Chapter 4: The Time Value of Money (Fundamentals of Corporate Finance)

Valuing a Stream of Cash Flows

Introduction

The concept of the time value of money is fundamental in financial accounting and corporate finance. It recognizes that a dollar today is worth more than a dollar in the future due to its earning potential. This section explains how to value a series of cash flows occurring at different points in time.

• Rule 1: Only values at the same point in time can be compared or combined. This means that cash flows must be adjusted to a common date before analysis.

• Rule 2: To calculate a cash flow’s future value, we must compound it. Compounding involves applying the interest rate over time to determine how much a present amount will grow.

• Formula for Future Value: $FV_n = C_0 \times (1 + r)^n$ Where $C_0$ is the initial cash flow, $r$ is the interest rate, and $n$ is the number of periods.

• Rule 3: To calculate the present value of a future cash flow, we must discount it. Discounting reverses compounding to determine the value today of a future amount.

• Formula for Present Value: $PV_0 = \frac{C_n}{(1 + r)^n}$ Where $C_n$ is the future cash flow, $r$ is the interest rate, and $n$ is the number of periods.

Valuing a Stream of Cash Flows

When dealing with multiple cash flows at different times, each must be discounted to the present and then summed.

• General Formula for Present Value of a Cash Flow Stream: $PV_0 = C_0 + \frac{C_1}{(1 + r)} + \frac{C_2}{(1 + r)^2} + \cdots + \frac{C_n}{(1 + r)^n}$

• Steps:

  1. Compute the present value of each cash flow individually.

  2. Sum all present values to obtain the total present value of the stream.

• Example: If you expect to receive $1000 in one year, $1100 in two years, and $1210 in three years at a 10% interest rate, the present value is:$PV_0 = \frac{1000}{1.10} + \frac{1100}{(1.10)^2} + \frac{1210}{(1.10)^3}$

Worked Example: Borrowing from Uncle Enriqué

Suppose you want to borrow money and can promise to pay $5000 in one year and $8000 each year for the next three years. If the lender could otherwise earn 6% per year, how much can you borrow?

• Cash Flow Timeline:

  Year	Cash Flow
  1	$5000
  2	$8000
  3	$8000
  4	$8000

• Present Value Calculation: $PV = \frac{5000}{1.06} + \frac{8000}{1.06^2} + \frac{8000}{1.06^3} + \frac{8000}{1.06^4}$ $= 4716.98 + 7119.97 + 6716.95 + 6336.75 = 24,890.65$ Interpretation: The maximum amount you can borrow is $24,890.65, which is the present value of the promised payments.

Key Terms

• Present Value (PV): The current worth of a future sum of money or stream of cash flows given a specified rate of return.

• Future Value (FV): The value of a current asset at a future date based on an assumed rate of growth.

• Discount Rate: The interest rate used to discount future cash flows to their present values.

• Compounding: The process of earning interest on both the initial principal and the accumulated interest from previous periods.

Additional info: These notes are based on textbook slides for Chapter 4 of "Fundamentals of Corporate Finance" (Canadian Edition), focusing on the time value of money, a core topic in financial accounting and finance courses.