Question

3. A stock index consists of the market value of a collection of stocks. The rate of return on stock of 1900-1999, the Standard & ollows an approximately normal distribution. In th oor's 500 index, abbreviated S & P 500, bes produced ar average annual return of approximately with a standard deviation of about 18% (a) In what range do the middle 50% of allstocks yearly returns lie? (As with other problems, it may help to araw a picture.) Answer: (-0.641%,23.641%) (b) The stock market is said to be doun on any year if the return on the S & P 500 is below zero. In what percent of years is the market down? Answer: 26.649% (c) Investors have "a lot to cheer about" if in any one year S & P 500 index returns in excess of 20%. In what percent of years does the index gain over 20%? Answer: 31.838% (5517)

Answer 1

Let x be the annual return of S&P 500 index

We are given x follows normal distribution with Mean $\mu$ = 11.5 and standard deviation $\sigma$ = 18

Part a) We are asked to find two limits ( a and b) such that area between them is 50%

The total area under the curve is 100% , then middle area between a and b is 50% therefore area to the left of a is 25% and area to the right of b is 25%

Now we have to find z score corresponding to limit "a" and limit "b" , we can use excel function =NORMSINV(area to the left)

For a,

Area to the left of a is 0.25 or 25%, So type function =NORMSINV( 0.25 ) in the excel,

We will get z score corresponding to "a"= -0.6745

Therefore a = z*σ + µ

= -0.6745*18 + 11.5

a = 0.641%

For b,

Total area to the left of b is ( 0.25+0.50) = 0.75 , So type function =NORMSINV( 0.75 ) in the excel,

We will get z score corresponding to "b"= 0.6745

Therefore b = z*σ + µ

= 0.6745*18 + 11.5

b = 23.641%

Part b) We are asked to find Probability x is less than 0 ; P( x < 0 )

First we need to find z score for x = 0

z = $\frac{x-\mu }{\sigma }$ = $\frac{0-11.5}{18 }$

z = -0.6389

To find area less than the z , we can use excel function =NORMSDIST( z )

So type =NORMSDIST( -0.6389 )in the excel.

We got 0.26145 or 26.145%

Part c)

We are asked to find Probability x is greater than 20 ; P( x > 20)

First we need to find z score for x = 20

z = $\frac{x-\mu }{\sigma }$ = $\frac{20-11.5}{18 }$

z = 0.4722

To find area greater than the z , we can use excel function = 1 - NORMSDIST( z )

So type =1 - NORMSDIST( 0.4722 )in the excel.

We got 0.31838 or 31.838%