Question

Calculate the arithmetic average return, the geometric average return, and standard deviation of yearly returns for each of the assets over the entire period. Which asset had the highest return? Highest risk?

			Year-by-Year Total Returns
Year	Large Company Stocks	Small Company Stocks	Long-term Corporate Bonds	Long-term Government Bonds	Intermed-term Government Bonds	U.S. Treasury Bills	Inflation	Political Party
1926	11.62	0.28	7.37	7.77	5.38	3.27	-1.49	R
1927	37.49	22.10	7.44	8.93	4.52	3.12	-2.08	R
1928	43.61	39.69	2.84	0.10	0.92	3.56	-0.97	R
1929	-8.42	-51.36	3.27	3.42	6.01	4.75	0.20	R
1930	-24.90	-38.15	7.98	4.66	6.72	2.41	-6.03	R
1931	-43.34	-49.75	-1.85	-5.31	-2.32	1.07	-9.52	R
1932	-8.19	-5.39	10.82	16.84	8.81	0.96	-10.30	R
1933	53.99	142.87	10.38	-0.07	1.83	0.30	0.51	D
1934	-1.44	24.22	13.84	10.03	9.00	0.16	2.03	D
1935	47.67	40.19	9.61	4.98	7.01	0.17	2.99	D
1936	33.92	64.80	6.74	7.52	3.06	0.18	1.21	D
1937	-35.03	-58.01	2.75	0.23	1.56	0.31	3.10	D
1938	31.12	32.80	6.13	5.53	6.23	-0.02	-2.78	D
1939	-0.41	0.35	3.97	5.94	4.52	0.02	-0.48	D
1940	-9.78	-5.16	3.39	6.09	2.96	0.00	0.96	D

Answer 1

First we find the average return of stocks i.e Sum of Returns / No of observations.

Standard deviation means how much the returns are dispersed from the mean value.

We calculate as : Square Root of (Sum of Square of Mean Dispersion / {No of observation - 1})

First Step : Finding Average

Year	Large Company Stocks	Small Company Stocks	Long-term Corporate Bonds	Long-term Government Bonds	Intermed-term Government Bonds	U.S. Treasury Bills
1926	11.62	0.28	7.37	7.77	5.38	3.27
1927	37.49	22.1	7.44	8.93	4.52	3.12
1928	43.61	39.69	2.84	0.1	0.92	3.56
1929	-8.42	-51.36	3.27	3.42	6.01	4.75
1930	-24.9	-38.15	7.98	4.66	6.72	2.41
1931	-43.34	-49.75	-1.85	-5.31	-2.32	1.07
1932	-8.19	-5.39	10.82	16.84	8.81	0.96
1933	53.99	142.87	10.38	-0.07	1.83	0.3
1934	-1.44	24.22	13.84	10.03	9	0.16
1935	47.67	40.19	9.61	4.98	7.01	0.17
1936	33.92	64.8	6.74	7.52	3.06	0.18
1937	-35.03	-58.01	2.75	0.23	1.56	0.31
1938	31.12	32.8	6.13	5.53	6.23	-0.02
1939	-0.41	0.35	3.97	5.94	4.52	0.02
1940	-9.78	-5.16	3.39	6.09	2.96	0
Total	127.91	159.48	94.68	76.66	66.21	20.26
Average	8.53	10.63	6.31	5.11	4.41	1.35

Then we find the Square of Mean Dispersion

For eg for Large Co Stocks, for year 1926 it would be (11.62 return of that year - 8.53 average of security ) ^2 = 9.56. Likewise we do for each  year and for each security . We sum the Square of mean dispersion.

Year	Large Company Stocks	Small Company Stocks	Long-term Corporate Bonds	Long-term Government Bonds	Intermed-term Government Bonds	U.S. Treasury Bills
1926	9.56	107.16	1.12	7.07	0.93	3.68
1927	838.84	131.52	1.27	14.59	0.01	3.13
1928	1,230.79	844.37	12.05	25.11	12.21	4.88
1929	287.21	3,843.01	9.25	2.86	2.55	11.56
1930	1,117.39	2,379.68	2.78	0.20	5.32	1.12
1931	2,690.22	3,645.99	66.62	108.59	45.35	0.08
1932	279.47	256.70	20.32	137.58	19.32	0.15
1933	2,066.85	17,486.89	16.55	26.84	6.68	1.10
1934	99.35	184.63	56.67	24.20	21.03	1.42
1935	1,532.15	873.68	10.88	0.02	6.74	1.39
1936	644.79	2,934.17	0.18	5.80	1.83	1.37
1937	1,897.24	4,711.72	12.69	23.82	8.15	1.08
1938	510.43	491.42	0.03	0.18	3.30	1.88
1939	79.88	105.72	5.48	0.69	0.01	1.77
1940	335.16	249.39	8.54	0.96	2.11	1.82
Total	13,619.32	38,246.05	224.45	378.50	135.54	36.45

SD = Square Root of (Sum of Square of Mean Dispersion / {No of observation - 1})

For Large Co = Square Root of (13619 / 14) = 31.19

Small Co = 52.27

Long Term Corp Bonds= 4

Long Term Gov = 5.2

Interm Gov Bonds = 3.11

US Treasury =1.61

SD is the disperion of actual return from the mean. Higher the dispersion, higher the risk.

Using the above calculations, Small Co Stocks has the highest return of 10.63% as well as highest risk of 52.27%