Cost Averaging with Random Walk

by Dr. Yves J. Hilpisch, Visixion GmbH, http://www.visixion.com

This simple example illustrates how to analyze financial problems with Python.

First we import the scientific stack.

In [1]:
from pylab import *

Some parameter values for the stimulation.

In [2]:
S0 = 100  # intial value
vol = 0.2  # volatility
T = 10.  # savings period in years
r = 0.025  # short rate
M = 20  # time steps
dt = T / M  # time interval

Simulate the Base Investment and plot the simulated path.

In [3]:
S = zeros((M + 1))  # vector for index level
S[0] = S0  # intialization
for t in range(1, M + 1):  # simulation loop
    S[t] = S[t-1] * exp((r - vol ** 2 / 2) * dt + vol * sqrt(dt) * standard_normal())
    # geometric Brownian motion
plot(S)  # show the random development
grid(True)
S
Out[3]:
array([ 100.        ,  131.86180161,  115.41549563,  109.09797105,
        125.853692  ,  107.21403887,   89.94434985,   90.32156587,
        102.93154117,   87.42385132,   82.38829348,   96.36388625,
         95.38434112,   76.52822176,   75.9340541 ,   63.35431946,
         75.21530176,   67.66162694,   75.43002615,   77.21285179,
         89.97773932])

Simulate the savings plan.

In [4]:
sparPA = 100.0  # money invested per year
sparEUR = sparPA / M  # money invested per time interval
sparANT = zeros_like(S)
sparVAL = zeros_like(S)
for t in range(0, M):
    sparANT[t] = sparEUR / S[t]
sparVAL = cumsum(sparANT) * S  # valuation of portfolio
perfS = S[-1] / S0 - 1  # performance of base investment in percent
perfV = sparVAL[-1] / (M * sparEUR) - 1  # performance of savings plan
print "Performance of Base Investment %4.3f" % perfS
print "Performance of Savings Plan    %4.3f" % perfV

Performance of Base Investment -0.100
Performance of Savings Plan    0.014

Show the evolution of the savings plan

In [5]:
plot(sparVAL)
grid(True)
sparVAL
Out[5]:
array([   5.        ,   11.59309008,   15.14715574,   19.31804239,
         27.28498783,   28.24392474,   28.69448511,   33.81482641,
         43.53578227,   41.97667122,   44.55884185,   57.11739418,
         61.53679218,   54.37184891,   58.94970407,   54.18370852,
         69.32779993,   67.36539143,   80.0997791 ,   86.99297664,
        101.37472187])

Plot the net value of the savings plan.

In [6]:
sparSUM = array(array([sparEUR,] * M + [0.0]))  # vector with investments
plot(sparVAL - sparSUM.cumsum())  # net value position (portfolio value - investments)
grid(True)