Before discussing the way in which market and economic risks can be modelled, it is worth considering some important characteristics of financial time series, par- ticularly in relation to equity investments.
In spite of the assumptions in many models to the contrary, market returns are rarely independent and identically distributed. First, whilst there is little obvious evidence of serial correlation between returns, there is some evidence that returns tend to follow trends over shorter periods and to correct for excessive optimism and
14.2 Market and Economic Risk 327 pessimism over longer periods. However, the prospect of such serial correlation is enough to encourage trading to neutralise the possibility of arbitrage. In other words, serial correlation does not exist to the extent that it is possible to make money from it – the expected return for an investment for any period is essentially independent from the return in the previous period, and for short periods is close to zero.
Whilst there is no apparent serial correlation in a series of raw returns, there is strong serial correlation in a series of absolute or squared returns: groups of large or small returns in absolute terms tend to occur together. This implies volatility clustering. It is also clear that volatility does vary over time, hence the development of ARCH and GARCH models.
The distribution of market returns also appears to be leptokurtic, with the degree of leptokurtosis increasing as the time frame over which returns are measured falls.
This is linked to the observation that extreme values tend to occur close together. In other words, very bad (and very good) series of returns tend to follow each other.
This effect is also more pronounced over short time horizons.
Given that exposure to equities usually comes from an investment in a portfolio of stocks, it is also important to consider the characteristics of multivariate return series. The first area of interest is correlation – or, more accurately, co-movement.
Correlations do exist between stocks, and also between asset classes and economic variables. However, these correlations are not stable. They are also not fully de- scriptive of the full range of interactions between the various elements. For exam- ple, whilst the correlation between two stocks might be relatively low when market movements are small, it might increase in volatile markets. This is in part a reflec- tion of the fact that stock prices are driven by a number of factors. Some relate only to a particular firm, others to an industry, others still to an entire market. The different weights of these factors at any particular time will determine the extent to which two stocks move in the same way.
Whilst correlations do exist (or appear to exist) between contemporaneous re- turns, there is little evidence of cross-correlation – in other words, the change in the price of one stock at timet does not generally have an impact on any other stock at time t+1. However, if absolute or squared returns are considered, then cross-correlation does appear to exist. This is a reflection of the fact that rising and falling levels of volatility can be systemic, affecting all stocks, and even all asset classes.
Similarly, extreme returns often occur across stocks and across time series mean- ing that not only are time series individually leptokurtic, but that they havejointly fat tails.
14.2.2 Modelling Market and Economic Risks
When modelling market and economic risks, the full range of deterministic and stochastic approaches can be used, although some asset classes require special con- sideration. A good example is in relation to bootstrapping. This is not necessarily appropriate for modelling the returns on bonds without some sort of adjustment.
The reason for this can be appreciated if a period of falling bond yields is con- sidered. This will lead to strong returns for bonds. However, since yields will be lower at the end of the period than at the start, the potential for future reductions in bond yields – and thus increases in bond prices – will be lower. A suitable ad- justment might be to base expected future returns on current bond yields and to use bootstrapping simply to model the deviations from the expected return.
The combination of impacts on the prices of individual securities and whole asset classes means that factor-based approaches are often used to model the returns on portfolios of stocks and combinations of asset classes. For example, a factor- based approach to modelling corporate bonds might start by recognising that the returns on this asset class can be explained by movements in the risk-free yield, movements in the credit spread, coupon payments and defaults. These can each be modelled and combined to give the return for the asset class. The interactions of these four factors and other financial variables would also need to be modelled.
For example, defaults could be linked to equity market returns. With a factor-based approach, complex relationships between asset class returns arise because of the linkages between the underlying factors. These models can also be specified to include heteroskedasticity through the use of ARCH or GARCH processes.
Rather than trying to determine which factors drive various securities or asset classes, it is possible instead to use a data-based multivariate distribution. The most common approach is to assume that the changes in the natural logarithms of asset classes are linked by a multivariate normal distribution. If this approach were being used to describe the returns on a range of asset classes – say UK, US and Eurozone and Japanese equities and bonds, the following process could be used to generate stochastic returns:
• decide the scale of calculation, for example daily, weekly, monthly or annual data;
• decide the time frame from which the data should be taken, choosing an appro- priate compromise between volume and relevance of data;
• choose the indices used to calculate the returns, ensuring that each is a total re- turn index, allowing for the income received as well as changes in capital values;
• calculate the returns for each asset class as the difference between the natural logarithm of index values;
14.2 Market and Economic Risk 329
• calculate the average return for each asset class over the period for which data are taken;
• calculate the variance of each asset class and the covariances between them; and
• simulate series of multivariate normal distributions with the same characteristics using Cholesky decomposition.
One issue with this approach is that it can involve calculating a very large num- ber of data series if the number of asset classes increases, or if individual securities are to be simulated. An alternative approach is to use a dimensional reduction tech- nique instead. In this case, principal component analysis (PCA) could instead be used to determine extent to which unspecified factors affect the returns. The fol- lowing procedure could then be followed instead:
• decide the scale of calculation, for example daily, weekly, monthly or annual data;
• decide the time frame from which the data should be taken, choosing an appro- priate compromise between volume and relevance of data;
• choose the indices used to calculate the returns, ensuring that each is a total re- turn index, allowing for the income received as well as changes in capital values;
• calculate the returns for each asset class as the difference between the natural logarithm of index values;
• calculate the average return for each asset class over the period for which data are taken;
• deduct the average return from the return in each period for each asset class, leaving a matrix of deviations from average returns;
• use PCA to determine the main drivers of the return deviations;
• choose the number of principal components that explains a sufficiently high pro- portion of the past return deviations;
• project this number of independent, normal random variables with variances equal to the relevant eigenvalues;
• obtain the projected deviations from the expected returns for each asset class by weighting these series by the appropriate elements of the relevant eigenvectors;
and
• add these returns to the expected returns from each asset class.
The PCA approach is particularly helpful if bonds of different durations are be- ing modelled, since a small number of factors drives the majority of most bond returns. In particular, changes in the level and slope of the yield curve explain most of the change.
A drawback with the PCA approach is that it essentially assumes that the un- derlying data are linked through by a multivariate normal distribution. This as- sumption comes about because PCA is performed on a covariance matrix, which ignores higher moments such as skew and kurtosis. Having already seen that the data series for many asset classes are not necessarily normal, it should be clear that this could pose problems. However, if all series are to be modelled instead, then it is also possible to use an approach other than a jointly normal projection of the natural logarithms of returns. This could mean using another multivariate distribu- tion, or taking more tailored steps. In particular, copulas could be used to model the relationship between asset classes, whilst leptokurtic or skewed distributions can be used to model each asset class individually. The extent to which the process moves away from a simple multivariate normal approach depends to a large extent on the volume of data that is available – it is difficult to draw any firm conclusions on the shape of the tails of a univariate or multivariate basis if only a handful of observations are available on which to base any decision.
14.2.3 Expected Returns
When carrying out stochastic or deterministic projections, assumptions for future returns are required. Whilst the returns experienced over the period of data analysis might be used, they typically reflect only recent experience rather than a realistic view of future returns. The forward-looking view can come from subjective, fun- damental economic analysis, or through quantitative techniques.
Whatever approach is used, it is important that tax is allowed for, either in the expected returns or at the end of the simulation process. This is particularly impor- tant if different strategies are being compared – such a comparison should always be made on an after-tax basis.
Government Bonds
For domestic government bonds that are regarded as risk free, a reasonable esti- mate for the expected return can be obtained from the gross redemption yield on a government bond of around the same term as the projection period. This represents the return received on a bond if held until maturity, subject to being able to invest any income received at the same rate. Better estimates of expected bond returns are discussed in Section 14.3. Note that the expected return is not based on the gross redemption yield on a bond of the same term as those held. The expected return on a risk-free bond for a given term is approximately equal to the yield on a bond of that term, and if a different return were available over the same period on a bond with a different term, then this would imply that an arbitrage opportunity existed.
Some adjustment might be made if the projection period and term of investments
14.2 Market and Economic Risk 331 differ significantly to allow for a term premium. If yields on long term bonds are consistently higher than those on short term bonds, then this might be due to in- vestors perceiving long-date assets as being riskier due to greater interest rate or credit risk, and requiring an additional premium to compensate. However, since long-dated bonds arelessrisky for some investors – in particular those with long- dated liabilities – term premiums cannot be taken for granted, and their presence will vary from market to market.
A better estimate of the expected return could be obtained by constructing a forward yield curve take into account the term structure of the bonds held in a port- folio, but this level of accuracy is spurious in the context of stochastic projections.
For overseas government bonds that are also regarded as risk free, a good ap- proximation for the expected return in domestic currency terms is, again, the gross redemption yield on adomesticgovernment bond of around the same term as the projection period. If it were anything else, then this would again imply an arbitrage opportunity. The fact that domestic and overseas government bond yields differ can be explained by an expected appreciation or depreciation in the overseas currency.
Corporate Bonds
For risky bonds where there is a chance of default, the expected return must be altered accordingly. Such bonds will usually be corporate bonds, although many government bonds are not risk free, particularly if there is any doubt over the ability of that government to be able to honour its debts. A starting point is the credit spread, which represents the additional return offered to investors in respect of the credit risk being taken. There are a number of ways in which the credit spread can be measured, the three most common being:
• nominal spread;
• static spread; and
• option adjusted spread.
The nominal spread is simply the difference between the gross redemption yields of the credit security and the reference bond against which the credit security is being measured (often a treasury bond). This is attractive as a measure because it is a quick and easy measure to calculate. It ignores a number of important features and should be regarded as no more than a rule of thumb if the creditworthiness of a particular stock is being analysed; however, for the purposes of determining an additional level of return for an asset class as a whole, it offers a reasonable approximation.
A more accurate measure is the static spread. This is defined as the addition to the risk-free rate required to value cash flows at the market price of a bond. On a basic level, this appears to be similar to the nominal spread mentioned above;
however, rather than just considering the yield on a particular bond, this approach considers the full risk-free term structure and the constant addition that needs to be added to the yield at each duration in order to discount the payments back to the dirty price.
Finally, the option adjusted spread (OAS) is similar to the static spread, but rather than looking at a single risk-free yield curve, it allows for a large number of stochastically generated interest rates, such that the expected yield curve is con- sistent with that seen in the market. The reason this is of interest is that by using stochastic interest rates it is possible to value any options that are present in the credit security that might be exercised only when interest rates reach a particular level.
One of the interesting features of credit spreads is that they have historically been far higher than would have been needed to compensate an investor for the risk of defaults, at least judging by historical default rates. One suggested reason is that the spread partly reflects a risk premium. This is a premium in a similar vein to the equity risk premium, effectively a ‘credit beta’, designed to reward the investor for volatility relative to risk-free securities.
Another suggestion has been that the spread is partly a payment for the lower liquidity of corporate debt when compared to government securities. This means that the reward is given for the fact that it might not be possible to sell (or at least sell at an acceptable price) when funds are required. Similarly, it costs more to buy and sell corporate bonds than to buy and sell government bonds, and part of the higher yield on corporate debt may be a reflection of this. The size of the effect would depend on the frequency with which these securities were generally traded.
Another argument used is that although credit spreads have historically been higher than justified by defaults, this does not reflect how bad markets could get.
In other words, the market could be pricing in the possibility of as-yet-unseen ex- treme events. Furthermore, the pay-off profile from bonds is highly skewed – the potential upside is limited (redemption yields cannot fall below zero), but the po- tential downside is significant (the issuer can default). If investors dislike losses more than they like gains, then this might mean that investors require additional compensation for this skewness.
Taxation can also play a part in the yield differences in some jurisdictions. Cor- porate bonds are sometimes treated less favourably than government bonds for some individuals. For example, capital gains on government bonds might be tax free whilst being taxed on corporate bonds. This effect might result in spreads be- ing higher than otherwise might be the case.
There is also evidence that the correlation between credit spreads and interest rates is typically negative. For example, when an economy is growing quickly and credit spreads might be lower, then the expectation might be that interest rates
14.2 Market and Economic Risk 333 would be raised. For investors focussing on absolute returns, this negative correla- tion might be reflected in a lower required credit spread as it offers diversification for an investor seeking absolute returns.
As valid as these arguments are, they suggest that the additional risk should be reflected in the volatility side of any modelling, or in liquidity planning, rather than in an adjustment to the additional return. This risk premium should be based on the credit spread, historical default rates and, if relevant, taxation.
Historical Risk Premiums
Whilst it is possible to derive expected returns from market prices for bonds – albeit with adjustments for corporate bonds – the uncertainty surrounding the income of other investments means that such an approach is not possible for them. So for eq- uities and property in particular, a different approach is needed. One approach is to consider the historical risk premium available for these asset classes. If this is being done on an annual basis, the return on a risk-free asset class should be deducted from the rate of return on the risky asset class each year for which historical data are available. The arithmetic average of these annual risk premiums should then be taken and this can be used as the annual expected return for any forward-looking analysis.
However, if past returns involve any changes as to the views on an asset class, then this will result in an historical premium that it is not reasonable to anticipate in the future. For example, if investors anticipated higher levels of risk at the start of the period than they did at the end, then prices would increase accordingly. This would raise the historical risk premium as prices would be higher at the end relative to their starting value, but the higher starting value would lower the expected future premium as a higher price would be paid for future earnings. This suggests that the historical risk premium should be altered for this re-rating.
The Capital Asset Pricing Model
One way of ensuring that risk premiums for different asset classes are consistent is to price them according to the capital asset pricing model (CAPM). This has already been discussed in Chapter 13 in the context of choosing a discount rate, but the model can also be used to calculate consistent expected returns across asset classes. To recap, the capital asset pricing model links the rate of return on an indi- vidual investmentX,rX, with the risk-free rate of returnr∗and the return available from investing in the full universe of investment opportunities,rU, as follows:
rX =r∗+βX(rU−r∗), (14.1) whereβX =σXρX,U/σU,σX and σU being the standard deviations of investment X and the universe of investment opportunities respectively. This approach still