First Semester Examination– June 2019 MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS EMET 7001 Reading Time: 15 Minutes Writing Time: THREE Hours Permitted Materials: A Non-programmable Calculator; One Double-Sided A4 Sheet Page 1 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) Answer all questions in this section using the answer booklet(s) provided. An- swers are expected to be succinct but complete. Answers that are too long and irrelevant will be penalized. Question 1 [10 marks] For each integral, determine whether it is proper and if so, compute it. 1. [3 marks] ∫ 1 −1 1 exp(x) dx. 2. [3 marks] ∫ 10 −10 x xdx. 3. [4 marks] ∫ pi 0 ( cos(2x) + 1 1+x ) dx. Page 2 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) Question 2 [20 marks] Each part of this question attempts to determine a function f on a given set A. For each part, first determine whether it indeeds defines a function and if it does, determine whether the function is increasing or decreasing or neither on A. 1. [4 marks] A = [0, 3.18], f(x) = x3 − 8×2 + x− 8.13. 2. [4 marks] A = [−5, 5], f(x) = ∫ x−10 ((sin t)2 + t4 exp(t)− log(t2 + 1) + log(30)) dt. 3. [4 marks] A = {−1, pi}, f(x) = −x2. 4. [4 marks] A = [0.2, 10], f(x) solves the equation x2 + √ 7x+ 4y2 = 1 (with unknown y). 5. [4 marks] A = [0.2, 10], f(x) solves the equation x2 + √ 7x + 4y3 + log y = 1 (with unknown y). Page 3 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) Question 3 [10 marks] An open rectangle in R2 is a set of the form {(x, y) ∈ R2 : x ∈ A and y ∈ B} for some (perhaps empty) open intervals A and B, and is denoted by A×B. Note that the empty set is an open rectangle by definition. 1. [5 marks] Show that the intersection of two open rectangles is an open rectangle. 2. [5 marks] Answer only ONE of the following two questions. If you attempt both, your answer to (a) will be marked. (a) Is the union of two open rectangles necessarily an open rectangle? Prove that it is or write down two open rectangles whose union is not an open rectangle. (b) Is the set difference between two open rectangles necessarily an open rectangle? Prove that it is or write down two open rectangles whose set difference is not an open rectangle. Page 4 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) Question 4 [15 marks] Each part of this question contains a claim, determine whether the claim is true and briefly explain. If the claim contains the phrase “if and only if”, evaluate the “if” part and “only if” part separately. As an example, consider the following claim: a differentiable function on an open interval is strictly increasing if and only if its derivative is positive everywhere. The “if” part asserts that if the derivative of the function is indeed positive everywhere, then the function must be strictly increasing. This is true by the Mean Value Theorem. The “only if” part asserts that if the function is strictly increasing, then its derivative must be positive everywhere; in other words, it asserts it that as soon as the derivative of the function fails to be positive at one point, the function cannot be strictly increasing. This assertion is false, as f(x) = x3 is differentiable on R and strictly increasing, but its derivative is zero when x = 0. The conclusion is that the “if” part of the claim is true while the “only if” part is false. 1. [3 marks] Let f be a function on [0, 1]. Claim: f is strictly increasing if and only if 1 is the unique maximum of f and 0 is the unique minimum. 2. [3 marks] Consider two bonds, Bond 1 and Bond 2, with the same maturity date. Claim: Bond 1’s yield-to-maturity is higher than Bond 2’s if Bond 1’s market (dirty) price is lower than Bond 2’s. 3. [3 marks] Claim: the geometric series limn→∞ ∑n j=0 a j converges (which means that the limit as the positive integer n approaches infinity exists and is finite) if −1 ≤ a < 1. 4. [3 marks] Let f be a continuous function on [0, 1]. Claim: f has a root in (0, 1) if and only if f(0)f(1) < 0. 5. [3 marks] Eating too much salt increases the risk of hypertension (high blood pres- sure). Scientists recommend that daily intake of sodium (the chemical element Na) should be no more than 2.3 grams. Calculation (known to be correct) shows that 5.93 grams of table salt contains 2.3 grams of sodium. Claim: an adult’s daily intake of sodium is at or below the recommended maximum if and only if he eats no more than 5.93 grams of table salt every day. Page 5 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) Question 5 [5 marks] If the value of a macroeconomic variable at period t (say Year t) is zt and zt is always positive, then the growth rate of the variable is defined as zt+1−zt zt . Part of the Solow model of economic growth postulates that Yt = AK α t L 1−α t , (1) where Yt is the output of the economy, Kt is the capital stock, and Lt is the labour force. The equation is a hypothesis, but Yt, Kt and Lt are observables. For the purpose of this question, assume that α is a known constant between 0 and 1. The Solow residual in Period (t+ 1) is defined as Yt+1 − Yt Yt − αKt+1 −Kt Kt − (1− α)Lt+1 − Lt Lt . If we assume that α is known, then the above expression only involves observables. In words, the Solow residual is the growth rate of output minus α times the growth rate of capital stock and the (1− α) times the growth rate of labour force. 1. [5 marks] Show that if Eq. (1) is valid with a time-independent A, then the Solow residual should be approximately zero in every period. Page 6 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) Question 6 [25 marks] Each part defines an interval A and a function f on A. Find all the maxima of f on A for each part, if any. 1. [5 marks] A = [−3, 3], f(x) = x− exp(x). 2. [5 marks] A = [−10, 10], f(x) = x4 − 8x2 + 2000. 3. [5 marks] A = R, f(x) = x4 exp(−x2). 4. [5 marks] A = (0,∞), f(x) = x−7 x2+x . 5. [5 marks] A = R, f(x) = − exp(x), if x < 0; 1, if x = 0; 2 exp(−x), if x > 0. Page 7 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) Question 7 [15 marks] In some applications, it is useful to consider matrices whose entries depend on a parameter, which means that each matrix entry is a function. Consider the following example: A(t) = ( cos θ(t) sin θ(t) − sin θ(t) cos θ(t) ) , for t ∈ R. (2) where θ : R→ R is a differentiable function and θ(0) = 0. We can differentiate A(t) with respect to t by differentiating each of its matrix entries, so A′(t) is a 2× 2 matrix whose (1, 1) entry is −θ′(t) sin θ(t), and so on. 1. [5 marks] Show that (A(t))TA(t) = I for every t ∈ R and A′(0) + (A′(0))T = 0, where the “0” on the right hand side is the zero 2× 2 matrix. (Hint: the following formula from trigonometry might be useful: (sinx)2 + (cosx)2 = 1 for every x ∈ R.) 2. [10 marks] Now consider a 3× 3 matrix B(t) which also depends on the parameter t ∈ R. Each of the nine matrix entries of B(t) is a differentiable function of t and B(0) = I. Assume that (B(t))TB(t) = I for every t ∈ R. Show that B′(0) + (B′(0))T = 0. (Hint: trigonometry is of little help here.) ——— End of Examination ——— ——————————————— ———————————— Page 8 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001)
