Análisis frontera eficiente

Se calcularán los indicadores de desempeño a los portafolios de la frontera.

Se utilizará una base de datos con los precios de cuatro acciones y los puntos del índice COLCAP con frecuencia diaria.

Importar datos

datos = read.csv("Cuatro acciones 2020 y COLCAP.csv", sep = ";", dec = ",", header = T)

head(datos)
tail(datos)
A data.frame: 6 × 6
FechaECOPFAVALISANUTRESACOLCAP
<fct><int><int><int><int><dbl>
126/03/20182775116513080257201469.04
227/03/20182645115513080257001450.00
328/03/20182615116513320259801455.52
42/04/2018 2690116513420259201470.88
53/04/2018 2730117513660259201492.84
64/04/2018 2740119013560258401497.59
A data.frame: 6 × 6
FechaECOPFAVALISANUTRESACOLCAP
<fct><int><int><int><int><dbl>
4953/04/2020 227090615500197001127.55
4966/04/2020 226095516140197201160.12
4977/04/2020 223093216680200201163.43
4988/04/2020 236093617200207001187.13
49913/04/2020225095117860213001193.98
50014/04/2020222095518000225001211.06

Matriz de precios

Se tendrá un objeto para los precios de las acciones precios y otro para el mercado COLCAP.

Los precios de las acciones están entre las columnas 2 y 5 de datos y el COLCAP está en la columna 6.

precios = datos[, 2:5]
precios = ts(precios)

COLCAP = datos[,6]
COLCAP = ts(COLCAP)

Nombres de las acciones

nombres = colnames(precios)
nombres
  1. 'ECO'
  2. 'PFAVAL'
  3. 'ISA'
  4. 'NUTRESA'

Matriz de rendimientos.

Matriz de rendimientos para las acciones rendimientos y matriz de rendimientos para el mercado rendimientos_mercado.

rendimientos = diff(log(precios))

rendimientos_mercado = diff(log(COLCAP))

Rendimientos esperado de cada acción y del mercado

Rendimientos esperados de las acciones rendimientos_esperados y rendimiento esperado del mercado rendimiento_esperado_mercado.

rendimientos_esperados = apply(rendimientos, 2, mean)
print(rendimientos_esperados)

rendimiento_esperado_mercado = mean(rendimientos_mercado)
rendimiento_esperado_mercado

          ECO        PFAVAL           ISA       NUTRESA
-0.0004471815 -0.0003983267  0.0006398545 -0.0002680433
-0.000387000234579747

Volatilidad de cada acción y del mercado

volatilidades para las acciones y volatilidad_mercado para el COLCAP.

volatilidades = apply(rendimientos, 2, sd)
print(volatilidades)

volatilidad_mercado = sd(rendimientos_mercado)
volatilidad_mercado

       ECO     PFAVAL        ISA    NUTRESA
0.03193244 0.02855772 0.02372920 0.01401047
0.0161376694676898

Frontera eficiente de Markowitz

library(fPortfolio)

frontera = portfolioFrontier(as.timeSeries(rendimientos), constraints = "longOnly")

Gráfico de la frontera

frontierPlot(frontera, cex = 2, pch = 19)
monteCarloPoints(frontera, col = "blue", mcSteps = 500, cex = 0.5, pch = 19)
minvariancePoints(frontera, col = "darkred", pch = 19, cex = 2)
equalWeightsPoints(frontera, col = "darkgreen", pch = 19, cex = 2)
../../../_images/output_24_02.png

Proporciones de inversión de la frontera

proporciones_frontera = getWeights(frontera)
print(proporciones_frontera)

              ECO      PFAVAL        ISA    NUTRESA
 [1,] 0.627055608 0.342516126 0.00000000 0.03042827
 [2,] 0.532194109 0.302671538 0.00000000 0.16513435
 [3,] 0.437332609 0.262826950 0.00000000 0.29984044
 [4,] 0.342471109 0.222982362 0.00000000 0.43454653
 [5,] 0.247609609 0.183137774 0.00000000 0.56925262
 [6,] 0.152748109 0.143293186 0.00000000 0.70395871
 [7,] 0.057886609 0.103448598 0.00000000 0.83866479
 [8,] 0.012448495 0.082369546 0.01244463 0.89273733
 [9,] 0.012089095 0.078406404 0.03623993 0.87326458
[10,] 0.011729696 0.074443262 0.06003522 0.85379182
[11,] 0.011370296 0.070480121 0.08383051 0.83431907
[12,] 0.011010897 0.066516979 0.10762581 0.81484632
[13,] 0.010651497 0.062553837 0.13142110 0.79537357
[14,] 0.010292098 0.058590696 0.15521639 0.77590082
[15,] 0.009932698 0.054627554 0.17901168 0.75642806
[16,] 0.009573299 0.050664412 0.20280698 0.73695531
[17,] 0.009213899 0.046701271 0.22660227 0.71748256
[18,] 0.008854500 0.042738129 0.25039756 0.69800981
[19,] 0.008495100 0.038774987 0.27419286 0.67853706
[20,] 0.008135701 0.034811846 0.29798815 0.65906430
[21,] 0.007776302 0.030848704 0.32178344 0.63959155
[22,] 0.007416902 0.026885562 0.34557874 0.62011880
[23,] 0.007057503 0.022922421 0.36937403 0.60064605
[24,] 0.006698103 0.018959279 0.39316932 0.58117330
[25,] 0.006338704 0.014996137 0.41696462 0.56170054
[26,] 0.005979304 0.011032996 0.44075991 0.54222779
[27,] 0.005619905 0.007069854 0.46455520 0.52275504
[28,] 0.005260505 0.003106712 0.48835049 0.50328229
[29,] 0.004411873 0.000000000 0.51217215 0.48341597
[30,] 0.001788542 0.000000000 0.53608946 0.46212200
[31,] 0.000000000 0.000000000 0.56017148 0.43982852
[32,] 0.000000000 0.000000000 0.58460640 0.41539360
[33,] 0.000000000 0.000000000 0.60904131 0.39095869
[34,] 0.000000000 0.000000000 0.63347623 0.36652377
[35,] 0.000000000 0.000000000 0.65791115 0.34208885
[36,] 0.000000000 0.000000000 0.68234607 0.31765393
[37,] 0.000000000 0.000000000 0.70678099 0.29321901
[38,] 0.000000000 0.000000000 0.73121590 0.26878410
[39,] 0.000000000 0.000000000 0.75565082 0.24434918
[40,] 0.000000000 0.000000000 0.78008574 0.21991426
[41,] 0.000000000 0.000000000 0.80452066 0.19547934
[42,] 0.000000000 0.000000000 0.82895558 0.17104442
[43,] 0.000000000 0.000000000 0.85339049 0.14660951
[44,] 0.000000000 0.000000000 0.87782541 0.12217459
[45,] 0.000000000 0.000000000 0.90226033 0.09773967
[46,] 0.000000000 0.000000000 0.92669525 0.07330475
[47,] 0.000000000 0.000000000 0.95113016 0.04886984
[48,] 0.000000000 0.000000000 0.97556508 0.02443492
[49,] 0.000000000 0.000000000 0.99999999 0.00000000

Media-varianza de la frontera

media_varianza_frontera = frontierPoints(frontera)
print(media_varianza_frontera)

   targetRisk  targetReturn
1  0.02757505 -4.249971e-04
2  0.02448843 -4.028126e-04
3  0.02157674 -3.806282e-04
4  0.01892091 -3.584438e-04
5  0.01664388 -3.362594e-04
6  0.01492009 -3.140750e-04
7  0.01395607 -2.918906e-04
8  0.01378202 -2.697062e-04
9  0.01373305 -2.475218e-04
10 0.01370275 -2.253374e-04
11 0.01369124 -2.031530e-04
12 0.01369858 -1.809686e-04
13 0.01372472 -1.587842e-04
14 0.01376957 -1.365998e-04
15 0.01383294 -1.144153e-04
16 0.01391457 -9.223094e-05
17 0.01401416 -7.004653e-05
18 0.01413131 -4.786212e-05
19 0.01426560 -2.567771e-05
20 0.01441655 -3.493303e-06
21 0.01458364  1.869111e-05
22 0.01476633  4.087551e-05
23 0.01496403  6.305992e-05
24 0.01517617  8.524433e-05
25 0.01540214  1.074287e-04
26 0.01564136  1.296131e-04
27 0.01589321  1.517976e-04
28 0.01615712  1.739820e-04
29 0.01643250  1.961664e-04
30 0.01671909  2.183508e-04
31 0.01701648  2.405352e-04
32 0.01732437  2.627196e-04
33 0.01764230  2.849040e-04
34 0.01796974  3.070884e-04
35 0.01830617  3.292728e-04
36 0.01865112  3.514572e-04
37 0.01900411  3.736416e-04
38 0.01936471  3.958260e-04
39 0.01973250  4.180105e-04
40 0.02010709  4.401949e-04
41 0.02048810  4.623793e-04
42 0.02087518  4.845637e-04
43 0.02126800  5.067481e-04
44 0.02166625  5.289325e-04
45 0.02206963  5.511169e-04
46 0.02247787  5.733013e-04
47 0.02289070  5.954857e-04
48 0.02330789  6.176701e-04
49 0.02372920  6.398545e-04
attr(,"control")
  targetRisk targetReturn         auto
       "Cov"       "mean"       "TRUE"

Tasa libre de riesgo

Tasa libre de riesgo de TES colombiano a 10 año con fecha del 14 de abril de 2020.

Rf = 0.06916 #E.A.
Rf = log(1+(1+ Rf)^(1/250) - 1)  #Continua diaria
Rf
0.000267493173736943

Beta de cada acción

Valores calculados con precios mensuales en un código anterior.

beta = c(1.23765583817092, 1.04762467584509, 0.696424844336778, 0.805687711895736)
print(beta)

[1] 1.2376558 1.0476247 0.6964248 0.8056877

Beta de cada portafolio de la frontera

1

1

beta_portafolios = apply(proporciones_frontera*beta, 1, sum)
print(beta_portafolios)

 [1] 1.1594231 0.9727072 0.8304460 0.8545292 0.9569550 1.1310742 1.0022669
 [8] 0.7467365 0.8259195 1.1692025 1.0425101 0.7714271 0.8110635 1.1369404
[15] 1.0649377 0.7961178 0.7962074 1.1046782 1.0873653 0.8208084 0.7813513
[22] 1.0724161 1.1097929 0.8454990 0.7664952 1.0401540 1.1322205 0.8701897
[29] 0.7516321 1.0057424 1.1540747 0.9017385 0.7391421 0.9640143 1.1726483
[36] 0.9360647 0.7284628 0.9217939 1.1912219 0.9703908 0.7177835 0.8795735
[43] 1.2097955 1.0047170 0.7071042 0.8373530 1.2283690 1.0390431 0.6964248

CAPM de cada portafolio de la frontera

2

2

CAPM_portafolios = Rf + beta_portafolios*(rendimiento_esperado_mercado - Rf)
print(CAPM_portafolios)

 [1] -0.0004913416 -0.0003691372 -0.0002760282 -0.0002917905 -0.0003588275
 [6] -0.0004727875 -0.0003884839 -0.0002212409 -0.0002730657 -0.0004977422
[11] -0.0004148228 -0.0002374008 -0.0002633425 -0.0004766268 -0.0004295015
[16] -0.0002535607 -0.0002536193 -0.0004555115 -0.0004441802 -0.0002697205
[21] -0.0002438961 -0.0004343961 -0.0004588589 -0.0002858804 -0.0002341729
[26] -0.0004132807 -0.0004735377 -0.0003020402 -0.0002244451 -0.0003907586
[31] -0.0004878411 -0.0003226887 -0.0002162705 -0.0003634478 -0.0004999974
[36] -0.0003451550 -0.0002092809 -0.0003358148 -0.0005121537 -0.0003676212
[41] -0.0002022914 -0.0003081819 -0.0005243100 -0.0003900875 -0.0001953018
[46] -0.0002805489 -0.0005364663 -0.0004125537 -0.0001883123

Indicador de diversificación de cada portafolio de la frontera

h = 1 - media_varianza_frontera[,1]/sum(volatilidades)
print(h)

        1         2         3         4         5         6         7         8
0.7192803 0.7507027 0.7803443 0.8073812 0.8305619 0.8481105 0.8579243 0.8596962
        9        10        11        12        13        14        15        16
0.8601948 0.8605032 0.8606203 0.8605457 0.8602795 0.8598230 0.8591779 0.8583468
       17        18        19        20        21        22        23        24
0.8573330 0.8561403 0.8547732 0.8532365 0.8515355 0.8496758 0.8476631 0.8455035
       25        26        27        28        29        30        31        32
0.8432030 0.8407678 0.8382038 0.8355172 0.8327138 0.8297962 0.8267688 0.8236344
       33        34        35        36        37        38        39        40
0.8203978 0.8170644 0.8136394 0.8101278 0.8065342 0.8028632 0.7991191 0.7953057
       41        42        43        44        45        46        47        48
0.7914269 0.7874864 0.7834874 0.7794331 0.7753266 0.7711707 0.7669679 0.7627209
       49
0.7584318

Indicadores de desempeño de cada portafolio de la frontera

* Ratio de Sharpe:

3

3

sharpe_portafolios = (media_varianza_frontera[,2] - Rf)/media_varianza_frontera[,1]
print(sharpe_portafolios)

            1             2             3             4             5
-0.0251129278 -0.0273723460 -0.0300379646 -0.0330817565 -0.0362747580
            6             7             8             9            10
-0.0389788785 -0.0400817586 -0.0389782896 -0.0375018727 -0.0359658211
           11            12            13            14            15
-0.0343757114 -0.0327378386 -0.0310590926 -0.0293468145 -0.0276086387
           16            17            18            19            20
-0.0258523298 -0.0240856217 -0.0223160651 -0.0205508916 -0.0187968976
           21            22            23            24            25
-0.0170603510 -0.0153469234 -0.0136616453 -0.0120088852 -0.0103923485
           26            27            28            29            30
-0.0088150937 -0.0072795619 -0.0057876177 -0.0043405937 -0.0029392983
           31            32            33            34            35
-0.0015842286 -0.0002755414  0.0009868800  0.0022034402  0.0033747989
           36            37            38            39            40
 0.0045018242  0.0055855525  0.0066271508  0.0076278856  0.0085890941
           41            42            43            44            45
 0.0095121609  0.0103984976  0.0112495260  0.0120666640  0.0128513145
           46            47            48            49
 0.0136048562  0.0143286365  0.0150239659  0.0156921141

* Ratio de Treynor:

4

4

treynor_portafolios =  (media_varianza_frontera[,2] - Rf)/beta_portafolios
print(treynor_portafolios)

            1             2             3             4             5
-5.972714e-04 -6.891137e-04 -7.804498e-04 -7.324934e-04 -6.309101e-04
            6             7             8             9            10
-5.141733e-04 -5.581186e-04 -7.193962e-04 -6.235655e-04 -4.215100e-04
           11            12            13            14            15
-4.514548e-04 -5.813404e-04 -5.255783e-04 -3.554214e-04 -3.586205e-04
           16            17            18            19            20
-4.518479e-04 -4.239344e-04 -2.854725e-04 -2.696158e-04 -3.301458e-04
           21            22            23            24            25
-3.184254e-04 -2.113150e-04 -1.842085e-04 -2.155518e-04 -2.088264e-04
           26            27            28            29            30
-1.325573e-04 -1.021847e-04 -1.074607e-04 -9.489590e-05 -4.886181e-05
           31            32            33            34            35
-2.335896e-05 -5.293751e-06  2.355546e-05  4.107329e-05  5.268387e-05
           36            37            38            39            40
 8.969899e-05  1.457157e-04  1.392208e-04  1.263554e-04  1.779713e-04
           41            42            43            44            45
 2.715110e-04  2.467906e-04  1.977648e-04  2.602119e-04  4.011060e-04
           46            47            48            49
 3.652081e-04  2.670147e-04  3.370187e-04  5.346756e-04

* Alfa de Jensen:

5

5

jensen_portafolios = (media_varianza_frontera[,2] - Rf) - (rendimiento_esperado_mercado - Rf)*beta_portafolios
print(jensen_portafolios)

           1             2             3             4             5
6.634452e-05 -3.367540e-05 -1.046000e-04 -6.665329e-05  2.256812e-05
           6             7             8             9            10
1.587124e-04  9.659328e-05 -4.846525e-05  2.554393e-05  2.724048e-04
          11            12            13            14            15
2.116698e-04  5.643224e-05  1.045584e-04  3.400271e-04  3.150862e-04
          16            17            18            19            20
1.613297e-04  1.835728e-04  4.076493e-04  4.185025e-04  2.662272e-04
          21            22            23            24            25
2.625872e-04  4.752716e-04  5.219189e-04  3.711247e-04  3.416016e-04
          26            27            28            29            30
5.428939e-04  6.253352e-04  4.760222e-04  4.206115e-04  6.091094e-04
          31            32            33            34            35
7.283763e-04  5.854083e-04  5.011745e-04  6.705362e-04  8.292702e-04
          36            37            38            39            40
6.966122e-04  5.829226e-04  7.316409e-04  9.301641e-04  8.078161e-04
          41            42            43            44            45
6.646706e-04  7.927455e-04  1.031058e-03  9.190200e-04  7.464187e-04
          46            47            48            49
8.538502e-04  1.131952e-03  1.030224e-03  8.281668e-04