Introduction & Learning Objectives
This guide is your interactive companion to understanding the core concepts of the **Time Value of Money (TVM)**, based on **Chapter 5 of Ross's textbook**. TVM is the fundamental principle that a dollar today is worth more than a dollar tomorrow. Mastering this concept is crucial for almost every financial decision, from valuing projects to planning for retirement.
Learning Objectives (Ross, page 130)
- Determine the future value of an investment made today.
- Determine the present value of cash to be received at a future date.
- Find the return on an investment.
- Calculate how long it takes for an investment to reach a desired value.
1. Future Value & Compounding
The Power of Compounding
**Future Value (FV)** is the value of an investment at a future date. **Compounding** is the process of earning interest on both the initial principal and the accumulated interest from previous periods.
Textbook Link: Ross, Chapter 5, Section 5.1 (Page 131)
Where $PV$ is the present value, $r$ is the interest rate per period, and $t$ is the number of periods.
Simple vs. Compound Interest
**Simple interest** is interest earned only on the original principal amount. **Compound interest** is interest earned on both the principal and the interest from previous periods. The difference between the two grows exponentially over time.
Textbook Link: Ross, Chapter 5, Section 5.1 (Page 132)
Exam Hint:
Be able to calculate both simple and compound interest over multiple periods and explain the difference. The power of compounding is a key theme.
2. Present Value & Discounting
The Core Concept of Present Value
**Present Value (PV)** is the current worth of a future cash flow. **Discounting** is the process of calculating a present value by taking a future cash flow and moving it back in time using a discount rate.
Textbook Link: Ross, Chapter 5, Section 5.2 (Page 138)
PV and FV have an inverse relationship: as the discount rate increases, the present value decreases.
Finding the Missing Variable
The basic TVM equation, $FV = PV \times (1+r)^t$, has four components: $PV$, $FV$, $r$, and $t$. If you know any three, you can solve for the fourth.
Textbook Link: Ross, Chapter 5, Section 5.3 (Page 142)
Exam Hint:
You'll need to be able to solve for any of these variables. **Question 7 from the May 2025 exam** asks you to find the interest rate for a given PV and FV, while **Question 5(a) from the January 2024 exam** asks you to solve for a compound growth rate to find which investment has a higher return.
3. Problems & Solutions
Problem Statement: Suppose you have just celebrated your 19th birthday. A rich uncle has set up a trust fund for you that will pay you $150,000 when you turn 30. If the relevant discount rate is 9 percent, how much is this fund worth today?
Solution:
This is a present value problem. We are given:
Future Value ($FV$) = $150,000
Years to maturity ($t$) = 30 - 19 = 11 years
Discount Rate ($r$) = 9%
Using the PV formula:
$PV = \frac{\$150,000}{2.5804} = \$58,130
The fund is worth $58,130 today.
Problem Statement: You've been offered an investment that will double your money in 10 years. What rate of return are you being offered? Check your answer using the Rule of 72.
Solution:
We are solving for the interest rate ($r$).
Future Value ($FV$) = $2 \times PV$. Let $PV = 1$. Then $FV = 2$.
Years ($t$) = 10
$(1+r)^{10} = 2 \implies 1+r = 2^{1/10} = 1.0718$
$r = 0.0718$ or 7.18%
Using the Rule of 72:
$r \approx \frac{72}{t} = \frac{72}{10} = 7.2\%$
The Rule of 72 provides a close approximation to the actual return.