An assignment where you should begin to appreciate mathematics/statistics and why inance professors may be smart but not necessarily rich. Answer the following questions for a total of 300 points and show all your work carefully.  You do not have to use Microsoft Excel for the assignments since all computations can also be done using programming environments such as EViews, GAUSS, MATLAB, Octave, Ox, Python, R, SAS, or S-PLUS. Please do not turn in the assignment in reams of unformatted computer output and without comments!   Make little tables of the numbers that matter, copy and paste all results and graphs into a document prepared by typesetting system such as Microsoft Word or TEX while you work, and add any comments and answer all questions in this document.

1.     a.  (1 point) By computing the necessary derivatives and evaluating them at 北 = 0, expand the functions e北 and ln(1 + 北) in a Taylor series about the point 北 = 0.

b.  (1 point) For each of the functions e北 and ln(1 + 北) compute the function val- ues at 北 = 1, 0.1, and 0.01. Keeping only up to second order terms in the Taylor expansions of these functions from previous part, compute the Taylor series approx- imations to the functions at those 北 values and determine the approximation error (absolute and relative) in each case.  Try to maintain as many decimal places in your answer as possible, e.g., 8 places. Note the improvement in the approximation due to the presence of the second order terms.

2.  (4 points) Given (北1 y1 ) . . . (北n yn ) from R2 , find the values of a1 , a2  and a3  that will maximize the function:

f(a1 a2 a3 ) = exp µ − (yi − a1 − a2北i )2 ¶

Verify your solution with the second derivative test.

3. Consider the matrix M = (mi,j ) :

⎛ 096 M = −0.3

(  1.5

0.6

16.04

1.18

−1.5

−0.3

1.18

4.1

−0.57

⎞⎟

−0.57 ⎟

a.  (2 points) Find the Cholesky decomposition of M.   Show your steps or the al- gorithm.   Then use the Cholesky decomposition to solve Mx  = b for x when b = (2.49 0.566 0.787 −2.209)> .

b.  (3 points) Find the eigenvalues of M and the corresponding eigenvectors with unit length.  Show your steps or the algorithm.  Is M positive definite?  Are the eigenvectors orthogonal?

4.  (2 points) Let W, X , Y and Z be random variables describing next year’s annual return on Weyerhauser, Xerox, Yahoo and Zymogenetics stock. The table below gives a discrete probability distribution for these random variables based on the state of the economy:

a. Plot the distributions for each random variable (make a bar chart). Comment on any difference or similarities between the distributions.

b. For each random variable, compute the expected value, variance, standard devi- ation, skewness and kurtosis and briefly comment.   Note:  You  cannot use the

Excel functions AVERAGE  VAR .P or VAR .S  STDEV .P or STDEV .S  SKEW .P or SKEW

and KURT for this problem.  These functions compute sample statistics which are different from the population moment calculations required for this problem.

5.  (3 points) Suppose a continuous random variable X has density function: f(北;a) = {  0(a北)2 (1 − 北 se(北)< 1,

a. Find value(s) of a such that f(北;a) is a density function.

b. Find the mean and median of X .

c. Find Pr(0.25 ≤ X ≤ 0.75).

6.  (3 points) Suppose a continuous random variable X has density function: f(北;a) = {  0(a北) 北 ≤ 1,

a. Find value(s) of a such that f(北;a) is a density function.

b. Find the mean and median of X .

c. For what value of a is the variance of X maximized?

7.  (1 points) Suppose X is a normally distributed random variable with mean 0.05 and variance (0.10)2 , i.e., X ∼ N(0.05, (0.10)2 ). Compute the following:

a. Pr(X > 0.10).

b. Pr(X < −0.10).

c. Pr(−0.05 < X < 0.15).

d. Determine the 1%, 5%, 10%, 25%, 50%, 75%, 90%, 95% and 99% quantiles of the distribution of X .

Hint: you can use the Excel functions NORM .DIST and NORM .INV to answer these ques- tions.

8.  (2 points) Suppose that fX (北) = 1/4 if |北| 想 1 and fX (北) = 1/(4北2 ) if |北| ≥ 1.  Show that R fX (北)d北 = 1 so that fX  really is a density, but that R∞ 北fX (北)d北 = −∞ and R0∞ 北fX (北)d北 = ∞ so that a random variable with this density does not have an expected value.

9.  (3 points) Let Z be a standard normal random variable, and let g be a differentiable function with derivative g\ . Note: Assume that g(北) ∈ o(exp(北2 )) with E(|g\ (北)|) 想 ∞ , where g(北) ∈ o(h(北)) means lim北→∞ h(g)北(北)  = 0.

a. Show that E(g\ (Z)) = E(Z g(Z)).

b. Show that E(Z几+1 ) = n E(Z几 −1).

c. Find E(Z4 ).

10. Suppose Xt i. . N(u, a2 ) with u > 0 for t = 1, . . . ,T. Define X =Q Xt .

a.  (2 points) Find E(X) and Var(X).

b.  (1 point) Does X follow a normal distribution or a skewed distribution?

11.  (4 points; Normal mixture models)

a. What is the kurtosis of a normal mixture distribution that is 95% N(0, 1) and 5% N(0, 10)?

b. Find a formula for the kurtosis of a normal mixture that is 100p% N(0, 1) and 100(1 − p)% N(0,a2 ) where p and a are parameter. Your formula should give the kurtosis as a function of p and a.

c. Show that the kurtosis of the normal mixtures in part (b) can be made arbitrarily large by choosing p and a appropriately. Find values of p and a so that the kurtosis is 10,000 or larger.

d. Let M > 0 be arbitrarily large.  Show that for any p0 想 1, no matter how close to 1, there is a p > p0  and a a such that the normal mixture with these values of p and a has a kurtosis at least M.  This shows that there is a normal mixture arbitrarily close to a normal distribution with a kurtosis above any M.

12.  (2 points) Let X be N(0,a2 ). Show that the CDF of the conditional distribution of X given that X > C is

Φ(北/a) − Φ(C/a)

1 − Φ(C/a)

where 北 > C, and that the PDF of this distribution is

巾(北/a)

a(1 − Φ(C/a))

where 北 > C.  Also show that if C = 0.25 and a = 0.3113, then at 北 = 0.25 this PDF equals the PDF of a Pareto distribution with parameters a = 1.1 and C = 0.25.  Note: The value of a = 0.3113 was originally found by interpolation.

13.  (7 points) Consider the following joint distribution of X and Y :

Y

2

1 0.1   0.2     0

X   2 0.1     0     0.2

3 0     0.1   0.3

a. Find the marginal distributions of X and Y.  Using these distributions, compute E(X), Var(X), SD(X), E(Y), Var(Y) and SD(Y).

b. Determine the conditional distribution of X given that Y equals 1, 2 and 3. Plot the marginal distribution of X along with the conditional distributions of X and briefly comment.

c. Determine the conditional distribution of Y given that X equals 1, 2 and 3. Plot the marginal distribution of Y along with the conditional distributions of Y and briefly comment.

d. Compute E(X|Y = 1), E(X|Y = 2), E(X|Y = 3) and compare to E(X). Compute E(Y |X = 1), E(Y |X = 2), E(Y |X = 3) and compare to E(Y).

e. Plot E(X|Y = y) versus y and E(Y |X = 北) versus 北 and briefly comment.

f. Are X and Y independent? Fully justify your answer.

g. Compute Cov(X》Y) and Corr(X》Y).

14.  (3 points) Let W, X , Y , Z be random variables and a, b, c, d be constants. Show that:

a. Var(aX + c) = Var(−aX − d).

b. Cov(aX》bY) = ab Cov(X》Y).

c. Cov(X》X) = Var(X).

d. Cov(aX+bY》cW+dZ) = ac Cov(X》W)+ad Cov(X》Z)+bc Cov(Y》W)+bd Cov(Y》Z).

e. Suppose W = 3 + 5X and Z = 4 − 8Y.

i. Is pYZ = 1? Prove or disprove.

ii. Is pwZ = pY? Prove or disprove.

15.  (4 points) Let X and Y be two random variables.

a. If Cov(X2》Y2 ) = 0, then Cov(X》Y) = 0. True/False/Uncertain. Explain.

b. If X and Y are independent, then Cov(X2》Y2 ) > Cov(X》Y). True/False/Uncertain. Explain.

c. If X and Y are independent and E() > 1, then > 1. True/False/Uncertain. Explain.

d. Prove that (Cov(X》Y))2  ≤ Var(X)Var(Y) and thus −1 ≤ pY ≤ 1.

16.  (4 points) Let X and Y be independent U(−a》a) random variables. Find (a) the prob- ability that the quadratic equation t2 + tX + Y = 0 has real roots, and (b) the limit of this probability as a → ∞ .

17.  (4 points) Let us assume that X1 and X2 are independent N(0, 1) random variables and let us define the random variable Y by

Y = {

|X2 |

|X2 |

if X1  > 0;

otherwise.

a. Prove that Y ∼ N(0, 1).

b. Say if (X1 ,Y) is bivariate Gaussian, and explain why.

18.  (6 points) The purpose of this problem is to show that lack of correlation does not imply independence, even when the two random variables are Gaussian!!!  We assume that X , e1   and e2   are independent random variables, that X  ∼  N(0, 1),  and that Pr(ei =    1) = Pr(ei = +1) = 1/2 for i = 1, 2. We define the random variables X1  and X2  by X1  = e1X and X2 = e2X .

a. Prove that X1  ∼ N(0, 1), X2  ∼ N(0, 1) and that p(X1 ,X2 ) = 0.

b. Show that X1  and X2  are not independent.

19. The goal of this problem is to prove rigorously a couple of useful results for normal and log-normal random variables.

a.  (2 points) Use the chain rule to differentiate

Z (北 )/a exp µ u2 ¶ du

with respect to 北 and hence find the density function of the random variable X such that Z = is a standard normal random variable with the distribution

Φ(X) = RX     exp( u2 )du.

i. pmin = (e a      1)/^(e    1)(ea2         1).

ii. pmax = (ea      1)/^(e    1)(ea2         1).

iii. lima →∞ pmin  = lima →∞ pmax = 0.

Do we have a problem interpreting the correlation between log-normal random variables as to their normal counterparts?

20. Suppose that X1 ,X2 , . . . ,X几 are independent real-valued random variables and that Xi has distribution function Fi for each i. The maximum and minimum transformations are very important in a number of applications. Specifically, let U = max{X1 ,X2 , . . . ,X几 }, V = min{X1 ,X2 , . . . ,X几 }, and let G and H denote the distribution functions of U and V respectively.

a.  (2 points) Show that:

i. G(北) = F1 (北)F2 (北) ··· F几 (北) for 北 ∈ R.

ii. H(北) = 1 − [1 − F1 (北)][1 − F2 (北)] ··· [1 − F几 (北)] for 北 ∈ R.

b.  (4 points) If Xi  has a continuous distribution with density function fi  for each i, then U and V also have continuous distributions, and the densities can be obtained by differentiating the distribution functions above.  Suppose that X1 ,X2 , . . . ,X几 are independent random variables, each uniformly distributed on (0, 1).

i. Find the distribution function, density function, expected value and variance of U. Hint: U has a beta distribution .

ii. Find the distribution function, density function, expected value and variance of V. Hint: V has a beta distribution .

21.  (4 points) Let X and Y be two independent N(0, 1) random variables. Find:

a. Cov(X, max[X, Y]) and Cov(X, min[X, Y])

b. Cov(max[X, Y], min[X, Y]), Var(max[X, Y]), and Var(min[X, Y])

22.     a.  (3 points) Suppose that X1  and X2  are independent random variables each uni- formly distributed over the interval (0, 1).  Define two random variables as Y1  = X1 + X2  and Y2  = X1 − X2 . Find the joint density function of Y1  and Y2 .

b.  (3 points) If X is uniform on (0, 2T) and Y , independent of X , is exponential with rate 1. Find the joint density function of U = ^2Y cosX and V = ^2Y sinX .

23.  (4 points; spurious correlation) Consider a sequence of random variables given any X0  : Xj  = Xj −1 + ej

where E(ej ) = 0, Var(ej ) =  1, and Cov(ej ,ek ) = 0 when k j, for j, k = 1, 2, . . . Suppose the series is placed in groups of m sequential non-overlapping observations, (X1 ,X2 , . . . ,Xm ), (Xm+1,Xm+2 , . . . ,X2m), . . . for m = 1, 2, . . . Note that Var(Xj+m  − Xj ) = m. Let Yi  be the average of the i-th group, for i = 1, 2, . . ., i.e.,

m

k=1

a. Find Var(Yi+1 − Yi ).

b. Find Corr(Yi+1 − Yi ,Yi − Yi −1). Give an intuition of your result.

24.  (4 points; stochastic volatility) Consider a sequence of random variables:

北t = ut + atet

for t = 1, 2, . . . ,n, where et  are serially independent and identically distributed random variables with a mean 0, variance 1, and kurtosis V. In general, et can follow a stochastic process and, if so, at is independent of et . However, for now, assume that future values of at are known (deterministic). Define yt as the de-meaned version of 北t so that yt ≡ 北t −ut .

a. Find the standardized kurtosis of yt , i.e., E(yt(4))/(E(yt(2))2 ).

b. Suppose you compute the i-th sample moment (denoted as mi ) using the series of realized values of yt  as follows:  mi  = P y /nt(i) .  What will be the (theoretical) standardized kurtosis of yt  based on such sample moments? How does it compare with that of the above?

25.     a.  (2 points) Suppose that a random variable X has the uniform distribution on the interval [0, 5] and the random variable Y is defined by Y = 0 if X ≤ 1, Y = 5 if X ≥ 3, and Y = X otherwise. Sketch the cumulative distribution function of Y .

b.  (2 points) Suppose X has a continuous distribution with probability density func- tion f . Let Y = X2 , show that the probability density function of Y is

g(y) = (f(^y)+ f(−^y))

c.  (2 points) Suppose that one can simulate as many i.i.d. Bernoulli random variables with parameter p as one wishes. Explain how to use these to approximate the mean of the geometric distribution with parameter p.

26.     a.  (10 points) Let X1  and X2  be random variables with CDF F1 (北) and F2 (北) with F1 (北) ≤ F2 (北) for all values of 北.

i. Which of these two distributions has the heavier lower tail? Explain.

ii. Which of these two distributions has the heavier upper tail? Explain.

iii. If these two distributions are proposed as models for the return of a given portfolio over the next month, and if you are asked to compute VaR0.01  for this portfolio over that period, which of these two distributions will give the larger value at risk?

b.  (10 points) Let W0 denote initial wealth to be invested over the month and assume W0 = $100, 000.

i. Let R denote the monthly simple return on Microsoft stock and assume that R ∼ N(0.04, (0.09)2 ). Determine the 1% and 5% value-at-risk (VaR) over the month on the investment. That is, determine the loss in investment value that may occur over the next month with 1% probability and with 5% probability.

ii. Let r denote the monthly continuously compounded return on Microsoft stock and assume that r ∼ N(0.04, (0.09)2 ).  Determine the 1% and 5% value-at- risk (VaR) over the month on the investment.  That is, determine the loss in investment value that may occur over the next month with 1% probability and with 5% probability. (Hint: compute the 1% and 5% quantile from the Normal distribution for r and then convert continuously compounded return quantile to a simple return quantile using the transformation R = er − 1.)

27.     a.  (10 points) The daily value-at-risk (VaR) under normal distribution for a bank is $8,500 (also called daily earnings at risk, DEAR).

i. What is the VaR for a 10-day period?

ii. What is the VaR for a 20-day period?

iii. Why is the VaR for a 20-day period not twice as much as that for a 10-day period? Explain.

b.  (10 points) Assume that the return density has a polynomial left tail, or equivalently that the loss density has a polynomial right tail.  That is, the return density f satisfies

f (y) ∼ Ay − (a+1)     as y → −∞

where A > 0 is a constant, a > 0 is the tail index, and “∼” means that the ratio of the left-hand to right-hand sides converges to 1.

i. What is the distribution function? That is, find F (y) = Pr(R ≤ y).

ii. Suppose an estimate of a is 3.1. If VaR(0.05) = $252, what is VaR(0.005)?

28.     a.  (10 points) Let P1  and P2  be two portfolios whose returns have a joint normal distribution with means u1  and u2 , standard deviations a1  and a2 , and correlation

p.  Suppose the initial investments are S1  and S2 .  Show that VaR(P1  + P2 ) ≤ VaR(P1 )+ VaR(P2 ) under joint normality of the returns.

b.    i.  (2 points) Suppose that stock A sells at $85 per share and stock B at $35 per share. A portfolio has 300 shares of stock A and 100 of stock B . What are the weight w and 1 − w of stocks A and B in this portfolio?

ii.  (3 points) More generally, if a portfolio has N stocks, if the price per share of the j-th stock is Pj , and if the portfolio has 几j  shares of stock j, then find a formula for wj  as a function of 几1 , . . . ,几N  and P1 , . . . ,PN .

iii.  (5 points) Let RP be a return on some type on a portfolio and let R1 , . . . , RN be the same type of returns on the assets in this portfolio.  Is RP  = w1R1 + ··· + wN RN  true if RP  is a net return? Is this equation true if RP  is a gross return? Is it true if RP  is a log return? Justify your answers.

29.     a.  (5 points) Stocks 1 and 2 are selling for $100 and $125, respectively. You own 200 shares of stock 1 and 100 shares of stock 2. The weekly returns on these stocks have means of 0.001 and 0.0015, respectively, and standard deviations of 0.03 and 0.04, respectively. Their weekly returns have a correlation of 0.35. Find the covariance matrix of the weekly returns on the two stocks, the mean and standard deviation of the weekly returns on the portfolio, and the one-week VaR(0.05) for your portfolio.