Monte Carlo Simulation ?

📊 What's This?

This ultra-advanced Monte Carlo simulation includes professional-grade features:

• 6 Distribution Models: Normal, Student-t, Log-Normal, Jump Diffusion, Regime Switching, GARCH
• 4 Strategies: DCA, Lump Sum, Comparison, Retirement Withdrawal
• Advanced Metrics: CVaR, Maximum Drawdown, Sharpe, Sortino, Calmar Ratios
• Professional Charts: Tornado Diagram, Heat Maps, Correlation Matrix, Efficient Frontier
• Up to 50,000 simulations for maximum accuracy

Available Templates

🛡 Conservative

• Monthly Investment: 500 EUR
• Expected Yield: 0.4% monthly (≈5% annual)
• Volatility: 8% (low risk)
• Ideal for: Risk-averse investors, near retirement

⚖ Balanced

• Monthly Investment: 300 EUR
• Expected Yield: 0.58% monthly (≈7% annual)
• Volatility: 12% (moderate risk)
• Ideal for: Diversified 60/40 portfolio

📈 Aggressive

• Monthly Investment: 200 EUR
• Expected Yield: 0.83% monthly (≈10% annual)
• Volatility: 18% (high risk)
• Ideal for: Long-term growth, young investors

🏖 Retirement

• Accumulation: 15 years @ 400 EUR/month
• Withdrawal: 4% annual rule
• Volatility: 10%
• Ideal for: Retirement planning

₿ Crypto

• Monthly Investment: 100 EUR
• Expected Yield: 1.5% monthly (≈20% annual)
• Volatility: 60% (extreme risk)
• Distribution: Jump Diffusion (captures crashes)

Why Different Distributions?

Real markets don't follow perfect bell curves. Advanced models capture fat tails (crashes), jumps (sudden shocks), and regime changes (bull/bear cycles).

1⃣ Normal (Gaussian)

Classic assumption: returns are symmetrically distributed around the mean.

• ✅ Simple, widely used
• ❌ Underestimates extreme events (2008 crash, COVID)

2⃣ Student-t

Heavier tails → captures rare but severe crashes.

• Parameter: Degrees of freedom (df). Lower df = fatter tails.
• ✅ More realistic for equities
• Recommended df: 5-10

3⃣ Log-Normal

Returns are skewed to the right. Common in finance since prices can't go below zero.

• ✅ Realistic for stock prices
• ✅ Prevents negative values

4⃣ Jump Diffusion (Merton Model)

Combines smooth drift + sudden jumps (e.g., market crashes).

• Parameters: Jump intensity (λ), jump magnitude
• ✅ Models flash crashes, circuit breakers

5⃣ Regime Switching

Markets alternate between bull (high return, low vol) and bear (low/negative return, high vol).

• ✅ Captures economic cycles
• Uses Markov chain for state transitions

6⃣ GARCH

Volatility changes over time (volatility clustering).

• ✅ Captures periods of high/low volatility
• ✅ More realistic than constant volatility

Save & Compare Simulations

The Scenario Manager lets you:

• Save up to 10 different simulation scenarios in Firebase
• Load saved scenarios to restore all parameters
• Compare 2-5 scenarios side-by-side in interactive charts
• Name each scenario for easy reference
• Delete unwanted scenarios

How to Load a Scenario

1. Click the "Load" button on any saved scenario card
2. All parameters (distribution, strategy, values) will be restored
3. Click "Run Simulation" to reproduce the exact same results

How to Compare Scenarios

1. Run 2-5 different simulations (e.g., Conservative vs Aggressive)
2. Click "Save Scenario" for each one
3. Check the boxes next to scenarios you want to compare
4. Click "Compare Scenarios"
5. A detailed comparison modal will open with charts and statistics

📈 What Is Dollar-Cost Averaging (DCA)?

Investing a fixed amount regularly (e.g., 200 EUR/month) regardless of market conditions.

✅ Advantages

• Reduces timing risk: You buy more when prices are low, less when high
• Discipline: Automates investing, removes emotions
• Accessible: No need for large lump sum
• Compound growth: Early investments grow the longest

📊 Example

Investing 200 EUR/month for 25 years at 8% annual return:

• Total Invested: 60,000 EUR
• Expected Value: ~200,000 EUR
• Gain: ~140,000 EUR (233% return)

📐 Definition

The Expected Yield is the average return you anticipate per month. It's the central tendency around which random returns fluctuate.

📈 Common Values

• 0.4%/month ≈ 5% annual (Conservative)
• 0.58%/month ≈ 7% annual (Balanced)
• 0.67%/month ≈ 8% annual (S&P 500 historical)
• 0.83%/month ≈ 10% annual (Aggressive)

📊 Definition

Volatility measures how much returns fluctuate around the average. High volatility = big swings up and down. Low volatility = stable returns.

📈 Typical Values (Annual)

• Cash/Bonds: 2-6%
• Balanced Portfolio (60/40): 10-12%
• Global Stocks (S&P 500): 15-20%
• Emerging Markets: 25-35%
• Cryptocurrencies: 60-100%+

⚠ Impact on Results

Higher volatility means:

• Wider range of possible outcomes
• Higher chance of extreme losses (and gains)
• Lower probability of hitting specific targets
• Higher risk-adjusted returns needed (Sharpe Ratio)

🎯 Why Multiple Simulations?

Each simulation is one possible future. Running many simulations creates a probability distribution of outcomes, giving you confidence intervals.

📊 How Many Do You Need?

• 100-500: Quick test, rough estimate (±5% accuracy)
• 1,000: Standard (±2% accuracy) → Recommended
• 5,000: High precision (±1% accuracy)
• 10,000: Professional use (±0.5% accuracy)
• 50,000: Academic/research (±0.2% accuracy)

⚡ Performance Impact

Higher simulation counts take longer to compute:

• 1,000 sims: ~1-2 seconds
• 10,000 sims: ~5-10 seconds
• 50,000 sims: ~30-60 seconds

What Is It?

The Degrees of Freedom (DF) is a parameter that controls the "thickness" of the tails in the Student-t distribution.

Why Student-t Instead of Normal?

Real financial markets have fat tails (more extreme events than the normal distribution predicts):

• Normal distribution underestimates crashes (2008, COVID-19)
• Student-t distribution captures extreme moves better
• More realistic for equity and crypto markets

Common DF Values & Their Meaning

DF = 2-3 (Extreme Fat Tails)

• Extremely heavy tails
• Use for: Cryptocurrencies, emerging markets
• Frequent large swings (±20% moves common)
• ⚠ High uncertainty

DF = 4-6 (Moderate Fat Tails) ⭐ RECOMMENDED

• Good balance for stocks
• Use for: Equities, ETFs, moderate portfolios
• Captures 2008-style crashes without being too extreme
• ✅ Default choice for most simulations

DF = 7-10 (Slightly Fat Tails)

• Closer to normal but still realistic
• Use for: Diversified portfolios, blue-chip stocks
• Conservative approach

DF = 30+ (Essentially Normal)

• Behaves like a normal distribution
• Use for: Bonds, cash-heavy portfolios
• Underestimates tail risk

Visual Comparison

How to Choose DF

1. For stocks/ETFs: Use DF = 5 (default)
2. For crypto: Use DF = 3
3. For bonds: Use DF = 10-15 or Normal
4. For diversified portfolios: Use DF = 6-8

Technical Note

The Student-t distribution was discovered by William Sealy Gosset (pen name "Student") in 1908 while working at Guinness Brewery. It's one of the most important distributions in statistics!

Warning Signs You Need Student-t

• Your asset has experienced sudden crashes
• Historical returns show outliers beyond ±3σ
• You want to stress-test for tail risk
• You're investing in volatile sectors (tech, energy, crypto)

What Is It?

The Jump Intensity (λ) is the average number of sudden jumps (crashes or spikes) per year in the Jump Diffusion model.

The Jump Diffusion Model (Merton 1976)

Financial assets don't move smoothly. They experience:

• Normal drift: Gradual day-to-day movements
• Jumps: Sudden large moves (crashes, flash rallies)

What Does λ Mean?

λ = 0.1 (Low Jump Frequency)

• Meaning: 1 jump every 10 years on average
• Use for: Stable markets, blue-chip stocks
• Example: Large-cap indices (S&P 500)

λ = 0.2-0.3 (Moderate) ⭐ RECOMMENDED

• Meaning: 1 jump every 3-5 years
• Use for: General equities, growth stocks
• Realistic for most markets
• Captures occasional crashes (2008, 2020, etc.)

λ = 0.5-0.7 (High Jump Frequency)

• Meaning: 1 jump every 1.5-2 years
• Use for: Small-caps, emerging markets
• Volatile sectors (biotech, energy)

λ = 1.0+ (Extreme Volatility)

• Meaning: ≥ 1 jump per year
• Use for: Cryptocurrencies, penny stocks
• ⚠ Very high uncertainty

Real-World Examples

How Jumps Are Simulated

In each month of simulation:

1. Calculate probability of jump: P = λ / 12 (monthly)
2. Roll a random number (0 to 1)
3. If random < P, a jump occurs
4. Add the jump magnitude to that month's return

Choosing the Right λ

Combined with Jump Magnitude

Jump Intensity (λ) and Jump Magnitude work together:

• High λ + Small magnitude: Frequent small jumps
• Low λ + Large magnitude: Rare but severe crashes (⚠ most realistic)
• High λ + Large magnitude: Extremely volatile (crypto-like)

What Is It?

The Jump Magnitude is the average size of sudden moves (in percentage) when a jump occurs in the Jump Diffusion model.

Understanding the Parameter

• Negative value: Average jump is a crash (down move)
• Positive value: Average jump is a spike (up move)
• Zero: Jumps can be up or down equally

Common Jump Magnitude Values

-5% to -8% (Moderate Corrections)

• Use for: Stable stocks, diversified portfolios
• Represents: Standard market corrections
• Real example: Tech selloffs, profit-taking

-10% to -15% (Significant Crashes) ⭐ RECOMMENDED

• Use for: General equities, growth stocks
• Represents: Major market events
• Real examples:
• 2020 COVID initial drop: -12%
• 2008 Lehman: -9%
• Flash crashes: -10% to -15%

-20% to -30% (Severe Crashes)

• Use for: High-risk assets, emerging markets
• Represents: Black Swan events
• Real example: 1987 Black Monday (-20%)

-40%+ (Catastrophic Events)

• Use for: Cryptocurrencies, extremely volatile assets
• Represents: Wipeout scenarios
• Real examples: Crypto crashes (BTC -50% days)

Positive Jumps (Rally Spikes)

You can also model positive jumps for sudden rallies:

+5% to +10% (Strong Rallies)

• Use for: Growth stocks, tech sector
• Real examples:
• 2020 vaccine rally: +9%
• Fed pivot rallies: +5% to +8%

+15%+ (Extreme Bull Moves)

• Use for: Meme stocks, crypto
• Real examples: GME (+100% days), crypto pumps

Historical Market Jumps

Asymmetric Jumps (Realistic Approach)

Markets tend to:

• Crash faster than they rally ("Stairs up, elevator down")
• Have larger negative jumps than positive
• This is called negative skewness

Calibrating Jump Magnitude

Method 1: Historical Data

1. Find the 5 largest down days for your asset
2. Calculate the average of these moves
3. Use this as your Jump Magnitude

Method 2: Stress Scenario

1. Ask: "What's a realistic worst-case single-day drop?"
2. For stocks: -10% to -15%
3. For crypto: -30% to -50%
4. For bonds: -2% to -5%

Combined with Jump Intensity

Advanced: Variable Jump Magnitude

In reality, not all jumps are the same size. The model assumes:

• Jump size follows a normal distribution
• Your input is the mean of this distribution
• Actual jumps vary around this mean

Recommended Settings by Asset Class

• S&P 500 / Large-Cap ETFs: -10% (realistic crashes)
• Tech / Growth Stocks: -12% to -15% (higher volatility)
• Small-Caps: -15% to -20% (wild swings)
• Emerging Markets: -20% to -25% (geopolitical risk)
• Bitcoin / Crypto: -30% to -50% (extreme moves)
• Bonds: -3% to -5% (stable, low jumps)

Testing Your Settings

After running the simulation, check:

1. Maximum Drawdown: Does it match historical crashes?
2. Worst Case scenario: Is it realistic or too extreme?
3. Frequency of large drops: Compare to actual market history

📈 Key Metrics Explained

1⃣ Median (50th Percentile)

The middle value when all outcomes are sorted. 50% of simulations are above this, 50% below.

2⃣ 10th Percentile (Pessimistic)

Only 10% of scenarios are worse than this. Think of it as a "bad luck" scenario.

3⃣ 90th Percentile (Optimistic)

Only 10% of scenarios are better than this. Think of it as a "good luck" scenario.

4⃣ Average

The arithmetic mean of all outcomes. Can be skewed by extreme values.

5⃣ Best/Worst Case

The extreme outcomes. Useful for understanding the full range of possibilities.

📊 Interpreting the Range

The difference between P10 and P90 shows your uncertainty:

• Narrow range (P90/P10 < 2x): Predictable outcome
• Wide range (P90/P10 > 3x): High uncertainty

📊 Key Risk Indicators

1⃣ Value at Risk (VaR) 5%

VaR 5% = The 5th percentile outcome. 95% of scenarios are better than this.

Example: VaR 5% = 50,000 EUR means there's only a 5% chance you'll end up with less than 50,000 EUR.

2⃣ Loss Probability

% of simulations where you end up with less than you invested (no gains, or losses).

Lower is better. Ideally <5%.

3⃣ Average/Median Return

Your expected return as a percentage of total invested capital.

Professional Risk Measures

1⃣ CVaR (Conditional Value at Risk)

Also called Expected Shortfall. Average loss in the worst 5% of cases.

More conservative than VaR (used by Basel III banking regulations).

2⃣ Maximum Drawdown

Largest peak-to-trough decline during the investment period.

Shows the worst loss you'd experience before recovery.

3⃣ Sharpe Ratio

Risk-adjusted return. Higher is better (>1 is good, >2 is excellent).

4⃣ Sortino Ratio

Like Sharpe Ratio but only penalizes downside volatility.

5⃣ Calmar Ratio

Annual return divided by maximum drawdown. Higher is better.

6⃣ Skewness & Kurtosis

• Skewness: Asymmetry (negative = more left-tail risk)
• Kurtosis: Tail thickness (>3 = fat tails, more extreme events)

Definition

Maximum Drawdown (MDD) is the largest percentage drop from a peak to a trough in your portfolio value.

Why It Matters

• Psychological Test: Can you stomach a -30% drop?
• Recovery Time: How long to get back to break-even?
• Strategy Viability: Too deep = strategy may fail

Example

What Is It?

A tornado diagram shows which input parameters have the biggest impact on your final portfolio value.

How to Read It

• Each bar shows the effect of ±10% change in that parameter
• Longer bars = more sensitive (higher impact)
• Bars are sorted by impact (largest first)
• Red = decrease (-10%), Green = increase (+10%)

Why It's Useful

• Identify key drivers of portfolio success
• Focus optimization efforts on high-impact parameters
• Understand which assumptions matter most

What Is It?

A heat map visualizes portfolio outcomes across different combinations of Expected Return (X-axis) and Volatility (Y-axis).

Color Legend

• 🔴 Red: Low portfolio values (worse outcomes)
• 🟡 Yellow: Moderate values
• 🟢 Green: High portfolio values (better outcomes)

How to Use It

• Find your current parameters on the map
• Explore "what-if" scenarios by moving to different cells
• Identify the optimal risk/return balance

What Is It?

A correlation matrix shows how different metrics are related to each other. Values range from -1 to +1.

How to Read It

• +1 (Green): Perfect positive correlation. When one goes up, the other goes up.
• 0 (White): No correlation. Metrics are independent.
• -1 (Red): Perfect negative correlation. When one goes up, the other goes down.

Metrics Analyzed

• Final Value: Your portfolio at the end of the simulation
• Max Drawdown: Largest peak-to-trough decline
• Volatility: How much returns fluctuate
• Sharpe Ratio: Risk-adjusted return

Why It's Useful

• Identify hidden relationships between risk metrics
• Understand which factors move together
• Validate your risk management assumptions

What Is It?

The Efficient Frontier is a curve showing the best possible return for each level of risk (volatility).

Key Concept

Invented by Harry Markowitz (Nobel Prize 1990), the Efficient Frontier represents portfolios that offer:

• Maximum return for a given level of risk
• Minimum risk for a given level of return

How to Read the Chart

• X-axis: Volatility (Risk) in %
• Y-axis: Annual Return in %
• The Curve: Points on the curve are "efficient" (optimal risk/return)
• Below the Curve: Suboptimal portfolios (can do better)

The Sweet Spot

Practical Use

• Find your risk tolerance on the X-axis
• Read the corresponding expected return on the Y-axis
• Portfolios to the right = higher risk, higher potential return
• Portfolios to the left = lower risk, lower return

What Is It?

Stress testing simulates how your portfolio would perform under extreme market conditions (crashes, recessions, bull markets).

The 4 Scenarios

1⃣ Crisis (-40%)

Severe market crash (like 2008 Financial Crisis or COVID-19 initial shock).

• Your portfolio value drops by 40%
• Tests worst-case resilience
• Historical example: S&P 500 fell 57% in 2008-2009

2⃣ Recession (-20%)

Moderate economic downturn (typical recession).

• Portfolio value drops by 20%
• More common than severe crises
• Historical example: 2022 bear market (-25%)

3⃣ Normal (0%)

Baseline scenario with no additional shock.

• Your expected median outcome
• Reference point for comparison

4⃣ Bull Market (+20%)

Strong economic growth (like 2020-2021 recovery).

• Portfolio value increases by 20%
• Upside potential scenario
• Historical example: 2019 (+31%), 2021 (+27%)

How to Use It

Why It Matters

• Risk Management: Know your downside before it happens
• Peace of Mind: Reduce panic-selling during crashes
• Strategic Planning: Adjust asset allocation if stress results are unacceptable

Regulatory Use

Banks and institutions are required by law (Basel III, Dodd-Frank) to perform stress tests to ensure financial stability. You're using the same professional techniques!

What Is It?

This chart shows how long it takes (in months/years) to reach your target portfolio value across all simulations.

How to Read It

• X-axis: Number of months
• Y-axis: Frequency (how many simulations)
• Peak of the curve: Most likely time to reach your goal
• Subtitle shows: Success rate & median time

Key Metrics

Success Rate

Percentage of simulations that reached the target within the time horizon.

Median Time

The middle value: 50% of successful simulations reached the target faster, 50% slower.

Interpreting the Shape

• Narrow peak: Predictable timeline (low uncertainty)
• Wide spread: High uncertainty (depends heavily on market conditions)
• Skewed right: Some simulations take much longer (tail risk)

What If Target Is Not Reached?

If many simulations don't reach your target, you can:

• Increase monthly investment amount
• Extend time horizon
• Lower target value
• Accept higher risk (higher expected return)

Real-World Example

What Is Sharpe Ratio?

The Sharpe Ratio measures risk-adjusted return. It answers: "How much return am I getting per unit of risk?"

Sharpe Ratio Scale

• < 0: Losing money (bad)
• 0 to 1: Suboptimal (could do better)
• 1 to 2: Good risk-adjusted returns
• 2 to 3: Excellent (top-tier funds)
• > 3: Outstanding (rare, verify if sustainable)

What Is "Rolling" Sharpe?

Instead of calculating one Sharpe Ratio for the entire period, we calculate it over moving 12-month windows.

Why Rolling?

• Shows how risk-adjusted performance changes over time
• Reveals periods of strong vs. weak performance
• Helps identify trends and regime changes

How to Read the Chart

• X-axis: Month number
• Y-axis: Sharpe Ratio
• Line going up: Improving risk-adjusted returns
• Line going down: Deteriorating performance (more risk for same return)

What to Look For

• Stable line around 1-2: Consistently good risk management
• Volatile line: Performance varies significantly (market-dependent)
• Downward trend: May indicate strategy decay or regime change
• Recovery after dips: Resilient strategy

Professional Use

Hedge funds and institutional investors use rolling Sharpe Ratio to:

• Monitor strategy performance in real-time
• Identify when to rebalance portfolios
• Compare different investment strategies
• Communicate risk-adjusted returns to clients

Limitations

• Assumes returns are normally distributed (not always true)
• Doesn't capture tail risk (extreme events)
• Sensitive to the chosen time window (12 months in our case)
• Historical metric (past performance ≠ future results)