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FIN 504: Financial Management Lecture 1: Introduction and Mathematical Review Course Material Instructor: Larry Schrenk (lschrenk@ubalt.edu) Course Web Page (http://home.ubalt.edu/ntsbschr/FIN%2050 4/fin_504.htm) Textbook WebTycho 2 FIN 504: Financial Management Calculations and Your Calculator Calculations and Your Calculator Required Functions: +, -, ×, ÷, (, ), e, ln, xy, 1/x, -/+ and a few memories A Possible Ambiguity: ln, log, logx, lng Use of Financial Functions 4 FIN 504: Financial Management Calculations and Your Calculator Working Financial Problems (Inside- Out) 100 1 PV 1 ? .08 1.08 10 5 FIN 504: Financial Management Calculations and Your Calculator Use of Calculator Parentheses Use of Calculator Memories 6 FIN 504: Financial Management Mathematical Review Summation Exponents Logarithms and e Equations in One Unknown Equations in Two Unknowns Suggestions for Solving Word Problems 7 FIN 504: Financial Management Summation Summation Single n x i 1 i Double n m x y i 1 j 1 i j 9 FIN 504: Financial Management Exponents Exponents Description aaaa a a n a a 1 a 1 0 a = base, n = exponent 11 FIN 504: Financial Management Exponents Inverses 1 a x x a a 1 12 FIN 504: Financial Management Exponents Rules for Exponent Operations a m a n a m n a n mn m a n 1 a n a 13 FIN 504: Financial Management Exponents Some Notational Equivalents: 1 a a a2 0.5 1 4 a a a4 0.25 etc. 14 FIN 504: Financial Management Exponents Solving an Equation for a Base (a) x an 1 1 n 1 x a n n n a a1 a a x n n 15 a 4 1 1 15 a a 15 1.97 4 4 15 FIN 504: Financial Management Exponents Caution! 2 5 2 5 10 3 3 3 2 5 250 3 10 1, 000 3 16 FIN 504: Financial Management Exponents Exponents in Financial Equations Discounting and Compounding Growth Forecasting Inflation 17 FIN 504: Financial Management Logarithms and e Logarithms and e Description Natural Logarithms versus Logarithms to a Base e = 2.718281828… 19 FIN 504: Financial Management Logarithms and e Inverse Operations ln(ea ) a ln a e a 20 FIN 504: Financial Management Logarithms and e Rules for Logarithmic Operations ln a n n ln a ln a b ln a ln b a ln ln a ln b b a FIN 504: Financial Management 21 ln ln a ln b b Logarithms and e Rules for e Since ex is an exponential expression, it follows the same rules as any other exponent function. a FIN 504: Financial Management 22 ln ln a ln b b Logarithms and e Solving an Equation for an Exponent b ax ln b ln a x ln b x ln a ln b x ln a 23 FIN 504: Financial Management Logarithms and e Solving an Equation for an Exponent 5 3x ln 5 ln 3 x ln 5 x ln 3 ln 5 x 1.46 ln 3 24 FIN 504: Financial Management Logarithms and e Caution! ln 5 5 ln ln 3 3 ln 5 1.465 ln 3 5 ln 0.511 3 25 FIN 504: Financial Management Logarithms and e Logarithms in Financial Equations Discounting and Compounding Growth Forecasting Inflation 26 FIN 504: Financial Management Equations in One Unknown Equations in One Unknown To solve for one unknown variable, use mathematical operations to isolate that variable alone on one side of the equation. 28 FIN 504: Financial Management Equations in One Unknown EXAMPLE: Solve for b. a b2 c a c b2 ac b b ac 29 FIN 504: Financial Management Equations in One Unknown EXAMPLE: Solve for b. 20 b 2 5 20 5 b 2 b 20 5 15 3.9 30 FIN 504: Financial Management Equations in One Unknown Note that you cannot always do this, i.e., find a ‘closed form solution’, in all cases, e.g., you cannot solve the following expression for r: 100 200 50 20 1 r 1 r 1 r 2 3 Excel has the ‘Goal Seek’ and ‘Solver’ functions for this type of problem. 31 FIN 504: Financial Management Equations in One Unknown Equations in One Unknown in Finance Many of the problems we have looked at already are equations in one unknown, and we shall use this frequently when we come to the sections on the time value of money. 32 FIN 504: Financial Management Equations in Two Unknowns Equations in Two Unknowns A system of two equations in two unknowns often has the form: 2 x 2 y 12 4 x y 16 34 FIN 504: Financial Management Equations in Two Unknowns You can think of these graphically: 35 FIN 504: Financial Management Equations in Two Unknowns To solve: 1) Solve either equation for one variable. 2) Substitute that into the other equation, so that it becomes one equation with one known. 3) Solve for that unknown. 4) Use that unknown to get an equation that solves for the other unknown. 36 FIN 504: Financial Management Equations in Two Unknowns To solve: 2 x 2 y 14 (1) 4 x y 16 (2) 1) Solve either equation for one variable. 2 x 2 y 14 2 x 14 2 y x 7 y (3 from 1) 37 FIN 504: Financial Management Equations in Two Unknowns To solve: 2 x 2 y 14 (1) 4 x y 16 (2) 2) Substitute that solution into the other equation, so that it becomes one equation with one known. 4 7 y y 16 (3 into 2) 38 FIN 504: Financial Management Equations in Two Unknowns To solve: 2 x 2 y 14 (1) 4 x y 16 (2) 3) Solve for that unknown. 4 7 y y 16 (3 into 2) 28 4 y y 16 28 3 y 16 3 y 16 28 3 y 28 16 (multiply by -1) 3 y 12 y 4 (4) 39 FIN 504: Financial Management Equations in Two Unknowns To solve: 2 x 2 y 14 (1) 4 x y 16 (2) 4) Use that unknown to get an equation that solves for the other unknown. 4 x 4 16 (4 into 2) 4 x 12 x3 x 3, y 4 40 FIN 504: Financial Management Equations in Two Unknowns Equations in Two Unknowns in Finance Portfolio Problems Immunization 41 FIN 504: Financial Management Suggestions for Word Problems Suggestions for Word Problems Two Stages Story Formula Use the Type of Data as Information Break the Larger Problem into Smaller Steps. Ask if Everything Makes Sense. Do You Have a Reasonable Answer? 43 FIN 504: Financial Management