Markov Chains: Marital Status in 25 Years

The business question

A demographer follows a group of 100 adults: 43 single, 29 married, 11 separated and 17 divorced. Each year a person may move between these states. The chances depend only on their current state (a Markov chain). A married person, for example, stays married with 88% probability, separates with 8% and divorces with 4%. Over 25 years these small yearly moves add up.

The model simulates the group two ways (in one step with VoseMarkovSample, and year by year with VoseMultinomial) and computes the exact expected numbers with VoseMarkovMatrix.

Who it is for

Anyone modelling a population that moves between states: customers between segments, patients between health states, loans between ratings, machines between conditions.

Main outputs

The number in each state after 25 years: expected, and at P10 / P90 · the same from two simulation methods · the mean number in each state year by year · the 25-year transition matrix.

How the model works

Four sheets after this one, read from left to right. Each one feeds the next.

1. 1

   Inputs sheet

   How many of the 100 people are in each state today.

2. 2

   Chain sheet

   The one-year transition matrix (rows: the state a person was in; columns: the state they move to). VoseMarkovMatrix raises it to 25 years, which gives the expected numbers; VoseMarkovSample draws the numbers in year 25 directly.

3. 3

   Years sheet

   The same with the multinomial method: each year VoseMultinomial splits each state's people across the four states, and the totals are next year's numbers. The outputs and checks are at the end.

4. 4

   Results sheet

   Expected, simulated and percentile numbers in each state in year 25 from both methods, and the mean numbers year by year.

ModelRisk techniques

• One-year transition matrix
• 25-year matrix VoseMarkovMatrix
• Population in one step VoseMarkovSample
• Year-by-year moves VoseMultinomial
• Output ranges and arrays

Find similar models

Product

• ModelRisk

Type

• Simulation

Application

• Forecasting
• Statistical techniques

Uses

• Multinomial
• Markov chains

Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Units and conventions

Transition matrix

Row: this year's state. Column: next year's. Each row adds up to 100%.

Year 0

Today.

Key assumptions and limitations

Markov

Next year's state depends only on this year's, not on how a person got there.

Fixed group

Nobody joins or leaves the group; the yearly probabilities do not change.

Data

The numbers and the matrix are those of ModelRisk's earlier Marital Status in 25 Years and Markov Chain Multinomial Method examples.

How to use it

1. Set today's numbers on the Inputs sheet and the transition matrix on the Chain sheet (each row must add up to 100%).
2. Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
3. Use VoseMarkovSample for the end state. Keep the year-by-year table (Years sheet) when you need the path, or when the probabilities change from year to year.

ModelRisk functions used

VoseMarkovMatrix

The 25-year transition matrix: the one-year matrix to the power 25.

VoseMarkovSample

The numbers in each state after 25 years, drawn in one step.

VoseMultinomial

Each year, how a state's people split across the states they move to.

VoseOutput

The numbers in year 25 from both methods.

VoseSimMean, VoseSimPercentile

Statistics of the simulation and the year-by-year means.

Further reading

• Monte Carlo simulation explained (Vose guide)
• Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: Markov chain models.