Chapter 5 Risk and Return Past and Prologue

Chapter 5 Risk and Return: Past and Prologue 1

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

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

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

Return over Multiple Periods – Arithmetic Average: r. A = (r 1+r 2)/2 – Geometric Average: r. G = [(1+r 1)(1+r 2)]1/2 – 1 5

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 period – Takes into account compounding effect – Provides an actual performance per year of the investment over the full sample period – Geometric returns <= arithmetic returns 6

Data from Table 5. 1 Quarter HPR 1. 10 2. 25 3 (. 20) 4. 25 What are the arithmetic and geometric return of this mutual fund? 7

Returns Using Arithmetic and Geometric Averaging Arithmetic ra = (r 1 + r 2 + r 3 +. . . rn) / n ra = (. 10 +. 25 -. 20 +. 25) / 4 =. 10 or 10% Geometric rg = {[(1+r 1) (1+r 2). . (1+rn)]} 1/n - 1 rg = {[(1. 1) (1. 25) (. 8) (1. 25)]} 1/4 - 1 = (1. 5150) 1/4 -1 =. 0829 = 8. 29% 8

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

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% 10

Risk and return • 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 11

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

HPR - Expected Return 13

HPR - Risk Measure Variance or standard deviation: 14

Problem 4, Chapter 5(p. 154) Suppose your expectations regarding the stock market are as follows: State of the economy Scenario(s) Probability(p(s)) Boom 1 0. 3 Normal Growth 2 0. 4 Recession 3 0. 3 Compute the mean and standard deviation of the HPR on stocks. HPR 44% 14% -16% 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% 15

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 16

Risk and return in the past • 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 17

Annual Holding Period Returns From Table 5. 3 of Text Historical Returns: 1926 -2003 Series World Stk US Lg Stk US Sm Stk Wor Bonds LT Treas T-Bills Inflation Arith. Mean% 11. 17 12. 25 18. 43 6. 13 5. 64 3. 79 3. 12 Stan. Dev. % 18. 38 20. 50 38. 11 9. 14 8. 19 3. 18 4. 35 Risk Premium 7. 38 8. 46 14. 64 2. 34 1. 85 0 18

Figure 5. 1 Frequency Distributions of Holding Period Returns 19

Figure 5. 2 Rates of Return on Stocks, Bonds and Bills 20

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) 21

Asset Allocation • 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. 22

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. 23

Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(r. P), 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% 24

Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(r. P), 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% 25

Complete Portfolio Risk Premium In general: 26

Portfolio Standard Deviation where sc - standard deviation of the complete portfolio s. P - 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 27

Portfolio Standard Deviation Example: Let the standard deviation on the risky portfolio, s. P, 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% 28

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

Capital Allocation Line Fig. 5. 5 Risk Tolerance and Asset Allocation: • More risk averse - closer to point F • Less risk averse - closer to P 30

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 31

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

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 riskfree rate 33

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 34

Figure 5. 5 Investment Opportunity Set with a Risk-Free Investment 35

Borrowing with rate > risk-free rate Example: Let the expected return on the risky portfolio, E(r. P), be 15%, the return on the risk-free asset, rf, be 7%, the borrowing rate, r. B, be 9% and the standard deviation on the risky portfolio, s. P, 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. 36

Figure 5. 6 Investment Opportunity Set with Differential Borrowing and Lending Rates 37

Risk aversion and allocation • 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 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. 38

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 39