# Probability for SFratio

What does the N stand for in the calculation of SFRatio probability? N(-SFRatio) and in practice the formula used is = 1-N(SFRatio).

Thanks

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Color me ignorant, but I have no idea what you mean by “SFRatio” nor its probability. If you would clarify that, I’d be happy to try to help.

Apologies- should have been more clear. referring to Safety-first ratio, pertaining to Roy’s criterion

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OK . . . now I’m trying to figure out where the N and probability come into this.

Roy’s Safety First criterion is just the ratio; you compare a number of investments, and the one that has the highest ratio is the best. I don’t see how probability comes into this.

Where did this arise?

SFRatio tells you simply whether you should invest in a security or a portfolio. It is the expected return - minimum required return divided by the standard deviation. The highest value is usually the best.

N(SFRatio) calculates the probability that the stock/security in question will have returns falling below the threshold of minimum required returns. So you assume a standard normal distribution and calculate the Z-value of each, then measure the CDF. Again, the highest value for SFR will always lead to the lowest probability of Pr(Rp

-N(SFRatio) is the probability that the above scenario will happen. Lower the better.

1-N(SFRatio) is the opposite.

Aha!

I hadn’t realized that kaia_p was using N(Z) to signify the cumulative standard normal probability function.

The scales fall from my eyes.

It confused me as well when I studied for L1, I had to look back again just now to recall.

sorry still a bit confused. let me give you an example, please help to explain what the N is

For example, suppose an investor’s threshold return, RL, is 2 percent. He is presented with two portfolios. Portfolio 1 has an expected return of 12 percent with a standard deviation of 15 percent. Portfolio 2 has an expected return of 14 percent with a standard deviation of 16 percent. The SFRatios are 0.667 = (12 − 2)/15 and 0.75 = (14 − 2)/16 for Portfolios 1 and 2, respectively. For the superior Portfolio 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.

thanks!

N is the cumulative standard normal distribution: N(Z) = P(z ≤ Z), where z’s distribution is standard normal (μ = 0, σ = 1).

For portfolio 1, the SF ratio is 0.67; i.e., the minimum acceptable return is 0.67 standard deviations below the mean. If the distribution of returns is normal, then the probability that the return will be below the minimum acceptable return is P(z ≤ -0.67) = N(-0.67). From table of cumulative standard normal distribution values, N(-0.67) = 0.2514, so there’s a 25% chance that portfolio 1 will have a return less than 2%.

For portfolio 2, the SF ratio is 0.75; i.e., the minimum acceptable return is 0.75 standard deviations below the mean. If the distribution of returns is normal, then the probability that the return will be below the minimum acceptable return is P(z ≤ -0.75) = N(-0.75). From table of cumulative standard normal distribution values, N(-0.75) = 0.2266, so there’s a 23% chance that portfolio 2 will have a return less than 2%.

Actually makes sense Thank you- much appreciated!

My pleasure.

Thank you.

You’re welcome.