Question

Use the data in the following table: Year End S&P 500 Index Dividends Paid* S&P 500 Realized Return Microsoft Realized Return 1-Month T-Bill Return 2004 1211.92 2005 1248.29 23.15 4.90% -0.90% 3.00% 2006 1418.3 27.16 15.80% 15.80% 4.80% 2007 1468.36 27.86 5.50% 20.80% 4.70% 2008 903.25 21.85 -37.00% -44.40% 1.50% 2009 1115.1 27.19 26.50% 60.50% 0.10% 2010 1257.64 25.44 15.10% -6.50% 0.10% 2011 1257.61 26.59 2.10% -4.50% 0.00% 2012 1426.19 32.67 16.00% 5.80% 0.10% 2013 1848.36 39.75 32.40% 44.30% 0.00% 2014 2058.9 42.47 13.70% 27.60% 0.00% 2015 2043.94 43.45 1.40% 22.70% 0.00% 2016 2238.83 49.56 12.00% 15.10% 0.20% 2017 2673.61 53.99 21.80% 40.70% 0.80% a) What was the average dividend yield for the S&P 500 from 2005-2017? b) What was the volatility of the dividend yield? c) What was the average annual return of the S&P 500 from 2005-2017 excluding dividends (i.e., from capital gains only)? d) What was the volatility of the S&P 500 returns from capital gains? e) Were dividends or capital gains a more important component of the S&P 500's average returns during this period? Which were the more important source of volatility?

Question 1[10 points] Use the data in the following table: Year End Dividends Paid" S&P 500 Microsoft Realized Realized Return Return 1-Month T-Bill Return 23.15 27.16 27.86 21.85 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 S&P 500 Index 1211.92 1248.29 1418.3 1468.36 903.25 1115.1 1257.64 1257.61 1426.19 1848.36 2058.9 2043.94 2238.83 2673.61 25.44 26.59 32.67 39.75 42.47 43.45 49.56 53.99 4.90% 15.80% 5.50% -37.00% 26.50% 15.10% 2.10% 16.00% 32.40% 13.70% 1.40% 12.00% 21.80% -0.90% 15.80% 20.80% -44.40% 60.50% -6.50% 4.50% 5.80% 44.30% 27.60% 22.70% 15.10% 40.70% 3.00% 4.80% 4.70% 1.50% 0.10% 0.10% 0.00% 0.10% 0.00% 0.00% 0.00% 0.20% 0.80% a) What was the average dividend yield for the S&P 500 from 2005-2017? b) What was the volatility of the dividend yield? c) What was the average annual return of the S&P 500 from 2005-2017 excluding dividends (i.e., from capital gains only)? d) What was the volatility of the S&P 500 returns from capital gains? e) Were dividends or capital gains a more important component of the S&P 500's average returns during this period? Which were the more important source of volatility?

Answer 1

Answer - a

Average dividend yield for the S&P 500 for a year can be calculated by taking simple average of the annual dividend yield calculated by dividing a year dividend amount by previous year index value because dividend paid in the current year is always for the previous year.

Statement showing annual dividend yield for the S&P 500 from 2005-2017

Year End	S&P 500 Index	Dividend Paid	Dividend Yield (Dividend /Previous year Index)
2004	1211.92
2005	1248.29	23.15	1.91%
2006	1418.30	27.16	2.18%
2007	1468.36	27.86	1.96%
2008	903.25	21.85	1.49%
2009	1115.10	27.19	3.01%
2010	1257.64	25.44	2.28%
2011	1257.61	26.59	2.11%
2012	1426.91	32.67	2.60%
2013	1848.36	39.75	2.79%
2014	2058.90	42.47	2.30%
2015	2043.94	43.45	2.11%
2016	2238.83	49.56	2.42%
2017	2673.61	53.99	2.41%
TOTAL	29.57%

Average dividend yield = Sum of annual dividend yield / Number of years

Where-

Sum of annual dividend yield is 29.57% calculated in the above statement

Number of years is 13 from 2005 to 2017

On putting these values in the formula, we get-

Average dividend yield = Sum of annual dividend yield / Number of years

Average dividend yield = 29.57% / 13

Average dividend yield = 2.27% (approx)

Answer - b

Volatility of the dividend yield is the standard deviation of the annual dividend yield from the average dividend yield (D) of 2.27%.

Statement showing standard deviation of the dividend yield

Year End	Dividend Yield (d)	Deviation (d - D)	Deviation2 (d - D)2
2005	1.91	-0.36	0.1295
2006	2.18	-0.09	0.0089
2007	1.96	-0.31	0.0934
2008	1.49	-0.78	0.6114
2009	3.01	0.74	0.5480
2010	2.28	0.01	0.0001
2011	2.11	-0.16	0.0242
2012	2.60	0.33	0.1074
2013	2.79	0.52	0.2660
2014	2.30	0.03	0.0008
2015	2.11	-0.16	0.0255
2016	2.42	0.15	0.0239
2017	2.41	0.14	0.0200
	29.57		1.8592

D = 29.57/13 = 2.27

Standard Deviation ($\sigma$) =

Deviation? Vn

Where-

Deviation2 is calculated in the above statement as 1.8592

n is the number of years under observation

On putting these figures in the formula, we get-

$\sigma$ =

/1.8592 V 13

$\sigma$ =

V0.1430

$\sigma$ = 0.38, or

Standard Deviation = 0.38%

Hence the volatility of the dividend yield is 0.38%

Answer - c

Average annual return of the S&P 500 for a year can be calculated by taking simple average of the annual return calculated from capital gains only

Statement showing annual return of the S&P 500 from 2005-2017

Year End	S&P 500 Index	Capital Gains (Current Index - Previous Index)	Return (Capital Gains / Previous Index)
2004	1211.92
2005	1248.29	36.37	3.00%
2006	1418.30	170.01	13.62%
2007	1468.36	50.06	3.53%
2008	903.25	-565.11	-38.49%
2009	1115.10	211.85	23.45%
2010	1257.64	142.54	12.78%
2011	1257.61	-0.03	0.00%
2012	1426.91	169.30	13.46%
2013	1848.36	421.45	29.54%
2014	2058.90	210.54	11.39%
2015	2043.94	-14.96	-0.73%
2016	2238.83	194.89	9.54%
2017	2673.61	434.78	19.42%
			100.52%

Average annual return = Sum of annual return / Number of years

Where-

Sum of annual return is 100.52% calculated in the above statement

Number of years is 13 from 2005 to 2017

On putting these values in the formula, we get-

Average annual return = Sum of annual return / Number of years

Average annual return = 100.52% / 13

Average annual return = 7.73% (approx)

Answer - d

Volatility of the return from capital gains is the standard deviation of the annual returns from the average annual return (R) of 7.73%.

Statement showing standard deviation of the return

Year End	Return (r)	Deviation (r - R)	Deviation2 (r - R)2
2005	3.00	-4.73	22.3632
2006	13.62	5.89	34.6854
2007	3.53	-4.20	17.6435
2008	-38.49	-46.22	2135.8996
2009	23.45	15.72	247.2503
2010	12.78	5.05	25.5299
2011	0.00	-7.73	59.7898
2012	13.46	5.73	32.8563
2013	29.54	21.81	475.4951
2014	11.39	3.66	13.4003
2015	-0.73	-8.46	71.5141
2016	9.54	1.81	3.2581
2017	19.42	11.69	136.6553
	100.52		3276.3408

R = 100.52/13 = 7.73

Standard Deviation ($\sigma$) =

Deviation? Vn

Where-

Deviation2 is calculated in the above statement as 3276.3408

n is the number of years under observation

On putting these figures in the formula, we get-

$\sigma$ =

3276.3408 13

$\sigma$ =

252.0262

$\sigma$ = 15.88, or

Standard Deviation = 15.88%

Hence the volatility of the returns from capital gains of S&P 500 is 15.88%

Answer - e

Average return from capital gains is 7.73% (calculated in Answer - c above) and average dividend yield is of 2.27% (calculated in Answer - a above), hence from we can conclude that the capital gains is more important component of the S&P 500's average returns during the period of year 2005 to 2017.

Volatility can be ascertained mostly from the average return from capital gains and not quitely from the average dividend yield since the volatility from the average return from capital gains is higher than that of from the average dividend yield.