Measuring risk and returns: Brief review of probability

1. Measuring risk and returns: Brief review of probability

2. The Investor’s Goal • Goal is to maximize what is earned relative to the amount put into an investment • Maximize either the • Rate of return • Investment’s terminal value Equivalent

3. Rate of Returns • One period rate of return is called a random variable • Returns tend to fluctuate randomly from period to period • Risk is associated with the variability of return • Total risk can be measured with variance or standard deviation • This chapter divides total risk into components

4. The Basic Random Variable • Ways to calculate one-period rate of return • Unmargined returns • Reflects price change and any cash flow income • Margined returns • Reflects price change, any cash flows and interest paid on borrowed funds • Transaction costs (TC) can include • Interest on borrowed funds • Taxes • Commissions

5. Wealth Indices for Average U.S. Investments in Different Asset Classes Compared to Inflation, 1926-99 • If you had invested $1 on December 31, 1925 in each of the following, you would have

6. Average Annual Rate of Return and Risk Statistics for Asset Classes and Inflation in the U.S., 1926-99

7. Uncertainty • Characterized by probability • How to interpret probability • Random variables • Expected value • Most likely value vs expected value • Variance • Covariance & standard deviation • Correlation

8. $144 (25%) $120 (50%) $108 (50%) $100 $90 (50%) $81 (25%) T=0 T=1 T=2 Example: • If we held the investment for 2 years, the following outcomes exist:

9. Historical Estimation • Histograpm • Average return: • Arithmetic average return • Geometric mean return • Variance/standard deviation • Correlation • Spreadsheet examples • IBM & MCD

10. GMA vs AMA • The geometric mean (GMR) differs from the arithmetic mean (AMR) in that the geometric mean • Considers the compounding of rates of return • GMR usually less than AMR

11. Geometric Mean Example • Example: Given the following asset prices, calculate the geometric mean of the annual returns

12. Contrasting AMR and GMR • GMR should be used for • Measuring historical returns that are compounded over multiple time periods • AMR should be used for • Future-oriented analysis where the use of expected values is appropriate

13. Example: GMR vs AMR • An investment costs $100 and it is equally likely to • Lose 10% or • Earn 20% • The probability distribution of such an investment is:

14. Example: GMR vs AMR • Expectations about the future should use the E(r) • If $100 is compounded at 5% annually for two years, the expected terminal value is $110.25 • If the investment actually grew to $108, the multi-period historical returns should be averaged using GMR • ($108/100)1/2 –1 = 0.03923 = 3.923%

15. Compounding Returns over Multiple Periods • Various periodic price relatives can be compounded to obtain a new rate of return for the entire period • 3 monthly returns can be compounded to determine 1 quarterly return • 12 monthly returns can be compounded to determine 1 annual return, etc.

16. Example:Compounding Returns over Multiple Periods • An investment earned the following returns over the last three years: GMR = (1.111)(0.978)(1.033)1/3 –1 = 1.12241/3 – 1 = 3.92% annual return. The total 3-year return is 12.24%. AMR = 11.1% + -2.2% + 3.3% = 12.2%  3 = 4.07%

17. Historical Estimation • Histograpm • Average • Variance/standard deviation • Correlation • Spreadsheet examples • IBM & MCD

18. Linear Regression • Brief review • Example