Series 7

Now that you understand variance, you need to become acquainted with covariance: the measure of correlation between two different quantities.

Let's say you have a theory that the stock of National Widget Corp. will rise and fall over time in tandem with that of American Widget Services. You want to explain this to your client because NWC's price has recently spiked, while AWS's has stayed in the doldrums. You believe that investing now in AWS stock will pay off soon when AWS's share price rises with the same tide that is now raising NWC's price.

Let's look at the data within a time-series chart. The data follows on the next page. For simplicity's sake, μ has been rounded to the nearest whole number:

Quarter	NWC	AWS
Mar - 00	10	35
Jun - 00	11	38
Sep - 00	12	40
Dec - 00	13	44
Mar - 01	14	48
Jun - 01	15	55
Sep - 01	14	50
Dec - 01	13	45
Mar - 02	12	41
Jun - 02	11	36
Sep - 02	10	29
Dec - 02	11	31
Mar - 03	12	37
Jun - 03	13	40
Sep - 03	10	49
Dec - 03	15	57
Mar - 04	10	58
Jun - 04	13	42
Sep - 04	12	33
Dec - 04	11	29
Mar - 05	10	23
Jun - 05	11	24
Sep - 05	12	25
Dec - 05	13	27
Mean (μ)	12	39

Here's how you prove your point to your client:

1. Collect the historical data, including the mean average.
2. Take the first series and compute its X -minus-μ values.
3. Take the second series and compute its X -minus-μ values.
4. Multiply each X -minus-μ value from step 2 by the value from step 3 for each observation.
5. Sum all values derived in step 4.
6. Divide by the number of observations.

On the following page our table has been expanded to perform the calculations for steps two to four. Note that, at this point, you should start using statistician's symbols for the two data sets. Instead of calling these two series NWC and AWS, which can get confusing, call them r₁ and r_{2 ,} which is more mathematically elegant. You can further distinguish r_1i and r_2i as the actual values and r_1ave and r_2ave as the μ averages of allr_1i and r_2i values.

Finally, push on to steps 4 and 5 - that is, multiply the values in the (r_1i - r_1ave) column by the values in the (r_2i - r_2ave) and sum up those products.

Quarter	r1i	r2i	Step 2 (r1i - r1ave)	Step 3 (r2i - r2ave)	Step 4 and 5 (r1i - r1ave) * (r2i - r2ave)
Mar - 00	10	35	-2	-4	8
Jun - 00	11	38	-1	-1	1
Se - 00	12	40	0	1	0
Dec - 00	13	44	1	5	5
Mar - 01	14	48	2	9	18
Jun - 01	15	55	3	16	48
Sep - 01	14	50	2	11	22
Dec - 01	13	45	1	6	6
Mar - 02	12	41	0	2	0
Jun - 02	11	36	-1	-3	3
Sep - 02	10	29	-2	-10	20
Dec - 02	11	31	-1	-8	8
Mar - 03	12	37	0	-2	0
Jun - 03	13	40	1	1	1
Sep - 03	10	49	-2	10	-20
Dec - 03	15	57	3	18	54
Mar - 04	10	58	-2	19	-38
Jun - 04	13	42	1	3	3
Sep - 04	12	33	0	-6	0
Dec - 04	11	29	-1	-10	10
Mar - 05	10	23	-2	-16	32
Jun - 05	11	24	-1	-15	15
Sep - 05	12	25	0	-14	0
Dec - 05	13	27	1	-12	-12
(r) average	12	39
Number (N)	24			SUM	184

You can see that the sum of the (r_1i - r_1ave)*(r_2i - r_2ave) values is 184. A quick count of the observations gives you an N of 24, so if you do the simple division you come up with a covariance (Cov) of 7.67.

Again, you might find it helpful to write this formula down on scrap paper when you write the exam:

Covariance = Cov (r1,r2) = 1/N * Σ (r1i - r1ave)*(r2i - r2ave)

One note: this example used hard numbers just to keep calculations simple. Generally, covariance is used to determine rates of return, which are expressed as percentages. However, the math is exactly the same.