mathematics

Profit at risk

Profit at risk is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Profit at risk rather than just read about it. In short: Profit-at-Risk (PaR) is a risk management quantity most often used for electricity portfolios that contain some mixture of generation assets, trading contracts and end-user consumption. It is used to provide a measure of the downside risk to profitability of a portfolio of physical and financial assets, analysed by time periods in which the energy is delivered.

Key takeaways

Profit at risk belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Profit at risk to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Profit at risk from memory before moving on to harder problems.

Reference excerpt

Profit-at-Risk (PaR) is a risk management quantity most often used for electricity portfolios that contain some mixture of generation assets, trading contracts and end-user consumption. It is used to provide a measure of the downside risk to profitability of a portfolio of physical and financial assets, analysed by time periods in which the energy is delivered. For example, the expected profitability and associated downside risk (PaR) might be calculated and monitored for each of the forward looking 24 months. The measure considers both price risk and volume risk (e.g. due to uncertainty in electricity generation volumes or consumer demand). Mathematically, the PaR is the quantile of the profit distribution of a portfolio.

Example If the confidence interval for evaluating the PaR is 95%, there is a 5% probability that due to changing commodity volumes and prices, the profit outcome for a specific period (e.g. December next year) will fall short of the expected profit result by more than the PaR value. Note that the concept of a set 'holding period' does not apply since the period is always up until the realisation of the profit outcome through the delivery of energy. That is the holding period is different for each of the specific delivery time periods being analysed e.g. it might be six months for December and therefore seven months for January.

History The PaR measure was originally pioneered at Norsk Hydro in Norway as part of an initiative to prepare for deregulation of the electricity market. Petter Longva and Greg Keers co-authored a paper "Risk Management in the Electricity Industry" (IAEE 17th Annual International Conference, 1994) which introduced the PaR method. This led to it being adopted as the basis for electricity market risk management at Norsk Hydro and later by most of the other electricity generating utilities in the Nordic region. The approach was based on monte-carlo simulations of paired reservoir inflow and spot price outcomes to produce a distribution of expected profit in future reporting periods. This tied directly with the focus of management reporting on profitability of operations, unlike the Value-at-Risk approach that had been pioneered by JP Morgan for banks focused on their balance sheet risks.

Critics As is the case with Value at Risk, for risk measures like the PaR, Earnings-at-Risk (EaR), the Liquidity-at-Risk (LaR) or the Margin-at-Risk (MaR), the exact risk measures implementation rule vary from firm to firm.

See also Value at risk Margin at risk Liquidity at risk

References

Worked examples

Example 1 — a first encounter with Profit at risk

Start with the simplest possible case. Write down what Profit at risk claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Profit at risk before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Profit at risk ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Profit at risk

In research: Profit at risk appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Profit at risk in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Profit at risk is common in secondary-school and first-year university syllabi. It links to neighbouring topics Financial risk management, Mathematical finance, Monte Carlo methods in finance, so understanding it makes those chapters shorter.
In everyday life: Look for Profit at risk outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Profit at risk in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Profit at risk means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Profit at risk out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Profit at risk in simple terms?

Why does Profit at risk matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Profit at risk?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Profit at risk.