Problem 2 It is a fact that \((1+0.12)^{3}... [FREE SOLUTION]

Chapter 35: Problem 2

It is a fact that $(1+0.12)^{3}=1.40 .$ Knowing that to be truc, what is the present value of $\$ 140$ received in three years if the annual interest rate is 12 percent? a. SI.40. b. SI2. c. S100. d. SII2.

Short Answer

Expert verified

The present value is $100, which corresponds to option (c).

Step by step solution

Understand the Problem

The problem asks for the present value (PV) of a future amount of $140, received in 3 years, with a 12% annual interest rate. We know from the problem statement that $(1+0.12)^{3}=1.40$, which gives us the future value factor.

Identify the Formula

The present value can be calculated using the formula: $ PV = \frac{FV}{(1 + r)^n} $, where $FV$ is the future value, $r$ is the interest rate, and $n$ is the number of periods. In this case, $FV = 140$, $r = 0.12$, and $n = 3$.

Substitute Known Values

Since we know $(1+0.12)^{3} = 1.40$, we can substitute these into the formula: $ PV = \frac{140}{1.40} $.

Calculate the Present Value

Perform the division: $ PV = \frac{140}{1.40} = 100 $.

Choose the Correct Answer

Based on the calculated present value, the closest option to 100 is option (c).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Future Value

Future Value is a core concept in financial mathematics. It refers to the amount of money an investment will grow to over a specific period, given a particular interest rate. In other words, it's the value of a current asset at a future date, considering interest or growth percentages.

Future Value Formula: The formula used to calculate future value is: \[ FV = PV \times (1 + r)^n \] where $FV$ represents the future value, $PV$ is the present value, $r$ denotes the interest rate, and $n$ is the number of periods.
Application Example: If you invest \$100 today at an interest rate of 5% for 3 years, the future value is calculated as:\[ FV = 100 \times (1.05)^3 = 115.76 \]Thus, after 3 years, your investment would grow to \$115.76.
Importance: Understanding future value helps in assessing how much an investment today is going to be worth tomorrow, guiding financial planning and decision-making.

Interest Rate

The Interest Rate is the percentage at which money grows over time. It serves as a tool to measure how much extra money will be generated from an investment or charged on a loan. The rate is usually expressed annually and affects both borrowed and saved sums.

Types of Interest Rates: There are various types of interest rates, such as simple interest and compound interest.
Simple Interest: Calculated using the formula: \[ binterest = P \times r \times t \] Where $P$ is the principal, $r$ the rate, and $t$ the time period.
Compound Interest: Unlike simple interest, this takes into account the interest from previous periods, calculated via: \[ \interest = P \times \left(1 + \frac{r}{n}\right)^{nt} \]where $n$ is the number of years.
Significance: Interest rates play a crucial role in economic activities, influencing borrowing and lending behaviors, as well as investment choices.

Time Value of Money

The Time Value of Money (TVM) is a principle in finance that explains why a sum of money today is worth more than the same sum in the future. This is mainly because the money today can earn interest over time, thus increasing in value.

Key Components: Future Value, Present Value, interest rates, and time.
TVM Concept: Essentially, receiving \$100 today is worth more than receiving \$100 in a year due to the potential interest that can be earned.
Present Value in TVM: Determines how much a future sum of money is worth today, helping in investment decisions and comparing the value of cash flows occurring at different times.\[ PV = \frac{FV}{(1 + r)^n} \]where $FV$ stands for future value, $r$ is the interest rate, and $n$ is the number of periods.
Utility: TVM is essential for comparing investments, loans, mortgages, and planning retirements.

Financial Mathematics

Financial Mathematics is a field that applies mathematical methods to solve problems in finance. It involves the calculation of interest, present value, future value, and risk, enabling better decision-making in investment and financial management.

Purpose: It helps manage wealth, retirement planning, and assess financial products.
Methods: It includes techniques such as time value of money calculations, interest rate analysis, and valuation of assets.
Real-life Applications: Financial mathematics is used by professionals and individuals alike for budgeting, investment analysis, calculating loan EMI, etc.
Importance: In a modern economy, understanding financial mathematics aids in interpreting the implications of financial transactions and market trends.

Financial mathematics thus plays a crucial role in both personal and corporate finance, ensuring informed and strategic financial planning and operations.

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91影视

It is a fact that \((1+0.12)^{3}=1.40 .\) Knowing that to be truc, what is the present value of \(\$ 140\) received in three years if the annual interest rate is 12 percent? a. SI.40. b. SI2. c. S100. d. SII2.