Jensen's Inequality

[Eveline]

Hi David,

Sorry, another weird question that I can't figure out:

Given the below data for the US dollar and Canadian dollar exchange rates, which of the following statement is true?

Level CAD/USD USD/CAD
Current 1.0 1.0
Up 1.1 0.9
Down 0.9 1.11111
Mean 1.0 1.0101010

1) E[CAD/USD]=1/E[CAD/USD]
2) E[CAD/USD]>=1/E[CAD/USD]
3) E[USD/CAD]<=1/E[CAD/USD]
4) E[CAD/USD]=E[USD/CAD]

Ans: B

I got the answer by observation. Can you tell me how the mean of USD/CAD is calculated? I assume it is arithmetic mean, maybe just a typing error...

The answer also quoted Jensen's inequality which I don't understand. What is this question really about?

Thks for your help!

 

[David Harper CFA FRM]

Hi Eveline,

Yea, it's not really related to the FRM and frankly would never be asked; the question likely pulls from some other financial exam...

It is precisely a (classic) manifestation of Jensen's inequality
(see: http://en.wikipedia.org/wiki/Jensen's_inequality
...sorry, the wikipedia entry is very unfriendly but i do not have a better pointer)
the function here is f(x) = 1/x, and Jensen's inequality applies because that it a convex function
(i.e., f(x) = 1/x, then second derivative = f''(x) = 2/x^3 is positive)

the mean of USD/CAD = average[1/1.1, 1/0.9] = 1.0101

...but, for purposes of FRM, i don't think this is relevant...keep in mind we do care about Jensen's alpha, which is not to be confused with this Jensen's inequality

David

 

[de]

This question also caused me problems however, in retrospect, it is more straight forward than the sample answer makes out.

All that must be remembered to answer the question is that E[] is equivalent to 'mean' and then to evaluate each of A-D to determine which one holds true.

So, stylistically, I think this is a type of question we should expect and be prepared for.

 

[David Harper CFA FRM]

de

I think you do have a point, "stylistically" probably a candidate should be sort of prepared for off-the-syllabus questions ... although, I may have a blind spot, even to revisit this (after 1.5 years) i find the question non-trivial. I still think it would be a bit unfair given Jensen's inequality is not assigned. (and i am not sure which assignment, if any, would quite prepare a candidate to do it ... but maybe i just don't see the easier way; i.e., the mean concept does not automatically resolve for me personally the issue of the direction of the inequalities).

But your general point is clearly valid, it's the (painful?) charm of the FRM that the testable domain is not narrowly predefined ....David