Investment Management – Capital Allocation Decision • Excess return = realised return – risk free rate • Risk premium = the expected value of the excess return • Risk averse investors reject investments that are fair games or worse • Equivalently, risk averse investors would consider only speculative investments with positive risk premiums >0 • Most investors are risk averse The mean-variance criterion • Variance = squared standard deviation = risk • High variance = high sd = higher risk • Risk averse investors would rank a portfolio as more attractive if it offers a higher risk premium but lower risk • In general, if the expected return and standard deviation of two portfolios, A and B, satisfy the following inequality • And at least one inequality is strict (to rule out indifference), then A dominates B Portfolio Choice • Suppose that an investor considers the following three alternative risky portfolios: • How might investors choose among these portfolios Risk Aversion and Utility Values • When risk increases along with return, the investor will rank portfolios based on the satisfaction (or the utility) from holding the portfolio • A portfolio with more attractive risk-return profile will be assigned higher utility scores (=higher levels of satisfaction) • Note that risk averse investors will assign higher scores for higher expected returns and lower scores for higher volatility (risk) • Investors, however, differ in their risk tolerance so that the utility scores that they assign to each portfolio will also be different Utility Function • A reasonable utility ‘scoring’ system that is commensurate with an investor’s tolerance for risk and is used by the CFA institute is • WHERE U is the utility score (value_ O2/p is the variance of the portfolio and A is an index of the investor’s risk aversion • Larger values of A = higher level of risk aversion • When A = 0, investors are risk-neutral • When A < 0, investors are risk lovers • U can be interpreted as a certainty equivalent rate of return • Assume the coefficient of risk aversion is 2 1 Indifference Curve • Note that though some portfolios may have different risk-return profiles, they will provide the investors the same amount of utility • Because of the same amount of utility, the investor will be indifferent to these portfolios • The set of all portfolios to which the investors is indifferent represents the indifference curve • The higher indifference curve that lies above and to the left of another indifference curve in the mean- standard deviation plane represent a higher utility • Rational investors would always prefer portfolios on the highest feasible indifference curve • Two indifference curves for the same investors cannot intersect • The steeper the indifference curve, the more risk averse the investor • Thus, investors with different levels of risk aversion will have indifference curves with different slopes Capital Allocation Decision • Implies an asset allocation choice among broad assets classes • To fix ideas, consider the basic asset allocation choice: how to spread the investment budget between a risk free asset (T-bill) and a portfolio of risky assets • The straight line that shows the standard deviation and the expected return for all possible portfolios of the risk-free and the risky portfolio is called the Capital Allocation Line (CAL) o The upward sloping straight line depicts the risk-return combinations available by varying capital allocation, that is,n by choosing different values of y o The slope, S, of the CAL equals the increase in expected return that an investor can obtain per unit of additional standard deviation, or extra return per extra risk – known as the Sharpe Ratio o The Sharpe Ratio plotted on the Cal are the same, whether it is a risky portfolio and risk-free asset, as even though e risk-return combinations differ according to the investor’s choice of y, the ratio of reward to risk is constant • If we know the risk aversion of the investor, we can determine the proportions of her capital that should be allocated to the risk-free asset and the risky portfolio, respectively • The optimal capital allocation to the two assets will be such that it maximises investor’s utility • If uP and oP are the mean and the sd for the risky asset, and rF is the risk-free rate, then the optimal capital allocation to the risky asset, wP is given by Investment Management – Efficient Diversification The Investors Objective • Remember that the variance/standard deviation on any investment reflects the disparity between actual and expected returns • Variance/standard deviation measures the total risk of the investment • Investors may have different objectives o Maximise the expected return of the portfolio o Minimise the portfolio’s risk o Or they can consider both – expected returns + risk The Role of Diversification • Diversification is a concept which relates to the reduction of risk • A naive diversification strategy build a risky portfolio that includes a large number of assets; the larger the number the higher the diversification 2 Example: Diversification • Stock A is cyclical: it does well, when the economy doe swell • Stock B is countercyclical: it does well when other firms do poorly • Using the results of the table; • The expected return of the portfolio is given by: • o Where E(rA) is the E(r) of Stock A and E(rB) is the E(r) of Stock B • The variance of the portfolio is given by: • • Where O2A is the variance of Stock A, O2B is the variance of Stock B and OA,B is the co-variance between Stock A&B • Covariance • Measure of co-movement • OA,B in the above example tells us how Stock A & B co-move, or equivalently, how close/different is the behaviour of Stock A and B • If OA,B > 0 , then Stock A & B tend to show similar behaviour • IF OA,B < 0, then Stock A & B tend to show opposite behaviour • The high of the OA,B therefore shows the tendency in the linear relationship between Stock A & B • Note that OA,B = 0 indicates that they are unrelated and that there are no relationship Covariance v Correlation • An important disadvantage of OA,B is that it depends on the units of measurement and is therefore difficult to compare • We cannot easily interpret the strength of the linear relationship between Stock A & B by the magnitude of OA,B 3 • In order to interpret the strength of the linear relationship, we need to scale OA,B to a unitless number = the correlation coefficient Correlation • OA,B lies between -1 & +1 • If > 0 - +ve relationship between A & B • If < 0 - -ve relationship between A & B • If = 0 – uncorrelated and no relationship between A & B • If = 1 – perfect positive correlation • If = -1 – perfective negative correlation • Larger positive values of OA,B indicates stronger positive correlation • Larger negative values indicate stronger negative correlation • The correlation between any variable and itself is 1 Correlation and Diversification • The concept of correlation is the building stone of efficient diversification • Efficient diversification I when we construct risky portfolios to provide the lowest possible risk for any given level of expected return • The lower the correlations between returns on assets in a portfolio, the lower the risk of the portfolios, and thus, the higher the diversification Investment Management – Efficient Diversification As an investor, you can build an infinite amount of portfolio, but only a number will be efficient, and out of those, only one will be your ideal perfect portfolio depending on your risk aversion and preferences • Depending on your risk aversion, you will allocate your capital according to your choices A realistic example • Consider two assets: the stock of Microsoft and the stock of Newmont Mining Corp, a gold mining company. Using three years of monthly data from Jan 2009 to Dec 2011, we find the following historical price paths When observing the movements of these two companies over time, you can see that there are volatility • When comparing, you need to look at the relationship of their trend, what happens when one goes up and the change of the other, do they move together, opposite direction etc • Using the time series of returns, can you spot the degree of association between the two companies • Take all of the individual return and look at it on a scatter plot and the returns can see the relationship, when they both go up or down and find a pattern • Look at the scatter plot, we find that the correlation between the returns of the two firms is slightly negative with a correlation of 0 4 • IMPORTANCE: of more or less a negative relationship and you want a relationship that is negative, (though it is usually quite rare) as usually they are positively correlated, so theoretically, when one goes up, other goes down, a huge reduction in risk • Consider a 50-50 portfolio for the two assets, we observe that the portfolio offers a reduction in the standard deviations and, at the same time, an average return To work out mean: As it’s 50:50, you just work out the mean accordingly (though likely not always the case) However, the standard deviation is lowest is A negative correlation will reduce the risk of portfolio and does not sacrifice much of the return, as the standard deviation of the portfolio of a 50:50 portfolio is not even between the two, but is significantly reduced From naïve to optimal • The reduction of the risk in the previous example illustrates the benefits of diversification which are larger when the correlations are negative • The 50-50 portfolio is however a very simple way to spread funds across two assets • Is there a better way to build a portfolio? Portfolio Statistics • Consider again a portfolio of two assets, if w is the percentage (weight) of the first asset in the portfolio, then 1-w is the percentage of the second asset.