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Chapter 5

Chapter 5. Risk and Return: Past and Prologue. Return over One Period: Holding Period Return (HPR). HPR: Rate of return over a given investment period. Ending Price = 110 Beginning Price = 100 Dividend = 4. Rates of Return: Single Period Example.

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Chapter 5

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  1. Chapter 5 Risk and Return: Past and Prologue

  2. Return over One Period: Holding Period Return (HPR) HPR: Rate of return over a given investment period

  3. Ending Price = 110 Beginning Price = 100 Dividend = 4 Rates of Return: Single Period Example

  4. What is the average return of your investment per period? Return over Multiple Periods $100 $50 $100 r1 r2 t = 0 1 2 r1, r2: one-period HPR

  5. Arithmetic Average: rA = (r1+r2)/2 Geometric Average: rG = [(1+r1)(1+r2)]1/2 – 1 Return over Multiple Periods

  6. Return over Multiple Periods • Arithmetic return: return earned in an average period over multiple period • It is the simple average return. • It ignores compounding effect • It represents the return of a typical (average) period • Provides a good forecast of future expected return • Geometric return • Average compound return per period • Takes into account compounding effect • Provides an actual performance per year of the investment over the full sample period • Geometric returns <= arithmetic returns

  7. Quarter 1 2 3 4 HPR .10 .25 (.20) .25 Data from Table 5.1 What are the arithmetic and geometric return of this mutual fund?

  8. Arithmetic ra = (r1 + r2 + r3 + ... rn) / n ra = (.10 + .25 - .20 + .25) / 4 = .10 or 10% Geometric rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1 rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29% Returns Using Arithmetic and Geometric Averaging

  9. Invest $1 into 2 investments: one gives 10% per year compounded annually, the other gives 10% compounded semi-annually. Which one gives higher return Quoting Conventions

  10. Quoting Conventions APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period)Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68%

  11. Risk in finance: uncertainty related to outcomes of an investment The higher uncertainty, the riskier the investment. How to measure risk and return in the future Probability distribution: list of all possible outcomes and probability associated with each outcome, and sum of all prob. = 1. For any distribution, the 2 most important characteristics Mean Standard deviation Risk and return

  12. Return distribution s.d. s.d. r or E(r)

  13. HPR - Expected Return

  14. HPR - Risk Measure Variance or standard deviation:

  15. Suppose your expectations regarding the stock market are as follows: State of the economy Scenario(s) Probability(p(s)) HPR Boom 1 0.3 44% Normal Growth 2 0.4 14% Recession 3 0.3 -16% Compute the mean and standard deviation of the HPR on stocks. E( r ) = 0.3*44 + 0.4*14+0.3*(-16)=14% Sigma^2=0.3*(44-14)^2+0.4*(14-14)^2 +0.3*(-16-14)^2=540 Sigma=23.24% Problem 4, Chapter 5(p.154)

  16. Historical Mean and Variance Data in the n-point time series are treated as realization of a particular scenario each with equal probability 1/n

  17. Year Ri(%) 1988 16.9 1989 31.3 1990 -3.2 1991 30.7 1992 7.7 Compute the mean and variance of this sample Risk and return in the past

  18. Arith. Stan. Risk Series Mean% Dev.% Premium World Stk 11.17 18.38 7.38 US Lg Stk 12.25 20.50 8.46 US Sm Stk 18.43 38.11 14.64 Wor Bonds 6.13 9.14 2.34 LT Treas 5.64 8.19 1.85 T-Bills 3.79 3.18 0 Inflation 3.12 4.35 Annual Holding Period ReturnsFrom Table 5.3 of Text Historical Returns: 1926-2003

  19. Figure 5.1 Frequency Distributions of Holding Period Returns

  20. Figure 5.2 Rates of Return on Stocks, Bonds and Bills

  21. Risk aversion and Risk Premium • Risk aversion: higher risk requires higher return, risk averse investors are rational investors • Risk-free rate • Risk premium • (=Risky return –Risk-free return)

  22. Historically, stock is riskier than bond, bond is riskier than bill Return of stock > bond > bill More risk averse, put more money on bond Less risk averse, put more money on stock This decision is asset allocation Asset allocation decision accounts for 94% of difference in return of portfolio managers. Asset Allocation

  23. Portfolios: Basic Asset Allocation • The complete portfolio is composed of: • The risk-free asset: Risk can be reduced by allocating more to the risk-free asset • The risky portfolio: Composition of risky portfolio does not change • This is called Two-Fund Separation Theorem. • The proportions depend on your risk aversion.

  24. Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if all of the funds are invested in the risk-free asset? What is the risk premium? 7% 0 What is the return on the portfolio if all of the funds are invested in the risky portfolio? 15% 8%

  25. Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? What is the risk premium? 0.5*15%+0.5*7%=11% 4%

  26. Complete Portfolio Risk Premium In general:

  27. Portfolio Standard Deviation where sc- standard deviation of the complete portfolio sP - standard deviation of the risky portfolio srf - standard deviation of the risk-free rate y - weight of the complete portfolio invested in the risky asset

  28. Portfolio Standard Deviation Example: Let the standard deviation on the risky portfolio, sP, be 22%. What is the standard deviation of the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? 22%*0.5=11%

  29. We know that given a risky asset (p) and a risk-free asset, the expected return and standard deviation of any complete portfolio (c) satisfy the following relationship: Capital Allocation Line Where y is the fraction of the portfolio invested in the risky asset

  30. Capital Allocation Line Fig. 5.5 • Risk Tolerance and Asset Allocation: • More risk averse - closer to point F • Less risk averse - closer to P

  31. Slope of the CAL S is the increase in expected return per unit of additional standard deviation S is the reward-to-variability ratio or Sharpe Ratio

  32. Slope of the CAL Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7% and the standard deviation on the risky portfolio, sP, be 22%. What is the slope of the CAL for the complete portfolio? S = (15%-7%)/22% = 8/22

  33. Borrowing • So far, we only consider 0<=y<=1, that means we use only our own money. • Can y > 1? • Borrow money or use leverage • Example: budget = 300,000. Borrow additional 150,000 at the risk-free rate and invest all money into risky portfolio • y = 450,000/150,000 = 1.5 • 1-y = -0.5 • Negative sign means short position. • Instead of earning risk-free rate as before, now have to pay risk-free rate

  34. Borrowing at risk-free rate The slope = 0.36 means the portfolio c is still in the CAL but on the right hand side of portfolio P

  35. Figure 5.5 Investment Opportunity Set with a Risk-Free Investment

  36. Borrowing with rate > risk-free rate Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%, the borrowing rate, rB, be 9% and the standard deviation on the risky portfolio, sP, be 22%. Suppose the budget = 300,000. Borrow additional 150,000 at the borrowing rate and invest all money into risky portfolio What is the slope of the CAL for the complete portfolio for points where y > 1, y = 1.5; E(Rc) = 1.5(15) + (-0.5)*9 = 18% Slope = (0.18-0.09)/0.33 = 0.27 Note: For y £ 1, the slope is as indicated above if the lending rate is rf.

  37. Figure 5.6 Investment Opportunity Set with Differential Borrowing and Lending Rates

  38. More risk-averse, the complete portfolio C is close to F Less risk averse, C is close to P Why we should invest in stock in practice even if we don’t like risk at all A portfolio of 100% T-bond and a portfolio of 83% T-bond, 17% blue chip stocks have the same amount of risk, but the mix portfolio gives higher return Typical portfolio: 60% stock, 40% bond. To reduce risk, we can increase the proportion of bonds However, a more effective way is 75% stock, 25% bill, give the same return but less risk In short term, stock can be bad In long term, stocks outperform other investments. Risk aversion and allocation

  39. Summary • Definition of Returns: HPR, APR and AER. • Risk and expected return • Shifting funds between the risky portfolio to the risk-free asset reduces risk • Examples for determining the return on the risk-free asset • Examples of the risky portfolio (asset) • Capital allocation line (CAL) • All combinations of the risky and risk-free asset • Slope is the reward-to-variability ratio • Risk aversion determines position on the capital allocation line

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