Safety-first and Normal Distribution someone, example, question, 446

lc26mizzou

Thanks for answer

After I woke up today look at it again I got the point.
Shortfall level is the level below which returns are unacceptable for the investor
as the probability rules says P(Z>=x) =1.0-N(x) thus to obtain that x must be negative
as inequation rule holds:

number Z >= x implies Z < -x

easy

ps example was:

$800 000 portolio, you are reponsible for research assets allocation, min liquidate level at the end of year $30 000. Data of portolios
A Er 25% stnd dev 27%
B Er 11% stnd dev 8%
C Er 14% stnd dev 20%

What is the probability that return on SFr portolio will be less that shortfall level

solution has three steps:
1)short fall level RL= 30 000 / 800 000 = 3.75%
2)Best Portolio is portfolio B with safety first ratio = (11%-3,75%)/8% = .90625
is the best alternative according to safery first ratio
then last one:
3) P(Rb<3.75) so we create inequality need to be standardized using equation

Z=(x-u)/stnd dev x normal rv and u mean

Z <= (3,75-11)/8
gives -.90625 rounded to -.91

look into N tables we find .8186 for .91 or ,1814 for negative substract from 1 gives
1-.8186 = .1814 is the probability that return on SFr portolio will be less that shortfall level

dkishore1

Hi
Could someone explain me the reason why in currc Reading 9 page 446
in example 9 question 3 the largest ratio .090625 has negative N(-0.91)?
question in task was:
Q:What is the probability that return on SFr optimal portoilio will be less that the shortfall level?

P(R<3.75) = N(- .90625)?

What is the reasoning here that we use negaitve sign N(-x)?

thanks
R