The Volatility of Investments

Transcription

1 The Volatility of Investments Adapted from STAT 603 (The Wharton School) by Professors Ed George, Abba Krieger, Robert Stine, and Adi Wyner Sathyanarayan Anand STAT 430H/510, Fall 2011

2 Random Variables in Finance Random variables (RV) are used in finance to represent returns on investments. Variance of such random variables measures the volatility (risk) of the investment. 2

3 Definitions S = amount you start with F = amount you finish with R = gross return = F/S r = net return = (F-S)/S = R - 1 3

4 Example Green, White and Red are three investments Probability distributions of R given by Die value Probability 1/6 1/6 1/6 1/6 1/6 1/6 Green Red White What can you say about the distributions of the three RVs? 4

5 Summaries Calculate the means and SDs of the three RVs Investment Mean SD Green Red White Are these more useful than the probability distributions? Less useful? Which of these investments is most appealing? What tradeoffs did you consider? 5

6 Multi-Period Simulation Suppose you begin with a $1,000 investment in each of Green, Red, and White. The outcome from rolling three dice determines the annual outcome of the investment of matching color. The value of the investment changes according to the gross return given in the appropriate column of the table. For example, suppose that on the first roll of all three dice, you obtain (Green 2) (Red 5) (White 3) Then the values of the investments after the first year are Green: $1,000 * 0.9 = $900 Red: $1,000 * 3 = $3000 White: $1,000 * 1 = $1000 6

7 Multi-period Simulation Suppose the second roll gives (Green 4) (Red 2) (White 6) By compounding, you get Green: $900 * 1.1 = $990 Red: $3000 * 0.2 = $600 White: $1000 * 1.1 = $1100 Note that Green went down by 10% and then up by 10%, but ended up losing value. Why does that happen? 7

8 A Class Experiment Form teams of 2 Start with $1,000 in each investment and carry out the simulation for 25 years of returns Each roll of all three dice represents one year Use the outcomes from rolling the dice to determine what happens to your investments Record the sequence of results on the results form as shown on the handout What happened? Are you surprised? 8

9 A Hybrid Investment Consider a fourth investment which puts half in Red and half in White - call it Pink. The gross return on Pink is just the average of the gross return in each round on Red and White. For example, if R(Red) = 3 and R(White) = 1 R(Pink) = (3+1)/2 = 2 What happens if you had invested $1,000 in Pink with the same series of dice outcomes? Use the fourth column in the handout 9

10 A Computer Simulation Let s do this experiment on the computer Each simulation is a possible realization of the investments What happens across multiple runs? 10

11 Computer Simulation Summary 11

12 Why Does Pink Seem Better? Volatility hurts by eating away at the average rate of return Example: Your starting salary was $100,000. You received a salary increase of 10% and then a salary reduction of 10%. What is your current salary? What would happen to your salary if this up/down bounce were repeated over and over? 12

13 Long-Run Returns Turns out Long-run multi-period gross return Expected single-period gross return - Variance/2 The quantity Variance/2 is called volatility drag Investment Mean SD Variance Green Red White Mean Var / Pink Pink wins! Mixing a loser and a poor performer produces a winner! 13

14 Background Green and White come from things you can actually buy Green: historical performance of the U.S. stock market (the value-weighted index) since 1925, adjusted for inflation White: historical performance of 30-day Treasury Bills, adjusted for inflation Investment Period N Mean SD Variance Stocks T-Bills Red was made up! 14

15 Volatility Drag Adjustment Initial wealth = S Final wealth = F Gross return from period t-1 to t = R t Total gross return after T periods S/F = R 1 R 2 R T The nominal annual return that satisfies the same gross return when compounded annually, r, is given by e rt = R 1 R 2 R T 15

16 Volatility Drag Adjustment Taking logarithms on both sides r = 1 T T t=1 log (R t ) = 1 T T t=1 log (1 + r t ) Using a Taylor s series expansion of log (1 + x) In the long-run 1 T T r 1 T T t=1 t=1 r t = E(r t ) and (r t r t2 /2) 1 T T r t 2 t=1 = Var(r t )/2 2 16

17 Volatility Drag Adjustment So, the long-run net return r = E r t Var(r t )/2 And the long-run gross return R = 1 + r = E R t Var(R t )/2 17

18 Summary Random variables are used to model returns on investments The long term value of an investment depends on its average rate of return and its variance Variance is also called volatility Volatility drag eats away at the return Mixing investments (or portfolio diversification) is a way to achieve higher long term returns 18