How can i tell what do i want to reject here? That the PM has a higher Std than the benchmark or lower Std than the benchmark?

Who knows?

Itâs a lousy question.

Itâs a poorly worded question as s2000 points out.

For the purpose of the CFA exam, unless they specifically say otherwise, I think a safe bet is to assume that the null hypothesis means âthe manager adds zero valueâ (no consistent ability to generate alpha or otherwise outperform the benchmark).

Then if you find you must reject the null, it means youâve actually got a good manager expected to add value (generate consistent alpha or otherwise outperform versus the benchmark).

This is a badly worded question but just based on the above rule of thumb, the null would be that the manager gets the same or higher std dev than the benchmark for the same return. They hence would add no value. If you reject the null, then it means youâve found a superior manager and should keep them hired to manage your money.

Cheers - good luck - you got thisđ

@S2000magician thanks S2000magician for confirming my feeling.

@Greybeard_The_Elder thanks Greybeard for confirming my intuitionâŠ Of course, my intuition was wrong since in the answer provided by the provider, they took the manager having lower or equal Std than the benchmark as the null :

So weâre trying to prove that he has a superior Std than the benchmark, such a pessimistic view of life. Why do people always approach things negatively and not hope for the positive

I think the question is wrong then. Because when you get to level 3 and do performance evaluation itâs always based on the null hypothesis being the manager adds zero value.

This level 1 null hypothesis directly contradicts the level 3 curriculum then. I wouldnât worry too much about it honestly.

Thanks for confirming. I meanât my intuition was correct and i thanked you for confirming it. But in the context of this question specifically, it was âwrongâ, which is normal since the question is freaking wrong.

Have a good night/day.

Even at the end of their answer they say âthe null that the manager did NOT achieve a level of risk LOWER than the benchmark.â

Which would mean their null is supposedly my rule of thumb for the null hypothesisâŠ and matching your intuition. I wouldnât sweat over it too much and focus on other questions

But thatâs an hypothesis about the mean return.

This question is about the standard deviation of returns.

Honestly, the null hypothesis depends on your viewpoint:

- If youâre the portfolio manager in question, you want to show that your risk is lower than the benchmarkâs, so your null hypothesis will be something like: H_0:\ Ï_{portfolio} â„ Ï_{benchmark}
- If youâre a competitor of the portfolio manager (or a coworker whoâs a jerk), you want to show that the managerâs risk is higher than the benchmarkâs, so your null hypothesis will be something like: H_0:\ Ï_{portfolio} â€ Ï_{benchmark}

But even in their model answer at the final sentence they in fact say the null hypothesis is the manager doesnât have a std dev (level of risk) below the benchmark. Thatâs verbatim the final sentence to their answer above.

Seems like a screwy question.

So . . . theyâre looking at it from the managerâs perspective: he would like to demonstrate that his risk is lower than the benchmarkâs.

But nothing in the question tells us that that has to be the correct viewpoint.

Youâre correct: screwy question.

PT: rest assured that on the real exam it will be clear which perspective the want you to take.

Based on the model answer above and looking at it, our intuitions are correct.

When they write âwe are hypothesizing that the manager obtained a lower level of risk than the benchmarkâ they are talking about the alternative hypothesis (H1).

Their null hypothesis (H0) is that the manager adds no value and does not generate a lower level of risk than the benchmark. As written in their final sentence.

Their model answer recites that they cannot reject the null. In other words they cannot say the manager constantly achieves a lower level of risk. The test didnât prove the manager to be superior to the benchmark.

Now the bonus round. If this test failed to reject the null hypothesis and it was in error, what type of error is it? It is a type 2 error. At level 3 you will call this a âfiring errorâ because it is an error that causes you to accidentally fire (or not hire) a good manager because your test says they canât outperform when they really can. You made a mistake by firing a superior manager.

A type 1 error, on the other hand, is a âhiring error.â It occurs if your test rejects the null hypothesis that the manager adds no value, but itâs an error and in fact the manager doesnât really add value in real life. You hired (or kept) a bum manager by mistake due to your hypothesis test giving an erroneous outcome.