Question

Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data.

x:	13	0	23	31	37	17	27	?18	?18	?8
y:	10	?9	18	24	25	25	20	?5	?10	?7

(a) Compute ?x, ?x², ?y, ?y².

?x		?x2
?y		?y2

(b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to two decimal places.)

	x	y
x
s2
s

(c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)

	x	y
Lower Limit
Upper Limit

Use the intervals to compare the two funds.

75% of the returns for the balanced fund fall within a narrower range than those of the stock fund. 75% of the returns for the stock fund fall within a narrower range than those of the balanced fund.     25% of the returns for the balanced fund fall within a narrower range than those of the stock fund. 25% of the returns for the stock fund fall within a wider range than those of the balanced fund.

(d) Compute the coefficient of variation for each fund. (Round your answers to the nearest whole number.)

	x	y
CV	%	%

Use the coefficients of variation to compare the two funds.

-For each unit of return, the stock fund has lower risk.

-For each unit of return, the balanced fund has lower risk.    

-For each unit of return, the funds have equal risk.

If s represents risks and x represents expected return, then s/x can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better? Explain.

-A smaller CV is better because it indicates a higher risk per unit of expected return.

-A smaller CV is better because it indicates a lower risk per unit of expected return.

Answer 1

a)

	X	Y	X^2	Y^2
	13	10	169	100
	0	-9	0	81
	23	18	529	324
	31	24	961	576
	37	25	1369	625
	17	25	289	625
	27	20	729	400
	-18	-5	324	25
	-18	-10	324	100
	-8	-7	64	49
Sum	104	91	4758	2905

; ; and

b) Mean:

Sample variance:

Sample standard deviation:

	X	Y
Mean	10.4	9.1
S.Var	408.49	230.767
S.Std	20.211	15.191

c) 75% chebyshev interval around the mean for x values and also for y values:

Chebhyshev's: and Z=k=2

Chebhyshev's Interval:

	X	Y
Lower	-30.0222	-21.282
Upper	50.82222	39.48201

Use the intervals to compare the two funds.

75% of the returns for the balanced fund fall within a narrower range than those of the stock fund.

d) Coefficient of variation:

CV of X= (20.211/10.4) *100= 1.943376

CV of Y=(15.191/9.1) *100= 1.669341

Use the coefficients of variation to compare the two funds:

For each unit of return, the balanced fund has lower risk. Because coefficient of variation is low.

If s represents risks and x represents expected return, then s/x can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better:

A smaller CV is better because it indicates a lower risk per unit of expected return