Fina 2802 Investments and Portfolio Analysis Spring 2010

Fina 2802: Investments and Portfolio Analysis Spring, 2010 Dragon Tang Lecture 8 Risk and Return: Past and Prologue February 4, 2010 Readings: Chapter 5 Practice Problem Sets: 3, 4, 5, 14, 15 -17 Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 1

Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 2

Risk and Return in Chinese Risk: 危机 Danger | Opportunity Return: 回报 Come back | Gratitude Fin Spring 08 - Tang Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 3

Risk and Return Objectives: 1. Characterize the risk and return on stocks (risky) and bonds (risk-free). 2. Historical risk and return of various securities Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 4

Return over One Period: Holding Period Return (HPR) HPR: Rate of return over a given investment period Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 5

Rates of Return: Single Period Example Beginning Price = Ending Price = Dividend = 100 110 4 HPR = ( 110 - 100 + 4 )/ ( 100) = 14% Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 6

Real vs. Nominal Rates • Notation: – R=nominal return – i =inflation rate – r =real return • Exact relationship • Approximate relationship • Example R = 9%, i = 6%: what is r? Fin 2802, Spring 10 - Tang Chapter Risk and Return Chapter 5: 5: Risk and Return 7

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% Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 8

Return over Multiple Periods $X t= 0 $Y r 1 1 $Z r 2 2 $X, $Y, $Z: Cash Flows; r 1, r 2: one-period HPR • Dollar-weighting: Internal Rate of Return (IRR) • Time-weighting: – Arithmetic Average: r. A = (r 1+r 2)/2 – Geometric Average: r. G = [(1+r 1)(1+r 2)]1/2 – 1 – r. A ? r. G always Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 9

Example 1 Assets(Beg. ) 1. 0 HPR. 10 TA (Before Net Flows) 1. 1 Net Flows 0. 1 End Assets 1. 2 Fin 2802, Spring 10 - Tang 2 1. 2. 25 3 2. 0 (. 20) 1. 5 0. 5 2. 0 1. 6 1. 0 (0. 8) 0. 0. 8 1. 0 Chapter 5: Risk and Return 4. 8. 25 10

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% Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 11

Dollar Weighted Average Example Net CFs $ (mil) 0 -1. 0 1 2 - 0. 1 - 0. 5 3 0. 8 4 1. 0 Solving for IRR 1. 0 = -. 1/(1+r)1 + -. 5/(1+r)2 +. 8/(1+r)3 +1. 0/(1+r)4 r =. 0417 or 4. 17% Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 12

Which One to Use? Dollar-weighted return(IRR): • Use if focus is total amount of money at some terminal date (wealth) Time-weighted return: - Arithmetic Average: , ignore compounding - Geometric Average: compounding over time. • Use if there is no control over timing • Used most by money management industry Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return , 13

HPR - Expected Return Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 14

Normal distribution Fin 2802, Spring 10 - Tang Chapter Risk and Return Chapter 5: 5: Risk and Return 15

HPR - Risk Measure Variance or standard deviation: Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 16

Why do we need the variance? • Two variables with the same mean. • What do we know about their dispersion? Fin 2802, Spring 10 - Tang Chapter Risk and Return Chapter 5: 5: Risk and Return 17

Example 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=0. 54 Sigma=0. 7348=73. 48% Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 18

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 Fin Spring 08 - Tang Fin 2802, Spring 10 - Tang Chapter Risk and Return Chapter 5: 5: Risk and Return 19

Annual Holding Period Returns Historical Returns: 1926 -2003 Series World Stk US Lg Stk US Sm Stk Wor Bonds LT Treas T-Bills Inflation Fin Spring 08 - Tang Fin 2802, Spring 10 - Tang Geom. Mean% 9. 41 10. 23 11. 80 5. 34 5. 10 3. 71 2. 98 Arith. Mean% 11. 17 12. 25 18. 43 6. 13 5. 64 3. 79 3. 12 Chapter Risk and Return Chapter 5: 5: Risk and Return Stan. Dev. % 18. 38 20. 50 38. 11 9. 14 8. 19 3. 18 4. 35 20

Skewed Distribution: Large Negative Returns Possible Median Negative Fin 2802, Spring 10 - Tang r Chapter Risk and Return Chapter 5: 5: Risk and Return Positive 21

Skewed Distribution: Large Positive Returns Possible Median Negative Fin 2802, Spring 10 - Tang r Chapter Risk and Return Chapter 5: 5: Risk and Return Positive 22

Table 5. 5 Risk Measures for Non-Normal Distributions Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 23

Summary ØDefinition of Returns: HPR, APR and AER. ØRisk and expected return ØNext: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 5: Risk and Return 24