Learn sensitivity and scenario analysis — Stock-Market notes

Core Concepts

Why One-at-a-Time vs. Multiple Inputs?

Sensitivity analysis isolates pure elasticity: "If WACC alone moves±2%, value moves ±$X." It's diagnostic—you discover that revenue growth dominates your model, so you should spend research time refining that forecast, not obsessing over the tax rate.

Scenario analysis captures correlation: In a recession, revenue growth and margins and default risk all move together. Varying them independently (sensitivity) mises the compound effect. A scenario bundles them into a realistic joint outcome.

Deriving the Sensitivity Framework

Start with a valuation function: $V = f(\theta_1, \theta_2, \ldots, \theta_n)$ where $\theta_i$ are inputs (growth $g$ , discount rate $r$ , terminal margin $m$ , etc.).

First-order sensitivity (elasticity): $\text{Elasticity}_i = \frac{\partial V / V}{\partial \theta_i / \theta_i} = \frac{\theta_i}{V} \cdot \frac{\partial V}{\partial \theta_i}$

WHY this formula? We want percentage change in output percentage change in input—that's the definition of elasticity. For a DCF: $V = \sum_{t=1}^{n} \frac{FCF_t}{(1+r)^t} + \frac{TV}{(1+r)^n}$

If we vary $r$ alone: $\frac{\partial V}{\partial r} = -\sum_{t=1}^{n} \frac{t \cdot FCF_t}{(1+r)^{t+1}} - \frac{n \cdot TV}{(1+r)^{n+1}}$

WHAT this tells us: Each cash flow is weighted by its time-distance; distant cash flows (like terminal value) are hyper-sensitive to $r$ because of the exponential denominator. A 1% higher discount rate compounds over10, 20 years.

HOW to implement:

Fix a base-case $V_0$ with all inputs at best estimates.
Perturb $\theta_i$ by ±10%, ±20% (or relevant ranges).
Recalculate $V$ each time. To show elasticity, plot the normalized change $\Delta V / V$ (fractional change in value) against $\Delta \theta_i / \theta_i$ (fractional change in input). Alternatively, for a tornado chart, plot absolute $\Delta V$ (in dollars) against the percentage change in $\theta_i$ —just be consistent about which axis is normalized.
Rank inputs by absolute impact → tornado chart (bars sorted by magnitude).

Figure — Learn sensitivity and scenario analysis

Scenario Analysis: The Three Worlds

Formula Constructing Scenarios

Identify corelated drivers: revenue growth, operating margin, capex intensity, terminal growth, cost of capital.
Build narrative logic:
- Bull: Strong economy → high growth (↑), expanding margins (↑), low default premium (WACC ↓).
- Base: Moderate growth, stable margins, normal WACC.
- Bear: Recession → low/negative growth (↓), margin compression (↓), rising risk premium (WACC ↑).
Assign probabilities (subjective): e.g., 20% bull, 60% base, 20% bear (these are the same weights used in the worked example below; you choose them to reflect your conviction and they must sum to 100%).
Compute expected value: $E[V] = \sum_{i} P_i \cdot V_i$ where $P_i$ is scenario probability, $V_i$ is intrinsic value in that scenario.

WHY probability-weight? A range alone (" $50 to$ 120") doesn't tell you where to anchor. If the bear case is unlikely and the bull case likely, you should skew toward the bull value. The expected value $E[V]$ is your risk-adjusted anchor.

Worked Example: Valuing a Retailer

Base Case:

Revenue growth: 5%/year for 5 years
EBIT margin: 8%
WACC: 9%
Terminal growth: 2%
Result: $V_0 =$ 75.00$

Sensitivity Table (One-Variable)

| Input | -20% | -10% | Base | +10% | +20% | |----|------|------|------|---| | Revenue growth (5%) | $62.30 |$ 68.50 | $75.00 |$ 82.10 | $89.80 | | EBIT margin (8%) |$ 69.00 | $72.00 |$ 75.00 | $78 |$ 81.00 | | WACC (9%) | $88.40 |$ 81.20 | $75.00 |$ 69.50 | $64.70 | | Terminal growth (2%) |$ 70.20 | $72.50 |$ 75.00 | $77.80 |$ 80.90 |

WHY these ranges? Revenue growth swings $27.50 ($ 89.80 - $62.30), WACC swings$ 23.70—these dominate. EBIT margin swings only $12, so it's less critical. Action: Spend time refining revenue forecasts and assessing business/financial risk (WACC components).

Scenario Analysis

Scenario	Rev Growth	EBIT Margin	WACC	Term Growth	Value	Probability
Bear	1%	6%	11%	1%	$48.20	20%
Base	5%	8%	9%	2%	$75.00	60%
Bull	8%	10%	8%	3%	$108.50	20%

Expected Value: $E[V] = 0.20 \times 48.20 + 0.60 \times 75.00 + 0.20 \times 108.50 = 9.64 + 45.00 + 21.70 = \$ 76.34$$

WHAT this tells us: Despite the wide range ( $48–$ 108), the probability-weighted fair value is $76.34, close to the base case. If the stock trades at$ 82, you're paying above the expected value but within the bull case—acceptable only if you have conviction the bull scenario is underpriced.

Common Mistakes

Practical Workflow

Build your base DCF with best-estimate inputs.
Identify 4–6 key inputs (usually: revenue growth, margin, WACC, terminal growth, capex).
Create sensitivity tables for each input (±20%, ±10%, base, +10%, +20%).
Plot a tornado chart: Rank inputs by $\Delta V$ range.
Construct 3 scenarios:
- Bear Coherent pessimistic story (recession, competition, margin pressure).
- Base: Your original model.
- Bull: Coherent optimistic story (market expansion, pricing power).
Assign probabilities (use market-implied vol, analyst consensus, or judgment; they must sum to 100%).
Calculate $E[V]$ and compare to current price.
Stress-test: "What input changes would make this stock attractive at current price?" (Goal-seek analysis.)

Recall Explain to a 12-Year-Old

Imagine you're guessing how much money your lemonade stand will make this summer. You think you'll sell100 cups at $2 each =$ 200. But what if it's super hot (150 cups) or rainy (50 cups)? What if your costs go up?

Sensitivity analysis is testing one thing at a time: "If I sell 150 cups but everything else stays the same, I make $300." "If my costs double but sales stay 100, I make$ 100." You figure out which guesses matter most.

Scenario analysis is imagining whole weather patterns: "Hot summer = more sales + I can charge more + lower costs (no spoilage)" vs. "Rainy summer = fewer sales + lower prices (discounts) + higher costs (wasted lemons)." You plan for different worlds, not just different numbers.

Connections

Discounted Cash Flow (DCF) Models – The valuation model you're stress-testing
Weighted Average Cost of Capital (WACC) – Often the highest-sensitivity input
Terminal Value Calculation – Dominates DCF sensitivity due to exponential discounting
Monte Carlo Simulation – Advanced extension: vary ALL inputs simultaneously with probability distributions
Option Pricing and Real Options – Captures asymetry (upside optionality) that scenarios approximate
Risk-Adjusted Return Metrics – Scenario probabilities feed into Sharpe ratio, downside deviation
Market-Implied Expectations – Reverse-engineer what scenarios the current stock price assumes

#flashcards/stock-market

What is sensitivity analysis in valuation? :: A one-variable-at-a-time technique that measures how much the intrinsic value changes when you vary a single input (growth, WACC, margin) across a range while holding all other inputs constant. It identifies which assumptions have the highest leverage on your conclusion.

What is scenario analysis?

Building complete alternative states of the world by changing multiple inputs simultaneously in a coherent narrative (Bull/Base/Bear cases). Each scenario produces a distinct valuation, giving you a range and expected value instead of a false-precision point estimate.

Why use tornado charts in sensitivity analysis?

To rank inputs by their absolute impact on valuation (

\Delta V

range). The chart visually shows which variables matter most—focus your research time on refining those high-impact inputs, not polishing low-impact ones.

How do you calculate expected value from scenarios?

E[V] = \sum_{i} P_i \cdot V_i

, where

P_i

is the subjective probability of scenario

i

and

V_i

is the intrinsic value in that scenario. This gives a probability-weighted anchor instead of just a range.

Why can't you just vary WACC and call it complete sensitivity analysis?

Because WACC might not be your highest-sensitivity input. A tornado chart often reveals that revenue growth or terminal growth dominate the variance. The 80/20 rule says focus on the 20% of inputs driving 80% of value swing—polishing a low-impact input is wasted effort.

What's the difference between base case and expected value?

The base case is your modal (most likely single outcome) estimate. The expected value is the probability-weighted average across all scenarios, which differs if the distribution is skewed. Use expected value for decisions, base case for storytelling.

Why do distant cash flows (terminal value) dominate discount rate sensitivity?

Because of exponential compounding:

\frac{TV}{(1+r)^{10}}

vs.

\frac{TV}{(1+r)^{11}}

. A 1% higher

r

gets raised to a large power, drastically shrinking present value. The derivative

\frac{\partial V}{\partial r} \propto -\frac{n \cdot TV}{(1+r)^{n+1}}

shows linear scaling with time

n

What makes a good scenario (vs. random input combinations)?

Coherent narrative logic. A recession scenario should bundle low growth + margin compression + rising WACC together (corelated effects). You can't have "high growth + high unemployment" without explaining the mechanism. Stick to 3–4 plausible stories, not 27 combinations.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Jab bhi aap kisi company ki valuation karte ho—DCF, multiples, jo bhi method—toh ek precise number milta hai: "Share price $87.32 hona chahiye." Par yeh number kitna **paka** hai? Agar apka discount rate assumption 9\% se 10\% ho gaya, toh value crash karke$ 62 ho jayega? Ya phir bas thoda sa budge karega? Sensitivity analysis yeh bata hai ki "Mere kaun se inputs sabse zyada dangerous hain—agar main galat hoon toh valuation kafi badal sakti hai." Aur scenario analysis complete picture banata hai: "Agar recession aye aur margins bhi gir jayein, toh kya hoga?" Bina iske, ap ek single number pe blind faith kar rahe ho, jo ki risky hai.

Sensitivity mein ap ek-ek input change karte ho (jaise revenue growth ko ±20% move karo, baki sab fixed rakho), aur dekhte ho value kitni swing karta hai. Resultek tornado chart hota hai jo dikhata hai ki konse variables high-impact hain. Jaise agar revenue growth ko change karne se value $62 se$ 90 tak jaa sakta hai, lekin tax rate change se sirf $73 se$ 77 tak jata hai, toh matlab revenue forecast pe research time spend karo, tax rate pe nahi.

Scenario analysis thoda alag hai—yahan aap pori kahani banate ho. "Bear case: Recession ayega, competition badhega, margins compress honge, aur risk premium bhi badhega—sab kuch ek sath." "Bull case: Market expand hoga, pricing power milega, efficiency badhegi." Har scenario ka ek value nikalta hai, aur phir probability assign karke expected value calculate karte ho. Yeh approach realistic hai kyunki real life mein ek chez badlne se dosri cheezon pe bhi asar padta hai—sensitivity ki tarah isolated nahi hote. Dono tools milke aapko confidence band dete hain: "Stock $50–$ 110 ke bech hai, lekin expected value $76 hai, toh current price$ 82 pe kharidne ka matlab hai main bull case ka bet laga raha hoon."

Test yourself — Valuation Methods

Connections