Standard Deviation - E-Commerce - Lecture Slides, Slides of Fundamentals of E-Commerce

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Standard Deviation:

Solution

-^ Step #1:

Calculate the Expected Return

(^

) (^ )

(^

) (^ )

(^

) (^ )

=^ = =^

+^

n ∑^ j^ = j j 1 ˆr^

r p 30%^.

15%

.^

0%^.

13.5%

Standard Deviation:

Solution

-^ Step #2:

Calculate the Variance ( ) ( )^

(^ )^

(^

)^ (^

)^ (^

)^ (^

∑^

$ n^

2 2

j^

j j=

2

2

2

σ^ =^

r^ - r^

p =^ 30-13.

.15 + 15-13.

.60 + 0-13.

.

= 87.

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Normal Probability Distribution •^ A symmetrical, bell-like curve where 50% ofpossible outcomes are greater than the expectedvalue and 50% are less than the expected value. •^ A normal distribution is fully described by justtwo statistics:^ •^ Mean^ •^ Standard deviation

Normal Probability Distribution

Mean- 1 σ 50% Probability - 3 σ^ - 2 σ

2 σ1 σ 3 σ 50% Probability 68.26% 95.44% 99.74%

Standard Normal Probability •^ Problem

:^ Standard deviation is correlated with size of the mean • Solution

:^ To allow for easy comparison among distributions with different means, standardizeusing a Z score • Z score measures the number of standarddeviations (

) a particular rate of return (r) is from the mean or expected value (

ˆ −r r = σ z

ˆr

σ

Standard Normal:

Example

-^ What is the probability of a loss on aninvestment with an expected return of 20% anda standard deviation of 17%? •^ Step #1:

Calculate the Z Score for the number of standard deviations from the mean for a 0%return

−^

− =^

=^

≅ −

ˆr r^ σ 0%^

20%

z^

17%

Probability of Earning Less then 0%

20% 3%

• 31%^

-14%^

37%^

54%^ 71%

1.18 st. dev.from the mean 11.9% Probability