PreSession Review. Part 2: Mathematics of Finance


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1 PreSession Review Part 2: Mathematics of Finance For this section you will need a calculator with logarithmic and exponential function keys (such as log, ln, and x y ) D. Exponential and Logarithmic Functions Exponential Functions are of the form f(x) = a x where the variable x appears as a power (exponent) in the equation. Typical application: Example: Growth rate problems Suppose the U.S. population is growing 3% per year. What will be the population in 0 years if the current population is P? Ans: in one year: P + (.03) P = ( +.03) P in two years: ( +.03) ( +.03) P = P ( +.03) 2 After 0 years: P ( +.03) 0 Properties of exponents: a 0 = by definition, for all a 0 (0 = 0 for all x > 0) a Y = a +Y example: = 2) = = 2 7 a / a Y = a!y so a Y = a 0!Y = a 0 / a Y = /a Y, also written as Y a ( a ) = a Y Y example: (2 3 ) 2 = 2) = 2 6 a a (a b) = a b so = for b 0 b b a Y = a so Y a = a, since a = a = a Y Y Y PreSession Review
2 Graphically:. f(x) = a if a > x f(x) = a a 2 = /a 2 a  = /a a a 2 2. f(x) = a if a = x f(x) = a 2 = / 2 =  = / = 2 =...so we have a series of similar graphs when a >, approaching a horizontal line at f(x) = when a decreases toward one. PreSession Review 2
3 f(x) = a when 0 < a < : x f(x) = a (½) 2 = /(½) 2 = 4 (½)  = /(½) = 2 ½ (½) 2 =...so we get a family of curves for different values of a when 0 < a < : What if a = 0? evaluate a if x = 2 : (0) 2 = /(0) 2 = /0 =?? (not defined) What if a < 0? evaluate a if x = ½ and a = 2 : (2) ½ = 2 =? ( imaginary number ) Moral: We only work with exponential functions f(x) = a when a > 0. PreSession Review 3
4 A very useful constant in finance and calculus applications is e = , the base of the natural log function (e is an irrational number). This leads to a special exponential function f(x) = e which is one of the family of exponential functions already graphed: Logarithmic Functions Log functions provide an inverse function for exponential functions. Rule: If a = b, then log a b = x (Since a = b, log a b = log a a = x) Examples: 0 2 = 00 so log 0 00 = log 0 (0) 2 = = 8 so log 2 8 = log 2 (2) 3 = 3 Conventions: Log 0 is called the common log and is usually written without the subscript: log Log e is called the natural log and is written: ln Examples: log 6 = x ö 0 x = 6 ö So x is greater than but much less then 2 (x..2042) ln 6 = y ö e y = 6 ö So y is greater than 2 but less than 3 (y ) PreSession Review 4
5 Properties of Logs log (ab) = log a + log b log (a/b) = log a log b log a b = b (log a) log c c a = a (ie, the log function is the inverse of the exponential function.) Graphically: (base 0) x /00 / f(x) = log x Note: log (a) is not defined if a # 0. Again, there is a family of curves for log functions with different bases: PreSession Review 5
6 Inverse Function Notation Since the inverse of (log c x) is c, shorthand notation has been developed to write the statement more compactly: Notation: antilog c x / c or log c / c antiln x / e or ln x / e Using antilogs: log c x = y ö log c log c x = log c y (apply antilog to both sides x = log c y to solve for x) x = c y ln a = b ö ln ln a = ln b (apply antiln to both sides a = ln b to solve for a) a = e b Examples and Applications: Solve for x:. log x = 2.5 ö log log x = log 2.5 x = x = ln x = 8.2 ö ln ln x = ln 8.2 x = e 8.2 x = e =.5 ö ln e = ln (.5) x = You have $A to invest and you want to double your money in 5 years. What interest rate must be earned? (Use compound interest formula $A ( + i) 5 = 2($A) and solve for i ) $A ( + i) 5 = 2($A) ö ( + i) 5 = 2 ( + i) = 2.2 (Both sides to /5 power) i = 2.2 (Subtract from both sides) =.487 =.487 So nearly a 5% return is needed. PreSession Review 6
7 5. 5 =.02 (Note: this is not a base 0 or base e problem, but base 5! You don t have a base 5 function key on your calculator, so how do you solve for x?) 5 =.02 ö ln 5 = ln.02 take ln of both sides (or log would work) (we do have ln key on the calculator) x (ln 5) = ln.02 use properties of exponential functions ln x = = =. 023 ln For you to try solve for x: 6. 0 log 5 = x 7. log = x 8. 7 = If log a x = 4.2 and if log a y =.4, show that log a xy = 28. PreSession Review 7
8 E. Mathematics of Finance Idea: Apply the exponential function tools of the last section to finance problems: Overview: Present value Future value Annuities. People are willing to accept payment in the future only if they will receive a greater amount. 2. The further in the future that payment is due, the smaller is the current equivalent amount. 3. The greater the rate of interest, the smaller is the equivalent value of a fixed future amount. Simple Interest Let P = principal (the amount invested or borrowed, to which interest is applied) I = amount of interest (in $) i = interest rate (annual rate, in decimal form) n = number of years Simple interest formula: I = n Example: How much interest is earned on an investment of $80 at 8.5% interest for ½ years? Problems: I = n = (80) (.085) (.5) = 0.2, or $0.20 The simple interest formula allows us to find any of I, P, i, or n if the other three values are known.. How long will it take a deposit of $000 at 2% to earn $50 simple interest? 2. How much interest is due on a $5,000,000 loan at 4.5% for day? (Think of a bank borrowing overnight from the Federal Funds market.) PreSession Review 8
9 Future and Present Value Future Value: Let P = current amount (present value) i = interest rate (annual rate, in decimal form) n = number of years F = future value Then the future value of present value P, n years from now, is F n = P ( + i) n Example: What is the future value of $5,000 invested at 2% for 5 years? F n = P ( + i) n = 5000 ( +.2) 5 = 5000 (.7623) = $8,8.7 Present Value: Just use the present value formula, but solve for F: P = F n ( + i) n = F ( + i) n n Example: What is the present value of $500, due in 3 years, if i = 0%? P = 3 = = $ ( +. 0) 33. Other than Annual Compounding Many applications require quarterly, monthly, or daily compounding. Still use the same type of formula, but let m = # of compounding periods r = per period interest rate F = P ( + r) m Example: What is the future value of $000 invested for 6 months at 2% rate, compounded monthly? Here, m = 6 r =.2/2 =.0 per month F = P ( + r) m = 000 ( +.0) 6 = 000 (.0) 6 = $,06.52 PreSession Review 9
10 Practice: What is the future amount of $20,000 invested for 90 days at.08, with daily compounding? Solving for the Number of Periods Suppose we need to solve for n in F = P ( + i) n? ö need logs! Example: How long will it take, compounding daily at 0%, for $0 to grow to $20? F = P ( + r) m where i =.0/365 = per period n =? P = 0 F = = 0 ( ) n ö 2 = ( ) n ln 2 = ln ( ) n ln 2 = ln ( ) ln n = = = ln so days, or about years. Solving for the Interest Rate Example: $,000 was invested for 4 years, it grew (compounded monthly) to $ What interest rate did it earn? Use F = P ( + r) m where r = unknown monthly interest rate n = 48 months P = 000 F = PreSession Review 0
11 Solution: = 000 ( + r) = ( + r) 48 ln (.3486) = ln ( + r) 48 ln (.3486) = 48 ln ( + r) ln (.3486)/48 = ln ( + r) e ln ln( + r) = e = + r r = (monthly rate, so the annual interest rate is 2(.00625) =.075, or 7.5%) Finding Present Value Use the same formula, but now solve for P: P = F ( + ) i n or P = F ( + i) n Practice Problem: I promise to pay your daughter $5,000 on her 8 th birthday. She is six years old today. How much do I have to set aside today to meet this future obligation, assuming I earn 6% per year simple interest? PreSession Review
12 Continuous Compounding Let i be the annual interest rate and t be the number of years. If m is the number of compounding periods per year, then i/m = per period interest rate = number of compounding periods in t years, and the future value formula can be written F = P + i m mt Now the number of periods per year (m) approach infinity. It can be shown that (+/m) m 6 e as m 6 4 so the formula above can be simplified to F = P e it where i = yearly interest rate t = number of years or, solving for P: P F = it or P = Fe e it Example: Find the present value of $000 due in ten years at a continuously compounded rate of 2% per year. P = F e it where F = 000, i =.2, t = 0 = (000) e (.2)(0) = $30.9 PreSession Review 2
13 Annuities An annuity is a series of equal amounts received (or paid) at the end of each period for a specified number of periods. Example: What is the future value of $00 to be deposited at the end of each of the next three years at 6% simple interest? One solution just add the individual future values for each deposit: F 3 = 00 ( +.06) ( +.06) + 00 = $ st deposit at 2 nd deposit final deposit the end of the at end of 2 nd at the end of st year accumulates year gains the 3 rd year interest during 2 nd interest for and 3 rd years. only one year In general, the formula for the future value of an amount R per period for n periods is F n = R( + i) n + R( + i) n R( + i) + R With a little manipulation, this can be written as: F = n ( + i) R i Redo the example above: R = 000, n = 3, I =.06 F = (. ) =. 06 $ (same as found above) The Present Value of an Annuity P = ( + i) R i n Practice Problem: You win a contest, and get $50,000 per year for 20 years. They claim you just won one million dollars. What is the actual present value of your winnings if the annual interest rate is 8%? PreSession Review 3
14 Amortization Amortization determines the equal future payments per period (R) needed to repay a specified present value (P): R = i P ( + i) n Practice Problem: Suppose you borrow $2,000 today for a 3 year car loan (36 months) at 2% interest, compounded monthly. What is your monthly payment? PreSession Review 4
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