Investment Management – Capital Allocation Decision

• Excess return = realised return – risk free rate • Risk premium = the expected value of the excess return • Risk averse reject 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 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 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 (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

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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 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 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

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Example: Diversification

• 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

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• 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

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• 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. We can derive the following portfolio statistics

• The variance squares the values and returns as if you condense them, you’re essentially cancelling out the positive with the negative which is not what you want, as all the information is important, therefore you want to square is all and make it all positive, therefore you can compare the values and the risk • And once you square root the value, you will get the standard deviation (risk)

Efficient Portfolios • In the absence of a risk-free asset, there is no single best investment portfolio. It depends on the willingness to trade off return for additional risk • However, there is a set of portfolios that are considered as good investments, these are known as efficient portfolios • An efficient portfolio offers the maximum expected return for a given level of standard deviation/variance (risk)

The Investment Opportunity Set • Suppose that we vary the composition of the portfolio, letting the weight (percentage) on Newmont to range from 0 – 100%. The set of available risk-return combinations is known as the investment opportunity set • As you get higher in the weightage of Newmont in your portfolio, the risk decreases then increases, therefore need to find the optimal point in which it is efficient (50%)

The • By plotting the standard deviation of the portfolio versus the mean return, it is easy to see that some portfolios are better than other

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• None of the portfolios on the bottom part of the graph are efficient as same sd, two different portfolios • Each is dominated by a portfolio on the top part of the graph, which has the same standard deviation and offers a higher return, so the same amount of risk provides higher return • The top part of the graph is the efficient frontier, portfolios of the frontier that have a higher return, also have a higher risk – represent the portfolios that maximise E(r) at each level of portfolio risk

The Minimum-Variance Portfolio • The portfolio on the left-hand corner of the efficient frontier is the minimum-variance portfolio

• The minimum variance portfolio offers the lowest standard deviation/variance among all available portfolios • It is the natural choice for investors seeking the minimum risk in their investment • In Excel – computed with Solver • We use Solver to minimise the portfolio variance by changing he percentage of Newmont stock in the portfolio • We find that the minimum variance portfolio consists of 36.63% in Newmont and 63.37% in Microsoft • Then, the efficient frontier consists of all portfolio that have more than 36.63% invested in Newmont

The Optimal Risky Portfolio • The capital allocation line is tangent to the efficient frontier, this means that it has a single common point with the efficient frontier that corresponds to the optimal risky portfolio • The optimal risky portfolio is the portfolio in the efficient frontier that gives the maximum excess return per unit of risk. In other words, it maximises the Sharpe Ratio

Separation Property • A portfolio manager will offer the same risky portfolio to all clients, no matter their degrees of risk aversion • More risk-averse clients will invest more in the risk0free asset and less in the risky portfolio than less risk- averse clients • Portfolio choice can be separated in two independent tasks o Determination of the optimal risky portfolio, which is a purely technical task o The personal choice of the best mix of the optimal risky portfolio and the risk-free asset

Complete Portfolio

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