Recursion: account value with distributions

Question

Update: not sure if this is possible without some form of a loop, but np.where will not work here. If the answer is, "you can't", then so be it. If it can be done, it may use something from scipy.signal.

---

I'd like to vectorize the loop in the code below, but unsure as to how, due to the recursive nature of the output.

Walk-though of my current setup:

Take a starting amount ($1 million) and a quarterly dollar distribution ($5,000):

dist = 5000.
v0 = float(1e6)

Generate some random security/account returns (decimal form) at monthly freq:

r = pd.Series(np.random.rand(12) * .01,
              index=pd.date_range('2017', freq='M', periods=12))

Create an empty Series that will hold the monthly account values:

value = pd.Series(np.empty_like(r), index=r.index)

Add a "start month" to value. This label will contain v0.

from pandas.tseries import offsets
value = (value.append(Series(v0, index=[value.index[0] - offsets.MonthEnd(1)]))
              .sort_index())

The loop I'd like to get rid of is here:

for date in value.index[1:]:
    if date.is_quarter_end:
        value.loc[date] = value.loc[date - offsets.MonthEnd(1)] \
                        * (1 + r.loc[date]) - dist
    else:
        value.loc[date] = value.loc[date - offsets.MonthEnd(1)] \
                        * (1 + r.loc[date]) 

Combined code:

import pandas as pd
from pandas.tseries import offsets
from pandas import Series
import numpy as np

dist = 5000.
v0 = float(1e6)
r = pd.Series(np.random.rand(12) * .01, index=pd.date_range('2017', freq='M', periods=12))
value = pd.Series(np.empty_like(r), index=r.index)
value = (value.append(Series(v0, index=[value.index[0] - offsets.MonthEnd(1)])).sort_index())
for date in value.index[1:]:
    if date.is_quarter_end:
        value.loc[date] = value.loc[date - offsets.MonthEnd(1)] * (1 + r.loc[date]) - dist
    else:
        value.loc[date] = value.loc[date - offsets.MonthEnd(1)] * (1 + r.loc[date]) 

In psuedocode, what is loop is doing is just:

for each date in index of value:
    if the date is not a quarter end:
        multiply previous value by (1 + r) for that month
    if the date is a quarter end:
        multiply previous value by (1 + r) for that month and subtract dist

The issue is, I don't currently see how vectorization is possible since the successive value depends on whether or not a distribution was taken in the month prior. I get to the desired result, but pretty inefficiently for higher frequency data or larger time periods.

Answer 1

You could use the following code:

cum_r = (1 + r).cumprod()
result = cum_r * v0
for date in r.index[r.index.is_quarter_end]:
     result[date:] -= cum_r[date:] * (dist / cum_r.loc[date])

You would make:

• 1 cumulative product for all monthly returns.
• 1 vector multiplication with scalarv0
• n vector multiplication with scalar dist / cum_r.loc[date]
• n vector subtractions

where n is the number of quarter ends.

Based on this code we can optimize further:

cum_r = (1 + r).cumprod()
t = (r.index.is_quarter_end / cum_r).cumsum()
result = cum_r * (v0 - dist * t)

which is

• 1 cumulative product (1 + r).cumprod()
• 1 division between two series r.index.is_quarter_end / cum_r
• 1 cumulative sum of the above division
• 1 multiplication of the above sum with scalar dist
• 1 subtraction of scalar v0 with dist * t
• 1 dotwise multiplication of cum_r with v0 - dist * t

Answer 2

Ok... I'm taking a stab at this.

import numpy as np 
import pandas as pd

#Define a generator for accumulating deposits and returns
def gen(lst):
    acu = 0
    for r, v in lst:
        yield acu * (1 + r) +v
        acu *= (1 + r)
        acu += v


dist = 5000.
v0 = float(1e6)
random_returns = np.random.rand(12) * 0.1

#Create the index. 
index=pd.date_range('2016-12-31', freq='M', periods=13)
#Generate a return so that the value at i equals the return from i-1 to i
r = pd.Series(np.insert(random_returns, 0,0), index=index, name='Return')
#Generate series with deposits and withdrawals
w = [-dist if is_q_end else 0 for is_q_end in index [1:].is_quarter_end]
d = pd.Series(np.insert(w, 0, v0), index=index, name='Movements')

df = pd.concat([r, d], axis=1)
df['Value'] = list(gen(zip(df['Return'], df['Movements'])))

now, your code

#Generate some random security/account returns (decimal form) at monthly freq:
r = pd.Series(random_returns,
          index=pd.date_range('2017', freq='M', periods=12))
#Create an empty Series that will hold the monthly account values:
value = pd.Series(np.empty_like(r), index=r.index)
#Add a "start month" to value. This label will contain v0.
from pandas.tseries import offsets
value = (value.append(pd.Series(v0, index=[value.index[0] - offsets.MonthEnd(1)])).sort_index())
#The loop I'd like to get rid of is here:

def loopy(value) :
    for date in value.index[1:]:
        if date.is_quarter_end:
            value.loc[date] = value.loc[date - offsets.MonthEnd(1)] \
                           * (1 + r.loc[date]) - dist
        else:
           value.loc[date] = value.loc[date - offsets.MonthEnd(1)] \
                           * (1 + r.loc[date]) 

   return value

and comparing and timing

(loopy(value)==list(gen(zip(r, d)))).all()
Out[11]: True

returns same result

%timeit list(gen(zip(r, d)))
%timeit loopy(value)
10000 loops, best of 3: 72.4 µs per loop
100 loops, best of 3: 5.37 ms per loop

and appears to be somewhat faster. Hope it helps.