Question

INUIR 2 MAT 147 S 20 - MAT 147, section 71, Spring 2020 WebAssign x2 reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percen Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Baland ck and 40% bond). For the past several years, we have the following data, 0 14 24 28 37 26 -33 - 6 15 25 28 x * - - 1,024 O =) Compute 2x, Ex, Xy, Ey2 196 57 Ex 1162x25456 28 841 1 369 Ey 10722 3521 676 119 b) Use the results of part (a) to compute the sample mearl variance, and standard deviation for x and to 494 our answers to two decimal places.) f 456 3521 DOT Do golt ONT- 3- 9-VII- ex - 9 2 Dolo 7 10/116 * .b 610. 1 3(xi-3)* c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your IS

Answer 1

Note- The sum of square of y is wrongly calculated as 3521 in the given calculation over the image of the problem. The correct sum of squares of y is calculated and the whole problem is solved with corrected value as follows.

(a)

Στ = 116
  
x2 = 5456

Σy = 107
  
Συ? = 3221

(b)

- 10/ r = 11.6
  
Ο – Συ= 10.7

6 = Σ( – 1)2 = 456.71
  
2 = 0 Συ - ) = 230.68

Sy = 456.71 = 21.37

Sy = V230.68 = 15.19

(c)

We know, Chebyshev's inequality is given by

P(|X – Hl> ko) <

= P(X -H 5ko) 21 -

We estimate population mean _$\mu$ by corresponding sample mean and population standard deviation $\sigma$ by sample standard deviation.

To obtain 75% interval we have

12 = 0.25 k= 2

So, 75% Chebyshev interval around mean is given by
_{(μ – 2σ, μ+ 2ο)}
.

	x	y
Lower limit	-31.14	-19.68
Upper limit	54.34	41.08

Range is $4\sigma$ in both cases. So, it is directly depending upon value of standard deviation. As
_{Sy <S}
we can conclude that 75% of the returns for the balanced fund fall within a narrower range than those of the stock fund.

(d)

Coefficient of variation is defined as

CV

We use sample standard deviation in place of population standard deviation and sample mean in place of population mean.

CV = 21.37 -* 100% = 184.22% 11.6

CV 15.19 10.7 * 100% = 141.96%

Since
_{CV > CV}
we can conclude that for each unit of return, the balanced fund has lower risk.

We know CV denotes coefficient of variation i.e. the average amount of variation. So, a smaller CV is better because it indicates lower risk per unit of expected return.

If two events A and B are independent, then P(A|B) = P(A) = 0.20 [As occurrence of A is independent of occurrence of B]

During computation of standard deviation, use of sample data or data comprising the entire population matters and the cause is as follows.

Yes. The formula for s is divided by n-1, while the formula for $\sigma$ is divided by N.