The Markov property makes the current state sufficient for predicting the next state: transition probabilities depend on the current state and chosen action, not earlier history. In goals-based investing, current wealth and asset allocation determine the possible wealth states at the next time step. section 6.2
Goals-based investing
If wealth at time t is the state and asset allocation is the action, a normal distribution can calculate the probabilities of reaching each possible wealth state at time t+1. The prior path of wealth does not enter that transition calculation. section 6.2
Listing 6.1 Creating a state space grid for goals-based investing
import numpy as np
import matplotlib.pyplot as plt
from math import log, sqrt, exp
W0 = 1000000
T = 10
nW = 101 #1
mu_stock = 0.08
mu_bond = 0.045
sig_stock = 0.20
sig_bond = 0.0
lnW = np.zeros((nW,T+1))
W = np.zeros((nW, T+1))
for t in range(T+1):
lnW_min = log(W0) + (mu_stock-0.5*sig_stock**2)*t - 2.5*sig_stock*sqrt(t)
lnW_max = log(W0) + (mu_stock-0.5*sig_stock**2)*t + 2.5*sig_stock*sqrt(t)
lnW[:,t] = np.linspace(lnW_min, lnW_max, nW) #2
W = np.exp(lnW)
plt.figure(figsize=(10, 6)) #3
plt.subplot(1,2,1) #3
for t in range(T+1): #3
plt.scatter(t*np.ones(nW), W[:,t], color='r') #3
plt.xlabel("time, $ t $") #3
plt.ylabel("Wealth, $ W $") #3
plt.subplot(1, 2, 2) #3
for t in range(T+1): #3
plt.scatter(t*np.ones(nW), lnW[:,t], color='r') #3
plt.xlabel("time, $ t$") #3
plt.ylabel("Log of Wealth, $ \symbol{92}ln(W) $") #3
A wealth-time state-space grid supplies the discrete states needed to represent Markov transitions in a financial planning model. section 6.2
