Slide 1 W. Suo

Measuring risk and returns: Measuring risk and returns:

Brief review of probabilityBrief review of probability

Slide 2 W. Suo

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 of ValuePresent

Funds Invested of Value Terminal return of rate 1

• Investment’s terminal value

return of rate 1 ValuePresent Value Terminal

Equivalent

Slide 3 W. Suo

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

Slide 4 W. Suo

The Basic Random Variable

Ways to calculate one-period rate of return Unmargined returns

Reflects price change and any cash flow income

Margined returnsReflects price change, any cash flows and

interest paid on borrowed funds Transaction costs (TC) can include

Interest on borrowed funds Taxes Commissions

Slide 5 W. Suo

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

Slide 6 W. Suo

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

Slide 7 W. Suo

Uncertainty

Characterized by probability How to interpret probability Random variables Expected value

Most likely value vs expected value Variance Covariance & standard deviation Correlation

Slide 8 W. Suo

Example:

If we held the investment for 2 years, the following outcomes exist:

T=0 T=1 T=2

$100

$120 (50%)

$90 (50%)

$81 (25%)

$108 (50%)

$144 (25%)

Slide 9 W. Suo

Historical Estimation

Histograpm Average return:

Arithmetic average return Geometric mean return

Variance/standard deviation Correlation Spreadsheet examples

IBM & MCD

Slide 10 W. Suo

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

Slide 11 W. Suo

Geometric Mean Example

Example: Given the following asset prices, calculate the geometric mean of the annual returns

Year PriceBegin PriceEnd

2001 $40 $60

2002 $60 $40

Slide 12 W. Suo

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

Slide 13 W. Suo

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:

Outcome Probability Rate of Return Product

Up 50% +20% 10%

Down 50% -10% -5%

Total 100% E(r) = 5%

Slide 14 W. Suo

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%

Slide 15 W. Suo

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.

Slide 16 W. Suo

Example: Compounding Returns over Multiple Periods

An investment earned the following returns over the last three years:

Year Return

1 11.1%

2 -2.2%

3 3.3%

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%

Slide 17 W. Suo

Historical Estimation

Histograpm Average Variance/standard deviation Correlation Spreadsheet examples

IBM & MCD

Slide 18 W. Suo

Linear Regression

Brief review Example