# Calculating Probability of Return using Z score

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I’m trying to use the Z-score to calculate the probability that an investment will experience a loss. I understand how you would do this with a project that has cash flows but how would you do this for equity return? In other words how would you go about calculating the IRR for an equity investment or what E(r) value would you use to calculate the Z score? Probably something simple but just humor me.

Being Born Wealthy > Being Jewish or WASPY > Born Pretty > Top 5 MBA > CFA > Avg MBA > Born middle class > Born lower class > Born in crack house > Working in IT but looking to switch to buyside

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Assuming you have mean and stdev, calculate the z-score where return=0 and take the CDF of that z-score.

Example:

mean = 10, stdev = 5.

z(0) = -2, so cdf(-2) = P[X <= z] ~= 2.2%

Yep, I did that using Normsdist but doesn’t the mean value to be an IRR value because where IRR=0 capital is returned or am I misinterpreting the Z score? Another question is it more accurate to use mean or CAGR for such a calculation (as you will get much different results)? Thanks!

Being Born Wealthy > Being Jewish or WASPY > Born Pretty > Top 5 MBA > CFA > Avg MBA > Born middle class > Born lower class > Born in crack house > Working in IT but looking to switch to buyside

That is arithmetic mean versus geometric mean. Thanks a bunch!

Being Born Wealthy > Being Jewish or WASPY > Born Pretty > Top 5 MBA > CFA > Avg MBA > Born middle class > Born lower class > Born in crack house > Working in IT but looking to switch to buyside

There’s no relation to IRR here. the Z score in this case tells you how many standard deviations your return is from the expected return. Like Justin’s example above, if your expected return is 10% and the standard deviation is 2%, the probability that a return of 0% would be returned is 2.2%. That’s all you use it for, IRR is not relevant in this case. If you did want to compute something like a breakeven return, that would be the expected return, generally computed using the CAPM for a simple equity.

DoubleDip Wrote:
——————————————————-
> There’s no relation to IRR here. the Z score in
> this case tells you how many standard deviations
> your return is from the expected return. Like
> Justin’s example above, if your expected return is
> 10% and the standard deviation is 2%, the
> probability that a return of 0% would be returned
> is 2.2%. That’s all you use it for, IRR is not
> relevant in this case. If you did want to compute
> something like a breakeven return, that would be
> the expected return, generally computed using the
> CAPM for a simple equity.

Just to clarify, this probability is a left tail probability, so we’re calculating the probability of a return less than 0% (not equal).

Also, DD’s example is different; it has a -5 z-score so its CDF is much less than 2%.

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