A.3.8 Separation Principle

If we now allow investors to borrow or lend, they will ‘mix’ the risk-free asset with the risky bundle represented by M, to get to their preferred position along the CML. Hence, investors’

preferences determine at which point along the CML each individual investor ends up. For example an investor with little or no risk aversion would end up at a point like B (Figure I.A.3.9), where she borrows money (at r) to augment her own wealth and then invests the borrowed money and all her own wealth in the risky bundle represented by M (but she still holds all her n risky assets in the fixed proportions w^*_i ).

The investor makes two separate decisions:

The PRM Handbook – I.A.3 Capital Allocation

1. Knowledge of expected returns, variances and covariances determines the efficient frontier.

The investor then determines point M as the point of tangency of the line from r to the efficient frontier. All this is accomplished without any recourse to the individual’s preferences. All investors, regardless of preferences (but with the same view about expected returns, etc.), will

‘home in’ on the portfolio proportions (w ) of the risky securities represented by M. All investors hold the market portfolio, or more correctly, all investors hold their n risky assets in the same proportions as their relative value in the market. Thus, if the value of Microsoft shares constitutes 10% of the total stock market valuation, each investor holds 10% of her own risky portfolio in Microsoft shares.

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2. The investor now determines how she will combine the market portfolio (consisting of a bundle of n risky assets) with the safe asset. This decision does depend on her subjective risk–

return preferences. At a point such as A (Figure I.A.3.9) the individual investor is reasonably risk-averse, and holds most of her (dollar) wealth in the safe asset and puts only a little into the market portfolio (in the fixed optimal proportions w ). In contrast Ms B is less risk-averse than Ms A and ends up at B (to the right of M), with a levered portfolio (i.e. she borrows to increase her holdings of the market portfolio in excess of her own initial wealth). An investor who ends up at M is moderately risk-averse, and puts all of her wealth into the market portfolio and neither borrows nor lends at the risk-free rate.

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Note that Ms A and Ms B both have the same Sharpe ratio as an investor at M – in part this is because all of these investors hold the same risky asset proportions. Hence, although Ms B holds more risk ƳB > ƳA than Ms A, she also expects a higher return ERB > ERA – as in Figure I.A.3.9.

These two effects just offset each other, so that the expected (excess) return per unit of risk is the same for both Ms A and Ms B and for the investor at M.

I.A.3.9 Summary

It should now be clear how portfolio theory can be used to determine the optimal asset allocation strategy for any investor. The portfolio manager first agrees with the investor on the likely future outcomes for expected returns ERi, variancesƳi and covariancesƳij (or correlationsƱij). These are likely to be based on historic values plus ‘hunches’ about particular markets. The analysis might initially be applied only to the industry asset allocation problem, namely the optimal proportions to

‘place’ in the major domestic industries (e.g. chemicals, engineering, services, electrical). The efficient frontier is constructed and, if the investor does not wish to borrow or lend, the analyst can suggest alternative risk–return combinations available along the ‘curved’ efficient frontier.

The investor will choose one of these (e.g. the minimum variance point) based on her own risk–

return preferences.

For Evaluation Only.

However, if the investor is willing to borrow or lend, she can expand her ‘opportunity set’ by moving up and down the CML. The optimal asset allocation among the risky assets is then given by her own ‘market portfolio’ M (and proportions ). She can then decide whether to put some of her wealth in the risk-free asset or whether to borrow money to provide additional funds to invest in her chosen ‘bundle’ of risky assets – as in the ‘baseline’ case, the latter decision depends on her own level of risk tolerance.

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wi

Mean–variance portfolio theory assumes investors can borrow or lend at the risk-free rate. It gives the optimal proportions in which risky assets are held. The ‘baseline’ case assumes all investors have the same view about the ‘market-determined variables’, expected returns, variances and covariances. Mean–variance portfolio theory makes the following predictions:

(i) All investors hold their risky assets in the same proportions regardless of their preferences for risk versus return. These optimal proportions constitute the market portfolio.

(ii) Investors’ preferences enter in the second stage of the decision process, namely the choice between the ‘fixed bundle’ of risky securities and the risk-free asset. The more risk-averse the individual, the smaller the proportion of her wealth that will be placed in the bundle of risky assets. A less risk-averse investor will borrow money at the risk-free rate to make additional investments in the risky-assets. All investors hold the same proportions in the risky assets but the dollar amount in each risky asset depends on the investor’s personal risk tolerance.

(iii) If investors cannot borrow or lend, they can only choose alternative combinations of risky-asset proportions, along the ‘curved’ efficient frontier. The chosen mix of risky assets will depend on the investor’s individual risk tolerance.

(iv) If investors have different forecasts of expected returns, variances and covariances, they will have their ‘own’ efficient frontier. If they are willing to borrow or lend, then they will choose their own particular optimal risky-asset proportions (i.e. their own ‘tangent portfolio) – these proportions will differ for different investors, because the efficient frontier differs for each investor.