Practice 1

Mathematics for Economics and Finance Practice Questions I Instructor: Norman Schürho¤ Teaching Assistants: Zhihua (Cissy) Chen, Natalia Guseva Session: Fall 2006 Date: November 1, 2006 1. The overall e¤ect of a change in the price of a good on the demand for it is the sum of two separate component e¤ects: the substitution e¤ect (demand for the good will increase when price falls because it is now cheaper relative to its substitutes), and the income e¤ect (a fall in the price of a good increases the consumer’s real income, leading to an increase in demand if the good is a normal good and a fall in demand if the good is an inferior good). Prove the following statements using deductive proof, proof by contraposition and proof by contradiction: (i) A su¢ cient condition for the demand for a good to increase when its price falls is that it is a normal good (ii) A necessary but not su¢ cient condition for the demand for a good to decrease when its price falls is that it is an inferior good. 2. Consider 10 years coupon bond. Coupon payments are annual and is $1, face value are $2, rate of return on this bond is 2%. Compute the price of the bond. What is the price of perpetual bond? 3. A and B are squared matrixes of order n > 2; det A and det B are their determinants. Which of the following statements are true: (i). if det A = det B then det(A

B) = 0

(ii) if A and B di¤ers only by interchange of 2 columns then det A = det B (iii) if det A 6= 0 then either det B = 0 or det(AB) = 0 4. There are n assets S1 ; S2 :::Sn : Expected returns on these assets are R1 ; R2 :::Rn : Variances of returns 21 , 22 ::: 2n and covariances between returns on assets Si ; Sj : ij ; 1 i; j n. Consider a portfolio P composed of these n assets in which weights of assets nP S1 ; S2 :::Sn are w1 ,w2 :::wn (0 wi 1; wi = 1): i=1

(i) Compute expected return Rp and variance (ii) Show that variance

2 p

2 p

of portfolio P

of portfolio P is a quadratic form in n variables w1 ; w2 ::::wn : 0 1 w1 B w1 C C Put this quadratic form in a matrix notation: 2p = w0 Aw, where w = B @ : A : A is wn called a variance-covariance matrix of assets S1 ; S2 :::Sn 1

(iii) De…ne the type of de…niteness of the quadratic form (P D; N D; P SD; N SD; ID). Note: No computations needed! (iv) Consider three assets: S1 ; S2 ; S3 : R1 = 7%; R2 = 9%; R3 = 14%: 21 = 0:02; 22 = 0:05; 23 = 0:09; 12 = 0:01; 13 = 0:02; 23 = 0:03w1 = 0:2,w2 = 0:5; wn = 0:3: Show that variance-covariance matrix of these assets is in fact positive de…nite by checking the signs of determinants of leading principle minors. 0 1 x11 ::::::x1n B x21 ::::::x2n C C 5. There are n di¤erent states of nature and m assets in economy: X = B @ A : xm1 ::::::xmn is a payo¤ matrix, where each column corresponds to each state and each row shows how much an asset gives in each particular state. Financial market is complete if by means of existing assets one can construct a portfolio which will give any desired combination of payo¤s in di¤erent states. If there are more assets than needed to construct such a portfolio then assets which are not used in a portfolio are called redundant. (i) If market is complete, what is the relation between m and n (n relation)

m; m

n or no

(ii) If market is complete what can you say about rank of matrix X? (iii) What is a condition for absence of redundant assets? (iv) Consider economy with 3 states and 4 assets. The payo¤ matrix is: X = 0 1 301 @ 5 1 4 A : Is market complete? Are there redundant assets? (v) Consider econ919 1 0 135 B 113 C C omy with 3 states and 4 assets. The payo¤ matrix is: X = B @ 3 9 15 A : Is market is 5 5 15 complete? Are there redundant assets? 6. Economy can be in three states: boom, recession and normal growth. There is 20% chance that when economy is in boom next period it will move to normal growth regime and 5% chance that the period of boom will lead to recession. If economy is in normal growth state there is 2% chance that it will boom next period and 8% chance that it will fall into recession. For economy which is currently in recession there is 20% chance it will become normal next year and 10% chance that the economy will boom. Today economy is in recession. What is the probability that in 10 years an economy will be in boom or recession? What happens with these probabilities as t ! 1 7. Show that for a matrix An n if the eigenvalues ( 1 ; :::; n ) are all distinct then the associated eigenvectors (x1 ; :::; xn ) are linearly independent. Pn 8. Let A = (aij )n n be a matrix where all columns sum to 1, i=1 aij = 1 for j = 1; :::n: Show that = 1 is an eigenvalue of A. 9. Determine the de…niteness of Q =

x21 + 6x1 x2 2

9x22

2x23

Following exercises are based on the book Sydsaeter et al., "Further Mathematics for Economic Analysis" 10. Page 39, Exercise 4. 11. Page 31, Exercises 1. 12. Page 31, Exercises 8. 13. Page 25, Exercise 3. 14. Page 20, Exercise 1.

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