Solution Sheet---Statistics 101 Problem Set #5

1. a) Ex 4.113 p303

If you sold only 12 policies, you would probably face a huge risk. Even though the average loss per person is only $250, that average comes from a distribution that has some losses equal to zero and a very few losses that are greater, perhaps $200,000 or more. One loss would be much more than the profit from 11 policies. If, instead, you sold thousands of polices, the gain from the many, many polices that paid out $0 would be much more than the few losses incurred

b) Ex 4.114 p303

Since the number of policies is 10,000, we can regard the average loss X is normal distributed with mean 250 and standard deviation 300/(sqrt(10,000))=3. Therefore,

P(X>260)=P((X-250/3)>10/3)=0.0004

2. Ex 5.42 p334

a). 509*0.116=59.044 so the mean number of home runs he will hit in 509 times at bat.

b). First, we notice the variance for a single hit is 0.116*(1-0.116)= 0.102544, therefore for 509 times hit the standard deviation will be sqrt(509*0.102544)=7.22.

P(X>=70)=P((X-59)/7.22>=1.522)=1-.9357=0.0643

c). The true probability is 0.0764, 1-0.0643/0.0764=15.8%, so using normal distribution does not give us a very good approximation this time.

3. Here are summary statistics on monthly returns for five U.S. common stocks, based on 60 observations from January 1991 through December 1995.

Company / Bristol Myers / Ford / IBM / Merck / US Airways
Mean / 0.017207 / 0.019168 / 0.002455 / 0.008826 / 0.009684
Std. dev. / 0.061725 / 0.077009 / 0.080709 / 0.049723 / 0.159719
Correlations
Bristol Myers / 1.0000 / 0.1237 / 0.0201 / 0.0933 / 0.2012
Ford / 0.1237 / 1.0000 / 0.2667 / 0.0432 / -0.0090
IBM / 0.0201 / 0.2667 / 1.0000 / 0.6612 / 0.2239
Merck / 0.0933 / 0.0432 / 0.6612 / 1.0000 / 0.2108
USAirways / 0.2012 / -0.0090 / 0.2239 / 0.2108 / 1.0000
Company / Bristol Myers / Ford / IBM / Merck / US Airways
Mean / 0.017207 / 0.019168 / 0.002455 / 0.008826 / 0.009684
Std. dev. / 0.061725 / 0.077009 / 0.080709 / 0.049723 / 0.159719
Covariance
Bristol Myers / 0.0038 / 0.00059 / 0.0001 / 0.00029 / 0.00198
Ford / 0.00059 / 0.00593 / 0.00166 / 0.00017 / -0.00011
IBM / 0.0001 / 0.00166 / 0.00651 / 0.00265 / 0.00289
Merck / 0.00029 / 0.00017 / 0.00265 / 0.00247 / 0.00167
USAirways / 0.00198 / -0.00011 / 0.00289 / 0.00167 / 0.02551

Assume that these summary figures will still apply to future investments. Also assume that you may not make negative investments (such as short sales) in any of the stocks.

  1. What portfolio (expressed as fractions of a dollar invested in each of the five stocks) would yield the highest expected monthly return (do NOT try to find the long run expected return)?

100% on Ford, it has the highest monthly return.

  1. What stock is the riskiest?

US Airway, std=0.159719

  1. Let P be a portfolio with half the money invested in Merck and half invested in the riskiest stock. Will P be riskier than a portfolio with all the money invested in Merck?

Var(0.5Merck+0.5USAirway)=0.00783269>Var(Merck)

  1. What pair of stocks has the highest covariance (not necessarily the highest correlation)? What pair of stocks has the lowest covariance?

Highest covariance: IBM and US airway, 0.00289

Lowest covariance: Ford and US airway, -0.00011

  1. Consider two portfolios, one with equal amounts invested in the two stocks with the highest covariance, the other with equal amounts invested in the two stocks with the lowest covariance. Find the monthly expected returns and standard deviations (per dollar invested) in each portfolio.

1)For IBM and US airway: monthly expected return is:

E(0.5IBM+0.5US airway)= 0.0060695;

standard deviation is:

sqrt(Var(0.5IBM+0.5US airway)) =sqrt(0.25V(IBM)+0.25V(US)+2*0.5*0.5Cov))=0.09721111

2) (By the same way)For Ford and US airway: monthly expected return is 0.014426; standard deviation is: 0.08834591

4, (Extra Credit).

In a portfolio, let X =return per dollar spent on Stock 1 and Y = return spent per dollar on stock 2.

One definition of an optimal portfolio is to invest P (fraction of $1) in Stock 1 and 1-P in Stock 2 so as to minimize Var(Z) where Z = PX+(1-P)Y.

a. Find V(Z) in terms of P, Var(X),Var(Y), and Cov(X,Y).

V(Z)=P2V(X)+(1-P)2V(Y)+2P(1-P)Cov(X,Y)

b. Find the value of P that minimizes Var(Z).

Note: P will be a function of Var(X), Var(Y), and Cov(X,Y).

Let , we get

So,

c. Assume that X and Y have the following joint probability distribution:

Y=0 / Y=1 / Y=2
X=0 / 0.2 / 0 / 0 / 0.2
X=1 / 0.05 / 0.50 / 0.05 / 0.6
X=2 / 0 / 0 / 0.2 / 0.2
0.25 / 0.50 / 0.25 / 1

i)Find Cov(X,Y), Var(X), Var(Y).

ii)Find the P that minimizes Var(Z) for this joint probability distribution.

i)E(X)=0*0.2+1*0.6+2*0.2=1,

E(Y)=0*0.25+1*0.5*+2*0.25=1

E(X2)=1.4, E(Y2)=1.5

E(XY)=0*0*0.2+0*1*0.05+1*1*0.5+2*1*0.05+2*2*0.2=1.4

So, Var(X)=0.4, Var(Y)=0.5, Cov(X,Y)=0.4

ii)

d. Suppose instead that our criterion was to maximize P(Z 2).

i)What would P(Z 2) be for the portfolio in C ii)?

ii)Is this the best that can be done in maximizing P(Z 2)? Explain.

(X,Y) / Z=PX+(1-P)Y / P(X,Y) / Is it possible that Z>=2 with positive probability
(0,0) / 0 / 0.2 / No
(0,1) / 1-P / 0 / No
(1,0) / P / 0.05 / No
(0,2) / 2(1-P) / 0 / No
(2,0) / 2P / 0 / No
(1,1) / 1 / 0.5 / No
(1,2) / 2-P / 0.05 / Yes, only when P=0
(2,1) / 1+P / 0 / No
(2,2) / 2 / 0.2 / Yes, for any P

i)Here P(Z>=2)=P(X>=2)=0.2

ii)No, this is not the best, since if we let P=0, according to the table the best P will be P=0, then P(Z>=2)=P(Y>=2)=0.25.