Pricing excess layers in casualty insurance and reinsurance requires precision, consistency and an understanding of how losses behave at different policy limits.

Increased Limits Factors (ILFs) are one of the fundamental tools actuaries use to achieve this – and behind them sits an elegant mathematical concept known as the power curve.

In this guide, we’ll explore what ILF curves are, how they work, and why power curves remain one of the most practical and widely used methods for excess pricing in insurance.

What you will learn:

• What Is an ILF in Insurance?
• Different Approaches to Excess Pricing
• From Exposure Curves to ILF Curves
• The Challenges of Building ILF Models
• Introducing Power Curves
• The Mathematics Behind the Power Curve (Simplified)
• Key Advantages of Power Curves
• Practical Application: Using ILF Curves in Casualty Pricing
• Limitations and Considerations
• Real-World Example: Layer Stacking in Reinsurance
• Why ILF Curves Still Matter in a Modern, Data-Driven Market

What Is an ILF in Insurance?

An Increased Limits Factor (ILF) is a multiplier used by insurers to estimate how expected losses change as the policy limit increases.

In simple terms, ILFs show the relationship between coverage limit and loss cost.

For example, an insured buying a £5 million limit instead of £1 million would not expect to pay five times the premium – the increase in loss potential isn’t linear. ILFs help insurers and reinsurers understand how much additional premium is justified for each incremental limit.

ILFs form the foundation of excess pricing models, especially in casualty and liability lines where claim severity can vary significantly. They are central to rating excess-of-loss reinsurance treaties and setting consistent premiums across layers.

Different Approaches to Excess Pricing

Although the core principle of excess pricing is universal – estimating the cost of losses above a given threshold – the methods differ between markets.

In the US, actuaries often use:

• ISO curves for property and casualty business
• Proprietary or “fitted” loss models
• The NCCI ELPPF for Workers’ Compensation
• Broker-developed curves and market benchmarks

In non-US markets, particularly in London, practitioners rely more on:

• Swiss Re first loss scales
• Internally developed exposure curves
• Power curves (also known as Riebesell’s curves or the German method)
• Underwriter judgement and minimum rate-on-line thresholds

Both approaches aim to achieve the same goal – accurate, consistent and transparent pricing of excess layers – but the tools and data sources vary depending on regulation, data maturity and market practice.

From Exposure Curves to ILF Curves

To understand ILFs, it helps to start with exposure curves – which describe how losses accumulate as a percentage of the sum insured.

An exposure curve effectively tells us what proportion of total losses fall below a certain limit.
By integrating that information, we can derive a Loss Elimination Ratio (LER) – the proportion of losses eliminated by a deductible or retained limit – and from there, calculate the ILF curve.

Exposure curves are built on several assumptions:

• The sum insured (or limit) is a reasonable proxy for the maximum possible loss
• The loss cost distribution is independent of the sum insured
• The relationship between loss size and insured value is consistent

While these assumptions work well for property insurance, they are more complex in liability classes, where the policy limit is chosen by the insured rather than dictated by the exposure itself. This is one reason why ILFs are trickier to model in casualty business.

The Challenges of Building ILF Models

Constructing accurate ILF models is far from simple. In practice, actuaries face several recurring challenges:

• Dependence between claim severity and policy limit – higher limits may attract larger or more complex claims
• Dependence between exposure base and limits purchased – for example, turnover or payroll may influence both exposure and limit selection
• Aggregate vs. any-one-claim limits – these affect how losses accumulate
• Incomplete policy data – many claims datasets lack information about policy-level limits or deductibles
• Mix of business changes year to year, introducing volatility
• Multiple claim records per event – can distort severity analysis if not aggregated

These complexities highlight why actuaries often turn to simpler, robust mathematical frameworks – and that’s where power curves come in.

Introducing Power Curves

Power curves, also known as Riebesell’s curves or the “German method”, have been widely used in the London Market for decades, particularly for non-US liability classes such as Employers’ Liability (EL), General Liability (GL), Directors & Officers (D&O), Professional Indemnity (PI) and Financial Institutions (FI).

The concept dates back to 1936, when Paul Riebesell introduced a method to describe how expected losses increase with policy limits using a simple exponential relationship.

At its heart, the power curve expresses how premiums grow as limits double, following a constant percentage increase (denoted by r). The relationship is governed by the alpha parameter (α), which defines the shape of the curve.

The Mathematics Behind the Power Curve (Simplified)

If we define:

• B as the basic limit (e.g. £1 million), and
• P(B) as the pure premium for that limit,

then the premium for twice that limit (2B) is given by:

P(2B) = P(B) × (1 + r)

where r is the proportional increase in expected loss when doubling the limit.

Extending this concept, for any limit L, we can generalise the relationship as:

P(L) = P(B) × (L / B)ᵅ,
where α = log₂(1 + r).

This elegant formula is what defines the ILF curve in the power curve model – a smooth, continuous relationship between limit and expected loss cost.

Key Advantages of Power Curves

Their main strengths include:

1. Familiarity and Comfort

Underwriters understand them and trust their logic, making it easier to apply consistently across teams.

2. Scale Invariance

3. Closed-Form Formula

The relationship between limit and loss cost is expressible in a simple equation, avoiding the need for interpolation.

4. Consistency Across Markets

Common alpha values can be applied across similar portfolios, supporting standardisation and benchmarking.

5. Alignment with Market Practice

They align closely with how underwriters naturally think about risk and excess layers.

Practical Application: Using ILF Curves in Casualty Pricing

In practice, underwriters and actuaries use ILF curves to price layers of coverage efficiently.

For example, if the basic premium for a £1m layer is £100,000 and r = 20% (α = 0.26), then:

• For £2m, the pure premium = £120,000
• For £4m, the pure premium = £144,000

This consistent pattern allows actuaries to project premiums for higher layers quickly, even when data is sparse.

Power curves are also useful in reinsurance pricing, helping treaty underwriters estimate expected loss costs across complex, multi-layer structures without manually adjusting for every deductible or attachment point.

Limitations and Considerations

While power curves are practical, they are not perfect. Actuaries should be aware of their limitations:

• Choice of Alpha: Selecting a single alpha value that fits all layers is difficult. Often, different alpha values are needed for different deductible levels or attachment points.
• Deductibles and Co-insurance: Ignoring these can lead to discontinuities when stacking limits.
• Inflation: Although mathematically inflation-neutral, in practice inflation affects claim severity trends.
• ALAE (Allocated Loss Adjustment Expenses): These are rarely included in the curves but can significantly affect pricing accuracy.
• Experience vs. Exposure Rating Differences: Real-world claim experience may deviate from exposure-based predictions.

Despite these nuances, power curves remain one of the most transparent and defensible frameworks for modelling ILF curves in casualty insurance.

Real-World Example: Layer Stacking in Reinsurance

Consider a £25m capacity spread across several excess layers, where the cedant retains the first £5m and purchases successive layers up to £25m.

By applying the power curve method with a rate-on-line of 30% and an alpha of 0.38, actuaries can calculate the expected loss for each layer (e.g. £5m xs £5m, £10m xs £10m, etc.) quickly and consistently.

This avoids complex manual adjustments and supports clear, auditable pricing logic – especially valuable when reinsurers need to justify decisions to regulators, brokers or cedants.

Why ILF Curves Still Matter in a Modern, Data-Driven Market

Today’s pricing tools may be powered by machine learning, but the principles of ILF insurance remain the same.

Whether used in Excel or embedded in advanced pricing platforms, ILF curves continue to provide the foundation for transparent, defensible and scalable pricing.

At MatBlas, we help insurers and reinsurers modernise traditional pricing methods – combining actuarial rigour with technology-driven efficiency.

By integrating ILF models, exposure curves and benchmarking tools into a single platform, our solutions make it easier to analyse data, standardise pricing assumptions and enhance portfolio management.

ILF curves and power curve models are more than mathematical constructs – they are the bridge between actuarial science and practical underwriting.

By capturing how risk scales with policy limits, they help insurers and reinsurers achieve pricing consistency, control volatility and maintain profitability.

As the market evolves, the fundamentals remain unchanged: understanding ILF curves is essential for anyone involved in casualty pricing, reinsurance or exposure modelling.