Risk vs Return, Made Obvious: Simple Simulations Anyone Can Run

Most debates about investing end the same way: someone says “higher return means higher risk,” and everyone nods without agreeing on what either word really means.

Why simulations beat slogans

Risk and return live in probability. You can read definitions all day, but a few quick simulations make the relationship feel real—because you’ll see how often things go right, how badly they go wrong, and what “average” hides.

A simulation is just repeated sampling from a model. In finance, that usually means generating many possible paths for an investment and studying the distribution of outcomes. That’s decision_models in action: you choose assumptions, run scenarios, and decide under uncertainty.

Before the experiments, a shared vocabulary:

• Return: the change in value, usually measured as a percentage over a period.
• Expected return: the probability-weighted average outcome (the “center of gravity”).
• Risk: not one thing. Common stand-ins include volatility, probability of loss, size of drawdowns, and the chance of failing a goal.
• Distribution: the spread of possible results; tails matter.
• Path dependence: the route taken matters, not just the start and end (especially with withdrawals).

The goal isn’t to “predict” markets. It’s to clarify trade-offs.

Simulation 1: The coin-flip portfolio (expected value vs lived experience)

Start with the simplest possible return generator:

• Each year, the investment either gains +20% or loses -15%.
• Probability is 50/50.
• You invest for 20 years.

What the math says

The arithmetic average return per year is:

• (20% + (-15%)) / 2 = 2.5%

So you might assume that after 20 years you should be comfortably ahead.

But compounding does not care about arithmetic averages. If you alternate up and down, you multiply:

• Up year: ×1.20
• Down year: ×0.85

A two-year cycle becomes ×(1.20 × 0.85) = ×1.02, or about 2% over two years, which is roughly 1% per year geometric growth in that alternating case.

And it gets more subtle: even with the same yearly outcomes, the order changes the depth of interim losses, which affects real behavior (panic selling) and any cash flows (contributions/withdrawals).

What the simulation reveals

If you run, say, 10,000 random 20-year sequences, you’ll typically observe:

• A wide range of ending wealth even though every year’s distribution is identical.
• Many paths cluster around a modest long-run gain, but a non-trivial chunk end flat or down.
• The median ending value can be meaningfully below what the arithmetic average “suggests.”

Decision model lesson: expected return is not a promise; it’s a weighted average across futures you will not live through. You will experience one path.

If you want to implement this quickly in a spreadsheet:

1. In a column, generate 20 random numbers (RAND()).
2. Convert each to a return: if RAND() < 0.5 then 0.20 else -0.15.
3. Compute cumulative wealth with a running product.
4. Copy the whole block down 10,000 rows.

You’ll have a distribution, not a point estimate.

Simulation 2: Same average return, different volatility (the volatility tax)

Now compare two assets with the same average yearly return but different variability:

• Asset A (smooth): +6% every year (obviously unrealistic, but clean).
• Asset B (bumpy): alternates between +18% and -6% in random order.

Both have an arithmetic average of about 6%. Many beginners expect similar long-term results.

Why they diverge

Compounding punishes volatility because losses require larger subsequent gains to recover. A -20% year needs +25% just to break even.

If you simulate 30-year outcomes:

• Asset A produces a single ending value: predictable compounding.
• Asset B produces a spread of endings, and the typical outcome (median) is often lower than Asset A—even though the average of yearly returns matches.

This is the volatility drag effect. In more formal terms, for small returns, the geometric mean is roughly:

• geometric ≈ arithmetic − (variance / 2)

The intuition is more important than the formula: variance leaks return when you compound.

Decision model lesson: when you choose “risk,” you might be paying for the chance of a higher outcome, but you’re also accepting a structural headwind on typical compounded growth.

Simulation 3: Two risky assets: diversification is a distribution-shaper

Risk isn’t only about each asset; it’s also about how they move together.

Set up two assets:

• Asset 1: expected return 8%, volatility 18%
• Asset 2: expected return 6%, volatility 10%

Now vary the correlation between them:

• Case X: correlation +0.9 (they move together)
• Case Y: correlation 0.0 (independent-ish)
• Case Z: correlation -0.3 (some offset)

If you simulate portfolio returns with different weightings (e.g., 60/40, 50/50), you’ll typically find:

• With high correlation (+0.9), diversification doesn’t help much. Bad years tend to be bad together.
• With low or negative correlation, the portfolio’s volatility drops more than you’d guess, and the left tail (big losses) often shrinks.
• Expected return is roughly a weighted average; risk is not.

This is where portfolio construction stops being a slogan and becomes a quantitative decision_models exercise: you’re not just picking “good assets,” you’re engineering the distribution of outcomes.

A practical way to visualize this without heavy math:

• Simulate annual returns for each asset (normal is a common toy assumption).
• For correlated returns, generate two random variables with shared components (e.g., r2 = corr*r1 + sqrt(1-corr^2)*noise after standardizing).
• Combine by weights and compound over time.

Then compare:

• Median ending wealth
• 10th percentile ending wealth (a “bad but plausible” future)
• Maximum drawdown distribution

The best portfolio isn’t necessarily the one with the highest average; it’s the one whose bad cases you can survive.

_{Photo by Sajad Nori on Unsplash}

Simulation 4: Risk of ruin (when “average return” becomes irrelevant)

Many personal finance goals aren’t about maximizing wealth; they’re about not failing.

Imagine a retiree with:

• $1,000,000 starting portfolio
• Withdraws $45,000 per year (inflation-adjusted)
• Invested in a portfolio with expected return 6% and volatility 12%
• Horizon: 30 years

Now simulate 10,000 return paths. Track whether the portfolio ever hits zero.

What you learn quickly:

• Two portfolios with the same long-run expected return can have dramatically different ruin probabilities depending on volatility and early returns.
• Sequence-of-returns risk is real: bad returns in the first 5–10 years can be fatal, even if later returns are strong.
• Median outcomes can look fine while the left tail includes genuine catastrophe.

This is one of the clearest “risk vs return” lessons because it changes the objective function. The decision model is not “maximize expected final wealth.” It’s “maximize the probability of funding spending” or “keep ruin probability below 5%.”

In that framing:

• A lower-return, lower-volatility mix may be better if it reduces early drawdowns.
• Cash buffers and dynamic spending rules can dominate small changes in expected return.

Simulation 5: Contributions flip the usual intuition

Most people invest while adding money (salary deferrals, monthly deposits). That changes what “risk” means.

Consider two investors saving $500 per month for 25 years:

• Investor Calm: invests in a low-volatility fund with lower expected return.
• Investor Bold: invests in a higher-volatility fund with higher expected return.

Run a simulation with realistic monthly volatility. You’ll often see something counterintuitive:

• When you’re contributing, volatility can sometimes help because you buy more shares when prices are low (dollar-cost averaging).
• But this benefit is not magic. If volatility comes with deep, prolonged drawdowns, you still might end with less—especially if the expected return isn’t meaningfully higher.

This is a good place to focus on distribution of outcomes, not a single number:

• What’s the 25th percentile final balance?
• How often does Bold beat Calm?
• What’s the worst-case drawdown along the way (the “staying invested” test)?

Decision model lesson: risk tolerance isn’t just emotional; it’s tied to cash-flow structure. Contributors can sometimes “afford” volatility more than withdrawers—but only if they truly stay in.

The hidden variable: what you call “risk” changes the answer

Investors talk past each other because they use different risk metrics. Simulations let you compute several at once:

• Volatility (standard deviation): common, but treats upside and downside equally.
• Max drawdown: captures the psychological gut punch and the forced-sale problem.
• Probability of loss over horizon: “What’s the chance I’m down after 10 years?”
• Shortfall risk: “What’s the chance I fail to reach $X by year Y?”
• Value at Risk (VaR): a percentile loss measure; easy to misuse.
• Conditional VaR / Expected Shortfall: average loss in the worst tail.

A portfolio can look “low risk” by volatility but still have ugly tail risk. Another can look scary by volatility but rarely produce catastrophic drawdowns if it mean-reverts and is diversified.

So when someone says “this is riskier,” the next question should be: riskier by which definition?

Building a simple decision model: choose the objective first

A clean decision model in investing has three parts:

1. Objective: maximize expected wealth, minimize ruin probability, hit a target with high confidence, etc.
2. Constraints: maximum drawdown tolerated, liquidity needs, time horizon, leverage rules.
3. Evaluation method: simulation that produces outcome distributions, plus metrics that match the objective.

Here’s a practical workflow you can use with basic tools:

• Pick 2–4 candidate portfolios (e.g., 80/20, 60/40, 40/60, or a stock/bond/cash mix).
• For each portfolio, simulate 10,000 paths with monthly returns for realism.
• Measure:
  • Median ending value
  • 10th percentile ending value
  • Max drawdown percentile (e.g., 90th percentile drawdown)
  • Probability of meeting your target
• Choose the portfolio that fits your constraints and has the best target success rate—not the one with the prettiest average.

This isn’t overkill. It’s the difference between “I hope this works” and “I chose a plan whose bad cases I can live through.”

Common simulation choices—and what they quietly assume

No simulation is neutral. Each one embeds assumptions that can understate or exaggerate risk.

Normal returns (the classic toy model)

Pros:

• Simple
• Easy to code
• Often “good enough” for intuition

Cons:

• Underestimates crashes (fat tails)
• Can generate impossible results (very large negative returns)

Bootstrapping historical returns

Pros:

• Preserves real-world skew and fat tails (to a degree)
• Intuitive: “resample what happened”

Cons:

• Assumes the future resembles the past
• Misses regime shifts unless your history includes them

Regime models (calm vs crisis)

Pros:

• Captures clustering of volatility (bad times come in bunches)
• Produces more realistic drawdown behavior

Cons:

• More parameters, more room to fool yourself

The educational trick is to run all three and see how sensitive your conclusions are. If your plan only works under the nicest assumptions, you don’t have a plan—you have a hope.

A hands-on mini lab you can run this weekend

Below is a set of “products” in the sense of small, runnable experiments. They’re not financial products; they’re simulation modules you can build in a spreadsheet, Python, or R.

1. Geometric vs arithmetic return lab

   • Simulate a two-outcome investment (+a% / -b%) and compare arithmetic mean to median compounded result.

2. Volatility drag lab

   • Create two assets with the same average return and different volatilities; compound 30-year paths and compare medians and tails.

3. Correlation & diversification lab

   • Generate two correlated return streams; test 100 different weightings; plot risk metrics vs expected return.

4. Sequence-of-returns retirement lab

   • Add withdrawals; compute probability of ruin and distribution of ending balances.

5. Target-based investing lab

   • Define a goal (e.g., $800k in 15 years). Compare portfolios by probability of hitting the target rather than by average return.

Run these and you’ll notice something that charts rarely show: risk isn’t a single dial. It’s a set of trade-offs that show up differently depending on the goal.

How professionals read the same simulation differently than amateurs

Two people can look at the same Monte Carlo chart and walk away with different conclusions.

A beginner sees: “The average outcome is great.”

A seasoned allocator looks for:

• Left-tail thickness: how ugly are the worst 5–10% of outcomes?
• Drawdown duration: not just how far it falls, but how long it stays underwater.
• Clustering: do bad years come in streaks?
• Sensitivity: if expected returns are 1–2% lower than assumed, does the plan collapse?
• Behavioral feasibility: can a human stick with this when it’s down 35%?

That last item is uncomfortable but central. The best decision model includes the decision-maker. If your strategy requires monk-like discipline you don’t actually have, the “optimal” portfolio is a paper tiger.

Risk vs return, stated plainly after the simulations

Return is what you want. Risk is the set of ways you might not get it—or might not be able to hold on long enough to realize it.

Simulations turn that into something concrete:

• Higher expected return usually requires accepting a wider distribution of outcomes.
• Volatility reduces typical compounded growth unless it is compensated by a higher expected return.
• Diversification can improve the distribution without sacrificing much expected return, but only when correlations cooperate.
• Cash flows (saving vs withdrawing) change which risks matter most.
• The “best” choice depends on the objective: maximize wealth, avoid ruin, or hit a goal with high confidence.

If you’re willing to run even a basic spreadsheet simulation, you stop asking “What’s the best investment?” and start asking the question that actually pays: Which distribution of outcomes can I live with, and which gives me the highest chance of meeting my goal?

External Links

Risk vs. Return Explained Like You’re 5 (But Smarter) - YouTube Risk and Return Explained - Financial Edge #KB Risk Analysis — Part 2- Simulation Models | by Prof. Frenzel Investing: Risk and return (article) | Khan Academy Risk-Return Tradeoff: How the Investment Principle Works

External References

• arithmetic return lab
• Volatility drag lab
• diversification lab
• Sequence of returns retirement lab
• Target based investing lab