# Logarithm/540 Week 4

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Problem Set — Review & Introductory Material Due in class. Email submissions will not be accepted.) NOTE: List all contributing members legibly on the ﬁrst page of your submission. 1. Math Background Simplify the following expressions (ln is the natural logarithm): (a) ln (b) b − ln(ex ) (a) ln(a) + ln(b) (c) eln(a)+ln(b)−ln(c) (d) ln ea × eb × ec (e) ex × e−x (f) ex+y − ex × ey (g) eln(x) 1 (h) ln(x) + ln( x ) ( ) 2. Math Background Sketch functions with (a) a positive ﬁrst derivative and a positive second derivative over the range x in (0, 1). (b) a negative ﬁrst derivative and a negative second derivative over the range x in (0, 1). 3. Math Background Consider the following function: 1 f (K, L) = K 1/2 × L1/2 2 This…show more content…
Plow stock over the coming Winter. (b) Suppose T-Bills pay 5% return over the Winter. Sketch the capital allocation line, i.e., the line in the standard deviation–expected return space representing all the portfolios achievable by combining the T-bill and the stock in a portfolio. (c) Now suppose you’ve just discovered Professor John Frink, an obscure scientist with a previously unknown, more accurate forecasting technology which allows you to predict that this year’s snowfall will deﬁnitely be between 7 and 11 inches. i. How much would you be willing to pay now for Mr. Plow stock? ii. Do you have a potential arbitrage opportunity? 7. More About Risk You are planning on investing in stocks. You decide to invest in Texaco and Delta Air Lines. Texaco stock is currently selling for \$60 per share while Delta sells for \$40 per share. You determine that the major factor that will impact your investment is the price of oil. After careful forecasting, you narrow the possible outcomes down to 3 major categories, each of which is equally likely to occur. Possibility Oil price increases Oil price unchanged Oil price decreases Texaco Stock pays a \$4 per share dividend; stock price is \$68 pays a \$3 per share dividend; stock price is \$60 pays a \$2 per share dividend; stock price is \$52 Delta Air Lines Stock pays no dividend; stock price is \$32 pays a \$2 per share dividend; stock price is \$42 pays a \$2 per share dividend; stock price is