Simulation

Most interesting financial questions have no closed-form answer. "Will this corpus last thirty years if I withdraw ₹6 lakh a year, rising with inflation, from a portfolio of equity and bonds?" is not an equation — it depends on the order the returns arrive in, which chapter 8 of the Risk subject called sequence risk.

Monte Carlo simulation answers such questions by generating thousands of possible futures and counting.

The method

1. Specify a process — a distribution for returns, inflation, and anything else varying.
2. Draw a path. Sample a value for each period and apply it in sequence.
3. Evaluate the outcome. Did the money last? What was the final balance?
4. Repeat ten or a hundred thousand times.
5. Read the distribution of outcomes, not the average of them.

Step 5 is the whole value. A formula gives one number; a simulation gives a shape, including the bad tail that the single number was hiding.

What it is genuinely good for

Path-dependent questions. Anything where the order matters. The arithmetic mean return cannot answer a withdrawal question because withdrawals interact with the sequence.

Combining several uncertainties. Returns, inflation, lifespan and spending all vary together; no formula aggregates them, and a simulation does so naturally.

Showing a range to someone who wants a number. A client asking "will this work" is better served by "in 90% of simulated futures, yes, and here is what the other 10% look like" than by a point estimate that was never going to be right.

What a 90% success rate is not

It is not a 90% probability that your retirement works.

It is: in the model specified, with the distributions assumed, 90% of generated paths did not run out of money. Every one of those qualifiers is doing work, and the output's precision — "90.3%" — hides that entirely.

PFRDA states the underlying reality plainly for the NPS: there is no implicit or explicit assurance of benefit, and investments are subject to market conditions. A simulation does not change that; it quantifies it under assumptions.

The four assumptions that decide the answer

The return distribution. Most simulations draw from a normal distribution. Chapter 6 showed that understates extreme losses by orders of magnitude, so a normal-based simulation is optimistic about exactly the scenario the exercise exists to find. Bootstrapping from historical returns is better; it still assumes the future resembles the sample.

Independence across periods. Drawing each year independently removes volatility clustering (chapter 13) and any mean reversion. Independent draws tend to understate the long bad stretches that actually break retirements.

The correlation structure. Equity and bond returns are usually drawn with a historical correlation. Chapter 6 of the Risk subject's point applies: correlations rise in crises, so the diversification the simulation credits you with is largest exactly when it is least available.

Fixed behaviour. Most simulations withdraw mechanically regardless of conditions. Real people cut spending in bad years, which chapter 5 of the Retirement subject identified as the single most effective lever. This assumption makes the result pessimistic, and it is the only one of the four that does.

Three of the four push the same way. A simulation reporting 90% is, on balance, telling you something closer to a ceiling than a central estimate.

Using it honestly

• Read the distribution, not the headline. What does the 10th percentile look like? Could you live in it?
• Vary the assumptions, not just the seed. Re-run with returns two points lower, with inflation a point higher, with correlations at one. If the conclusion survives, it is robust; if it moves a lot, the model was reporting its inputs.
• Prefer a stress test for the tail. "What if equities fall 40% in year one" is a more useful question than a tail probability from a distribution that cannot model tails.
• Treat precision as a warning. An output of "87.4%" is reporting arithmetic, not knowledge. Say "roughly nine in ten".

The honest summary of the subject

This chapter closes the foundation, and the thread running through all fourteen is a single caution: every number in finance is produced by a procedure that assumed something. Compounding assumes a reinvestment rate. A mean assumes which mean. A standard deviation assumes a distribution. A p-value assumes one test. A regression assumes independence. A simulation assumes all of them at once.

Knowing the procedure is what lets you say how much to trust the number, and that — rather than any particular formula — is what the rest of the course is built on.

The point

Monte Carlo generates thousands of paths and reads the distribution of outcomes, which is the only way to answer path-dependent questions like whether a corpus survives a given withdrawal. A "90% success rate" is a property of the assumed model, not a probability about your life — and three of its four main assumptions, normality, independence and stable correlations, all push the number up. Vary the assumptions rather than the seed, and read the bad decile rather than the headline.