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James Clark FM212 - Principles of Finance 1

FM212 - Principles of Finance

Asset Pricing

Lecture 5 – Risk, Return and the Cost of

Capital

Key Topics

✓ History of US Stock Market Risk and Return

✓ Expected/Mean Return

✓ Variance and Standard Deviation

✓ Covariance and Correlation

✓ Expected Return and Variance of a Portfolio

✓ Diversification

✓ Beta

James Clark (^) FM212 - Principles of Finance 2

History of US Stock Market Risk and Return Equity Returns - Annual, Nominal Returns Since 1900 James Clark

History of US Stock Market Risk and Return Equity Return Histogram - Annual, Nominal Returns Since 1900 James Clark

History of US Risk and Return Risk and Return Across Asset Classes in the US, 1926 - 2008 Clear positive correlation between mean returns and risk (as measured by standard deviations). James Clark

Population Vs. Sample James Clark A population is the complete set of all items from a system or process that is being studied. Typically populations are very large and often unobservable. A parameter is a characteristic of a population. Population Sample A sample is an observed subset of population values of manageable size. A statistic is a characteristic of a sample.

Sample Mean Return James Clark Statistics Definition Sample Average/Mean 1 2 ........ N R R R R N

= 1 1 N i i R R N (^) = =

R 1 ……., R N is the data series of returns from our sample. The sample average/mean (pronounced R bar) N is the number of observations in our sample.

Variance and Standard Deviation James Clark Probability Definition ( ) 2 2 1 N R i i i  p R E R = =  − 

( ) ( ) ( ) 2 2 2 2 1 1 2 2 ........... R N N  = p  R − E R  + p  R − E R  + + p  R − E R        Variance of Returns One measure of the risk of a financial asset is the variance of its returns. The variance is the expected value of the squared deviations of the random variable from its population mean: Remember that the units of the variance are not in the same units as R because they have been squared.

Variance and Standard Deviation Example James Clark Given the following information calculate the standard deviation for Stock A. This question gives the probabilities for the random variable where the random variable is the return on stock A. We therefore will use the probability definition of standard deviation.

Variance and Standard Deviation Example James Clark First calculate the expected return on stock A. Next calculate the variance. ( )

=

3 i 1 A i i E R p R ( ) = 0. 2 ( 0. 14 ) + 0. 4 ( 0. 075 ) + 0. 4 (− 0. 02 ) = 0. 05 A E R ( ( ))  2 2 R A A E R E R A  = −

Sample Variance and Standard Deviation James Clark Statistics Definition

2 2 2 2 1 2 .......... 1 N R R R R R R s R N − + − + + − = − Sample Variance

2 2 1 1 1 N i i s R R R N (^) = = − −

Remember that the units of the sample variance are not in the same units as R because they have been squared.

Sample Variance and Standard Deviation James Clark Statistics Definition Sample Standard Deviation

2 1 1 1 N i i s R R R N (^) = = − −

2 s R = s R The sample standard deviation is simply the square root of the sample variance and hence the units of the sample standard deviation are in the same units as R.

Sample Variance and Standard Deviation Example James Clark First calculate the sample mean return. 1 1 N i i R R N (^) = =

( ) (^) ( ) 2 2 1 1 1 N i i s R R R N (^) = = − −

0.10 0.08 0.06 0.01 0.

5 R

• − + + = = Next calculate the sample variance.

Sample Variance and Standard Deviation Example James Clark The question asks for the sample standard deviation so we need to square root the sample variance. ( ) ( ) 2 s R = s R s R ( (^) ) = 0.0055 = 0. The sample standard deviation is 7.42%. ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 0.1^ 0.05^ 0.08^ 0.05^ 0.06^ 0.05^ 0.01^ 0.05^ 0.12^ 0.^

5 1 s R − + − + − − + − + − = = −