Covariance problem I don't get his, government, generated, following

suxiaoxiao

Hey all,

Kenny James, CFA, is calculating the covariance of his large-cap mutual fund returns against the returns generated by intermediate government bonds over the past five years. The following information is provided: (A-a) is the annual return minus the mean return for the large-cap mutual fund; (B-b) is the annual return minus the mean return for the intermediate government bonds):

-------- || (A-a) || (B-b)
Year 1 || -23.4 || 4.2
Year 2 || -13.2 || -1.6
Year 3 || -10.4 || 4.8
Year 4 || 19.7 || -12.2
Year 5 || 27.2 || 4.7

Which of the following is closest to the covariance between the mutual fund and government bonds?

A. -47.9
B. -59.9
C. -239.6

--------ANSWER IS SHOWN BELOW--------

The answer is B, can anyone explain why isn't it A??

cchang

that's correct answer:

take a sumproduct of (A-a) and (B-b). divide the total by 4. note, the division is by 4 not by 5

kamara5

why is it 4 not 5?

pacmandefense

It's a sample.

dcfox83

believe Wrote:
-------------------------------------------------------
> why is it 4 not 5?


I believe it has to do with the fact that the data in this problem is historical and/or a sample from a larger population. Same idea when when you divide by degrees of freedom -1.

If you open SN book 1 (quant) to page 202-204....the covariance formula is not divided by anything while SN book 4 (corp fin., port mgt, equity) page 117-118 it is divided by n-1.

So if you're doing a covariance problem in the quant section then don't divide by anything and if you're doing a covariance problem in portfolio mgt and they are talking about a sample then divided by n-1.

I wouldn't get too hung up over why.....its just the way the math works out to be in statistics.



Edited 2 time(s). Last edit at Tuesday, December 1, 2009 at 03:21PM by cjb1010.

bluejazzy

I agree that in this situation it's not important to know the theory behind why it is n-1, but you do need to be very clear on when to use n and when to use n-1.

Micholien

".....its just the way the math works out to be in statistics."

Sorry, studying hard right now (procrastinator) or I would expand on this topic a lot more. Implicit in the calculation above is the a priori equal weighting of each year in contribution to the sample or population covariance. 5 data points of equal value, so in a popluation sense, each contributes 1/5 to the covariance (this is not mathematically rigorous but it will do). Since you can factor out 1/5 from each term, the sum of pairs of products becomes the product of a scalar with a sum. In Portfolio Theory, returns are assigned a probability (and thereby a weighting towards the covariance - again not rigorous, really a joint probability functoin P(x,y) which simplifies in our case to the weights), since the probability weights (all less than or equal to 1) are multiplied by each corresponding return rather than factored out, there is no scalar fraction that divides the resulting sum - it has already been distributed among the addends. And just like in the first example, the sum of the weights = 1 (this is mandated by a bigger topic about probability measures having p(omega) = 1 where omega is the probability space - i think this was briefly discussed in book.) This a rather crude explanation but I hope it helps. Best of luck to you on Saturday.