.0201	0.0012	0.07635
2019-12-31	72.4825	73.4200	72.3800	73.4125	100805600	71.17213	0.09878	0.0277	0.0074	0.0177	0.0014	0.09738
2020-01-31	80.2325	80.6700	77.0725	77.3775	199588400	75.01614	0.05401	-0.0011	-0.0312	-0.0628	0.0013	0.05271
2020-02-28	64.3150	69.6025	64.0925	68.3400	426510000	66.41167	-0.11470	-0.0813	0.0107	-0.0380	0.0012	-0.11590
2020-03-31	63.9000	65.6225	63.0000	63.5725	197002000	61.77869	-0.06976	-0.1338	-0.0482	-0.1383	0.0013	-0.07106
2020-04-30	72.4900	73.6325	72.0875	73.4500	183064000	71.37748	0.15537	0.1365	0.0246	-0.0134	0.0000	0.15537
2020-05-29	79.8125	80.2875	79.1175	79.4850	153532400	77.45129	0.08509	0.0558	0.0249	-0.0486	0.0001	0.08499
2020-06-30	90.0200	91.4950	90.0000	91.2000	140223200	88.86655	0.14739	0.0246	0.0269	-0.0221	0.0001	0.14729
2020-07-31	102.8850	106.4150	100.8250	106.2600	374336800	103.54124	0.16513	0.0577	-0.0231	-0.0141	0.0001	0.16503
2020-08-31	127.5800	131.0000	126.0000	129.0400	225702700	125.96510	0.21657	0.0763	-0.0024	-0.0297	0.0001	0.21647
2020-09-30	113.7900	117.2600	113.6200	115.8100	142675200	113.05035	-0.10253	-0.0363	0.0002	-0.0270	0.0001	-0.10263
2020-10-30	111.0600	111.9900	107.7200	108.8600	190272600	106.26595	-0.06001	-0.0210	0.0438	0.0423	0.0001	-0.06011
2020-11-30	116.9700	120.9700	116.8100	119.0500	169410200	116.41364	0.09549	0.1247	0.0580	0.0206	0.0001	0.09539
2020-12-31	134.0800	134.7400	131.7200	132.6900	99116600	129.75159	0.11457	0.0463	0.0488	-0.0151	0.0001	0.11447
2021-01-29	135.8300	136.7400	130.2100	131.9600	177523800	129.03775	-0.00550	-0.0003	0.0737	0.0304	0.0001	-0.00560
2021-02-26	122.5900	124.8500	121.2000	121.2600	164560400	118.75188	-0.07971	0.0278	0.0206	0.0709	0.0000	-0.07971
2021-03-31	121.6500	123.5200	121.1500	122.1500	118323800	119.62349	0.00734	0.0308	-0.0236	0.0740	0.0000	0.00734
2021-04-30	131.7800	133.5600	131.0700	131.4600	109839500	128.74092	0.07622	0.0493	-0.0319	-0.0097	0.0000	0.07622
2021-05-28	125.5700	125.8000	124.5500	124.6100	71311100	122.23987	-0.05050	0.0029	-0.0021	0.0708	0.0000	-0.05050
2021-06-30	136.1700	137.4100	135.8700	136.9600	63261400	134.35495	0.09911	0.0275	0.0166	-0.0784	0.0000	0.09911
2021-07-30	144.3800	146.3300	144.1100	145.8600	70440600	143.08565	0.06498	0.0127	-0.0397	-0.0184	0.0000	0.06498
2021-08-31	152.6600	152.8000	151.2900	151.8300	86453100	149.16527	0.04249	0.0291	-0.0041	-0.0019	0.0000	0.04249
2021-09-30	143.6600	144.3800	141.2800	141.5000	89056700	139.01660	-0.06804	-0.0437	0.0068	0.0515	0.0000	-0.06804
2021-10-29	147.2200	149.9400	146.4100	149.8000	124953200	147.17090	0.05866	0.0665	-0.0235	-0.0045	0.0000	0.05866
2021-11-30	159.9900	165.5200	159.9200	165.3000	174048100	162.63591	0.10508	-0.0155	-0.0131	-0.0044	0.0000	0.10508
2021-12-31	178.0900	179.2300	177.2600	177.5700	64062300	174.70816	0.07423	0.0310	-0.0168	0.0325	0.0001	0.07413
2022-01-31	170.1600	175.0000	169.5100	174.7800	115541600	171.96310	-0.01571	-0.0625	-0.0595	0.1288	0.0000	-0.01571
2022-02-28	163.0600	165.4200	162.4300	165.1200	95056600	162.66579	-0.05407	-0.0229	0.0220	0.0311	0.0000	-0.05407
2022-03-31	177.8400	178.0300	174.4000	174.6100	103049300	172.01472	0.05747	0.0305	-0.0162	-0.0181	0.0001	0.05737
2022-04-29	161.8400	166.2000	157.2500	157.6500	131747600	155.30682	-0.09713	-0.0946	-0.0139	0.0620	0.0001	-0.09723
2022-05-31	149.0700	150.6600	146.8400	148.8400	103718400	146.84317	-0.05450	-0.0034	-0.0184	0.0848	0.0003	-0.05480
2022-06-30	137.2500	138.3700	133.7700	136.7200	98964500	134.88580	-0.08143	-0.0843	0.0210	-0.0601	0.0006	-0.08203
2022-07-29	161.2400	163.6300	159.5000	162.5100	101786900	160.32979	0.18863	0.0957	0.0280	-0.0405	0.0008	0.18783
2022-08-31	160.3100	160.5800	157.1400	157.2200	87991100	155.32623	-0.03121	-0.0377	0.0140	0.0029	0.0019	-0.03311
2022-09-30	141.2800	143.1000	138.0000	138.2000	124925300	136.53531	-0.12098	-0.0935	-0.0082	0.0005	0.0019	-0.12288
2022-10-31	153.1600	154.2400	151.9200	153.3400	97943200	151.49295	0.10955	0.0783	0.0007	0.0802	0.0023	0.10725
2022-11-30	141.4000	148.7200	140.5500	148.0300	111380900	146.48955	-0.03303	0.0460	-0.0350	0.0138	0.0029	-0.03593
2022-12-30	128.4100	129.9500	127.4300	129.9300	77034200	128.57790	-0.12227	-0.0641	-0.0070	0.0136	0.0033	-0.12557
2023-01-31	142.7000	144.3400	142.2800	144.2900	65874500	142.78842	0.11052	0.0665	0.0500	-0.0402	0.0035	0.10702
2023-02-28	147.0500	149.0800	146.8300	147.4100	50547000	146.09869	0.02318	-0.0258	0.0117	-0.0080	0.0034	0.01978
2023-03-31	162.4400	165.0000	161.9100	164.9000	68749800	163.43311	0.11865	0.0251	-0.0552	-0.0887	0.0036	0.11505
2023-04-28	168.4900	169.8500	167.8800	169.6800	55209200	168.17061	0.02899	0.0061	-0.0335	-0.0005	0.0035	0.02549
2023-05-31	177.3300	179.3500	176.7600	177.2500	99625300	175.91628	0.04606	0.0035	0.0160	-0.0772	0.0036	0.04246
2023-06-30	191.6300	194.4800	191.2600	193.9700	85069600	192.51045	0.09433	0.0646	0.0154	-0.0025	0.0040	0.09033
2023-07-31	196.0600	196.4900	195.2600	196.4500	38824100	194.97179	0.01279	0.0321	0.0207	0.0409	0.0045	0.00829
2023-08-31	187.8400	189.1200	187.4800	187.8700	60794500	186.70815	-0.04238	-0.0239	-0.0312	-0.0112	0.0045	-0.04688
2023-09-29	172.0200	173.0700	170.3400	171.2100	51814200	170.15115	-0.08868	-0.0524	-0.0249	0.0151	0.0043	-0.09298
2023-10-31	169.3500	170.9000	167.9000	170.7700	44846000	169.71388	-0.00257	-0.0318	-0.0386	0.0017	0.0047	-0.00727
2023-11-30	189.8400	190.3200	188.1900	189.9500	48794400	189.02396	0.11378	0.0884	-0.0001	0.0161	0.0044	0.10938
2023-12-29	193.9000	194.4000	191.7300	192.5300	42628800	191.59138	0.01358	0.0487	0.0634	0.0493	0.0043	0.00928
2024-01-31	187.0400	187.1000	184.3500	184.4000	55467800	183.50102	-0.04223	0.0070	-0.0510	-0.0233	0.0047	-0.04693
2024-02-29	181.2700	182.5700	179.5300	180.7500	136682600	180.09834	-0.01854	0.0506	-0.0016	-0.0354	0.0042	-0.02274
2024-03-28	171.7500	172.2300	170.5100	171.4800	65672700	170.86176	-0.05129	0.0283	-0.0246	0.0416	0.0043	-0.05559
2024-04-30	173.3300	174.9900	170.0000	170.3300	65934800	169.71590	-0.00671	-0.0467	-0.0234	-0.0054	0.0047	-0.01141
2024-05-31	191.4400	192.5700	189.9100	192.2500	75158300	191.81670	0.13022	0.0434	0.0061	-0.0137	0.0044	0.12582
2024-06-28	215.7700	216.0700	210.3000	210.6200	82542700	210.14528	0.09555	0.0278	-0.0302	-0.0331	0.0041	0.09145
2024-07-31	221.4400	223.8200	220.6300	222.0800	50036300	221.57947	0.05441	NA	NA	NA	NA	NA
2024-08-30	230.1900	230.4000	227.4800	229.0000	52990800	228.74834	0.03235	NA	NA	NA	NA	NA

Calculate annualized average excess return.

\[ \begin{aligned} r^A &= \big(1+\text{HPR}(T)\big)^{\frac{12}{T}}-1 \\ &= \left[\prod_{t=1}^T (1+r_{t}) \right]^{\frac{12}{T}}-1 \end{aligned} \] Note that the formula indicates \(\text{HPR}(T) = \left[\prod_{t=1}^T (1+r_{t}) \right]-1\), which is the holding period return for \(T\)-period.

n <- nrow(reg_data)-1 # number of months
r_A <- prod(reg_data$eRi+1, na.rm = TRUE)^{12/n}-1
r_A

## [1] 0.2315527

Plot excess return and ERP.

library(xts)
plot_data <- xts(reg_data[,c("eRi","rmrf")], order.by = reg_data$Date)
plot_data

##                     eRi    rmrf
## 2014-08-29  0.000000000  0.0424
## 2014-09-30 -0.017073105 -0.0197
## 2014-10-31  0.071960286  0.0252
## 2014-11-28  0.105978772  0.0255
## 2014-12-31 -0.071890915 -0.0006
## 2015-01-30  0.061424124 -0.0311
## 2015-02-27  0.100776504  0.0613
## 2015-03-31 -0.031371661 -0.0112
## 2015-04-30  0.005786313  0.0059
## 2015-05-29  0.045339376  0.0136
##        ...                     
## 2023-11-30  0.109380167  0.0884
## 2023-12-29  0.009282557  0.0487
## 2024-01-31 -0.046927173  0.0070
## 2024-02-29 -0.022743109  0.0506
## 2024-03-28 -0.055586355  0.0283
## 2024-04-30 -0.011406350 -0.0467
## 2024-05-31  0.125822330  0.0434
## 2024-06-28  0.091452593  0.0278
## 2024-07-31           NA      NA
## 2024-08-30           NA      NA

col_vec <- c("black", "red")
plot.xts(plot_data, col = col_vec, main = "Excess Returns on Asset and Market")
addLegend("topright", 
          legend.names = c("META", "Equity Risk Premium"), 
          lty = c(1, 1), 
          lwd = c(2, 2),
          col = col_vec,
          bg = "white",
          box.col = "white")

CAPM

CAPM as a regression can be expressed as

\[ r_{t} - r_{f,t} = \alpha + \beta (r_{m,t}-r_{f,t}) + \varepsilon_{t} \] where \(\alpha\) (intercept) and \(\beta\) (slope) are the parameters to estimate.

We use the following data to estimate the parameters.

\(r_t\) is the return on the asset at time \(t\);
\(r_{m,t}\) is the return on the market portfolio;
\(r_{f,t}\) is the risk free rate of interest;
\(r_t-r_{f,t}\) is the excess return on the asset;
\(r_m-r_{f,t}\) is the Equity Risk Premium.

capm_ml <- lm(eRi~rmrf, data=reg_data)
summary(capm_ml)

## 
## Call:
## lm(formula = eRi ~ rmrf, data = reg_data)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.212236 -0.036970  0.002496  0.041311  0.148802 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.009168   0.005620   1.631    0.106    
## rmrf        1.187585   0.120201   9.880   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05994 on 117 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.4548, Adjusted R-squared:  0.4502 
## F-statistic: 97.61 on 1 and 117 DF,  p-value: < 2.2e-16

FF 3-factor

The Fama-French three-factor model as a regression can be expressed as

\[ r_{t} - r_{f,t} = \alpha + \beta^{RMRF} (r_{m,t}-r_{f,t}) + \beta^{SMB}SMB_t + \beta^{HML}HML_t + \varepsilon_{t} \]

FF_ml <- lm(eRi~rmrf+SMB+HML, data=reg_data)
summary(FF_ml)

## 
## Call:
## lm(formula = eRi ~ rmrf + SMB + HML, data = reg_data)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.210858 -0.028634 -0.001951  0.034289  0.128891 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.007119   0.005292   1.345    0.181    
## rmrf         1.235424   0.117441  10.520  < 2e-16 ***
## SMB         -0.214322   0.196916  -1.088    0.279    
## HML         -0.569461   0.137374  -4.145 6.52e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05596 on 115 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.5329, Adjusted R-squared:  0.5207 
## F-statistic: 43.74 on 3 and 115 DF,  p-value: < 2.2e-16

Merge the two models in one table.

library(stargazer)
stargazer(capm_ml, FF_ml, type="html", 
          title="Regression Results for AAPL", 
          align = TRUE)

**Regression Results for AAPL**

	Dependent variable:

	eRi
	(1)	(2)

rmrf	1.188^***	1.235^***
	(0.120)	(0.117)

SMB		-0.214
		(0.197)

HML		-0.569^***
		(0.137)

Constant	0.009	0.007
	(0.006)	(0.005)


Observations	119	119
R²	0.455	0.533
Adjusted R²	0.450	0.521
Residual Std. Error	0.060 (df = 117)	0.056 (df = 115)
F Statistic	97.614^*** (df = 1; 117)	43.737^*** (df = 3; 115)

Note:	p<0.1; p<0.05; p<0.01

Excel options

Excel can achieve the same results. You use Analysis ToolPak \(\rightarrow\) Regression to conduct the regression. It will generate similar regression tables for you.

How to load Analysis ToolPak:
https://www.goskills.com/Excel/Resources/Excel-data-analysis-toolpak