Imagine you’re sailing a financial ship through unpredictable markets — sunshine one moment, a storm the next. How do you estimate how bad the next wave could be? Value at Risk $($VaR$)$ is your early-warning radar. It answers the critical question: “What’s the most I could lose, with a certain confidence, over a specific time period?” Whether you’re a seasoned trader or just someone trying not to sink their savings, VaR gives a statistical snapshot of potential losses — like a financial weather report that warns: “Take cover, you might maximum lose \$1 million with 95% probability.”
VaR isn’t about predicting your worst day — it’s about setting expectations for most days. Saying ‘you’ll lose $1 million’ is misleading; what it really means is: with 95% confidence, your losses won’t exceed $1 million — but there’s still a 5% chance they might exceed $1 million.
To navigate this risk-filled ocean, we have three trusty compasses — Historical, Parametric $($Normal$)$, and Lognormal approaches. Each one offers a unique way to map out potential danger zones. So, let’s dive into each and see how they help keep your portfolio afloat when markets get rough.
📚 The Historical Simulation – Grandma’s Recipe for Risk
You know how grandma never measures anything — she just knows that three spoonfuls of sugar make it perfect? The historical simulation approach to VaR is a lot like that. It doesn’t assume anything fancy. No equations. Just pure, “let me see what happened before and assume it’ll happen again.”
All you need is:
- A list of past returns $($hopefully not from 2008$)$,
- A confidence level $($like 95%$)$,
- And a calculator that won’t judge you for scrolling past 999 rows in Excel.
Here’s how it goes:
- Line up all return observations from worst to best — the horror to the hero.
- Pick a significance level $($$\alpha$$)$, like 5%, and compute the index:
- $\text{Index} = (\alpha \cdot n) + 1$
- If $n = 1000$, the 51st $($5% of 1000$)$ worst return tells you: “Buddy, this is the loss you could face once every 20 months.”
Suppose the 51st return is $-$15.5
$1,000,000 \cdot($\$15.5%$)$=\$155,000
This is the amount your portfolio could lose in a month if luck hits snooze.
Historical VaR is like checking last year’s weather to decide whether to carry an umbrella today. It might rain, or you might just look ridiculous.
🧮 Parametric VaR – The Suit-and-Tie Statistician
Historical VaR is the guy in flip-flops showing you photos from last summer. Parametric VaR walks in wearing a tailored suit, holding a bell curve, and says, “Let’s assume your returns are normally distributed.”
He doesn’t care about your messy history — he believes in clean equations:
$VaR=−μ+σ⋅z_α$
Where:
- $\mu$ = expected return $($your optimistic friend$)$
- $\sigma$ = standard deviation $($your moody cousin$)$
- $z_\alpha$ = the z-score for your chosen confidence level
If your expected annual return is \$15M and your volatility is \$10M:
- At 95% confidence: VaR
- VaR=−15+10⋅1.65=1.5M
That means: “With 95% confidence, your losses won’t exceed \$1.5 million.” $($And with 5% confidence, you’re on your own.$)$
Parametric VaR trusts the model. He believes the past is noise and the future is a smooth curve with predictable bumps — even if those bumps sometimes turn out to be mountains.
🌀 Lognormal VaR – The Optimist Who Thinks Prices Can’t Go Below Zero
Imagine someone who says, “Yes, stocks can go up 200%, but they can’t go below zero, right?” That’s the lognormal believer.
Lognormal VaR models asset prices, not just returns. Because asset prices don’t do negative numbers — unless you count oil futures in 2020, but let’s not open that can of crude.
If geometric returns are normally distributed:
$\text{VaR}_{\text{lognormal}} = P_{t-1} \cdot \left(1 – e^{\mu_R – \sigma_R z_\alpha} \right)$
This approach is more realistic for long-term holdings, because it reflects the fact that your asset isn’t going to magically become worth \$-100. $($Unless it’s your old Blackberry stock.$)$
🥊 The Final Duel: Historical vs Parametric
| Attribute | Historical VaR | Parametric VaR |
|---|---|---|
| Based on actual returns | ✅ Yes, full of scars and glory | ❌ No, lives in a statistical fantasy |
| Easy to implement | ✅ Even Excel agrees | ⚠️ Needs model assumptions |
| Handles fat tails | ✅ Naturally shows them | ❌ Not without special tweaks |
| Reacts to new regimes | ❌ Still quoting data from 2002 | ✅ Flexible, if you tweak assumptions |
Historical VaR is the street-smart survivor. Parametric VaR is the polished analyst with a calculator for a brain. They both predict trouble, but they don’t always agree on when and how much.
🎯 So, What’s the Moral of This Tale?
VaR is not a psychic. It won’t tell you when the storm hits, but it shows how wet you might get if it does.
Use Historical VaR when you trust your past more than your models.
Use Parametric VaR when you believe math has the answer to everything — except love.
But remember: whether you look back or run simulations forward, neither can protect you from the risk of your intern trading options on a Friday.