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Optimal Portfolio Choice

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Rijish Chandran

This document discusses concepts related to portfolio optimization and the capital asset pricing model (CAPM). It provides information on calculating expected portfolio returns and volatility based on the returns and correlations of individual stocks. It also discusses how adding risk-free assets such as Treasury bills affects the efficient frontier and introduces the capital allocation line. The Sharpe ratio is presented as a measure of portfolio return relative to risk. The separation theorem and using the tangent portfolio to maximize returns for a given risk level are also summarized.

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Optimal Portfolio Choice
Portfolio Expected Return Suppose you invest Rs.10,000 in Ford and Rs.30,000 in Johnson and Johnson stock. You expect a return of 10% for Ford and 16% for J & J. What is the expected return for your portfolio?
Combining Risks Portfolio Returns Year Stock Returns North Air West Air Tex oil ½ RN + ½ RW ½ RW + ½ RT 1998 21% 9% -2% 15% 3.5% 1999 30% 21% -5% 25.5% 8.0% 2000 7% 7% 9% 7.0% 8.0% 2001 -5% -2% 21% -3.5% 9.5% 2002 -2% -5% 30% -3.5% 12.5% 2003 9% 30% 7% 19.5% 18.5% Average Returns 10.0% 10.0% 10.0% 10.0% 10.0% Volatility 13.4% 13.4% 13.4% 12.1% 5.1%
Optimal Portfolio Choice
Year Deviation from Mean (RN – RˉN) (Rw – Rˉw) (Rt –Rˉt) North West & West Air (RN – RˉN) (Rw – Rˉw) West air & Tex oil (Rw – Rˉw) (Rt – Rˉt) 1998 11% -1% -12% -0.0011 0.0012 1999 20% 11% -15% 0.0220 -0.0165 2000 -3% -3% -1% 0.0009 0.0003 2001 -15% -12% 11% 0.0180 -0.0132 2002 -12% -15% 20% 0.0180 -0.0300 2003 -1% 20% -3% -0.0020 -0.0060 Covariance 0.0112 -0.0128 Correlation 0.624 -0.713 Computing Covariance and Correlation between Pairs of Stocks
Interpreting Covariance • Positive covariance signifies the fact that the stocks move together in the same direction. • If the stocks move in opposite directions, the covariance is negative. • Correlation quantifies the relationship between the stocks and is between +1 and -1.
Interpreting Correlation
Historical Annual Volatilities and Correlation (based on monthly returns) Microsoft Dell Delta Air Lines American Airline General Motors Ford Motors Anheuser - Busch Volatility 42% 54% 50% 72% 33% 37% 18% Correlation with Microsoft 1.00 0.65 0.27 0.19 0.22 0.26 -0.07 Dell 0.65 1.00 0.19 0.18 0.32 0.32 0.10 Delta Air 0.27 0.19 1.00 0.69 0.31 0.38 0.19 American 0.19 0.18 0.69 1.00 0.35 0.58 0.11 General Motors 0.22 0.32 0.31 0.35 1.00 0.64 0.11 Ford 0.26 0.32 0.38 0.58 0.64 1.00 0.10 Anheuser - Busch -0.07 0.10 0.19 0.11 0.11 0.10 1.00
Calculating the Portfolio returns and Portfolio Volatility • Let’s say you have 2 stocks: I-flex and HCL. Assume that I-flex’s average return over the last 5 years has been 20% per year and that of HCL has been 25%. Also assume that the standard deviations of those returns were 30% and 40% respectively. • If the correlation coefficient for these two stocks is 0.8, what would be the standard deviation of a portfolio invested 40% in I-fled and 60% in HCL? • If the correlation coefficient were 0.5 instead, would the portfolio standard deviation be greater than or less than in (a)? Why?
Calculation of Expected Return and Portfolio Volatility The common stocks of Bajaj and Colgate have expected returns of 15% and 20% respectively, while their deviations from the expected returns are 20% and 40%. The expected correlation coefficient between the stocks is 0.36. 1. What is the expected value of return and the portfolio volatility of a portfolio consisting of (a) 40% Bajaj and 60% Colgate? (b) 40% Colgate and 60% Bajaj? 2.How should the correlation coefficient move to bring the portfolio risk still lower?
Efficient Portfolio with Two Stocks Consider Intel Corporation and the Coca-Cola Company. The returns of these two companies were uncorrelated. Suppose an investor wants to invest in these two portfolios for which the details are given as under:
Choosing a Portfolio Correlation with Stock Expected Return Volatility Intel Coca-Cola Intel 26% 50% 1.0 0.0 Coca-cola 6% 25% 0.0 1.0 How should the investor choose a portfolio of these two stocks? Calculate a Portfolio for its returns and volatility if you invest 40% in Intel and 60% in Coca-Cola?
Expected Returns and Volatility for Different Portfolios of Two Stocks Portfolio Weights Expected Return(%) Volatility (%) X(Intel) X(Coca-cola) Expected return of Portfolio SD [R p] 1.00 0.00 26% 50% 0.80 0.20 22% 40.3% 0.60 0.40 18% 31.6% 0.40 0.60 14% 25% 0.20 0.80 10% 22.3% 0.00 1.00 6% 25%
0% 40%20% 30%10% 0% 25% 20% 15% 10% 5% 60%50% 30% Coca-Cola Intel Inefficient Portfolios Efficient Portfolios (0 , 1) (0.2 , 0.8) (0.4 , 0.6) (0.6 , 0.4) (0.8 , 0.2) (1 , 0) Volatility (Standard Deviation) ExpectedReturn
Improving Returns with an Efficient Portfolio Sally has invested 100% of her money in Coca- Cola and is seeking investment advice. She would like to earn the highest expected return possible without increasing her volatility. Which portfolio would you recommend?
Effect of Correlation
0% 40%20% 30%10% 0% 25% 20% 15% 10% 5% 60%50% 30% Coca-Cola Intel Volatility (Standard Deviation) ExpectedReturn Correlation = -1 Correlation = +1 Effect of Correlation on Volatility and Expected Return
Suppose you have Rs 20,000 in cash to invest. You decided to short sell Rs 10,000 worth of Coca-Cola stock and invest the proceeds from your short sale, plus your Rs 20,000 in Intel. At the end of the year, you decide to liquidate your portfolio. If the two stocks have the following realized returns, what is the return on your portfolio? PO Div1 + P1 Return Intel 25.00 31.50 26% Coca - Cola 40.00 42.40 6% Calculation of Returns from Short Selling
Volatility with Short Sales Suppose Intel stock has a volatility of 50%, Coca- Cola stock has a volatility of 25% and the stocks are uncorrelated. • What is the volatility of a portfolio that is short Rs.10,000 of Coca-Cola and long Rs.30,000 of Intel?
Risk-Free Saving and Borrowing Consider an arbitrary risky portfolio with returns Rp. Lets look at the effect on risk and return of putting a fraction x of our money in the portfolio, while leaving the remaining fraction (1-x) in risk-free Treasury bills with a yield of rf . E[Rxp] = rf + x(E[Rp] – rf)
Volatility of the Portfolio • Next let’s compute the volatility. The volatility of the risk-free investment is zero. • Therefore the volatility of this investment is only equal to the fraction of the volatility of the portfolio. That is it is equal to xSD(Rp).
25% 20% 15% 10% 5% 0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% P Buying P on Margin X = 200% X = 150% X = 100% X = 50% Risk – Free Investment Investing in P and the Risk-free environment Efficient Frontier of Risky Investments Volatility (standard deviation) ExpectedReturn Buying Stocks on Margin
Margin Investing Suppose you have Rs.10,000 in cash and you decide to borrow another Rs.10,000 at a 5% interest rate to invest in the stock market. You invest the entire Rs.20,000 in Portfolio Q with a 10% expected return and a 20% volatility. 1.What is the expected return and volatility of your investment? 2.What is your realized return if Q goes up 30% over the course of the year? 3. What return do you realize if Q falls by 10% over the course of the year?
F ExpectedReturn D B σS σm Changing efficient frontier with risk-free asset Capital Allocation Line P C A M Rs Rm
Capital Allocation Line Mr.Xing is considering an investment of Rs.4,00,000. He wants to allocate a part of the funds to a portfolio of risk-free securities providing a return of 8% and the remaining part in a share portfolio, that has an expected return and the standard deviation of 24% each. a.If Mr.Xing is prepared to assume a maximum risk of 15%, what is the best return he can hope for? b.In case Mr.Xing needs a return of 32%, what minimum risk would he have to assume?
Identifying the Tangent Portfolio • By forming a portfolio out of risk- free asset and a portfolio which is effectively on the efficient portfolio at a level higher than the Portfolio P, we get a line tangent to the efficient portfolio which is steeper than the line that is through P. • If the line is steeper, then for any level of volatility, we will earn a higher expected return.
P Risk-Free Investment Efficient Frontier Including Risk-Free Investment Tangent or Efficient Portfolio Efficient Frontier of Risky Investments Volatility (standard deviation) ExpectedReturn
Sharpe Ratio The slope of the return through a given portfolio P is often referred to as the Sharpe ratio of the portfolio. Sharpe Ratio = Portfolio Excess Return Portfolio Volatility = E[Rp] – rf σ p The Sharpe ratio measures the ratio of reward- to-volatility provided by a portfolio.
Separation Theorem • The portfolio managers offer the same set of assets on the efficient portfolio to all their clients. • Generally the don’t conduct separate research for identifying suitable portfolios for each individual client, thereby saving cost, time and effort. • To accommodate individual preferences of the clients all they need to do is to adjust the proportion of investment in the two assets.
Optimal Portfolio Choice Your uncle calls and asks for investment advice. Currently, he has Rs.1,00,000 invested in a portfolio P as shown in the above graph. This portfolio has an expected return of 10.5% and a volatility of 8%. Suppose the risk-free rate is 5% and the tangent portfolio has an expected return of 18.5% and a volatility of 13%. a.Recommend a portfolio to maximize his return without increasing the risk. b.Recommend a portfolio to minimize his risk but maintaining the same expected return.
Cost of Capital for Investment i • The cost of capital of investment (i) is equal to the expected return of the best available portfolio in the market with the same sensitivity to systematic risk. • And the best available portfolio is the Tangent Portfolio or the portfolio that has the highest Sharpe ratio of any portfolio in the economy.
Cost of Capital for Investment i • Because all other risk is diversifiable, it is an investment’s beta with respect to the efficient portfolio that measures its sensitivity to systematic risk and therefore its cost of capital. rі = rf + β

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Optimal Portfolio Choice

  • 2. Portfolio Expected Return Suppose you invest Rs.10,000 in Ford and Rs.30,000 in Johnson and Johnson stock. You expect a return of 10% for Ford and 16% for J & J. What is the expected return for your portfolio?
  • 3. Combining Risks Portfolio Returns Year Stock Returns North Air West Air Tex oil ½ RN + ½ RW ½ RW + ½ RT 1998 21% 9% -2% 15% 3.5% 1999 30% 21% -5% 25.5% 8.0% 2000 7% 7% 9% 7.0% 8.0% 2001 -5% -2% 21% -3.5% 9.5% 2002 -2% -5% 30% -3.5% 12.5% 2003 9% 30% 7% 19.5% 18.5% Average Returns 10.0% 10.0% 10.0% 10.0% 10.0% Volatility 13.4% 13.4% 13.4% 12.1% 5.1%
  • 5. Year Deviation from Mean (RN – RˉN) (Rw – Rˉw) (Rt –Rˉt) North West & West Air (RN – RˉN) (Rw – Rˉw) West air & Tex oil (Rw – Rˉw) (Rt – Rˉt) 1998 11% -1% -12% -0.0011 0.0012 1999 20% 11% -15% 0.0220 -0.0165 2000 -3% -3% -1% 0.0009 0.0003 2001 -15% -12% 11% 0.0180 -0.0132 2002 -12% -15% 20% 0.0180 -0.0300 2003 -1% 20% -3% -0.0020 -0.0060 Covariance 0.0112 -0.0128 Correlation 0.624 -0.713 Computing Covariance and Correlation between Pairs of Stocks
  • 6. Interpreting Covariance • Positive covariance signifies the fact that the stocks move together in the same direction. • If the stocks move in opposite directions, the covariance is negative. • Correlation quantifies the relationship between the stocks and is between +1 and -1.
  • 8. Historical Annual Volatilities and Correlation (based on monthly returns) Microsoft Dell Delta Air Lines American Airline General Motors Ford Motors Anheuser - Busch Volatility 42% 54% 50% 72% 33% 37% 18% Correlation with Microsoft 1.00 0.65 0.27 0.19 0.22 0.26 -0.07 Dell 0.65 1.00 0.19 0.18 0.32 0.32 0.10 Delta Air 0.27 0.19 1.00 0.69 0.31 0.38 0.19 American 0.19 0.18 0.69 1.00 0.35 0.58 0.11 General Motors 0.22 0.32 0.31 0.35 1.00 0.64 0.11 Ford 0.26 0.32 0.38 0.58 0.64 1.00 0.10 Anheuser - Busch -0.07 0.10 0.19 0.11 0.11 0.10 1.00
  • 9. Calculating the Portfolio returns and Portfolio Volatility • Let’s say you have 2 stocks: I-flex and HCL. Assume that I-flex’s average return over the last 5 years has been 20% per year and that of HCL has been 25%. Also assume that the standard deviations of those returns were 30% and 40% respectively. • If the correlation coefficient for these two stocks is 0.8, what would be the standard deviation of a portfolio invested 40% in I-fled and 60% in HCL? • If the correlation coefficient were 0.5 instead, would the portfolio standard deviation be greater than or less than in (a)? Why?
  • 10. Calculation of Expected Return and Portfolio Volatility The common stocks of Bajaj and Colgate have expected returns of 15% and 20% respectively, while their deviations from the expected returns are 20% and 40%. The expected correlation coefficient between the stocks is 0.36. 1. What is the expected value of return and the portfolio volatility of a portfolio consisting of (a) 40% Bajaj and 60% Colgate? (b) 40% Colgate and 60% Bajaj? 2.How should the correlation coefficient move to bring the portfolio risk still lower?
  • 11. Efficient Portfolio with Two Stocks Consider Intel Corporation and the Coca-Cola Company. The returns of these two companies were uncorrelated. Suppose an investor wants to invest in these two portfolios for which the details are given as under:
  • 12. Choosing a Portfolio Correlation with Stock Expected Return Volatility Intel Coca-Cola Intel 26% 50% 1.0 0.0 Coca-cola 6% 25% 0.0 1.0 How should the investor choose a portfolio of these two stocks? Calculate a Portfolio for its returns and volatility if you invest 40% in Intel and 60% in Coca-Cola?
  • 13. Expected Returns and Volatility for Different Portfolios of Two Stocks Portfolio Weights Expected Return(%) Volatility (%) X(Intel) X(Coca-cola) Expected return of Portfolio SD [R p] 1.00 0.00 26% 50% 0.80 0.20 22% 40.3% 0.60 0.40 18% 31.6% 0.40 0.60 14% 25% 0.20 0.80 10% 22.3% 0.00 1.00 6% 25%
  • 14. 0% 40%20% 30%10% 0% 25% 20% 15% 10% 5% 60%50% 30% Coca-Cola Intel Inefficient Portfolios Efficient Portfolios (0 , 1) (0.2 , 0.8) (0.4 , 0.6) (0.6 , 0.4) (0.8 , 0.2) (1 , 0) Volatility (Standard Deviation) ExpectedReturn
  • 15. Improving Returns with an Efficient Portfolio Sally has invested 100% of her money in Coca- Cola and is seeking investment advice. She would like to earn the highest expected return possible without increasing her volatility. Which portfolio would you recommend?
  • 17. 0% 40%20% 30%10% 0% 25% 20% 15% 10% 5% 60%50% 30% Coca-Cola Intel Volatility (Standard Deviation) ExpectedReturn Correlation = -1 Correlation = +1 Effect of Correlation on Volatility and Expected Return
  • 18. Suppose you have Rs 20,000 in cash to invest. You decided to short sell Rs 10,000 worth of Coca-Cola stock and invest the proceeds from your short sale, plus your Rs 20,000 in Intel. At the end of the year, you decide to liquidate your portfolio. If the two stocks have the following realized returns, what is the return on your portfolio? PO Div1 + P1 Return Intel 25.00 31.50 26% Coca - Cola 40.00 42.40 6% Calculation of Returns from Short Selling
  • 19. Volatility with Short Sales Suppose Intel stock has a volatility of 50%, Coca- Cola stock has a volatility of 25% and the stocks are uncorrelated. • What is the volatility of a portfolio that is short Rs.10,000 of Coca-Cola and long Rs.30,000 of Intel?
  • 20. Risk-Free Saving and Borrowing Consider an arbitrary risky portfolio with returns Rp. Lets look at the effect on risk and return of putting a fraction x of our money in the portfolio, while leaving the remaining fraction (1-x) in risk-free Treasury bills with a yield of rf . E[Rxp] = rf + x(E[Rp] – rf)
  • 21. Volatility of the Portfolio • Next let’s compute the volatility. The volatility of the risk-free investment is zero. • Therefore the volatility of this investment is only equal to the fraction of the volatility of the portfolio. That is it is equal to xSD(Rp).
  • 22. 25% 20% 15% 10% 5% 0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% P Buying P on Margin X = 200% X = 150% X = 100% X = 50% Risk – Free Investment Investing in P and the Risk-free environment Efficient Frontier of Risky Investments Volatility (standard deviation) ExpectedReturn Buying Stocks on Margin
  • 23. Margin Investing Suppose you have Rs.10,000 in cash and you decide to borrow another Rs.10,000 at a 5% interest rate to invest in the stock market. You invest the entire Rs.20,000 in Portfolio Q with a 10% expected return and a 20% volatility. 1.What is the expected return and volatility of your investment? 2.What is your realized return if Q goes up 30% over the course of the year? 3. What return do you realize if Q falls by 10% over the course of the year?
  • 24. F ExpectedReturn D B σS σm Changing efficient frontier with risk-free asset Capital Allocation Line P C A M Rs Rm
  • 25. Capital Allocation Line Mr.Xing is considering an investment of Rs.4,00,000. He wants to allocate a part of the funds to a portfolio of risk-free securities providing a return of 8% and the remaining part in a share portfolio, that has an expected return and the standard deviation of 24% each. a.If Mr.Xing is prepared to assume a maximum risk of 15%, what is the best return he can hope for? b.In case Mr.Xing needs a return of 32%, what minimum risk would he have to assume?
  • 26. Identifying the Tangent Portfolio • By forming a portfolio out of risk- free asset and a portfolio which is effectively on the efficient portfolio at a level higher than the Portfolio P, we get a line tangent to the efficient portfolio which is steeper than the line that is through P. • If the line is steeper, then for any level of volatility, we will earn a higher expected return.
  • 27. P Risk-Free Investment Efficient Frontier Including Risk-Free Investment Tangent or Efficient Portfolio Efficient Frontier of Risky Investments Volatility (standard deviation) ExpectedReturn
  • 28. Sharpe Ratio The slope of the return through a given portfolio P is often referred to as the Sharpe ratio of the portfolio. Sharpe Ratio = Portfolio Excess Return Portfolio Volatility = E[Rp] – rf σ p The Sharpe ratio measures the ratio of reward- to-volatility provided by a portfolio.
  • 29. Separation Theorem • The portfolio managers offer the same set of assets on the efficient portfolio to all their clients. • Generally the don’t conduct separate research for identifying suitable portfolios for each individual client, thereby saving cost, time and effort. • To accommodate individual preferences of the clients all they need to do is to adjust the proportion of investment in the two assets.
  • 30. Optimal Portfolio Choice Your uncle calls and asks for investment advice. Currently, he has Rs.1,00,000 invested in a portfolio P as shown in the above graph. This portfolio has an expected return of 10.5% and a volatility of 8%. Suppose the risk-free rate is 5% and the tangent portfolio has an expected return of 18.5% and a volatility of 13%. a.Recommend a portfolio to maximize his return without increasing the risk. b.Recommend a portfolio to minimize his risk but maintaining the same expected return.
  • 31. Cost of Capital for Investment i • The cost of capital of investment (i) is equal to the expected return of the best available portfolio in the market with the same sensitivity to systematic risk. • And the best available portfolio is the Tangent Portfolio or the portfolio that has the highest Sharpe ratio of any portfolio in the economy.
  • 32. Cost of Capital for Investment i • Because all other risk is diversifiable, it is an investment’s beta with respect to the efficient portfolio that measures its sensitivity to systematic risk and therefore its cost of capital. rÑ– = rf + β
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