# Alternative Approach

# An Alternative Approach: The Downside Risk Tolerance Model

## Introduction

As discussed in Chapters 3 to 5, the foundation of investment theory rests on the belief that there exists a relationship between risk and return. Risk and its assessment play an important part in everyday life. Without risk

'… financial markets would consist of the exchange of a single instrument each period, the communications industry would cease to exist, and the profession of investment banking would reduce to that of accounting.'

(Machina & Rothschild [1990, p. 227])

An increase in 'risk' is assumed to be compensated by an increase in return, and so with a decrease in return, assumed to be compensated by a concomitant decrease in risk. According to Markowitz [1959, p. 129]

'A portfolio is inefficient if it is possible to obtain (a) higher expected (or average) return with no greater variability of return, or obtain (a) greater certainty of return with no less average or expected return.'

The existence of a positive relationship between risk and return would be difficult to deny. However, the premise underlying the above quote, and much of investment theory, is that risk is captured in the second moment ─ the variance ─ of a portfolio. An assumption with which this chapter takes issue.

Building and fire insurance are used to remove rare but large downside risks corresponding to negative returns. However it is rarely, if ever, that there is available the offer of 'insurance' to deal with unexpected large positive returns, such as winning the National Lottery or a Nobel Prize.^{1}

These types of uncertainty are generally not thought of as risk. Machina & Rothschild [1990] provided a detailed discussion of risk as seen from the viewpoint of economic theory. They concluded that when returns are drawn from alternative distribution functions, a risk-averse investor always experiences a decline in expected utility when moving from an asset with a certain return, *ω*, to an asset with an uncertain return *E*[*r*] = *ω*.

The basis of mean-variance analysis dates back to Hicks [1946], but did not attain its full potential and recognition until it was formalised by Markowitz [1952; 1959]. Since then, the mean-variance approach has dominated portfolio analysis. Its basic premise is that investors should limit consideration to those portfolios that are mean-variance efficient. That is, those portfolios for which there does not exist an alternative portfolio having at least as high a mean and a lower variance. Such a procedure limits the choice to portfolios in the mean-variance efficient set. To determine an optimal portfolio from amongst this set requires, in addition, that the investor express preferences between mean and variance trade-offs. As previously discussed, if:

- utility functions are quadratic; or
- the utility function is concave and asset returns are multivariate normally distributed

the mean-variance approach provides the set of optimal portfolios as the expected utility approach.

Thus, mean-variance portfolio theory relies on the assumption that, *inter alia*, security returns are normally distributed or that investors have a quadratic utility function. Research has suggested, however, that the shape of an investor's utility function may change dependent on the form of expected returns, with an investor showing greater risk-aversion in the lower part of a returns distribution (Moser [1990] and section 5.3.6). In addition, as discussed there is doubt over the assumption of multivariate normal distribution of returns. This may therefore lead to concern over downside risk, with the minimisation of that becoming the objective.

## Normality

As highlighted in sections 5.3.6 and 6.4.1, the returns on certain types of assets are neither normal nor log normally distributed.^{2} The most obvious type is options. For example, a call option allows an investor, for a price, to participate in the positive returns on an underlying asset, but avoid the asset's negative returns. Essentially, a call option truncates the underlying asset's return distribution at the point where losses begin. The returns on a call option, or on a plethora of other derivative securities^{3} are almost never normally distributed.

Moreover, some assets have options embedded in them. For example, callable bonds can be redeemed at the discretion of the issuers, within a previously specified period. However, issuers will do so only if interest rates move in their favour. Mortgage-backed securities have similar provisions for early repayment.^{4} As discussed in section 6.4.4, it could be argued that real estate returns behave as if they possess an embedded option. However, even if this is not the case, it would be difficult to argue that they are normally distributed.

## Quadratic Utility

Domar & Musgrave [1944] associated the risk involved in investing under conditions of uncertainty with the possibility of sustaining a loss, rather than with the measure of total variance. They suggested that investors should measure risk solely on the basis of that possibility:

'Of all possible questions which the investor may ask, the most important one, it appears to us, is concerned with the probability of actual yield being less than zero, that is, the probability of loss. This is the essence of risk. '

Domar & Musgrave developed a quantitative index of risk affected both by the probability of obtaining a result less than zero, and by the size of the possible loss. According to their model, which emphasises the negative segment of the probability distribution, a larger dispersion does not necessarily involve a greater risk.

Baumol [1963] argued that variance *per se* does not indicate the degree of risk. According to Baumol, risk mainly reflects the possibility that the random variable may have extremely low values. If the expected return from an investment is high relative to its standard deviation, he suggested that the risk index be the difference between the standard deviation multiplied by a factor representing an investor's attitude to risk, and the expected return.

It is therefore suggested that investors should be more concerned with the probability or chance of incurring a loss, or a smaller than expected return from an investment decision, than total variance of returns. This is applicable to investing institutions, due to the structure of their liabilities; see sections 2.2 and 2.3. Institutions should be less concerned with achieving a higher than expected return. Their main concern should be with downside risk i.e. that of achieving a smaller return than expected. According to Gitman [1987, p. 333] institutional risk may then be defined as:

'the chance of achieving a less than expected rate-of-return.'

This leads us to consider alternative models that are specifically concerned with down-side risk, and which may better describe the risk/return relationship. The simplest of these is 'shortfall probability'. It measures the chance of an investment's return falling short of a target rate-of-return. Essentially, it is the proportion of the probability distribution lying to the left of the target return.

As outlined in section 5.3.6 semi-variance and downside-variance are more complex downside risk measures, and are superior to shortfall probability as they take into account the distance of each downside return from the expected or target rate-of-return.

Downside-variance, as discussed by Mao [1970], Hogen & Warren [1972], and Porter, Bey & Lewis [1975], has several positive features, including a close correspondence with fund managers criteria of choice. Swalm [1966] and Mao [1970] provided evidence that downside-variance as a measure of portfolio risk is more consistent with risk, as understood by fund managers, than is variance. The matching of long-term liabilities is fundamental to the management of pension funds. Accordingly, the arguments in favour of using the downside-variance model as a method of controlling risk are particularly relevant to fund managers. According to Nantell & Price [1979];

'Semi-variance has the advantage of capturing the financial manager's intuitive notion of risk as a failure to meet some minimum target. Unlike variance, it is influenced only by returns below the target rate.'^{5}

Downside-variance as a risk measure is defined as follows:^{6}

with *τ* being the target rate-of-return as suggested by Markowitz [1959, p. 188].

This risk measure is expounded on the notion that investors discern risk as the failure to achieve a target rate-of-return, *τ*, an investor being risk-averse below, and risk neutral above, the target return. This notion may be particularly relevant to life and pension funds, given the structure of their liabilities. In addition, the Pensions Act 1995 has imposed a MFR which increases the pressure on pension fund trustees to attain a minimum rate-of-return.^{7}

Figure 7.1 illustrates a dichotomous utility function, which is quadratic below, and linear above, *τ*; the target rate-of-return. This provides the theoretical foundation in expected utility/return space, of the downside-variance model. As discussed, for this class of utility function, expected utility maximisation and downside risk-minimisation objectives have been shown to be equivalent.^{8} Nantell & Price [1979] demonstrated that under the assumption of the multivariate normal distribution of returns, equilibrium rates-of-return are equal whether a variance or downside-variance definition of portfolio risk is used or not. This follows from a proof in their paper that measures of relative asset risk - asset risk relative to efficient portfolio risk - are equal in the two models.

If returns are not normally distributed, however, then the two measures are not equivalent. Although as highlighted in section 5.3.6, the downside-variance model suffers from the unacceptable assumption, that investors are concerned exclusively with downside risk.^{9} Consequently, the variability of returns above the target return is not a determinant of systemic risk. This differs from the CAPM, which places equal importance on the returns distribution above and below the target return.

In contrast with these two models, it is suggested that return variability above the target return may not be as crucial in determining risk as the variability below the target. However, it does offer useful information, especially if the assumption of quadratic utility is relaxed. It could be argued that a utility function, mapping an institution's attitude towards risk and return, should be concave up to the target rate-of-return, and 'risk-tolerant' thereafter. To paraphrase, institutions may be risk-averse up to a target rate-of-return, and risk-tolerant - where they require an extra unit of return for each additional unit of risk - above it. This is consistent with the hypothesis of Friedman & Savage [1948], which stated that as an investor's wealth increases, his marginal utility of wealth decreases i. e. as wealth increases, absolute risk-aversion decreases. ^{10}

As mentioned, the critical factor missing from the standard CAPM is the concept of a target rate-of-return and its role in determining systemic risk. A shortcoming of the downside-variance model is that it fails to take account of return distributions above the target rate-of-return.^{11} The remainder of this chapter is therefore devoted to proposing an alternative model.

Table 7.1 contains a summary of the asset category symbols used in this chapter, which proceeds as follows. The next section outlines the general form of the proposed Downside Risk Tolerance ("DRT") model, deriving a closed form solution. The model is then used to construct and rebalance portfolios composed of UK equities, commercial real estate, long-term government bonds and a risk-free asset over the period 1977-1994. The results are then compared with those from the traditional MPI model, contained in Chapter 3. The DRT model is then applied to the performance evaluation of a sample of UK quoted real estate companies, with its behaviour discussed in relation to the Sharpe performance measure (Sharpe [1966]).^{12} The last section contains a summary and concluding comments.

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^{1}This is an absurd concept, due both to the issues of adverse selection and moral hazard. An individual would be selling the right to a possible future gain that depended to an extent upon his future actions. However, it serves as an illustration.

^{2}See section 4.3.1 on page 89 for a discussion on measuring return.

^{3}See for example, Hull [1993].

^{4}See for example, Bartlett [1988] for a discussion on the behaviour of mortgage-backed securities.

^{5}It is inferred from Nantell & Price's paper that the discussion is on downside-variance, as defined in this thesis, rather than on semi-variance. See footnote 22 on page 147.

^{6}From equation (5.25), on page 147.

^{7}See section 2.3.

^{8}However, Fishburn [1977] pointed out that risk-neutrality above the target return is limited to few returns distributions.

^{9}See footnote 25 on page 149.

^{10}For a complete review of decisions under certainty, risk, and uncertainty, see Moser [1990].

^{11}It should be noted here that the theoretical justification for alternative asset-pricing models, as discussed in footnote 8, argues that the above models do not take account of other valuable sources of information.

^{12}Discussed in section 4.4.1 on page 93.