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Exponents and Logarithms in Finance, Summaries of Mathematics

Ton Duc Thang UniversityMathematics

A comprehensive introduction to exponents and logarithms in the context of finance. It covers concepts such as present value, annuities, optimal holding time, and logarithmic derivative. Examples, formulas, and applications of these concepts in finance, as well as exercises for practice. It is intended for students studying finance or economics, and can be used as study notes, lecture notes, summaries, or exercises to prepare for exams.

Typology: Summaries

2023/2024

Uploaded on 04/15/2024

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7/29/2020
1
7/29/2020 B02005 – Chapter 0: Introduction
CHAPTER 2:
EXPONENTS AND
LOGARITHMS
COURSE CODE: B03013
1
PREPARED BY: FINANCE DEPARTMENT
LEARNING OBJECTIVES
Introduce the present value.
Define the form of the annuities.
Identify and explain the optimal holding
time.
Introduce the logarithmic derivative.
7/29/2020 B03013 – Chapter 2: Exponents
and Logarithms
2
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7/29/2020 B02005 – Chapter 0: Introduction

CHAPTER 2:

EXPONENTS AND

LOGARITHMS

COURSE CODE: B

1

PREPARED BY: FINANCE DEPARTMENT

LEARNING OBJECTIVES

Introduce the present value. Define the form of the annuities. Identify and explain the optimal holding time. Introduce the logarithmic derivative. 7/29/2020 B03013^ –^ Chapter^ 2: Exponents and Logarithms 2

CONTENT

  • Present value 2
  • Annuities 3
  • Optimal holding time 4
  • Logarithmic Derivative. 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 3

CONCEPTS

  • Exponent and exponential function.
  • Logarithm and logarithmic function 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 4

Logarithm

  • To solve mathematical expressions that include exponents.
  • Ex: 10x=
  • Could be iterated as “To what power must the number 10 be raised to equal 100?”. 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 7

Logarithm

  • But in logarithms: + 1st^ : to solve the question as a logarithm. + Then : solve for x using a logarithm table.
  • The relationship 10 x=100 can be expressed using logarith notation: x=log 10 100=
  • => « log base 10 of 100 is 2 ». 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 8

Logarithm

  • Logarithmic derivative :
  • The derivative of the logarithmic function y =lnx
  • Derivative of y = ln u (where u is a function of x ) 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 9

Logarithm

  • Example Y= 𝑥^2 − 1 4 𝑥^2 + 1 To compute the derivative 7/29/2020 B03013^ – and Logarithms^ Chapter 2: Exponents 10

Applications

  • Annuities: 7/29/2020 B03013^ –^ Chapter 2: Exponents and Logarithms 13

Applications

Example: You have started your first job and decide to put $200 a month into an annuity. The annuity earns 7.2% interest per year, compounded monthly. How long (in months and years) will it take for the account to be worth $1,000,000? 7/29/2020 B03007^ –^ Chapter 2: Exponents and Logarithms 14

Applications

7/29/2020 B03007^ –^ Chapter 2: Exponents and Logarithms 15

Applications

  • Optimal holding time : a decision to hold an asset in a certain period.
  • There are many assets: appreciate/depreciate over time- vintage wine, real estate, forestry plantation and mining to name a few.
  • Compare an investment to others by the present value. 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 16

Applications

  • Example 3 : You have 500 USD, you sent it in your bank account. It the interest rate is 8% per year. How much your account will increase to after 5 years? 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 19

Exercise

  1. At 10 percent annual interest rate, which of the following has the largest present value a) $215 two years from now b) $100 after each of the next two years, or c) $100 now and $95 two years from now
  2. Assuming a 10 percent interest rate compounded continuously, what is the present value of an annuity that pay $500 a year a) For next five years b) Forever? 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 20

Exercise

3) Suppose you own a rare book whose value at time t years from now will be B(t) = 2 √𝑡^ dollars. Assuming a constant interest rate of 5%, when is the best time to sell the book and invest the proceeds 7/29/2020 B03007^ – and Logarithms^ Chapter 2: Exponents 21

Summary

  • Reading the law of exponents and logarithmic.
  • Applying the forms of those functions in economics (interest compounding, optimal holding time…).
  • Your discussions: how can we calculate some macro-factors in Vietnam. 7/29/2020 B03007^ –^ Chapter 2: Exponents and Logarithms 22