26 May 2023 5 min read

Which LPI model is right for my scheme?

By John Southall

A common problem faced by trustees of defined benefit (DB) schemes is what model to assume when valuing and hedging their LPI (Limited Price Indexation) liabilities.


As we have seen in previous blogs as part of our mini series, a common way to hedge LPI benefits is via ‘delta hedging’. This requires an appropriate model to value LPI.

There are two key questions DB schemes need to consider:

  1. Market-consistent, ‘real-world’ or a blend? Market-consistent models are fitted to LPI swap prices[1] whereas real-world models move away from this to be more realistic[2]
  2. If a real-world model, which one and with what parameters?

The choice between a market-consistent or a real-world approach depends on a combination of scheme circumstances and beliefs.

In terms of beliefs, one view is that a market-consistent approach could lead to suboptimal hedging due to the highly illiquid and distorted LPI swap market, particularly for 0% floors. But as unrealistic as it might appear, a market-consistent approach remains the most objective way to set assumptions.

In terms of circumstances, many schemes now find themselves close to potential buy-ins or buyouts. They may therefore be looking to stabilise the value of their assets relative to insurer pricing. As insurers tend to price LPI cashflows on a market-consistent basis, it can make sense to hedge on that basis, or at least gradually shift towards market consistency as the endgame nears.

Welcome to the real world

The most common real-world approach is the ‘Black’ model. This normally assumes the Retail Price Index takes a random walk in line with that implied from gilt or swap markets, and volatility equal to a chosen parameter. A common assumption is 1.5% per annum.

There are some potential concerns around this model. In its most common form (a single volatility parameter) it neglects features such as skew, ‘fat tails’ and term structure.

A key advantage of the model, however, is its simplicity. It is much easier for scheme stakeholders to coalesce around a single approach if it is transparent, easy to calculate and cheap.


Although it’s easy to criticise the Black model from a theoretical perspective, there is substantial ‘model and parameter uncertainty’ – code for ‘nobody knows the right answer’! This largely stems from a lack of data to calibrate to. Data on breakeven inflation only stretches back to the 1980s and although there is far more data on realised inflation, you can question the relevance of data before inflation targeting began (1992), or the independence of the Bank of England (1997).

However, there are some more sophisticated real-world models. One of these is the Jarrow-Yildrim model, which offers some advantages over the Black model, such as a term structure. Deciding whether it is worth it comes down to factors such as the size of LPI risks relative to other scheme risks, and governance budget given the extra complexity involved.

Has volatility increased?

A common approach in practice is to use the Black model with a 1.5% p.a. volatility. Given the high inflation we’re experiencing, should that number be higher?

Perversely for a real-world model, looking at some market-implied numbers can help. The chart below looks at market-implied pricing for the 5% cap (considered relatively undistorted compared with the 0% floor), expressed as a volatility.


Implied volatilities at March 2023 were broadly the same as they were two years previously. While there might have been a good case to increase volatility around Spring 2022, particularly at short tenors, this has since died away, consistent with the markets’ view that inflation will get under control.

Short tenors tend to be unimportant as they contribute relatively little to the inflation sensitivity of a scheme’s liabilities. As such, it isn’t crazy for real-world models to look past the near term. The chart below shows how historic volatilities change once you include more recent experience (an extra two years):


The blue bars represent a common, but flawed, estimate of volatility based on realised inflation.  The grey bars improve on it[3] by looking at realised volatility relative to market expectations over rolling yearly periods; this volatility could be suitable at short tenors. The orange bars look at realised volatility relative to expectations over rolling 10-year periods; these numbers are likely to be most suitable at longer tenors and deserve the most attention if picking a single volatility number. 

What’s interesting is that the orange bar is only slightly higher when including the last two years of data. Overall, a number around 1.5%-1.7% looks like it could make sense.

Wrap up

Despite the illiquidity of the LPI swap market, we believe there can be some good reasons to stick with a market-consistent approach, not least that buy-ins and buyouts are on the cards for many schemes. However real-world models also have a potential role to play, with the ‘simple’ Black model likely to remain a favourite for practical reasons. Although realised inflation has spiked recently there isn’t a strong case to increase inflation volatility numbers substantially relative to a couple of years ago.


[1] The model typically used is just a flexible way to interpolate and extrapolate between prices from banks. At every tenor point a different set of parameters is fitted to replicate LPI swap prices.

[2] They usually are chosen to be consistent with gilt markets or inflation swap markets, however. Broadly speaking, this means they are consistent with what the market says about how inflation is expected to evolve, but not the uncertainty of it.

[3] A further refinement is to allow for stochastic interest rates

John Southall

Head of Solutions Research

John works on financial modelling, investment strategy development and thought leadership. He also gets involved in bespoke strategy work. John used to work as a pensions consultant before joining LGIM in 2011. He has a PhD in dynamical systems and is a qualified actuary.

John Southall