Scenario analysis · RiskAtoms Academy

By the end you will be able to

1Tell historical scenarios from hypothetical ones, and give a use for each.
2Build a scenario whose moves are consistent with each other.
3Revalue a portfolio under a scenario, position by position, and compare with the sensitivity estimate.
4Explain what a scenario result tells you and what it does not.

Why it matters

The fund's worst 10 trading days in our data, days 385 to 394, cost it 5.93%: about $593,314 on $10,000,000. A board member asks the obvious next question: “And what if something worse happens?” Volatility and correlation cannot answer that. A scenario can.

Core explanation

What if this happened tomorrow?

A scenario is a short list of moves in risk factors applied together: equities down 15%, yields up 150 bp, option volatility up 8 points. You apply them to today's positions and ask what the portfolio would be worth. The answer is a single loss figure, broken down by position, that anyone in the room can understand.

Scenarios come from two places:

Historical

Replay a real period

Take the moves from a real crisis, such as 19 October 1987, September 2008 or March 2020, and apply them to today's positions. Nobody can say it could never happen: it did. But the next crisis will not be a copy of the last.

Hypothetical

Write a story

Imagine a plausible event, such as an inflation shock, a sovereign default or a cyber attack, and decide how each risk factor would move. It can cover what has never happened, but it rests on judgement.

Sort itHistorical or hypothetical?

5 items

Apply the moves of 19 October 1987, when the Dow Jones Industrial Average fell 22.6% in a day.
A large euro area government misses a bond payment and its yields jump 400 bp.
Replay the weeks after Lehman Brothers filed for bankruptcy on 15 September 2008.
A cyber attack shuts the main payment system for three days.
Replay days 385 to 394 of the fund's own data.

Consistency matters. The moves in a scenario must hang together. When equities crash, option volatility rises, and investors often run to government bonds so yields fall. A scenario that crashes equities but lowers volatility will understate what put options earn. And every factor the scenario leaves unchanged is also an assumption: “the dollar stays put” is a statement about the world, not a neutral default.

Your callDoes the scenario hang together?

0 of 3

Equities fall 30% and option volatility falls 5 points.
Equities fall 10%, government bond yields fall 50 bp and corporate credit spreads widen 80 bp.
A UK inflation shock: gilt yields rise 150 bp, while sterling, UK shares and credit spreads stay exactly where they are.

Revalue, do not just multiply. Sensitivities (L2.5.04: Sensitivities) give a quick first answer, but scenarios are about large moves, exactly where straight lines fail. So scenario analysis normally uses full revaluation: each position is repriced with its own model at the new market levels. Scenarios are usually instantaneous: the moves arrive at once and nobody gets to trade out.

Illustrative dataThe Module 2.5 data set

500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.

In this lesson the bond fund behaves like a 7-year 4% bond, and the optional hedge is the 12 three-month index put contracts from L2.5.04: Sensitivities, costing $77,215.

The formal version

Write the portfolio as positions $V_1, \dots, V_n$ , each a function of the risk factors $\mathbf{x}$ (index level, yield, volatility). A scenario is a vector of moves $\Delta\mathbf{x}$ . Its P&L under full revaluation, and the sensitivity estimate it replaces, are:

\begin{aligned} \text{P\&L}_{\text{full}} &= \sum_{i=1}^{n} \big[\, V_i(\mathbf{x} + \Delta\mathbf{x}) - V_i(\mathbf{x}) \,\big] \\[4pt] \text{P\&L}_{\text{sens}} &\approx \sum_{i=1}^{n} \sum_{k} \frac{\partial V_i}{\partial x_k}\,\Delta x_k \end{aligned}

The two agree for small moves. The gap between them is the curvature that sensitivities leave out, plus any factor (here, volatility) the sensitivity estimate does not include.

Conventions on this page

Moves are applied instantaneously to today's positions, with no time decay and no trading. Equity moves are percentage changes in the index; rate moves are parallel shifts in yield in basis points; volatility moves are in percentage points of implied volatility. A loss is shown as a negative P&L. A scenario carries no probability.

Try it

The builder opens on the inflation shock without the hedge. Switch the puts on and watch their row. Then try the flight to quality, where bonds help, and the historical replay. Finally build your own and ask whether it hangs together.

Try itScenario builder

Start from a scenario

Inflation surprises badly: central banks are expected to raise rates hard, yields jump 150 bp, equities fall 15% and volatility rises 8 points.

Equity index−15.0%

Bond yields+150 bp

Option volatility+8 pts

Position	Value now	After the scenario	P&L	Sensitivity estimate
Equity index fund	$6,000,000	$5,100,000	−$900,000	−$900,000
Bond fund	$4,000,000	$3,659,022	−$340,978	−$360,123
Total	$10,000,000	$8,759,022	−$1,240,978	−$1,260,123

P&L as a share of the $10,000,000 fund: −12.4%. Full revaluation reprices the bond fund (a 7-year 4% bond) and the puts (Black-Scholes, new index level and volatility). The sensitivity estimate uses DV01, beta and delta only, so it ignores convexity, gamma and the volatility move.

Worked example

The inflation shock

Hypothetical scenario: equities −15%, yields +150 bp, option volatility +8 points. Revalue the fund without and then with the puts. Try each step before revealing it.

Revaluing the fund

Step 1 of 5

1
Equity index fund
$\$6{,}000{,}000 \times (-0.15) = -\$900{,}000$ . An index fund is linear, so full revaluation and the beta estimate agree.

All three presets, full revaluation:

Scenario	Equity	Yields	Vol	Without puts	With puts
Flight to quality (hypothetical)	−10.0%	-50 bp	+10	−$477,709	−$110,027
Inflation shock (hypothetical)	−15.0%	+150 bp	+8	−$1,240,978	−$687,982
Days 385–394 replayed (historical)	−10.1%	-6 bp	+10	−$591,563	−$220,292

The ranking is the lesson. The flight to quality costs $477,709 without the hedge because falling yields lift the bond fund while shares drop. The inflation shock costs $1,240,978: the shares fall further and the bonds fall with them. The historical replay of days 385 to 394 ($591,563) could never have shown that, because in our data yields fell (-6 bp) on those days. History tells you what has happened; only a hypothetical scenario can show you what has not.

Regulatory lens

Regulators write scenarios too. The Solvency II standard formula for European insurers sets its equity capital charge with an instantaneous scenario: listed equities in developed markets fall 39%, other equities 49%, each adjusted by up to 10 points up or down by a “symmetric adjustment” that depends on recent market moves. Interest rates, spreads and currencies have their own prescribed shocks (M4.6: Solvency II and insurance regulation). Banks run scenario analysis as part of their own capital assessment, and supervisors publish common scenarios for system-wide stress tests (L2.5.09: Stress testing). The detail is in Level 4: Regulation, Capital & Accounting.

Sparks

SparkShell and the oil shock

In the early 1970s a planning team at Royal Dutch Shell led by Pierre Wack wrote scenarios for the oil market, including one in which producers cut supply and prices jumped. When the oil embargo of October 1973 came, Shell's managers had already thought through what to do. Wack told the story in the Harvard Business Review in 1985, and scenario planning spread from there.

SparkThe scenario nobody ran, 1994

In February 1994 the US Federal Reserve began raising its policy rate from 3%. By February 1995 it was 6%. Bond prices fell around the world, and investors who had assumed rates would stay low, including Orange County (L2.5.04: Sensitivities), took large losses. A simple “rates up 300 bp” scenario would have shown the danger in advance.

Key takeaways

A scenario applies a set of risk-factor moves together to today's positions and asks: how much would we lose?
Historical scenarios replay real periods; hypothetical ones are stories built by judgement. Use both.
Moves must be consistent with each other, and factors left unchanged are assumptions too.
Large moves need full revaluation: sensitivities miss curvature and can badly understate option hedges.
A scenario tells you “how much if”, never “how likely”. That is what value at risk adds.

Reaction

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Glossary

Scenario: A set of moves in risk factors applied together to today's portfolio, answering “what would we lose if this happened?”
Historical scenario: A scenario that replays the market moves of a real past period on today's positions.
Hypothetical scenario: A scenario built by judgement: moves that have not happened together, but plausibly could.
Full revaluation: Repricing every position with its full pricing model at the scenario's market levels, rather than estimating from sensitivities.
Instantaneous shock: A scenario applied as if all the moves happened at once, today, with no time to trade or rebalance.
Consistent scenario: A scenario whose moves hang together economically, including the factors it leaves unchanged.

Deep Dives & Origins

Deep Dives

J. C. Hull, Risk Management and Financial Institutions, Wiley, the chapter on stress testing. Intermediate.
R. Rebonato, Coherent Stress Testing: A Bayesian Approach, Wiley, 2010. Advanced.

Origins

P. Wack, “Scenarios: uncharted waters ahead”, Harvard Business Review, September–October 1985.
Commission Delegated Regulation (EU) 2015/35 (Solvency II), Article 169: the standard equity risk shocks.
Board of Governors of the Federal Reserve System, FOMC statements, February 1994 to February 1995.

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