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Multiple choice questions (15) and short answer/open response questions (3) - both calculation and discussion questions

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Multiple choice questions (15) and short answer/open response questions (3) - both calculation and discussion questions. FINC5001 Module 4 – Investor Preferences, Diversification, and Portfolios Tutorial Solutions Question 1 You have just finished your studies at university and begin investing your savings as a way of applying what you have learnt in your finance course. You have collected the following information for two large stocks in the Australian market: Expected return Standard Deviation Commonwealth Bank of Australia 14.91% 5.33% CSL Limited 16.85% 7.14% You decide to invest half your money in each share. a) Given the correlation between CBA and CSL is 0.55, what is the expected return and standard deviation of your portfolio? ??? = ?? = ?1 ?1 + ?2 ?2 + ? + ?? ?? = (0.5 × 0.1491) + (0.5 × 0.1685) = 0.1588 = 15.88% ??2 = ?12 ?12 + ?22 ?22 + 2?1 ?2 ?1 ?2 ?1,2 = (0.52 × 0.05332 ) + (0.52 × 0.07142 ) + (2 × 0.5 × 0.5 × 0.0533 × 0.0714 × 0.55) = 0.003031258 ?? = √0.003031258 = 0.055056861 = 5.506% b) What would be your portfolio’s expected return and standard deviation if the correlation between the stocks is +1? What about if the stocks are perfectly negatively correlated? Perfect positive correlation (ρij = +1) ??? = 15.88% (???? ?? ??????????) ??2 = (0.52 × 0.05332 ) + (0.52 × 0.07142 ) + (2 × 0.5 × 0.5 × 0.0533 × 0.0714 × 1) = 0.003887522 ?? = √0.003887522 = 0.06235 = 6.235% ??, ?? = (0.5 × 0.0533) + (0.5 × 0.0714) = 0.06235 = 6.235% Perfect negative correlation (ρij = -1) ??? = 15.88% (???? ?? ??????????) ??2 = (0.52 × 0.05332 ) + (0.52 × 0.07142 ) + (2 × 0.5 × 0.5 × 0.0533 × 0.0714 × −1) = 0.000081902 ?? = √0.000081902 = 0.00905 = 0.905% c) What conclusions can you draw about the impact of correlation between assets on portfolio risk and return? Correlation Portfolio risk Conclusion Perfectly positive, +1 6.235% No benefits to diversification. Perfectly negative, -1 0.905% Maximum benefits to diversification. Positive, +0.55 5.506% Some benefits to diversification. Question 2 You can form a portfolio of two assets (A and B) that have the following the characteristics: Asset A B Expected Return 12% 16.50% Standard Deviation 18% 37% Covariance 0.013986 If you require an expected return of 14.12%, what weighting should you hold each asset? ??? ?? = (1 − ?? ) ??? = ?? = ?1 ?1 + ?2 ?2 + ? + ?? ?? 0.1412 = [?? × 0.12] + [(1 − ?? ) × 0.165] 0.1412 = 0.12?? + 0.165 − 0.165?? −0.0238 = −0.045?? ?? = 0.52888888 ?? = 0.52889 = 52.889% ??? ?? = 1 − 0.52889 = 0.47111 = 47.111% Question 3 Mr Jorge Sorrows is considering retirement and investing some funds into a portfolio of investments with the following returns and standard deviations: Return (%) Standard Deviation (%) $ Value S&P/ASX200 index fund 10 14 450,000 International shares fund 14.5 28 2,250,000 Emerging market fund 21 26 800,000 Treasury bonds 6 0 1,500,000 a) Based on this proposed initial portfolio composition, what would be the expected return on Mr Sorrows’ portfolio? ????? ????? = $450 + $2,250 + $800 + $1,500 = $5,000 ??? = ?? = ?1 ?1 + ?2 ?2 + ? + ?? ?? 450 2250 800 1500 =( × 0.100) + ( × 0.145) + ( × 0.210) + ( × 0.060) 5000 5000 5000 5000 = 0.12585 = 12.585% b) Now he is considering a re-allocation of his investments to include more Treasury bonds. If he transfers all his money out of the emerging market and international share funds into Treasury bonds, what would be the expected return and standard deviation of this resulting portfolio? ??? = ?? = ?1 ?1 + ?2 ?2 + ? + ?? ?? 450 4550 =( × 0.100) + ( × 0.060) = 0.06360 = 6.360% 5000 5000 ??2 = ?12 ?12 + ?22 ?22 + 2?1 ?2 ?1 ?2 ?1,2 ??2 = [( 450 2 4550 2 450 4550 ) × 0.142 ] + [( ) × 02 ] + (2 × × × 0.14 × 0) 5000 5000 5000 5000 450 2 ) × 0.142 ] + (0) + (0) = 0.00015876 = [( 5000 ?? = √0.00015876 = 0.01260 = 1.260% c) Mr Sorrows now wants to increase his expected rate of returns. He decides to invest 75% of all his money into the international shares fund and the rest into the emerging market fund. What would be expected return and standard deviation of this portfolio if the funds have a covariance of 0.049504? ??? = ?? = ?1 ?1 + ?2 ?2 + ? + ?? ?? = (0.75 × 0.145) + (0.25 × 0.21) = 0.16125 = 16.125% ??2 = ?12 ?12 + ?22 ?22 + 2?1 ?2 ?1 ?2 ?1,2 ??2 = (0.752 × 0.282 ) + (0.252 × 0.262 ) + (2 × 0.75 × 0.25 × 0.28 × 0.26 × 0.049504 ) 0.28 × 0.26 = (0.752 × 0.282 ) + (0.252 × 0.262 ) + (2 × 0.75 × 0.25 × 0.049504) = 0.0441 + 0.004225 + 0.018564 = 0.066889 ?? = √0.066889 = 0.25863 = 25.863% d) If Mr Sorrows would like to diversify his holdings and have equal amounts invested in Treasury bonds, the international shares fund and emerging market fund. What would be the expected return and standard deviation of this portfolio? ??? = ?? = ?1 ?1 + ?2 ?2 + ? + ?? ?? 1 1 1 = ( × 0.145) + ( × 0.21) + ( × 0.06) = 0.13833 = 13.833% 3 3 3 ??2 = ?12 ?12 + ?22 ?22 + ?32 ?32 + 2?1 ?2 ?1,2 + 2?1 ?3 ?1,3 + 2?2 ?3 ?2,3 Note: because the Treasury bonds have a standard deviation of 0, its covariance with the other assets is also 0. ??2 1 2 1 2 1 1 2 = [( ) × 0.28 ] + [( ) × 0.262 ] + 0 + (2 × × × 0.049504) + 0 + 0 3 3 3 3 = 0.027223 ?? = √0.027223 = 0.16499 = 16.499% Question 4 Distinguish between total risk, systematic risk and unsystematic risk. What risk are you exposed to if your money is invested entirely in the shares of one company? What if you invest in a fully diversified portfolio? An asset’s total risk is the sum of it systematic and unsystematic risk and can be measured by its variance (σ2) or standard deviation (σ). Systematic risk (also known as market risk, or non-diversifiable risk) refers to risk factors that affect the whole market, such as major changes in government economic policy, war, the state of the economy (growing or recession) and share market booms and crashes. An asset’s beta (β) measure its systematic risk. Unsystematic risk (also known as firm-specific risk, diversifiable risk, or unique risk) refers to risk which is particular to the asset (or a small group of assets). These factors include positive NPV factors, such as successful new products, and factors that tend to reduce share prices such as industrial accidents, strikes, and lawsuits. If we held shares from only a single company, then the value of our investment would fluctuate because of company specific events (unsystematic risk factors) as well as market wide, systematic risk factors. Hence, we are exposed to the total risk of the asset. Whereas, if we hold a large portfolio of shares, some of the shares in the portfolio will go up in value because of positive company specific events and some will go down in value because of negative company specific events. The net effect is these effects will tend to cancel each other out, and so unsystematic risk can be eliminated by holding the asset as part of a diversified portfolio. However, we will still be exposed to systematic risk because a systematic risk affects almost all assets to some degree. Question 5 The following portfolios are plotted in the risk-return space: A B C D E F G H I E(r) (%) 4.25 5 6.25 7.75 8.25 8.75 9.5 9.5 11.25 SD (%) 12.25 11.5 10.5 12.5 14.5 14.5 16 17.5 21 a) Which of these portfolios are efficient? Portfolios A, B, E and H are inefficient. The rest are efficient portfolios. b) Suppose you can also borrow and lend at a risk-free interest rate of 3.75%. Which of the above portfolios has the highest Sharpe ratio? ?????? ????? = ??? − ?? ?? Portfolio G has the highest Sharpe ratio = c) 0.095−0.0375 0.16 = 0.359375 Suppose you are prepared to tolerate a standard deviation of 14.5%. What is the maximum expected return that you can achieve if you cannot borrow or lend? You should select the portfolio up to the highest amount of risk tolerable, which is Portfolio F. This has an expected return of 8.75%. Portfolio E (with the same amount of risk) would not be chosen by a rational investor as it is inefficient. d) What is your optimal strategy if you can borrow or lend at 3.75% and are prepared to tolerate a standard deviation of 14.5%? What is the maximum expected return that you can achieve with this risk? You should select some combination of Portfolio G and the risk-free asset. 2 2 ??2 = ??? ??? + ??2 ??2 + 2??? ?? ??? ?? ???,? ??2 = 0 + ??2 ??2 + 0 ?? = ?? ?? 0.145 = ?? 0.16 ?? = 14.5 = 0.90625 ??? ??? = 1 − 0.90625 = 0.09375 16 ??? = (0.90625 × 0.095) + (0.09375 × 0.0375) = 0.08961 = 8.961% FINC5001 Module 4 Slides Written by Mike Shin The University of Sydney Business School The University of Sydney Page 1 Purpose of the Slides 1. These are supplementary slides to the modules. 2. Keep these as a reference and do not rely solely on them. The University of Sydney Page 2 Investor Preferences The University of Sydney Page 3 Present Value of Risky Cashflows – We will incorporate risk into our analysis in two ways: – 1. Expected Cashflows – 2. Discount Rate (Required Rate of Return, Cost of Capital) – We will discuss the choice of discount rate in Module 5. Module 4 will provide the foundations of a fundamental concept in Finance: "There is a risk-return tradeoff." We will show what the risk-return tradeoff looks like and how to make risk corrections in Module 5 The University of Sydney Page 4 Present Value of Risky Cashflows – Then we extend our formula to be: – ?? = ! "#! $%& + ! "#" $%& " + ?+ ! "## $%& # – ??' = Cashflow in period i – ? = Discount rate – ? ??' = Expected cashflow in period i The University of Sydney Page 5 Investor Decisions Under Uncertainty – An investor can choose to invest in a risk-free asset: – Return is certain across all possible states of the world – Otherwise, they will choose risky assets: – Return is not certain across all possible states of the world – The range of possible future cash flows will impact on their wealth The University of Sydney Page 6 Investor Decisions Under Uncertainty – If a risky asset has a higher expected return than a risk-free one, we call the difference, the risk premium of the asset. That is: – ?( = ? ?' − ?) – ?( = Risk premium – ? ?' = Expected return on asset i – ?) = Risk−free rate – We call this a fair gamble. This is where a risky investment whose expected return equals the risk-free rate of return. In other words, a risky investment that has zero risk premium. The University of Sydney Page 7 Risk Preferences – How an investor responds to a fair gamble indicates which group of investors they belong to: – Risk-averse – Risk-neutral – Risk-seeking – For Modules 4 and 5, we will assume investors are risk-averse. That is, they like expected returns and dislike risk. The University of Sydney Page 8 Risk Preferences – A risk-averse investor will reject a fair gamble. They require an appropriate risk premium to accept a risky investment. In particular, the more risk-averse an investor is, the higher the risk premium needs to be for the investor to accept the risky investment. – A risk-neutral investor will be indifferent between a fair gamble and a riskfree alternative. – A risk-seeking investor will select a fair gamble. They prefer a fair gamble over a risk-free alternative. Risk-seeking investors will pay a premium to take on risk. The University of Sydney Page 9 Portfolio Theory: An Overview The University of Sydney Page 10 Portfolio Theory: An Overview – Portfolio - A portfolio is a combination of assets – Portfolio weights: – Add up to 1 (100%) – Can be zero, positive, or negative – A common assumption in portfolio theory is that investors: – Like return – Hate risk The University of Sydney Page 11 Portfolio Theory: An Overview – Then, we can say that investors want: – The highest returns for a given risk or – The same returns for a lower risk – In this module, we learn about the key ideas of Portfolio Theory, which is a series of ideas that answer questions such as: – How much does a stock contribute to the risk and return of a portfolio? – Is there a way to create a portfolio that achieves the highest return for a given risk? – Do portfolios of stocks do better than the individual stocks? The University of Sydney Page 12 Portfolio Mean and Variance The University of Sydney Page 13 Portfolio Mean and Variance (2-Stocks) – The expected return of a portfolio with 2-stocks is: ??( = µ( = ω$ µ$ + ω* µ* – – – – ω$ = Portfolio weight on stock 1 ω* = Portfolio weight on stock 2 µ$ = Expected return of stock 1 µ* = Expected return of stock 2 – The expected return of a portfolio is the weighted average of the expected returns of the individual stocks. The University of Sydney Page 14 Portfolio Mean and Variance (2-Stocks) – The expected return of a portfolio with n-stocks is: ??( = µ( = ω$ ω$ + ω* µ* + ? + ω+ µ+ – ω' = Portfolio weight on stock i – µ' = Expected return of stock i The University of Sydney Page 15 Portfolio Mean and Variance (2-Stocks) – The variance of a portfolio is NOT the weighted average of the variance of the individual stocks – The variance of a portfolio with 2-stocks is: σ*( = ω$* σ$* + ω** σ** + 2ω$ ω* σ$ σ* ρ$* – – – – – σ$* = Variance of stock 1 σ** = Variance of stock 2 σ$ = Std dev of stock 1 σ* = Std dev of stock 2 ρ$* = Correlation coefficient for stocks 1 and 2 The University of Sydney Page 16 Portfolio Mean and Variance (2-Stocks) – The standard deviation of a portfolio with 2-stocks is: σ( = ω$* σ$* + ω** σ** + 2ω$ ω* σ$ σ* ρ$* = σ*( – Portfolio variance depends on the correlation (or covariance) between stocks. – If correlation: – Goes up, portfolio variance goes up – Goes down, portfolio variance goes down The University of Sydney Page 17 Portfolio Mean and Variance (3-Stocks) – The variance of a portfolio with 3-stocks is: σ"! = ω#" σ#" + ω"" σ"" + ω"$ σ"$ + 2ω# ω" σ# σ" ρ#" + 2ω# ω$ σ# σ$ ρ#$ + 2ω" ω$ σ" σ$ ρ"$ – σ*' = Variance of stock i – σ' = Std dev of stock i – ρ', = Correlation coefficient for stocks i and j The University of Sydney Page 18 Portfolio Mean and Variance (3-Stocks) – The standard deviation of a portfolio with 3-stocks is: σ( = σ*( – The variance formula gets more and more complicated with more stocks – There are more correlation (covariance) terms than variance terms – As you increase the number of stocks, only the correlation terms matter. The University of Sydney Page 19 Diversification The University of Sydney Page 20 Diversification – Diversification happens because with more correlation terms, adding more and more stocks, dampens more and more risk. – With many stocks, the correlation matters for the portfolio variance not the variance of individual stocks. – Why? – The variance of a portfolio with n-stocks and equal weights is: 1 1 σ*( = ∗ ??????? ???????? + 1 − ∗ ??????? ?????????? ? ? – As you add more and more stocks (increase n), only the average covariance matters for volatility. The University of Sydney Page 21 Systematic and Unsystematic Risk – Can you eliminate all risk? – No. Diversification has its limits. – Systematic Risk - Systematic risk or market risk is a risk that is common to all stocks and cannot be diversified away. – Unsystematic Risk - Unsystematic risk or diversifiable risk is a risk specific to a stock that can be diversified away. The University of Sydney Page 22 Systematic and Unsystematic Risk – Here is a diagram of systematic versus unsystematic risk: – Financial experts state that around 25 stocks seem to diversify away all unsystematic risk. Systematic risk exists because there are some factors that are common to all stocks. – The University of Sydney Page 23 Optimal Portfolios I The University of Sydney Page 24 Optimal Portfolios I: Efficient Frontier – Efficient Frontier - The efficient frontier is the curve of portfolios that give the highest return for a given risk. – Which portfolio on the efficient frontier is the best? – Depends on investor preferences: – Conservative investors prefer less risk and less return – Riskier investors prefer more risk and more return – The University of Sydney Page 25 Optimal Portfolios I: Risk-free Rate – Risk-free rate - The risk-free rate is the return on an investment that is riskless. – We usually think of government bonds such as a U.S. Treasury Bill as a proxy for the risk-free rate. – Once we introduce the risk-free asset to our analysis on optimal portfolios, we find that just as before, we can create any combination of portfolios on the efficient frontier, together with the risk-free asset. Any combination between the risk-free asset and any portfolio can be depicted by a straight line. The University of Sydney Page 26 Optimal Portfolios I: Market Portfolio – What is M? – M is the market portfolio. – Market Portfolio - The market portfolio is a theoretical portfolio that contains every stock with weights proportional to the total presence in the market. – This is why many financial experts recommend investors to hold index funds. – In practice, we use a large market index like the S&P 500 as a proxy for the market portfolio. The University of Sydney Page 27 Optimal Portfolios II The University of Sydney Page 28 Optimal Portfolios II: CML – Capital Market Line (CML) - The CML is a line that represents the risk-return tradeoff between the market portfolio and the risk-free rate. The University of Sydney – Page 29 Optimal Portfolios II – We can measure the portfolio's risk-return tradeoff by dividing the portfolio's risk premium by the volatility. – Risk Premium - The risk premium is the return on an asset minus the risk-free rate. – Sharpe Ratio - The Sharpe Ratio is a measure of a portfolio's risk-return trade-off equal to the portfolio's risk premium divided by its volatility. The University of Sydney Page 30 Optimal Portfolios II – The formula for the Sharpe Ratio is: ??( − ?) ?????? ?????( = σ( – ??( = Expected return of portfolio – ?) = Risk-free rate – σ( = Standard dev of portfolio The University of Sydney Page 31 Optimal Portfolios II – The higher the Sharpe Ratio the better the portfolio. – The slope of the capital market line (CML) is the Sharpe Ratio. – The market portfolio M has the highest possible Sharpe Ratio of any portfolio on the efficient frontier. The University of Sydney Page 32 Optimal Portfolios II – The equation for the capital market line (CML) is: