For a portfolio with a given safety-first ratio, the probability that its return will be less than RL is N(−SFRatio). Why the negative SFRatio give us the probability that its return will be less than RL?

For example,

Suppose an investor’s threshold return, RL, is 2 percent.

Portfolio 2 has an expected return of 14 percent with a standard deviation of 16 percent. The SFRatio is 0.75 = (14 − 2)/16 for Portfolios 2

the probability that portfolio return will be less than 2 percent is N(−0.75) = 1 − N(0.75) = 1 − 0.7734 = 0.227 or about 23 percent, assuming that portfolio returns are normally distributed.

I have also found that the probability that portfolio return will exceed 14 percent is P(Z ≥ 0.75) = 1.0 − P(Z ≤ 0.75)=N(−0.75) which mean the the probability that portfolio return will exceed 14 percent is same as the probability that portfolio return will be less than 2 percent. Am I correct?

The portfolio return that will exceed 14% should be 50% because then z = (x- X-bar)/SD = (14- 14)/16 = 0. And the N(0) equals 50%.

The negative SFR gives us the probability because SFR ratio is defined as (Expected return - Threshold return)/Standard deviation whereas the z-value is minus of SFR ratio i.e. (Threshold return - Expected return)/S.D.

The probability that the portfolio return will be less than 2% will be equal to the probabiity that the portfolio return is greater than 26%, amigo.

Thank you so much! I have fully understood my problem that I have mixed up the order of the mean return and shortfall level in the numerator.