2.2.1 Sponsors with traded corporate debt

For firms that have traded corporate debt, a market-implied (risk-neutral) default rate can be derived from the market price of the debt. However, even with this observable market price of debt and the assumption that corporate debt default and deficit contribution default always occur jointly, there are still several practical issues to consider in implementing the default risk assumptions required for the valuation model. These are discussed further below.

The derivation of the risk-neutral default probability requires an assumption about the loss that the corporate debt-holder experiences when default occurs. This is expressed as a percentage of the promised payments and is known as the Loss Given Default (LGD). Alternatively, market practitioners may refer to the Recovery Rate, which is simply (1- LGD). Empirical data suggests typical recovery rates of 35%–40% of the promised amount are experienced when unsecured corporate debt defaultsFootnote ².

A term structure of default probabilities may be required that exceeds the observable corporate bond yield term structure of the employer. The sponsor covenant may entail deficit contributions being paid over many years or even decades. Valuation of the covenant therefore requires estimation of market-consistent default probabilities for each year that deficit contributions may be payable under the covenant. But the market prices from which these default probabilities are derived may only be observable for a limited term of corporate debt, and so the longer-term default probabilities need to be ‘extrapolated’ from the observable market prices.

This extrapolation problem is not unique to estimating the market-consistent cost of credit default risk – it is a recurring theme in the market-consistent valuation of long-term insurance liabilities. For example, the issue of extrapolation of risk-free interest rates for the purposes of market-consistent valuation of long-term insurance liabilities has been a topic of significant debate in the development of the Solvency II frameworkFootnote ³. Such extrapolation problems can generally be described in three components: what is the longest reliable market price that can be observed? What ultra long-term assumption should be used? How should these two points be joined up? All three of these components of the extrapolation process may involve some subjectivity. Arguably, this issue is less important in the context of credit default probabilities for sponsor covenants, as most sponsors’ deficit contribution plans will be measured over periods of several years rather than several decades.

A worked example will be useful to illustrate how market data can be used to develop the risk-neutral default probabilities that will be crucial inputs into the sponsor covenant valuation process. Figure 1 displays the credit spread term structure implied by Marks & Spencer credit default swaps as at 31^st December 2011, as provided by Bloomberg.

Figure 1 Credit Spread Term Structure for Marks & Spencer as derived from Credit Default Swaps at 31/12/11

The above credit spread term structure can be used to derive a term structure for the risk-neutral cumulative default probabilities of the issuer. This requires one further assumption – the Recovery Rate or Loss Given Default that is embedded in the credit default swap pricingFootnote ⁴. This calculation can be applied whether using market credit spreads derived from credit default swaps or corporate bond prices. Source: Bloomberg

Figure 2 shows the risk-neutral default probabilities implied when we assume a Recovery Rate of 35%, i.e. a Loss Given Default of 65%.

Figure 2 Risk-Neutral Cumulative Default Probabilities for Marks & Spencer as at 31/12/11

As these default probabilities are risk-neutral, they have no direct economic interpretation - it does not imply that market participants’ best estimate of the 10-year cumulative default probability on 31^st December 2011 was 37%. However, these probabilities are fundamental parameters for the market-consistent valuation of the credit-risky cashflows promised to Marks & Spencer debt-holders. That is, the calculation tells us that the market price of 10-year credit default risk can be derived from making the joint assumption that the 10-year cumulative default rate was 37% and the corporate debt-holders do not require a risk premium for bearing credit risk. This observation can be used in the market-consistent valuation of other more complex forms of cashflows that are a function of whether or not Marks & Spencer defaults over the next ten years.

A further and important topic naturally arises in the above discussion. In general, any market-consistent valuation process requires a definition of the risk-free interest rate (and its term structure where relevant). Across the financial sector, differences can arise in consideration of what market prices should be used in the specification of the risk-free interest rate. In particular, should government bond prices or interest rate swap rates be used to derive the risk-free interest rates that are fundamental to market-consistent valuation of any given cashflow? This distinction between government bond yields and interest rate swap rates has been especially relevant in recent years, where the difference between the two have been at historical highs in economies where sovereign default risk is viewed as low.

In the context of the sponsor covenant valuation, this assumption can be important. In particular, should the market-consistent default probabilities be derived to be consistent with credit spreads over government bond yields or swap rates? In the above example, the credit spreads have been derived in excess of swap rates. If the default probabilities were derived with reference to the credit spread over government bond yields, the default probabilities would be higher. For example, the 10-year swap spread over UK government bond yields on 31^st December 2011 was 0.22%. The 10-year risk-neutral default rate for Marks & Spencer above would increase from 37% to 40% if we used the credit spread over government bonds rather than over swaps. Swap spreads can vary considerably over time and historically it has not been unusual to see swap spreads over 100 basis points. In such circumstances the difference in risk-neutral default probabilities generated by the choice of swaps or government bonds will be significantly greater than in this example.

It should also be noted that the choice of risk-free interest rate will impact on both sides of the DB pension fund holistic balance sheet – a government bond yield basis will result in smaller asset values (sponsor covenant) and larger liabilities (present value of promised pension payments) than an interest rate swap. It is therefore likely to be a fundamentally important assumption in the construction of holistic balance sheets.

Finally, assumptions about how the sponsor default probability varies with economic conditions are required. In particular, the correlation between sponsor default and the size of the deficit is important to the covenant valuation. Put another way, the risk-neutral 1-year default rate for a given employer may be observed to be, say, 1%. But the default rate conditional on equities falling, say, 25%, may be significantly higher.

The size of the deficit is generally a function of two economic variables: the returns of the risky asset portfolio and the behaviour of risk-free interest rates. We can generally assume a very strong correlation between a firm's equity returns and its probability of default (i.e. when the firm defaults, the value of its equity is conditionally expected to be significantly lower than in the circumstance where it has not defaulted). So the correlation of other economic variables with the sponsor default rate can be equivalently considered as their correlation with the sponsor's equity return. This is usually helpful as there will be more relevant data available on the behaviour of firms’ equity returns.

This correlation effect is well-documented both theoretically and empiricallyFootnote ⁵. Typically market-implied correlation numbers between a single firm's equity return and the broader equity market index will be of the order of +0.5Footnote ⁶.

2.2.2 Sponsors with no traded corporate debt

Many UK DB pension fund sponsors are smaller firms that do not have market-traded corporate debt. In this case, the market-consistent default modelling is further complicated by the lack of an obvious market price to be consistent with. For such sponsors, it will be necessary to estimate the risk-neutral default probabilities of the sponsor using other market information – most obviously, the market prices of the corporate debt of firms of similar levels of credit worthiness.

Deciding which other firm's corporate debt prices to use in the calibration requires estimation of a sponsor's credit worthiness, which is a generally complex undertaking. A robust estimation would require sophisticated analysis of the sponsor's financial condition and business prospects. This would be a daunting task to perform for the many thousands of small DB pension funds in the UK. Fortunately, there are well-established commercial service providers that deliver such analysis of corporation's default risk: credit ratings agencies. Credit ratings agencies’ ratings provide a standardised way of describing credit default risk, and are provided for both public and private businesses. These ratings are so prevalent that standard market credit spread term structures for different credit ratings will be publically available from market data providers such as Bloomberg. So once we have assigned an estimated credit rating to a sponsor, we have an obvious set of reference market prices to use in deriving the market-consistent risk-neutral default probabilities.

The breadth of publically available ratings-based spread curves means that this ratings-based approach can be implemented to varying degrees of granularity – for example, a B-rated sponsor in the energy sector could be proxied by using the general B-rated credit spread curve, or the energy sector-specific public credit spread curve could be used as the reference set of credit-risky prices.

It can also be noted that the UK Pension Protection Fund (PPF) has experience of building an internal model that includes estimation of the credit riskiness of the sponsors of all eligible occupational pension funds, and recently published a paper describing their methodologyFootnote ⁷. Whilst the PPF internal model does not attempt to perform market-consistent valuation of sponsor covenants, it does require sponsor-specific credit risk estimates and their paper describes how it uses a credit ratings-based approach to categorise the level of sponsor risk.

2.4.1 Modelling the promised deficit contribution schedule

The valuation of the employer covenant requires a description of what deficit contributions will be paid when in all possible future circumstances. This valuation process can be considered in two components: modelling the behaviour of the promised deficit contribution; and modelling the employer's ability to carry through with their promises (credit default risk). Or, put another way, the sponsor covenant valuation requires a projection of the contributions paid up to the point of default, and an assumption about what will be paid at the point of default.

It may often be possible for the promised deficit contribution assumption to be based on existing deficit contribution arrangements that exist between an employer and the pension fund. A typical arrangement could be that every three years the deficit contribution level for the next three years is set based on the level of the deficit at that point in time and a target horizon for funding the deficit shortfall. This type of arrangement could be easily modelled within the risk-neutral simulation framework. As will be seen in the examples in section 2.5 and particularly 2.6, the simulation framework provides virtually unlimited flexibility in what form of contribution pattern can be valued in the sponsor covenant valuation.

This type of flexible modelling framework may create an incentive for the sponsor to ‘assume risk away’ by encouraging unrealistic assumptions about the scale and pattern of future deficit contributions in the valuation exercise. It will generally be necessary to ensure that what is assumed in the model is an accurate representation of the ‘real-life’ behaviour of the employer. A similar issue arises in insurance valuation, where the current valuation of contingent liabilities can be highly sensitive to assumed future management actions. For example, in recent years UK with-profit funds developed a Principles and Practices of Financial Management document that had a number of uses including providing statements of future management actions that the regulatory capital assessment's valuation modelling had to be consistent with. Similarly, Solvency II requires a number of criteria to be met by the future management action assumptions used in the valuation such as verifiability and realism before the assumptions can be incorporated into the valuation modelFootnote ⁸.

2.5.1 Risk-free sponsor and no asset risk

We begin with the simplest possible example. This trivial case may appear of limited value, but it will provide a useful ‘baseline’ result that can allow us to measure how increasing complexity impacts on the valuation. The example assumes:

• • The pension fund has a portfolio of assets that have a market value of 90 today.

• • The pension fund liability consists of a single liability cashflow of 100 that is fixed and certain and is due in 1 year.

• • The sponsor has committed to paying a contribution at the point the liability cashflow falls due if the pension fund assets are not sufficient to fully pay the liability cashflow. There is no credit risk associated with this sponsor commitment, i.e. it is assumed to be certain that the sponsor will pay any required deficit contribution at the end of the year.

• • The pension fund assets are invested in a 1-year risk-free zero-coupon bond.

• • The 1-year risk-free interest rate is 2% (annually compounded).

Under the above assumptions, the end-year asset portfolio value (before any required deficit contribution) is certain to be 91.8. The end-year pension fund deficit is therefore certain to be 100 − 91.8 = 8.2. The market-consistent value of this risk-free cashflow is simply found by discounting the cashflow by the risk-free interest rate, giving a value of 8.04.

Similarly, the current market-consistent value of the pension fund liability is simply found by discounting the 1-year certain cashflow of 100 at the risk-free interest rate of 2% to give a value of 98.04.

This market-consistent balance sheet of the pension fund is presented in table 1.

Table 1 Market-consistent balance sheet of pension fund (risk-free sponsor, no asset risk)

In this example, the total end-year value of assets (asset portfolio + deficit contribution) always exactly equals the value of the end-year liability cashflow: there is no circumstance in which there is an excess or shortfall of assets (gross of the required contribution) less liabilities. The obvious result is obtained: the current market-consistent values of assets and liabilities are equal.

2.5.2 Risk-free sponsor and a risky asset portfolio

We now introduce the first element of complexity to the above case. It is assumed that the end-year asset value is no longer invested in a risk-free portfolio but is invested in a portfolio of risky assets. For the purposes of the example, it is assumed that the end-year asset portfolio value is lognormally distributed with a volatility of 15%. As we are working with risk-neutral probabilities, the expected return of the risky asset portfolio is not required and it can be assumed that the portfolio has an expected return equal to the risk-free return.

The assumption that the sponsor commitment is default risk-free is retained. That is, the sponsor has committed to paying a contribution at the point the liability cashflow falls due, if the pension fund assets are not sufficient to fully pay the liability cashflow. There is no credit risk associated with this sponsor commitment, i.e. it is assumed to be certain that the sponsor will pay any required deficit contribution at the end of the year.

In this example, the sponsor commitment can be described in the following statement:

If the end-year pension fund asset value exceeds 100, pay zero; if the end-year pension fund asset value is less than 100, pay (100 − asset value).

This statement is a pay-out function of a 1-year put option on the value of the assets with a strike price of 100 and a current asset price of 90. Under our assumption that the end-year asset portfolio value is lognormally distributed, the sponsor covenant can be valued using the standard Black-Scholes formula. This calculation produces a current value for the sponsor covenant of 10.53. Note the value of the sponsor covenant has increased by 2.3 from the value in the previous case. This is reasonably intuitive and there are two (related) drivers of this increase in market-consistent value:

1. 1. The variability in the end-year asset portfolio value means that the sponsor's commitment could now require a payment of any value between 0 and 100. The average value of the required sponsor cashflow will therefore be higher, even after allowing for an increase in expected return of the pension fund assets.

2. 2. From a market-consistent valuation perspective, the valuation of the large contribution cashflows will tend to be disproportionately more costly than the (real-world) probability that is attached to them. This is because their replication cost is greatest (it requires a greater asset investment at time 0 to finance a cashflow shortfall at time 1 that arises after a fall in the value of assets).

The new market-consistent balance sheet is set out in table 2:

Table 2 Market-consistent balance sheet of pension fund (risk-free sponsor, risky asset portfolio)

The balance sheet now has a positive net asset value. The source of this surplus is again intuitive: in this example, there are no scenarios where the end-year total assets (inclusive of the contribution) will be less than the liability cashflow (as there is no sponsor default risk in this example); but there are scenarios where the (risky) asset portfolio has an end-year portfolio value that exceeds the fixed liability cashflow of 100. Note this has made the important assumption that any terminal surplus of the pension fund is not distributed to pension fund members and hence is not included as a component of the pension fund liability value. Under this assumption, the end-year net asset value of the pension fund can be written as:

If asset portfolio is less than 100, net asset value equals zero; if asset portfolio is greater than 100, net asset value equals (asset portfolio − 100).

The above statement is the pay-out function of a 1-year call option with a strike price of 100 (and an underlying asset price of the current asset portfolio value of 90). We again use the standard Black-Scholes formula to value this call option, and the Black-Scholes calculation does indeed produce a value of 2.49. This simple case is an example of put-call parity.

Note we have made no assumptions in the construction of the balance sheet about where any surplus assets go in the event that an end-year surplus exists, other than that it does not need to be considered as a form of liability to the pension fund members. If there was an obligation to distribute this ‘trapped surplus’ to pension fund members, it could be considered as a component of the liability and in this case the net asset value would equal zero in all cases.

Figure 3 shows how the market-consistent value of the sponsor covenant varies as a function of the starting asset portfolio in our two examples.

Figure 3 Value of Sponsor Covenant as a function of starting asset portfolio value

The above chart is merely showing the difference between the intrinsic value and total value of a put option on a risky asset. But it highlights some basic insights that are germane to the sponsor covenant valuation subject:

• • The sponsor covenant increases in value as the pension fund deficit increases (prior to allowing for any sponsor credit risk).

• • Increases in asset portfolio risk will generally result in an increase in the value of the sponsor covenant (prior to allowing for any sponsor credit risk).

• • When the pension fund has a very large surplus or deficit, the asset portfolio strategy (and its volatility) will not have a major impact on the value of the sponsor covenant; rather, the asset portfolio risk will have the most significant impact on the sponsor covenant valuation when the pension fund's funding level is around 100%.

2.5.3 Credit-risky sponsor and no asset risk

The next two cases develop the single-period example to address the crux of the valuation problem: how to allow for the impact of sponsor credit risk in the valuation of the sponsor covenant. We first consider the case where the asset portfolio is risk-free (as per the 2.5.1 case). However, it is no longer assumed that the sponsor's ability to meet the required deficit contribution is certain. Instead, the following assumptions are used:

• • The sponsor's corporate bonds are traded and the credit spread of the 1-year corporate bond over the 1-year risk-free interest rate is 3%. The market assumes a 30% recovery rate in the pricing of the corporate bond (i.e. in the event of default, the bond will pay 30% of the promised amount at the end of the year). This is sometimes referred to as a 70% Loss Given Default assumption.

• • It is assumed that the sponsor will fully pay any required contribution in scenarios where it does not default on its 1-year corporate bond. And that in the scenarios where it does default on its 1-year corporate debt, the sponsor will pay 30% of any required contribution.

Section 2.5.1 showed that the required end-year deficit contribution is a fixed amount of 8.20. However, the valuation of this cashflow must now take into account the impact that default risk has on the market-consistent valuation of the sponsor covenant. This can be done by assessing the sponsor covenant valuation conditional on default occurring or not, and then applying market-consistent (risk-neutral) probabilities to those values.

We know from section 2.5.1 that the value of the sponsor covenant conditional on no default occurring is 8.04. In the event of default, the sponsor is assumed to pay 30% of the promised contribution. The value conditional on default is therefore 30% of 8.04 = 2.41.

To complete the calculation, the risk-neutral default probability is required. In a risk-neutral setting, the expected default loss of the corporate must equal the credit spread implicit in the bond price. The expected default loss is the multiple of the probability of default (PD) and the loss given default (LGD). The LGD has been assumed to be 70%. The credit spread has been observed in the market at 3%. The risk-neutral 1-year PD is therefore 3%/70% = 0.043.

The value of the sponsor covenant can therefore be calculated as:

$${{\rm{(1 - PD)}}\,\times \,{\rm{Value}}\,{\rm{given}}\,{\rm{no}}\,{\rm{default}}\, + \,{\rm{PD}}\,\times \,{\rm{Value}}\,{\rm{given}}\,{\rm{default}} \cr {\rm{(1 - 0}}{\rm{.043)}}\,\times \,{\rm{8}}{\rm{.04}}\, + \,{\rm{0}}{\rm{.957}}\,\times \,{\rm{2}}{\rm{.41}}\, = \,{\bf {{7}}}{\bf {{.80}}} $$

The market-consistent balance sheet applicable in this case is set out in table 3. It shows that, after allowing for sponsor default risk and its impact on the market-consistent value of the sponsor covenant, the total asset value of the balance sheet is now lower than the market-consistent liability value.

Table 3 Market-consistent balance sheet of pension fund (credit-risky sponsor, risk-free portfolio)

You may note that the reduction in value of the sponsor covenant from 8.02 to 7.80 is exactly 3% of the value of the no-default value. It is no coincidence that this reduction in value is equal to the assumed size of the credit spread. If we assume that the LGD or recovery rate applicable for the pension fund is the same as experienced by a corporate bondholder, then the recovery rate/LGD assumption cancels out of the calculation. In particular, we can note that:

$${\rm{PD}}\, = \,{\rm{Credit}}\,{\rm{Spread}}\,{\rm{/}}\,{\rm{LGD}}$$

and

$${\rm{Value}}\,{\rm{given}}\,{\rm{default}}\, = \,{\rm{Value}}\,{\rm{given}}\,{\rm{no}}\,{\rm{default}}\,\times \,{\rm{LGD}}$$

Equations 2.2 and 2.3 can be substituted into 2.1 to obtain the relationship:

$${\rm{Value}}\,{\rm{of}}\,{\rm{sponsor}}\,{\rm{commitment}}\, = \,{\rm{Value}}\,{\rm{given}}\,{\rm{no}}\,{\rm{default}}\,\times \,{\rm{(1}}\,{\rm{ - }}\,{\rm{Credit}}\,{\rm{Spread)}}$$

This is a useful result as the LGD assumption is generally not directly observable from corporate bond prices, and so removing the need to estimate it is helpful. As discussed in section 2.3, we think that the assumption that the LGD on a sponsor's corporate bonds and the sponsor covenant will generally be reasonable.

2.5.4 Credit-risky sponsor and a risky asset portfolio

We now turn to the most complex single-period example: valuing the sponsor covenant in the presence of both sponsor credit risk and asset portfolio risk. In the example, the same asset portfolio risk assumptions as in 2.5.2 above are used – the end-year asset portfolio value is assumed to be lognormally distributed with a proportional volatility of 15%. The sponsor's commitment is again the same as in 2.5.2:

If the end-year pension fund asset value exceeds 100, pay zero; if end-year pension fund asset value is less than 100, pay (100 − asset value).

To incorporate sponsor credit risk into the valuation, we now need to solve the valuation equation:

$$no{{\rm{Value}}\, = \,{\rm{E \ [max}}\,{\rm{(0,}}\,{\rm{100}}\,{\rm{ - }}\,{\rm{asset}}\,{\rm{value)}}\ {\rm{given}}\,{\rm{No}}\,{\rm{Default]}}{\rm{.}}\,{\rm{Prob}}\,{\rm{(no}}\,{\rm{Default)}}\, \cr + \,{\rm{E}}\,{\rm{[max}}\,{\rm{(0,}}\,{\rm{100}}\,{\rm{ - }}\,{\rm{asset}}\,{\rm{value)}} \\ \ {\rm{given}}\,{\rm{default]}}{\rm{.}}\,{\rm{Prob}}\,{\rm{(Default)]\, \ast\,(1 - LGD)]}}\,{\rm{/}}\,{\rm{(1}}\, + \,{\rm{r)}}$$

where expectations are taken in the risk-neutral measure.

If we assume that the change in the asset portfolio value is statistically uncorrelated with sponsor default, then this function simply becomes:

$${\rm{Value}}\, = \,{\rm{E[max\ (0,}}\,{\rm{100 - asset}}\,{\rm{value)]}}{\rm{.[Prob\ (no}}\,{\rm{Default)}}\, + \,{\rm{Prob\ (Default)\,\ast\,(1 - LGD)]}}\,{\rm{/}}\,{\rm{(1}}\, + \,{\rm{r)}}$$

Noting that:

$${\rm{Prob}}\,{\rm{(No}}\,{\rm{Default)}}\, + \,{\rm{Prob\ (Default)}}\,{\rm{\ast}}\,{\rm{(1 - LGD)}}\, = \,{\rm{1}}\,{\rm{ - }}\,{\rm{Credit}}\,{\rm{Spread}}$$

and

E [max (0, 100 − asset value)/(1+r) is the Black-Scholes value obtained in 2.5.2, the following result is obtained:

Sponsor Covenant Value (with credit risk and asset portfolio risk) = (1 − Credit Spread).Sponsor Covent Value (no credit risk, with asset portfolio risk)

So this result is consistent with 2.5.3, i.e. the value in the presence of sponsor default risk is simply the value in the absence of sponsor default risk multiplied by a factor of (1 − Credit Spread).

Continuing with the numbers assumed in our examples above, this means that the sponsor covenant value is (1 − 0.03) × 10.53 = 10.21. The impacts of the asset portfolio risk (increasing the sponsor covenant value) and the sponsor credit risk (reducing the value) are largely offsetting – this valuation is quite similar to the value produced under the risk-free sponsor/risk-free portfolio case of section 2.5.1 (value of 10.53).

The updated market-consistent balance sheet is given in table 4.

Table 4 Market-consistent balance sheet of pension fund (credit-risky sponsor, risky asset portfolio, no correlation)

This result only applies under the assumption that sponsor default risk and the asset portfolio risk are statistically uncorrelated. As discussed in section 2.2, generally, sponsor default and the asset portfolio value will be correlated – a general economic shock could be expected to impact on both general financial market prices and the likelihood of the sponsor being able to re-pay its corporate debt. Assuming a non-zero correlation between the sponsor default event and the asset portfolio value creates the requirement to find the two conditional expectations in equation 2.5:

$${\rm{E\ [max\ (0,}}\,{\rm{100}}\,{\rm{ - }}\,{\rm{asset}}\,{\rm{value)}}\ {\rm{given}}\,{\rm{No}}\,{\rm{Default]}}$$

and

$${\rm{E\ [max\ (0,}}\,{\rm{100}}\,{\rm{ - }}\,{\rm{asset}}\,{\rm{value)}}\ {\rm{given}}\,{\rm{Default]}}$$

No exact analytical solutions are available for the above expectations, even in this relatively simple single-period example. However, a numerical approximation is readily obtainableFootnote ⁹. As discussed earlier, Monte-Carlo simulation has emerged as the standard actuarial methodology for valuation of complex liabilities in insurance and is also a natural choice for this challenge of valuing the complex pension fund asset that arises in the form of the sponsor covenant. This is a natural point to introduce a simulation approach to the valuation problem.

It is fairly straightforward to build a simulation model of the above asset and default dynamics. In the results below, 25,000 1-year market-consistent simulations of the asset portfolio value and sponsor default event were produced using an Excel spreadsheet. The following algorithms were used in the modelling:

$${{\rm{End{\hbox-}year}}\,{\rm{Asset}}\,{\rm{Portfolio}}\,{\rm{Value}}\, = \,{\rm{Start {\hbox-} Year}}\,{\rm{Value}}\,\times \,{\rm{(1}}\, + \,{\rm{risk{\hbox-}free}}\,{\rm{rate)}}\, \cr \times \,{\rm{exp( - \,{\vskip-2pt\hskip1\hat}\!\sigma \,2/2}}\, + \,{\rm{\sigma }}{\rm{.}}{{{\rm{z}}}_{\rm{1}}}{\rm{)}} $$

Sponsor default occurs if z₂ < Default Probability

$${{{\rm{z}}}_{\rm{2}}}\, = \,\rho {{{\rm{z}}}_{\rm{1}}}\, + \,{\rm{sqrt(1}}\,{\rm{ - }}\,{{\rho }^{\rm{2}}} {\rm{)}}{\rm{.}}{{{\rm{z}}}_{\rm{3}}}$$

where σ is the assumed 1-year standard deviation of the asset portfolio total return, and ρ is the correlation between the asset portfolio total return and the sponsor default.

z₁ and z₃ are independent standard normal variables that were sampled using the inverse normal function of a uniform random variable (produced using the rand() Excel function). This approach to random number generation is not sufficiently robust for industrial applications, but has sufficient accuracy for this simple simulation problem.

Figure 4 shows the valuation results that are obtained under different assumptions for the correlation between the asset portfolio value and sponsor default. A high correlation means that more defaults occur when the asset portfolio value falls. As a result, the average size of the deficit that arises when default occurs increases as the strength of the correlation increases. This increases the credit risk of the sponsor commitment and reduces its value. The chart also shows that the sponsor support value tends towards the risk-free value as the correlation approaches −1Footnote ¹⁰.

Figure 4 Value of Sponsor Commitment as a function of correlation between asset portfolio value and sponsor default

Figure 5 End-December 2011 UK Government Bond Spot Yield Curve

In practice, we would generally expect positive correlation between the asset portfolio value and sponsor default risk factors in the above model specification. As described in section 2.2, an assumption of around 0.5 is typically used in the joint modelling of a diversified equity index and the single-name exposure that drives the default event. At these levels of correlation, the impact of non-zero correlation is fairly modest in our example: it reduces the market-consistent value of the sponsor covenant by 1%–2.5%.

And finally, the market-consistent balance sheet with a asset portfolio/sponsor default risk correlation of +0.5 is given in table 5.

Table 5 Market-consistent balance sheet of pension fund (credit-risky sponsor, risky asset portfolio, 0.5 correlation)

2.5.5 Summary of results obtained from the single-period examples

These simple single-period examples have produced some intuitive results. To recap, we have found that:

• • In the absence of sponsor credit risk, asset portfolio risk or liability cashflow risk, the value of the sponsor covenant is simply equal to the current market-consistent deficit (and zero if the pension fund asset portfolio market value exceeds the market-consistent liability present value);

• • With sponsor credit risk and no asset or liability risk, the value of the sponsor covenant is the credit risk-free valuation, multiplied by a factor of (1 − Credit Spread). This assumes the Loss Given Default for the sponsor commitment is the same percentage value as the Loss Given Default for the sponsor's corporate bonds.

• • With no sponsor credit risk, the presence of asset portfolio risk transforms the sponsor covenant into a put option on the value of the assets. This means that increases in asset portfolio risk will increase the value of the sponsor covenant. This impact will be greatest when the pension fund is close to 100% funded on a market-consistent valuation basis. Under suitable assumptions about the asset portfolio value behaviour, we can value this option using the standard Black-Scholes formula.

• • If we assume that sponsor credit default risk and asset portfolio risk are statistically uncorrelated, we again obtain the result that the sponsor commitment value in the presence of sponsor credit risk is simply the sponsor commitment value in the absence of sponsor credit risk multiplied by a factor of (1 − Credit Spread).

• • Positive correlation between the asset portfolio value and the sponsor's credit quality increases the credit riskiness of the sponsor covenant and reduces its market-consistent value.

2.6.1 The risk-neutral simulation model for interest rates, asset portfolio returns and sponsor credit quality

In the stylised examples below, we include three key sources of risk in the market-consistent modelling:

1. 1. The risk-free yield curve. We will assume this yield curve is used to value expected pension fund liability cashflows, and to value risk-free bonds that are included in the pension fund's asset portfolio. For the purposes of this example, we will use a commonly known stochastic yield curve model – the Black-Karasinski modelFootnote ¹¹.

2. 2. The risky asset portfolio. The portion of the pension fund asset portfolio that is not invested in risk-free bonds is modelled as a single risky asset portfolio using a lognormal total return model.

3. 3. The sponsor credit risk. We will use a credit rating-based approach to simulating the behaviour of the sponsor credit risk.

This clearly is not an exhaustive list of the risk exposures of pension funds. Inflation risk and longevity risk are other obvious risk factors that could be incorporated into the sponsor covenant valuation process. However, for the purposes of developing illustrative examples, these three risk factors will provide a useful starting point.

Again note the requirement here is for risk-neutral joint paths for the above variables to be generated – so the implementation of the above modelling components will be in a risk-neutral setting. Below we provide some details of the modelling approaches outlined above.

The Black-Karasinski model for risk-free interest rates

The Black-Karasinski model is one of a class of arbitrage-free yield curve models known as short rate models. Other well-known short rate models include the Vasicek model and the Cox-Ingersoll-Ross model. In a short rate model, a stochastic process for the short-term interest rate is specified (different short rate models differ by specifying different stochastic dynamics for the short-term interest rate model). The fundamental idea of short rate models is that the stochastic process for the future behaviour of the short-term interest rate uniquely defines the arbitrage-free prices today of risk-free bonds of any term. So in an arbitrage-free setting, the specification of the stochastic process for the short-term interest rate also implies a bond pricing formula for the valuation of a risk-free cashflow of any duration. In a similar way, the model dynamics also implies arbitrage-free pricing formulae for interest rate derivatives such as swaptions, caps and floors. The parameters of the stochastic process for the short-term interest rate can therefore also be calibrated to derivatives such as these in order to generate market-consistent levels of interest rate volatility from the model. The mathematics of the specific model used in our examples is described in more detail in Appendix B.

In the following examples we use a calibration of this model to UK government bonds and swaption-implied volatilities as at end-December 2011.

The Lognormal Model for the risky asset portfolio returns

It is assumed that the continuously-compounded total return in excess of the risk-free short rate is normally distributed. As we are modelling in a risk-neutral setting, it must be assumed that the (arithmetic) mean excess total return is zero. So the only parameter remaining to be specified is the volatility of the excess total return. In market-consistent modelling, this parameter can be calibrated to the market prices of relevant option contracts, where they are available. For the purposes of the examples in this section, it is assumed that the option-implied volatility of the excess total return is 20%.

The Rating-Based Model for the Credit Quality of the Sponsor

As discussed in section 2.4, the market-consistent, risk-neutral stochastic modelling of sponsor default is primarily driven by reference to the market price of default risk, as implied by the level of credit spread on the traded debt of the sponsor firm. We may also be interested in projecting intermediate changes in the credit quality of the sponsor, as there may be management actions that are triggered by falls in sponsor credit quality.

In the following examples a stochastic credit rating model developed by Jarrow, Lando and Turnbull (JLT)Footnote ¹² is used. In particular, a proprietary implementation of this model as found in Barrie & Hibbert's Economic Scenario Generator is used, though any JLT implementation that is calibrated to market credit spreads would be expected to deliver similar results for the example valuations considered below.

The model is calibrated to the general level of credit spreads found in corporate bond prices in the UK as at end-December 2011. In this calibration of the credit model, the credit spread of corporate bonds in excess of government bonds is used.

As discussed in section 2.5, if credit spreads were defined as the bond yield in excess of swap rates, the implied risk-neutral default rates would be lower. A calibration of credit spread to swap rates would therefore generally result in an increase in the value of the credit-risky sponsor covenants.

A recovery rate of 35% is assumed when deriving risk-neutral default probabilities from corporate bond prices.

Correlation between the variables

For the purposes of our examples, it is assumed that risk-free interest rate changes are uncorrelated with the other variables; and the credit quality of the sponsor and the risky asset portfolio return have a correlation of +0.6.

Below a series of multi-period examples are incrementally developed using the stochastic modelling set-up described above.

2.6.2 No risky assets; no credit risk

As in the series of single-period examples of section 2.5, this section introduces a series of multi-period examples of increasing complexity. The initial examples will therefore by highly stylistic, but this approach again can provide some transparency on how more realistic and complex features incrementally impact on the sponsor covenant valuation.

The first example assumes a default risk-free sponsor, and assumes that the asset portfolio is entirely invested in government bonds. Figure 6 shows the schedule of promised liability cashflows. The cashflows are assumed to arise on a yearly basis over the next 60 years (with the next cashflow due one year from the valuation date). When discounted using the UK government bond yield curve given in section 2.6.1, the present value of this liability cashflow schedule is exactly £1 billion. The duration of the liability cashflows is 16.3 years.

Figure 6 Schedule of Promised Liability Cashflows

The current market value of the pension fund's asset portfolio is assumed to be £800 m and is initially assumed to be entirely invested in a risk-free government bond portfolio. This bond portfolio is represented in the model by a 10-year bond that pays a 5% annual coupon. The portfolio is assumed to be re-balanced every year such that the bond portfolio at the start of every future year is invested in a 10-year bond with a 5% coupon.

Whilst this example has no risky assets and no sponsor credit default risk, there is still significant interest rate risk that arises from the asset portfolio's smaller market value and shorter duration relative to the promised liability cashflows. This risk will be captured by the stochastic interest rate paths produced by the interest rate model. The contribution strategy that will be applied in this example has not yet been specified, but almost any contribution strategy in this example will result in some variability arising in the size of contributions ultimately required to fund the liability cashflows as a result of the interest rate risk inherent in the asset-liability position.

Throughout the following examples two contribution strategies will be modelled:

• • Contribution Strategy 1: A contribution is only paid when the asset portfolio is exhausted and is insufficient to fund the immediately-required liability cashflow.

• • Contribution Strategy 2: A required contribution is calculated annually as Market-Consistent Deficit/5 (subject to a minimum of zero).

These two contribution strategies are intended to represent the two ends of the spectrum in terms of the pace of deficit funding. We expect most DB pension funds’ deficit contribution strategy will sit somewhere between these strategies, and hence the results of the respective strategies should provide an indication of the range of outcomes produced by variation in the deficit contribution strategy.

We now consider the sponsor covenant valuation results obtained under the two contributions strategies under the risk-free sponsor/risk-free asset portfolio case. It may be recalled from the single-period case in section 2.5.1 that the value of the sponsor covenant in this case was found to simply be the market-consistent value of the current deficit (i.e. liability cashflows discounted at the risk-free yield curve less the market value of the asset portfolio, subject to zero). This general result will also hold in the risk-free case multi-period case, with one exception that is discussed below.

Even in the risk-free asset portfolio case, the multi-period example has significant asset-liability mismatches that will generate variations in contributions payable over time under both of the above contribution strategies. This creates the possibility that the contribution strategy will result in a surplus of assets arising after all liability cashflows have been paid (if higher-than-expected future risk-free interest rates subsequently arise it can result in some of the paid contributions ultimately not being required to fund the promised liability cashflows).

If these surplus assets were assumed to be distributed to shareholders as a ‘negative contribution’ and added to the contribution cashflow schedules, the sponsor covenant valuations in the risk-free sponsor case would always be equal to the market-consistent deficit value. Similarly, if they were assumed to be distributed to pension fund members as ‘bonus payments’ and treated as liabilities, the liability valuation would increase to offset the increase in the value of the sponsor covenant, and hence again the result of a net asset value of zero for the market-consistent balance sheet would be obtained. However, if we assume these are undistributed surplus assets, there will be scope for the market-consistent sponsor covenant value to be greater than the current size of the deficit. How much greater will depend on the asset and contribution strategies, and how likely they are to result in a terminal surplus asset level arising.

When Contribution Strategy 1 is run using the risk-neutral simulation model described in section 2.6.1, an analogous result to the no asset risk/no sponsor default risk case in section 2.4 is found: the present value of the contributions was found to be £200 m (which equates to the £200 m difference between the liability cashflow present value of £1bn and the market value of the asset portfolio of £800 m). However, Contribution Strategy 2 results in a higher value of £209 m. The difference in sponsor covenant valuation between Contribution Strategies 1 and 2 reflects the higher probability of terminal surplus assets arising under Strategy 2 than Strategy 1.

In each of these cases, 10,000 risk-neutral simulations of the 60-year path of contribution cashflows have been simulated and discounted using the risk-neutral discount functions. Analysing these simulation outputs, we also find that the average duration of the simulated contribution cashflow stream under Contribution Strategy 1 is 32.4 years, and under Contribution Strategy 2 it is 5.6 years (we will use these statistics below).

2.6.3 With risky asset portfolio; no sponsor credit risk

Mirroring the progression of the single-period examples, we now consider the multi-period case when some of the asset portfolio is moved from risk-free assets into the risky asset portfolio described in section 2.6.1. In particular, two cases are considered: when the asset portfolio is invested 50%/50% in the risky asset portfolio and the risk-free bond portfolio (with annual rebalancing back to 50% allocations); and a 100% allocation to the risky asset portfolio.

Figure 7 shows the results for sponsor covenant valuation of these two cases, and compares these results with the valuation produced above with the risk-free bond portfolio. Again, both contribution strategies are analysed.

Figure 7 Sponsor covenant valuation under different asset allocations (no sponsor default risk)

These results are consistent with the single-period examples. Moving into riskier assets creates a more option-like commitment that incurs a greater cost for the sponsor (again assuming the shareholder cannot recover any terminal surplus, or at least that they are not considered as negative contributions in the covenant valuation). And in all cases this effect is more pronounced under Contribution Strategy 2 as the possibility that a surplus pool of assets arises after all liability cashflows have been paid is greater under Contribution Strategy 2 than Strategy 1.

2.6.4 Credit-risky sponsor; no risky assets

This section considers the impact of changes in the sponsor credit rating on the valuation of the sponsor covenant in the multi-period case when the asset portfolio is invested in the risk-free bond portfolio. It is assumed that 35% of the required future contributions are recovered from the sponsor in the event of default (i.e. a cash lump sum of 35% of the market-consistent deficit is paid into the pension fund).

You can recall from section 2.6.2 that the value of the sponsor covenant in this case when the sponsor is assumed to have no default risk was calculated to be £200 m for Contribution Strategy 1 and £209 m for Contribution Strategy 2. The 35% recovery rate implies that a sponsor who immediately defaults will pay £70 m (i.e. 35% of the £200 m deficit) under both contribution strategies. So we know that different starting sponsor credit ratings must result in covenant valuations in the range of £70 m to £200 m for Contribution Strategy 1 and a range of £70 m and £209 for Contribution Strategy 2.

Figure 8 presents the valuation results obtained by the model for eight credit ratings under both contribution strategies.

Figure 8 Sponsor covenant valuation and starting sponsor credit ratings (asset portfolio = £800m; no risky assets)

One of the most immediately striking aspects of the figure 8 is how much the value of the sponsor covenant is reduced under Contribution Strategy 1 when the sponsor credit rating is changed from risk-free to AAA – this reduces the covenant valuation by more than 40%. A few back-of-the-envelope calculations can help provide some intuition for this result.

Recall from section 2.6.2 that the average duration of the contribution cashflow stream produced in the risk-free example under Contribution Strategy 1 is 32.4 years. The starting 32-year AAA credit spread over government bonds at the end-2011 valuation date is assumed to be 2.0% in the model. Using the 35% recovery rate, the 32-year risk-neutral annualised default rate is 1.9%/(1 − 0.35) = 2.9%. This implies a 32-year risk-neutral default probability of 61%. This default probability is broadly consistent with the reduction in sponsor covenant valuation – i.e. assuming a 35% recovery rate is obtained in default, the 61% default probability implies an expected default loss of 0.61*0.65 = 40%.

As mentioned in 2.6.1, using credit spreads over swap rates rather than government bond yields would reduce the risk-neutral default probabilities, and hence increase the covenant valuations for the credit-risky sponsor cases. Also, the size of the above impact arises in part because of the very long duration of the contribution cashflow stream that is produced by Contribution Strategy 1. The proportional reduction in sponsor covenant valuation that is generated by the move from risk-free to AAA sponsor under Contribution Strategy 2 is significantly smaller than for Contribution Strategy 2: 18% instead of 43%.This is mainly because the duration of the contribution cashflow stream is much shorter under Contribution Strategy 2 (5 years instead of over 30 years), and the probability of a AAA sponsor defaulting over 5 years is much lower than over 30 years.

2.6.5 Credit-risky sponsor; with risky asset portfolio

We now assess the sponsor covenant valuation behaviour for credit-risky sponsors when some of the asset portfolio is allocated to risky assets. Figure 9 shows the results for the sponsor covenant valuation produced by the risk-neutral multi-period simulation model for the three asset strategies considered in section 2.6.3 and Contribution Strategy 1. Figure 10 shows the corresponding results under Contribution Strategy 2.

Figure 9 Sponsor covenant valuation as function of starting sponsor credit rating and asset strategy (Asset portfolio = £800m; Contribution Strategy 1)

Figure 10 Sponsor covenant valuation as function of starting sponsor credit rating and asset strategy (Asset Portfolio = £800m; Contribution Strategy 2)

These results are generally consistent with the results obtained throughout section 2: more asset risk generally results in an increase in the sponsor covenant; lower sponsor credit quality results in lower covenant valuations; Contribution Strategy 2 produces higher valuations than Contribution Strategy1.

However, an interesting feature of the above charts is that the value of the sponsor covenant in the risky asset strategy case falls below the risk-free strategy case when the sponsor credit rating is low (BB to CCC).This is driven by the wrong-way risk effect that is captured by the model. The model assumes a strong correlation between the risky asset portfolio and the sponsor default occurrence – defaults and falls in asset portfolio value tend to occur together, hence increasing the effective exposure to default risk. This is true in all cases, but the impact of this effect is greater for the low credit qualities. In these cases, the increase in default probability that occurs when assets perform poorly dominates the option value increase that is created by the sponsor's commitment to (try to) fund deficit increases.

4.2.1 Yield curve stress

This section considers how a 99.5^th percentile change in the risk-free yield curve can impact on the items of the holistic balance sheet and its net asset value.

In theory, the holistic balance sheet is impacted by a change in any point of the yield curve. So, a full approach to assessing the sensitivity of the balance sheet to changes in the yield curve could consider its sensitivity to changes in each of, say, 60 annual points on the yield curve, and then stochastically model how each of those 60 points behave.

In practice, simpler approaches can be developed by recognising that these 60 points on the yield curve will generally be quite strongly correlated. This means that a smaller number of statistical risk factors can be used to accurately represent the joint behaviour of all 60 points. Typically, a statistical technique such as Principal Components Analysis (PCA) is used to represent the joint behaviour of all points on the yield curveFootnote ¹⁸. PCA typically shows that the vast bulk of empirical variations in yield curves can be explained by three or four factorsFootnote ¹⁹. Whilst PCA factors are derived purely statistically rather than from any descriptive model of yield curve behaviour, the factors can generally be given intuitive explanations: the first PCA factor will generate shifts in the level of the yield curve; the second factor will produce changes in the slope of the yield curve; and the third factor will generate twists in the shape of the yield curve.

In a principle-based approach to capital assessment, the methodology should focus on the risk factors that have greatest impact on the capital assessment of the balance sheet. (A key feature of a principle-based risk assessment system should be that the risk assessment methodology should be driven by the balance sheet's risk exposures, rather than the other way round.) This will be a function of how the balance sheet is managed, and consequently which risk factors it has greatest net exposure to. For example, when a balance sheet is duration-matched but not cashflow-matched, its exposure to the first PCA factor (yield curve level) should be low, but the exposure to the second and third factors (slope and twist risk) will be comparatively significant. But a balance sheet that has a substantial duration mis-match will tend to find that its risk exposure to the first PCA factor will dominate the yield curve risk and capital assessment, and relatively little will be added by considering its sensitivity to ‘higher-order’ forms of interest rate risk.

In the example described above, the holistic balance sheet's assets (both the asset portfolio and the future contribution cashflow stream) have a significantly shorter duration than the promised liability cashflows. This is likely to be currently representative of a significant portion of UK DB pension funds. The yield curve risk assessment in this example will therefore focus on the first yield curve risk factor: changes to the level of the yield curve. More complex changes to the shape of the yield curve will be omitted from the example, but it could be naturally extended to include additional PCA factors as risk factors in the capital assessment. It can be noted that this is also the approach proposed for the Solvency II Solvency Capital Requirement's Standard Formula – the Standard Formula's yield curve stresses only considers specified changes in the level of interest rates; it does not consider the impact of changes in slope or twists in yield curve shape.

As the example holistic balance sheet has a net shortfall in duration, it can be intuitively deduced that the net assets of the balance sheet will be exposed to a fall in the level of the yield curve rather than interest rate rises. The yield curve fall stress specified in the latest draftFootnote ²⁰ of Solvency II's Standard Formula is used as our example 99.5^th percentile yield curve stress. Figure 27 shows the base and stressed yield curve produced by the Solvency II stress test.

Figure 27 Base and stressed risk-free yield curve

The change in the value of the promised liability cashflows and the asset portfolio's government bond holdings can be directly calculated by discounting these cashflows using the stressed risk-free yield curve. The liability valuation increases by 17.7% from £1,000m to £1,177m under this change in the yield curve. The bond holding increases by 8.3% from £400m to £433m. The value of the risky asset sub-portfolio is unchanged by this stress test.

The above numbers show the liability valuation has increased by over £140m more than the asset portfolio. What about the market-consistent value of the sponsor covenant? The contribution strategy will now generate, on average, more contributions than in the base case, and those cashflows will be discounted at a lower interest rate, so it can be expected that the sponsor covenant asset will generally increase in value under this stress test. In the case of the risk-free sponsor, the increase in the value of the contributions will exactly offset the change in the market-consistent deficit. But in the other cases, the risk-neutral simulation model must be re-run with the new yield curve in order to assess the increase in the sponsor covenant value (unless, of course, we are using approximate valuation functions such as those produced in section 3).

The stressed holistic balance sheets are set out in Tables 9–11.

Table 9 Holistic balance sheet of pension fund; including yield curve stress results (Risk-free sponsor)

Table 10 Holistic balance sheet of pension fund; including yield curve stress results (A-rated sponsor)

Table 11 Holistic balance sheet of pension fund; including yield curve stress results (BB-rated sponsor)

In the risk-free sponsor case, the change in the value of the risk-free contribution stream entirely offsets the change in the liability cashflow valuation (net of the asset portfolio value change); but in the presence of credit risk, the increase in promised contributions has a more limited impact on the valuation of the sponsor covenant as the new contribution stream is riskier in the sense that it will fall in value by a greater amount when sponsor default occurs. As a result, the net asset value of the balance sheet falls under the stress test when the sponsor covenant is credit-risky.

For the A-rated sponsor, the increase in covenant value (£113m) offsets around three-quarters of the increase in market-consistent deficit (£144m). The BB-rated sponsor offers less protection, and the covenant increase (£91m) is only three-fifths of the deficit increase.

4.2.2 Risky asset portfolio value stress

This section considers a stress to the starting value of the risky asset sub-portfolio. The sub-portfolio has a starting value in the base case of £400m (a 50% allocation of the £800m total asset portfolio). It was assumed in the valuation calculations in section 2 that the risky asset portfolio return is lognormally distributed with a volatility of 20%. Using this assumption to set the 1-year 99.5% stress testFootnote ²¹, we obtain a stress test of a 37.7% fall in the risky asset portfolio valueFootnote ²².

This results in a stressed risky asset portfolio value of £249m. The risk-free bond portfolio value is unaffected by this stress, and so the starting total asset portfolio value is £649m (£400m of risk-free bonds plus £249m of risky assets). The value of the promised liability cashflow stream is also unaffected by this stress.

The sponsor covenant valuation in this stress test can be calculated by re-running the risk-neutral simulation model with the revised starting value for the risky asset portfolio. Intuitively, an increase in the value of the sponsor covenant can be expected as the contribution strategy will generate an increased amount of contributions, acting as an ‘absorber’ of some of the asset portfolio value loss. As in 4.2.1, in the risk-free sponsor case the sponsor covenant will absorb the entire deficit impact. But in the credit-risky sponsor cases, the credit risk associated with the increased contributions acts to limit the capacity of the covenant to absorb the entire mark-to-market impact. Tables 12 and 13 present the stressed balance sheets for the A-rated and BB-rated sponsors respectively.

Table 12 Holistic balance sheet of pension fund; including risky asset portfolio stress results (A-rated sponsor)

Table 13 Holistic balance sheet of pension fund; including risky asset portfolio stress results (BB-rated sponsor)

In these results around two-thirds of the asset portfolio value loss has been offset by the increase in value in the sponsor covenant that arises in the stress. Naturally, the A-rated sponsor is able to absorb more of the impact than the BB-rated sponsor.

4.2.3 Sponsor credit rating stress

This section considers the impact on the holistic balance sheet of a 99.5^th percentile stress of the credit rating of the sponsor. This first requires us to define what would happen to the credit ratings of the A-rated and BB-rated sponsors after applying a 99.5^th percentile stress to their rating. Table 14 shows the long-term historical frequencies of 1-year changes in credit ratings of A-rated and BB-rated corporate sponsorsFootnote ²³.

Table 14 Historical frequency of 1-year change in credit rating of corporate bonds (1920–2010)

Table 14 shows that the probability of the A-rated sponsor being downgraded to B or lower is 0.3%, and the probability of a downgrade to BB or lower is 1%. So the 99.5^th percentile 1-year event for the A-rated sponsor is a downgrade to BB. By chance, the example is already quite familiar with the BB-rated sponsor covenant valuation: table 8 showed that the sponsor covenant valuation for a BB-rated sponsor under Contribution Strategy 2 and the 50/50 asset strategy is £155m. No other items of the holistic balance sheet are impacted by this stress test. Table 15 shows the stressed balance sheet for the A-rated sponsor.

Table 15 Holistic balance sheet of pension fund; including sponsor credit rating stress results (A-rated sponsor)

For the sponsor that is currently rated BB, table 14 shows that there is a 1.5% probability of defaulting within 1 year, and so the 1-year 99.5% stress test is that the sponsor defaults. Section 2.6.5 showed that the result for this asset strategy and contribution strategy where the sponsor has defaulted is a sponsor covenant valuation of £70m. The results for this case are shown in table 16.

Table 16 Holistic balance sheet of pension fund; including sponsor credit rating stress results (BB-rated sponsor)

4.2.4 Credit spread level stress

The final of the four risk factor stresses left to consider is an overall change in the market level of credit spreads. In considering how to define the credit spread stress, a similar discussion arises as in the yield curve risk section. The sponsor covenant valuation will in theory be a function of the size of the sponsor credit spread at every term at which a contribution cashflow may arise. However, variation in credit spreads will be highly correlated across different points on the term structure of a given sponsor or credit rating. Moreover, the overall level of market credit spreads of different credit ratings will also tend to be highly correlated.

So in a similar spirit to the yield curve risk stress test, the example uses a single factor to produce a stressed increase in the overall level of credit spreads (across both term and credit rating). This factor is a parameter in the market-consistent credit model that is used to specify the level of credit spreads (and hence risk-neutral default rates).

The parameter has been stressed to produce a shift in credit spreads that is broadly consistent with the level of spread stress in the Solvency II QIS5 specificationFootnote ²⁴. The QIS5 specification requires a 1.4% increase in the A-rated spread curve, and a 4.5% increase in the BB curve. The QIS5 specification of a shift of the same size in the credit spread at all terms is not consistent with the theory and empirical reality that credit spreads will be less volatile at longer terms. The credit model factor therefore tends to struggle to fit exactly to this specification. Figures 28 and 29 show the stressed credit spread term structures produced by the model.

Figure 28 Stress test of credit spreads: impact on A-rated credit spread term structure

Figure 29 Stress test of credit spreads: impact on BB-rated credit spread term structure

Figure 28 shows that the stress applied in the model produces a 1.4% shift in the 13-year A-rated credit spread, with a higher stress produced at shorter terms and a smaller stress produced at longer terms. Similarly, a 4.5% shift in the B-rated credit spread was produced at a term of 2–3 years, and a smaller stress was produced for longer terms.

The stress to the level of credit spreads does not impact on the value of the asset portfolio as its bonds holdings are assumed to be risk-free. So this stress only impacts on the market-consistent valuation of the credit-risky sponsor covenant asset. The valuation impact can be assessed by re-running the risk-neutral simulations with the new credit model calibration (note the assumed higher level of spreads will result in the risk-neutral model producing a higher level of sponsor defaults). The balance sheet results for the A-rated and BB-rated sponsors are shown in tables 17 and 18 respectively.

Table 17 Holistic balance sheet of pension fund; including credit spread stress results (A-rated sponsor)

Table 18 Holistic balance sheet of pension fund; including credit spread stress results (BB-rated sponsor)

4.2.5 Summary of the stress test results

Having calculated the capital requirements generated by each risk factor on a stand-alone basis, the following section will turn to their aggregation to produce a total solvency capital requirement for the holistic balance sheet. Before doing that, table 19 summarises the capital requirement results produced in sections 4.2.1–4.2.4.

Table 19 Summary of individual capital requirements by risk type and sponsor credit rating

The stressing of the holistic balance sheet in these various ways highlights a few general observations that can be summarised as follows:

• • The higher the credit quality of the sponsor, the more the sponsor covenant will act as a loss-absorber for the impact of unexpected risks on the balance sheet – the sponsor covenant will increase in value to offset some of the increase in pension fund deficit. Taken to the limiting case of a risk-free sponsor, the sponsor covenant will absorb all risks and the net assets of the balance sheet will be immunised from the impact of all risks on the balance sheet (perhaps except from the risk that the assumption the sponsor is risk-free turned out to be wrong!).

• • With lower quality sponsors, the sponsor covenant is not able to absorb as much of the impact of other risks: whilst promised and expected contribution cashflows may increase after a shock to say, equity markets, the credit-riskiness of the contribution stream will mean that the impact on the valuation will be less than proportional to the increase in expected contributions.

• • At a 99.5% confidence level, a credit rating stress of sponsors with a credit rating of BB or lower will produce a sponsor default stress. This implies that a large number of corporate pension fund sponsors will be able to obtain only limited credit for the presence of the sponsor covenant within a 99.5% VaR framework – the sponsor covenant asset will be largely offset by the need to hold capital against the risk that the sponsor immediately defaults. But there are two effects that will slightly improve this picture: a recovery rate from the sponsor can be assumed in the event of default; the capital required for sponsor default risk will diversify to some degree with the other capital requirements. This latter point is the focus of the next section.

4.2.6 Aggregated risk-based capital assessment

The aggregation of the above stand-alone capital requirements requires assumptions about the joint behaviour of the four risk factors. The illustrative correlation assumptions used in this example are summarised in table 20.

Table 20 Assumed correlations between stand-alone capital requirements

Note this illustrative example assumes all risk factors are correlated with a Gaussian copula (i.e. the strength of the correlation does not vary as a function of the size of the marginal shock to the risk factor).

Under these assumptions the aggregate capital requirement can be calculated as the square root of the sum of squared capital requirements plus covariance terms. The aggregation calculation generates an aggregate capital requirement of £122m for the A-rated sponsor and £175m for the BB-rated sponsor. The pre-diversification capital requirement (i.e. simply the sum of the individual capital requirements produced for each risk factor) is £161m for the A-rated sponsor and £237m for the BB-rated sponsor. The correlation assumptions have generated a diversification benefit of £39m for the A-rated sponsor and £62m for the B-rated sponsor. In both cases the diversification benefit is a reduction in the aggregate capital requirement of around 25% of the pre-diversification capital requirement.

Figures 30 and 31 chart the stand-alone capital requirements and diversification benefit for the A-rated and BB-rated sponsors respectively.

Figure 30 Aggregate capital requirement by risk type and diversification benefit (A-rated)

Figure 31 Aggregate capital requirement by risk type and diversification benefit (BB-rated)

And finally, the holistic balance sheets with their solvency capital requirements are set out in tables 21–23 for each of the risk-free, A-rated and BB-rated sponsors.

Table 21 Holistic balance sheet and capital requirement of pension fund (Risk-free sponsor)

Table 22 Holistic balance sheet and capital requirement of pension fund (A-rated sponsor)

Table 23 Holistic balance sheet and capital requirement of pension fund (BB-rated sponsor)

In the examples produced in this section, there is insufficient available capital to meet the 1-year 99.5% VaR capital requirement in both the A-rated and BB-rated sponsor cases. In the credit-risky sponsor cases, the Solvency Capital Requirement has been calculated at 12% to 18% of the liability valuation. It should also be noted that the example has only considered a sub-set of all the risks that would likely need to be considered in reality in the implementation of this capital assessment process. Longevity risk and inflation risk are perhaps the most obvious examples of omitted risks whose inclusion would further increase the Solvency Capital Requirement.

So this analysis could lead to an expectation of a Solvency Capital Requirement of 15%–20% or more of the liability valuation for DB pension funds with characteristics similar to those of the examples used in this section. Given the liability cashflows have a duration of around 16 years in the example, this magnitude of capital buffer is roughly equivalent to the impact of reducing the discount rates used in the valuation of the liability cashflows by around 100 basis points. In other words, total assets of the holistic balance sheet are required to have a market-consistent value equal to at least the value of the liability cashflows when discounted at risk-fee minus 100 basis points. The quid pro quo is that those total assets now explicitly include the market-consistent value of the sponsor covenant. As we have seen in the examples of section 2.6, the sponsor covenant valuation will be highly sensitive to sponsor credit rating, to the pace of deficit contribution funding that is assumed in the valuation.

The following section discusses what risk management actions could be incentivised for a Defined Benefit pension fund attempting to meet the capital adequacy approach developed in this paper.

4.3.1 Strategies to increase the holistic balance sheet's total assets

In general, the value of the sponsor covenant on the holistic balance sheet can be increased by increasing the size, improving the credit quality and/or accelerating the timing of the future sponsor contributions produced by the assumed contribution strategy. It is self-evident that promising to pay more and to pay it sooner will generally increase the value of the sponsor covenant. The simulation framework provides the flexibility to investigate a wide range of contribution strategies that may differ in terms of how much they promise to pay in what circumstances. It may also be possible to improve the credit quality of the promised contributions by creating some form of collateral arrangement with the sponsor, for example, through the use of contingent assets. We have already seen some activity of this kind in the UK DB pension fund sector, and a risk-based capital framework could accelerate the use of this type of mechanism. Of course, such approaches will not be free lunches for the sponsor: they essentially put the pension fund members up the pecking order of corporate debt, and in general this will increase the cost of raising corporate debt for the sponsor business.

4.3.2 Strategies to reduce the holistic balance sheet's liabilities

In some cases the holistic balance sheet framework might highlight the financial status of DB pension funds and their sponsors, making the possibility that not all promised pension payments are going to be made by the sponsor increasingly visible to trustees and members. This may mean members are more inclined to support a reduction in the cost of the promised pension payments by accepting cash values that recognise the credit risk of the sponsor and are hence lower than the value of the pension promise when valued on a risk-free basis.

4.3.3 Strategies to reduce the solvency capital requirement

In the examples of section 4.2, the capital requirement was a significant 13%–19% of the liability valuation for the credit-risky sponsors. There may be a number of financial market hedging activities that the pension fund could pursue that would significantly reduce this capital requirement. For example, the yield curve risk capital requirement could be significantly mitigated by lengthening the duration of the bond portfolio or through the use of interest rate swaps; the risky asset portfolio risk capital requirement could be significantly mitigated by switching into lower risk assets or through hedging strategies such as the use of futures contracts or put options; the general credit spread capital requirement could be mitigated through the use of index credit default swaps. Similarly, for non-market risks not considered in these examples such as longevity risk, a growing market in longevity insurance or hedging has been gradually emerging over many years that would perhaps be further incentivised by this risk-based framework.

All of these risk mitigation strategies will generally reduce the total expected return on the pension fund assets. And whilst the market-consistent 1-year VaR capital adequacy framework is unaffected by changes in the ‘real-world’ expected asset return, the sponsor may nonetheless view it as a relevant metric.

The mitigation of the sponsor-specific credit downgrade/default risk is particularly interesting. At a conceptual level, this risk exposure is very inefficient – for under-funded pensions funds, the pension fund members may be bearing very significant exposure to a single undiversified credit issuer. There is an obvious risk diversification benefit to be generated by sharing this risk with someone else.

This credit risk reduction could be achieved through holding some credit default swaps on the sponsor within the pension fund asset portfolio. Another, perhaps more radical, approach could be to take the promised contribution cashflow strategy as assumed in the valuation of the sponsor covenant and convert it into a financial asset through securitisation. In theory, this would convert the sponsor covenant valuation on the holistic balance sheet into a cash asset (which would not be reduced in value under the various stresses discussed in section 4.2). Or there may be a sponsor risk mutualisation strategy that the pension fund sector could pursue that does not directly involve securitising the contribution commitments on the financial markets, but that entails pension funds exchanging each others’ sponsor covenant ‘securities’. The Pension Protection Fund could be viewed as a government-mandated approach to this debt mutualisation idea.

In summary, there is a rich and complex set of risk management possibilities that are created from the economic insights provided by the holistic balance sheet framework, and this is a natural area for further research.