Question

Assume you have N stocks σ²=16. Correlations between each pair of stocks is rho = 0.125 = 1/8

a. what would be pairwise covariances?

b. what would be the variance of an equally weighted portfolio of N stocks?

c. Using excel calculate the variance you found in the portfolio in part b for N=1,2,3...30 stocks and plot it

d. Find the percentage of the total portfolio variance coming from pairwise covariances and plot as a function of N for N=1,2,3...30

Answer 1

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c).

Variance 18.00 16.00 Variance 0 5 10 25 30 35 15 20 Number of stock

d).

Of (14+2n)/n, 16/n is because if the variance of ith stock. So we remove it to get total portfolio variance coming from pairwise covariances

So  total portfolio variance coming from pairwise covariances = (14+2n)/n - 16/n = (2n-2)/n

So, the percentage of the total portfolio variance coming from pairwise covariances = ((2n-2)/n)/((14+2n)/n)

Plotting it on excel:

Percentage The presentage of the total portfolio variance coming from pairwise covariance 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 0 5 10 15 20 25 30 Number of stock 35

Variance and the percentage of the total portfolio variance coming from pairwise covariances values in excel:

Number of stock	Variance	the percentage of the total portfolio variance coming from pairwise covariance
1	0.00	0.00%
2	9.00	11.11%
3	6.67	20.00%
4	5.50	27.27%
5	4.80	33.33%
6	4.33	38.46%
7	4.00	42.86%
8	3.75	46.67%
9	3.56	50.00%
10	3.40	52.94%
11	3.27	55.56%
12	3.17	57.89%
13	3.08	60.00%
14	3.00	61.90%
15	2.93	63.64%
16	2.88	65.22%
17	2.82	66.67%
18	2.78	68.00%
19	2.74	69.23%
20	2.70	70.37%
21	2.67	71.43%
22	2.64	72.41%
23	2.61	73.33%
24	2.58	74.19%
25	2.56	75.00%
26	2.54	75.76%
27	2.52	76.47%
28	2.50	77.14%
29	2.48	77.78%
30	2.47	78.38%