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Risk and Return

Risk and Return. BKM: Chapter 5. The question . . . How much do you punch the accelerator? Greater speed can be thrilling but it can also get pretty ugly. Risk and Return. The Framework. Only two assets to choose: A risk-free bond Some other risky asset Can be some portfolio

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Risk and Return

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  1. Risk and Return BKM: Chapter 5

  2. The question . . . • How much do you punch the accelerator? • Greater speed can be thrilling but it can also get pretty ugly.

  3. Risk and Return

  4. The Framework • Only two assets to choose: • A risk-free bond • Some other risky asset • Can be some portfolio • Can go long or short • Shorting the risk-free bond is like borrowing to buy more of the risky asset.

  5. The Question Again • What fraction of investment equity • do you put in THE RISK-FREE BOND • do you put in THE RISKY ASSET • How much do you punch the accelerator? • How ugly can it get?

  6. The Portfolio Decision • The more you put in risky asset: • Expect, on average, total position to earn higher returns • HOW MUCH higher? • Expect Risk to be greater • HOW MUCH GREATER? • Combine Stat Rules #1 and #2

  7. Portfolio of Risk-Free Asset and One Risky Asset • Expected Return: • Standard Deviation:

  8. Example • E[rs]=8% • ss=.12 • rf=4% • E[rp]=wE[rS]+(1-w)rf • sp=wss RiskyRisk-Free A: 0% 100% B: 100% 0% C: 50% 50% D: 150% -50% E[rp] D 10% B 8% C 6% A 4% sp .06 .12 .18

  9. Capital Allocation Line: • What is the equation for the line? • Clearly, the intercept is the risk-free rate • What is slope? • Rise-over-run for any two points • Point 1: sp=0, E[rp]=rf • Point 2:sp=ss, E[rp]=E[rs]

  10. Capital Allocation Line • This is just the equation for a line! Slope Y variable X variable Intercept

  11. CAL Example • E[rs]=8% • ss=.12 • rf=4% • E[rp]=wE[rS]+(1-w)rf • sp=wss RiskyRisk-Free A: 0% 100% B: 100% 0% C: 50% 50% D: 150% -50% E[rp] D 10% B 8% C 6% A 4% sp .06 .12 .18

  12. Capital Allocation Line • How much do you punch the accelerator? • You make the call . . . . • Risk-averse investors will invest more in the risk-free asset. • Risk-tolerant investors will invest more in the risky asset.

  13. Targeting E[r] with Vanguard • Expected return on Vanguard fund: 12% • Stdev of risky portfolio: 0.16 • Risk free rate: 7% • Assume you can borrow and lend at this rate. • Suppose you want an expected return of 17%. • What are portfolio weights?

  14. Targeting E[r] with Vanguard • Use stat rule 1 • Invest 200% of investment equity in risky portfolio by borrowing at risk-free rate.

  15. Targeting E[r] with Vanguard • With weight of 2.0 in risky asset, stdev of portfolio is 2.0*0.16=0.32 by stat rule #2 • What is intercept and slope of CAL line? • Intercept=0.07 • Slope=(0.12-0.07)/0.16=0.3125 • Note: 0.07+.3125*.32 = 17%

  16. Targeting E[r] with Vanguard Position Stdev 100% in risky portfolio 0.16 Desired position 0.32 E[r] .17 .12 .07 sp .32 .16

  17. Targeting E[r] with Vanguard • Suppose • Investment equity = $1000 • Price of risky asset is $18/share • Then • Want $2000 in Vanguard Fund • Buy 2000/18 =111 shares • Borrow $1000 at 7%

  18. Targeting s with Vanguard • Suppose you want stdev of portfolio to be 0.25. • Use stat rule 2 • Invest 156% of investment equity in risky portfolio by borrowing at risk-free rate.

  19. Targeting s with Vanguard • With weight of 1.56 in risky asset, • We can also use the equation for the CAL line: 0.07+.3125*.25 = 14.8%

  20. Targeting s with Vanguard Position Stdev 100% in risky portfolio 0.16 Desired position 0.25 E[r] .148 .12 .07 sp .25 .16

  21. Targeting s with Vanguard • Suppose • Investment equity = $1000 • Price of risky asset is $18/share • Then • Want $1560 in Vanguard Fund • Buy 1560/18 =87 shares • Borrow $560 at 7%

  22. Passive Investing • Select passive portfolio with highest Sharpe Ratio possible • Invest a fraction of your wealth in the portfolio according to your level of risk aversion, and the rest in a risk free asset. • Benefits: • No need to spend time researching stocks • No need to pay someone else to research

  23. Improving Client’s Position • A client asks your advice about her investments. She has invested $70,000 in a Mosaic mutual fund and $10,000 in risk-free bonds. She asks you whether she should re-allocate her assets. • Mosaic fund • Expected return: 15% • Standard deviation: 30%. • Vanguard Fund • Expected return: 12% • Standard deviation: of 16%. • The risk-free rate is currently 7% (borrowing and lending)

  24. Improving Client’s Position • Question 1: What is the expected return and the standard deviation of her current portfolio? • The expected return is: E(r)=(7/8)*(0.15) + (1/8)*(0.07) = 14% (stat rule 1) • The standard deviation is: Stdev(r)=(7/8)*0.30 = 0.2625 (stat rules 2)

  25. Increasing E[r], without Increasing s • Question 2: Assume that she can borrow and lend at an interest rate of 7%. • Can you find a portfolio with same total risk but higher expected return? • If yes, indicate the dollar amounts invested in each security and the expected return and the standard deviation of this portfolio. • If not, please explain why a dominating portfolio is not possible.

  26. Increasing E[r] without Increasing s • Find Sharpe ratios: • Sharpe Mosaic=(0.15-0.07)/0.30 = 0.267 • Sharpe Vanguard=(0.12-0.07)/0.16 = 0.3125 • So Vanguard Fund has a higher-sloped CAL. • Vanguard is a better investment vehicle.

  27. Increasing E[r] without Increasing s

  28. Increasing E[r] without Increasing s • To find a portfolio with same total risk, use stat rules 2: 0.2625=w(.16) w=1.64 (1-w)=-0.64 • The total equity = $80,000, • Invest 164% of equity in Vanguard Fund ($131,200) • Borrow 64% of equity ($51,200) at 7% • Portfolio expected return= 1.64*.12-0.64*0.07=0.152>0.14

  29. Lowering s without lowering E[r] • Suppose we wanted to create a portfolio with the same expected return, but lower total risk

  30. Lowering s without lowering E[r] • To find portfolio with same expected return, use stat rules #1: • Total equity = $80,000, • Invest 140% of equity in Vangaurd ($112,000) • Borrow 40% of equity ($32,000) at 7%. • Portfolio stdev=1.4*0.16=0.224 < 0.2625

  31. Different Borrowing and Lending Rates • Assume now that borrowing and lending rates differ. In particular, the borrowing rate equals 11% and the lending rate equals 7%.

  32. Different Borrowing and Lending Rates • Question - Assuming now that borrowing and lending rates differ: • Can you find a portfolio with same total risk but higher expected return? • In this case, we can’t simply turn to the Sharpe Ratio, because there are two risk-free rates.

  33. Different Borrowing and Lending Rates • To find a portfolio with same total risk, use stat rule 2: 0.2625=w(.16) w=1.64 (1-w)=-0.64 • Expected return • Which risk free rate do you use? • Since my weight in Vanguard is above 1, this means I am borrowing. • E[r]=1.64*0.12-0.64*0.11 = 0.1264 < 0.14

  34. Different Borrowing and Lending Rates • Now, if we switch to Vanguard we are worse off if we create a portfolio with same total risk as the client’s current portfolio. • Your client is better of sticking with her Mosaic fund.

  35. Different Borrowing and Lending Rates • Question - Assuming now that borrowing and lending rates differ: • Can you find a portfolio with same expected return but lower total risk? • In this case, we can’t simply turn to the Sharpe Ratio, because there are two risk-free rates.

  36. Different Borrowing and Lending Rates • To find a portfolio with same total risk, use stat rule 1: • Which risk free rate do you use? • I need to get an expected return of 14% using Vangaurd which I expect to only earn 12%. • To get a higher expected return, I must borrow.

  37. Different Borrowing and Lending Rates • What is the total risk of this portfolio? • Use stat rule 2: • sp=3*0.16=0.48>0.2625 • Again, your client is better of sticking with her Mosaic fund.

  38. Different Borrowing and Lending Rates • What do CAL line look like when borrowing and lending rates differ? • See Excel Spreadsheet

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