# Downside Model

# The Downside Risk Tolerance Model

The general validity and applicability of any criterion depends on the appropriateness of the underlying assumptions. The review of mean-variance portfolio theory, equilibrium asset pricing and the characteristics of real estate undertaken in this thesis, together with the empirical work pertaining thereto, suggest that there are three principal objectives that should be consider when constructing a model, namely:

- to accommodate the notion of downside-risk-aversion;
- to avoid Roll's [1977; 1978; 1980] criticism by not requiring the specification of a market portfolio; and
- aim to limit any restrictions placed on the assumed distribution of returns.

The methodology proposed here aims to accommodate these objectives. In its general form, the proposed Downside Risk Tolerance ("DRT") model -a risk dominance model in which risk is measured by a probability-weighted function - accommodates all three objectives. In this initial study, however, a special case is considered which relies on the assumption that returns are normally distributed. This assumption was made in order to simplify the problem, and enable a closed form solution to be obtained.

Specifically, three assumptions were made, that:

- returns are normally distributed for all asset categories, except real estate, where it was assumed that returns were generated by a discrete mixture of two normal distributions;
- there are no transaction costs; and
- investors reassess and consequently rebalance their portfolios at the end of each period.

Many of the positive features of the downside-variance model are shared by the proposed model, in which an investor's utility function is defined as^{13}

assuming continuity, where

the following is obtained:

therefore, given

the following general form can be derived:

where *α* is an investor's target rate-of-return; and *κ* is a measure of his risk-aversion (which must be 1). The DRT model is based on the notion that investors define risk over the entire returns distribution, placing greater importance on returns below a specified target rate-of-return. The utility function specified is risk-averse below *α*, and risk tolerant above. Figure 7.2 illustrates the proposed utility function for an investor with a target rate-of-return, *α*, equal to 10%, and a risk-aversion factor, *κ*, of either 1, 2, 4 or 8.

There is an essential difference between those models that rely on the traditional mean-variance criterion, and the proposed DRT model. The former places equal weight on returns either side of the mean, while the DRT model places greater importance on returns below a specified target rate-of-return; the degree of which is dependent on* κ*.

Under the assumptions outlined, the DRT model can be applied to a choice between:

- mutually exclusive investment opportunities; and/or
- portfolios having uncertain returns.

A closed form solution for the above general form, equation (7.3), may now be derived.

Given (7.2), where

and the above utility function, (7.1), the following form can be derived for

In order to solve for the above, let

### Solving for A

and as the following is known:

where

to solve the above, however, a change of variable must be made

y = -*κx ,* dy = -*κdx*

therefore

where

where

and

hence

However, the form is required, thus

Therefore, from equation (7.4)

substituting for *μ*' and *σ*', the following is obtained

### Solving for B

and as is known

### Solving for C

Therefore, the first requirement must be to solve for Q, where

Integrating by parts, the following is obtained

letting

To solve for q, as the following is known

where

thus

therefore as

### Solving for D

and as the following is known

therefore bring (7.5) .... (7.8) together

which simplifies to

Therefore, by varying *κ* the risk attitude of an investor may be altered. As *κ* tends to infinity, the emphasis placed on expected returns below the target rate, *α*, increases. The investor thereby becomes progressively more risk-averse in downside-variance/risk space. Alternatively, as *κ* approaches 1, the investor becomes increasingly risk tolerant. Note that if the entire returns distribution lies to the right of *α*, then an investor's level of risk-aversion, *κ*, will have no effect on expected utility. *κ* may be used to approximate a wide variety of attitudes towards the risk of falling below a target return.

By varying the required rate-of-return, *α*, emphasis is again placed on the downside-variance of the portfolio. However, its effects are relative rather than absolute. For example, if one of two pension funds has a higher actuarially defined MFR (in this case, *α*), then the DRT model would reduce the variance of expected returns, and push the distribution to the right. This would increase the density of the distribution around the target rate, and would reduce prospective returns.

The incorporation of *α* and *κ -* as required by the DRT model - begs the question as to how an investor selects them. Possible choices for the target rate-of-return in this model might be

- the nominal risk-free rate;
- the real risk-free rate (or that defined in real terms);
- duration-matched bond yield; or
- an actuarially defined rate.

The target rate-of-return chosen will depend on an investor's objectives. If a pension fund is underfunded, then trustees should be guided by the MFR. However, if the pension fund is solvent, then it may consider using the yield on bonds with a similar duration to that of the fund. In this case, *α* has been defined as the risk-free rate. *κ*, defined as an investor's degree of risk aversion against the return on the portfolio falling below the target rate-of-return, must be fixed by the model builder following discussions with the investor. Plausible values for *kappa*, as used in this model, might be 2,4, or 8.

Due to the exploratory nature of this work,^{14} analysis has been restricted to examination of the DRT model's performance and behaviour compared to Grauer & Hakansson's [1986] multi-period investment model; utilised in Chapter 3. It is also compared to Sharpe's performance measure, which is outlined in section 4.4.1. Although the DRT model's sensitivity to a was tested, to facilitate comparison with the MPI model and Sharpe measure the risk-free rate was assumed to be the required rate-of-return a . The DRT model specified in each case will be given in the following format; DRT(*α*, *κ*).

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^{13}Note that in the model's general form, the utility function may be specified to take any form.

^{14}Note that the DRT model is computationally time consuming. Each of the models outlined here

was run on a Sun SparcCentre 2000, and often required over 18 hours to complete; see Appendix D for details.