Analysis of VaR-iance

(Mar 13, 2018)

In recent years we have published a number of papers on stochastic mortality models.  A particular focus has been on the application of such models to longevity trend risk in a one-year, value-at-risk (VaR) framework for Solvency II.  However, while a small group of models has been common to each paper, there have been changes in the calculation basis, most obviously where updated data have been used.  Sometimes these changes stemmed from more data being available, but, as Richard Willets covered in his blog, the ONS also restated the population estimates following the 2011 census.  This makes it tricky to compare results between papers. We therefore thought it would be instructive to do a step-by-step analysis…

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Tags: Lee-Carter, value-at-risk, longevity trend risk, Solvency II

Twin Peaks

(Jun 15, 2017)

If you are over forty, the title of this blog will call to mind an iconic, sometimes disturbing, television series of the same name from 1990.  If you clicked on the link expecting murder, surreal horror and an undercurrent of sleaze, however, then this posting is as far away from all that as you are ever likely to get: setting capital requirements for life insurers.  Take a deep breath to recover from your crushing disappointment and let's get back to the day job…

Insurers in the EU operate under Solvency II, which is a one-year, value-at-risk regulatory regime.  The idea is that an insurer needs to hold reserves which will be sufficient to cover 99.5% of adverse scenarios over the coming year and still have enough…

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Tags: value-at-risk, bimodal distribution, Solvency II, model risk

Division of labour

(Jan 10, 2017)

At this time of year insurers have commenced their annual valuation of liabilities, part of which involves setting a mortality basis.  When doing so it is common for actuaries to separate the basis into two components: (i) the current, or period, mortality rates and (ii) the projection of the future path of mortality rates (usually mortality improvements).  This sub-division is carried over into the regular Solvency II assessment of capital requirements, where there is always a minimum of two sub-risks for longevity:

  1. Mis-estimation risk, i.e. the uncertainty over the current level of mortality.
  2. Trend risk, i.e. the uncertainty over the future direction of improvements.

In practice a Solvency II assessment…

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Tags: Valuation, Solvency II, mis-estimation risk, trend risk

What — and when — is a 1:200 event?

(Nov 12, 2015)

The concept of a "one in two hundred" (1:200) event over a one-year time horizon is well established as a reserving standard for insurance in several territories: the ICA in the United Kingdom, the SST in Switzerland and the forthcoming Solvency II standard for the entire European Union.  The basic idea is simple: insurers must be capitalised to withstand 99.5% of events which could arise over the coming year.  Other territories use concepts like conditional tail expectations.

There is room for debate as to what constitutes a 1:200 event, however.  For example, Figure 1 shows the elevated mortality caused by the 1918 influenza pandemic, which for many people would be a starting point for calibrating a modern…

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Tags: Spanish influenza pandemic, mortality shocks, longevity shocks, Solvency II, ICA, SST, VaR, value-at-risk

Quantiles and percentiles

(Aug 20, 2014)

Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. They are generally described as q-quantiles, where q specifies the number of intervals which are separated by q−1 points. For example, the 2-quantile is the median, i.e. the point where values of a distribution are equally likely to be above or below this point.

A percentile is the name given to a 100-quantile.  In Solvency II work we most commonly look for the 99.5th percentile, i.e. the point at which the probability that a random event exceeds this value is 0.5%.  The simplest approach to estimating the 99.5th percentile might be to simulate 1,000 times and take the 995th or 996th largest…

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Tags: quantile, percentile, Solvency II, Excel, R

A benchmark for longevity swap prices

(Aug 1, 2013)

What should a reinsurer charge for longevity risk? Is it possible to benchmark longevity swap prices? One answer to these questions comes from an unlikely source: European Solvency II.

In recent years, longevity swaps have become a popular instrument in the U.K. for managing the longevity risk inherent in portfolios of pensions and annuities. The insurance company or pension scheme replaces uncertain future benefits paid out to pensioners or annuitants with a schedule of fixed reinsurance premiums. The reinsurer takes on the responsibility to pay the actual benefits and receives the fixed reinsurance premiums in return. These premiums will reflect the reinsurer's best estimate of future cash flows,…

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Tags: longevity swap, reinsurance, Solvency II, Solvency Capital Requirement, SCR, risk margin, cost of capital

Discounting longevity trend risk

(Nov 12, 2012)

Establishing the capital requirement for longevity trend risk is a thorny problem for insurers with substantial pension or annuity payments.  In a previous posting I looked at the link between capital requirement and age, as well as the importance of model risk.  However, another important factor is the discounting function used for future cashflows.  This is illustrated in Figure 1, which shows the capital requirements implied by stressing the longevity trend over the lifetime of the annuitant.  This is done for the same projection model at various discount rates.  Besides the pattern with age, the most obvious feature is the strong dependence of the capital requirement on the discount rate used.

Figure…

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Tags: Solvency II, ICA, longevity trend risk, yield curve

Following the thread

(Sep 18, 2012)

Gavin recently explored the topic of threads and parallel processing.  But what does this mean from a business perspective?  Well, parallel processing can result in considerable speed increases for certain actuarial and statistical calculations. If done well, spreading the workload over four threads (say) can reduce the execution time to almost a quarter of its single-threaded equivalent. Many complicated actuarial calculations lend themselves well to multi-threading, and thus considerable reductions in run-times.  A good example of this is simulation, which plays a major role in Solvency II work.  To illustrate, Table 1 shows the execution time for 10,000 run-off simulations of a large annuity…

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Tags: threads, parallel processing, simulation, Solvency II, technology

Canonical correlation

(Jul 28, 2012)

At our seminar earlier this year I looked at the validity of assumptions underpinning some stochastic projection models for mortality.  I looked at the assumption of parameter independence in forecasting, and examined whether this assumption was borne out by the data.  It transpires that the assumption of independence is a workable assumption for some models, but not for others.  This has important consequences in a Solvency II context - an internal model must be shown to have assumptions grounded in fact.

To illustrate, we contrast two models which share a number of features. First, the Lee-Carter model:

Lee-Carter model

and second the age-period-cohort (APC) model:

Age-Period-Cohort model

Forecasting in these models depends on an assumption…

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Tags: mortality projections, canonical correlation, cohort effect, Solvency II

Trend risk and age

(May 12, 2012)

There are several ways of looking at longevity trend risk, as covered in our recent seminar. However, regardless of how you choose to look at this risk, there are some pitfalls to watch out for. By way of illustration, we will consider here the capital requirements under the stressed-trend approach to longevity risk, although the basic points apply to most approaches.

Figure 1 shows the capital requirements implied by stressing the longevity trend over the lifetime of the annuitant. This is the so-called run-off approach, as opposed to the one-year, value-at-risk approach required by Solvency II (or ICA in the United Kingdom). The first aspect of Figure 1 is how different the various capital requirements are…

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Tags: Solvency II, ICA, longevity trend risk, model risk

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