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Asset pricing

Asset pricing relates to the systematic determination of the value of risky securities such as equities, bonds and derivatives. Modern asset pricing models all derive from the notion that price equals the expected discounted payoffs from an asset. This is also the basis of the discounted cashflow approach that actuaries have traditionally used to value sets of cashflows.


Approaches used include:

 the capital asset pricing model (CAPM)
●  arbitrage pricing theory 

●  multifactor models 

●  option pricing models, and 

●  term structure models 
all of which were considered in subject A205. All these models are attempts to handle the effects of delay and risk in evaluating a sequence of anticipated paymen


The models therefore attempt to allow for the fact that payments:

  occur in the future 

  cannot usually be predicted with certainty.


Asset pricing models help us to perform two functions:

1. To determine whether an observed asset price is “wrong” – whether an asset is mispriced (and represents a trading opportunity for the shrewd investor). 
Thus we could use CAPM, for example, to price an asset. Then if the price predicted by CAPM is greater than the actual market price of the asset, CAPM suggests that the asset is cheap. We might therefore decide to buy the asset in the expectation that its price will rise once other investors also realise that is cheap and start buying it too. 
This is an example of quantitative analysis, to which we will refer again in Chapter 20. 

2. To determine what the price of an asset should be (where it cannot be observed). Examples here might be capital projects or unquoted investments. 


Absolute pricing

Absolute pricing prices assets by reference to exposure to fundamental sources of macroeconomic risk – ie unpredictable macroeconomic variables such as inflation, economic growth and interest rates. It is the unpredictability of these variables that leads to the unpredictability of returns on risky assets
The consumption-based and general equilibrium models (such as CAPM) are examples of this approach, which can be used to predict how prices might change if policy or economic structures change. The prices so obtained are therefore capable of an economic interpretation.


Relative pricing

Relative pricing, as exemplified by Black-Scholes option pricing and Arbitrage Pricing Theory, considers the value of an asset given the price of some other assets. Here we use as little information about fundamental risk factors as possible, and we do not ask where the prices of the other assets came from.
Thus, the Black-Scholes option pricing formula tells us what the price of a call option on a share should be given the current share price and a particular set of assumptions – most importantly that markets are arbitrage-free. This makes sense as the option price must be closely related to the share price, but it does not tell us where the share price itself comes from. More generally the predictions obtained by relative pricing may be more accurate than those obtained by absolute pricing.


general asset pricing formula

Asset pricing can be summarised in two equations:
. (1)  pt =E(mt+1xt+1) 

. (2)  mt +1 = f (data, parameters) 

  pt = asset price 

  xt +1 = asset payoff 

  mt+1 =stochasticdiscountfactor(orstatepricedeflator). 

In this way, we can separate the specification of the economic assumptions of the model (2) from the step of deciding which kind of empirical representation to pursue (1). Thus the first equation can lead to predictions stated in terms of returns, price-dividend ratios, expected return (beta) representations, moment conditions and so on.
Equation (1) is therefore consistent with the discounted cashflow approach that actuaries have traditionally used to value sets of cashflows. It is a general equation that holds for many different asset pricing models, including the risk-neutral approach used to price share options in Subject A205. It is Equation (2) that both differs between, and distinguishes between, different asset pricing models.


Asset liability modelling 2.1 Background

The process of asset liability modelling (ALM) was introduced in Subject A205 and considered further in Subject A301.
We also discussed it again in the previous chapter of this course.
Recall that an investor’s objectives will often be stated with reference to both assets and liabilities. In setting an investment strategy to control the risk of failing to meet the objectives it is therefore necessary to take account of the simultaneous variation in both the assets and the liabilities. This can be done by constructing an appropriate model to project the asset proceeds and liability outgo into the future.


In banking, ALM is used to

describe a process that actuaries would regard as closer to matching – ensuring that any mismatches between assets and liabilities are not so large as to expose them to serious risk if there is a sudden sharp market movement in the near future. Bankers have developed techniques such as Value at Risk (VaR) to help assess such risks – see Chapters 17 and 21.
So the aim here is to assess the market risk that the bank faces on its trading in bonds, equities, derivatives and commodities. For example, the Basel Regulations concerning the financial soundness of international banks encourage banks to develop market risk measurement systems based on Value at Risk, calculated using a 99% confidence interval over a 10-day holding period. The 10-day period is designed to reflect the length of time over which a bank could realise the assets in order to limit any further losses. We will discuss Value at Risk in more detail in


Actuaries, on the other hand, see ALM

as being an investigation of longer-term issues – the carrying-out of projections of assets and liabilities (and of related characteristics such as solvency levels) over periods of several years. This is because actuaries have traditionally been involved in the financial management of institutions with longer time horizons, such as life insurance companies and pension funds.


ALM is commonly used by institutional investors when

ALM is commonly used by institutional investors when setting their investment strategy. Historically an ALM study has resulted in an investment strategy following one of the two techniques:
  a benchmark consisting of specific (fixed) percentages allocated to each asset class which the investment manager is expected to follow (within suitable ranges), or 

  extraction of a “core” portfolio, typically of bonds, with the remaining assets invested in a “balanced” fashion. 
The use of ranges provides the investment manager with the freedom to undertake short-term tactical deviations away from the benchmark asset allocation in an attempt to boost investment returns based on current market conditions. 
The percentage of the total market value of the portfolio invested in each asset category (or in the core / non-core split) is typically rebalanced back to the benchmark weighting from time to time. The benchmark used for investing the assets is typically static and not directly linked to the performance of the underlying liabilities. 
A more recent development within ALM is the use of stochastic modelling to evaluate the performance of investment strategies under different scenarios. 
Investors, and pension funds in particular, are becoming increasingly focussed on hedging liabilities which has resulted in approaches such as Liability Driven Investing (LDI) gaining popularity. Under a LDI approach the underlying benchmark is more directly linked to the actual liabilities. 
LDI is discussed in Section 3 of Chapter 19. 
The benchmark for investing the assets changes as the underlying liabilities change. This type of benchmark is often referred to as a ‘dynamic liability’ benchmark and is a better reflection of the underlying liabilities than the static benchmark which have been prevalent during the 1990s and early 2000s.


Stages in an ALM exercise

The main stages in an ALM exercise are usually as follows. They are described in the context of a pension benefit scheme, but can readily be adapted to many other situations:
1. The key objectives that investment and funding policy should aim to achieve need to be clarified. These involve objectives such as:
  future ongoing funding levels 

  future solvency levels 

  future company contribution rates 

  the level of risk (performance mismatch between assets and liabilities) that is prepared to be taken. 
Here we have in mind a pension fund, although the ideas apply equally to other investors. The objectives might also specify:
  appropriate time horizons and probability levels 

  how both assets and liabilities are to be valued. 

A typical objective might be to:
. (i)  maximise the expected surplus of assets over liabilities at the end of a five-year period, with both asset and liability values calculated on a consistent discounted cashflow basis, subject to: 

. (ii)  ensuring that the value of the assets exceeds 110% of the value of the liabilities at all points during that five-year period with a probability of 99%. 

2. Suitable assumptions to use in the study need to be agreed. 

3. Data on the liabilities needs to be collected to carry out the projections. For detailed liability analysis data on individual members is required to build up an accurate assessment of the future cashflow projections. 

4. The overall nature of the liabilities is considered — an analysis of current funding level, maturity and cash flow projections under different scenarios is considered. 

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An analysis would be carried out to identify how the pension fund might progress in the future if different investment strategies were adopted.
In other words, the projected possible future values of the important outputs of the model – ie values of assets and liabilities, surplus/funding levels, contribution rates etc – would be analysed according to the assumed mix of bonds and equities.
Different asset mixes would then be analysed in more detail to assess the risks (relative to the liabilities) and the rewards of each alternative under consideration – in order to assess which look most likely to meet the investor’s investment objectives.
For example, at this stage we might consider the split of equities between domestic and overseas and the split of bonds between different durations. Of course, Stages 5 and 6 might be combined into a single stage in practice.
The results would be summarised and presented – often in graphic format, so as to make them easier to understand for those such as directors or trustees with ultimate responsibility for the investment decision.
Compared to a traditional actuarial valuation, ALM provides much more information in three (or more) extra dimensions:
1. providing projections into the future (time dimension) 

2. providing some estimate of the range of likely outcomes (probabilistic 

3. indicating the effect of changing investment strategy (asset mix dimension). 

In steps 5 and 6, ALM uses an asset model to produce stochastic simulations of returns on asset classes and other relevant economic data (such as inflation). For pension funds looking to hedge their liabilities or adopt an LDI approach to investment strategy an in-depth analysis on the impact of changes to the liability profile is undertaken. The output from the ALM model is used to calculate (approximate) liability values at different time horizons.
The models in common use are all designed to be used in Monte Carlo simulation exercises.


Presentation of ALM results

The presentation of ALM results is usually in graphic format, and looks at the distribution of a target objective (such as level of solvency or funding) resulting from an investment strategy over the projection period.
Figure 18.1 is an example of this graphic format. This graph shows how the Favourable (90th percentile), Median and Unfavourable (10th percentile) outcomes might develop through time for a single potential investment strategy. All three outcomes start from the same current solvency level.


Where an investor has decided to not fully hedge the assets and liabilities the results often show

a range of results (funding levels) based on projected asset and liability performance based on different economic scenarios. Thus, for any given output variable and any given time horizon, the results resemble an expanding “funnel of doubt” where the uncertainty associated with the output increases the further into the future projections are made. As a result, asset liability models are essentially a qualitative method for explaining risks.
For strategies that are designed to hedge the liabilities, either partially or fully, the funnel of doubt is narrower than an investment strategy that has significant mismatch between the assets and liabilities.


Choosing between different investment strategies

The final stage in any ALM exercise is to use the results obtained to choose the appropriate investment strategy to pursue. Some form of objective criterion is therefore needed to decide between the alternative possible strategies. For example, if the investor has clearly stated investment objectives specified in terms of the mean and variance of end-of-period surplus, then ALM could be used to estimate the values of the mean and variance and a choice of investment strategy subsequently made.


Using shortfall or ruin probabilities

ALM can also provide an assessment of the likelihood of a shortfall. For example, a common use of ALM is to assess “ruin probabilities” of insolvency for a fund.
However, these probabilities cannot be translated directly into assessments of the value of different strategies except by using the concept of “risk-neutral probabilities” or by adopting deflator (stochastic discounting) methodologies.
Just because the ALM results suggest that Investment Strategy A (high equity proportion) yields a higher ruin probability than Investment Strategy B (low equity proportion), it does not necessarily follow that A should be rejected in favour of B. This is because, in making a sensible choice, we need to take account of the full range of possible financial outcomes associated with each strategy. For example, Investment Strategy A might also lead to a higher average surplus than B.


We may also need to take account of

other factors, such as the general economic circumstances corresponding to any particular outcome. For example, unfavourable solvency outcomes are likely to be associated with unfavourable economic conditions, which may affect the investor in other ways than just directly via the solvency level of the fund. The value the investor places on a particular solvency outcome may therefore vary according to the prevailing economic conditions. So, we may require a different approach to simply estimating ruin or shortfall probabilities.


Asset liability mismatch reserving

Asset liability mismatch reserving is an example of the use of modelling in actuarial work. The emerging asset and liability position is projected under a range of possible conditions (economic, environmental, etc) in order to establish the extent to which assets and liabilities are mismatched. Appropriate supplementary reserves can then be set up to cover the possible levels of shortfall identified.
Here we are assessing the extent to which we are mismatched relative to a perfectly or absolutely matched position.
The modelling can, as usual, be carried out using either deterministic or stochastic methodologies.


In a deterministic framework, it is up to the modeller to

decide the nature and extent of the scenarios to be tested for the purpose of setting the reserves. At its simplest, the investigation may be restricted to the current portfolio of assets and liabilities only, and consider the impact of an immediate change in conditions, rather than involve projections of the emerging state of the fund.
For example, we might consider the impact of factors such as:
  an immediate fall of 25% in equity prices 

  an immediate increase of 1% in bond yields. 
Such an approach is often referred to as resilience testing – as it assesses the resilience of the investor to sudden changes in market conditions. 
However, with modern computer modelling power readily available, more dynamic approaches are typically adopted. These assess the resilience of the fund to changes in market conditions through time. 


Stochastic modelling

These would include the use of stochastic techniques, where multiple projections would be made in order to generate many possible future scenarios.
Certainly, stochastic modelling allows the assessment of many more scenarios and a wider range of scenarios than could be quickly and easily tested using a deterministic approach.
Most often, the stochastic element of the projections would apply to the asset portfolio and investment returns, in order to assess exposure to systematic risk. Given that a finite number of projections must be performed, assessment of the results is often carried out in the form of ruin probability, that is, the outcomes are ranked in terms of a target measure (such as the shortfall of assets relative to liabilities at a specified future date). Additional reserves are then set up at a level sufficient to cover all but a specified proportion of such shortfalls.
For example, suppose that the 10th worst outcome out of 1,000 simulations produced a shortfall of $10 million. This suggests that an extra $10 million of reserves would be required if the aim of the investor is to ensure that the probability of ruin is no greater than 1%.