Appendix F: Time Value of Money – Financial Accounting Study Notes

Time Value of Money

Introduction

The time value of money is a foundational concept in financial accounting and investment analysis. It recognizes that a sum of money has different values depending on when it is received or paid, due to its potential to earn interest over time.

Key Concepts

• Interest: The cost of using money. For borrowers, interest is a fee paid for the use of funds; for lenders, it is revenue earned.

• Future Value (FV): The amount an investment will grow to at a specified time in the future, given a certain interest rate.

• Present Value (PV): The current worth of a future sum of money, discounted at a specific interest rate to reflect the time value of money.

• Compound Interest: Interest calculated on both the initial principal and the accumulated interest from previous periods.

• Discounting: The process of determining the present value of a future amount.

Calculating Future Value

Definition and Formula

Future value is the sum of money that an investment will be worth at a specified time in the future. It is calculated using the following formula:

$FV = PV \times (1 + r)^n$

Where:

• FV = Future Value

• PV = Present Value (initial investment)

• r = Interest rate per period

• n = Number of periods

Example: If you invest $4,545 in corporate bonds at 10% annual interest, after one year the investment grows to $5,000.

Compound Interest

Compound interest is earned on both the principal and the interest that has already been added to the account. This accelerates the growth of the investment over time.

Table: Interest Revenue Over Five Years

End of Year	Interest	Future Value
0	-	$4,545
1	$4,545 \times 0.10 = $455	$5,000
2	$5,000 \times 0.10 = $500	$5,500
3	$5,500 \times 0.10 = $550	$6,050
4	$6,050 \times 0.10 = $605	$6,655
5	$6,655 \times 0.10 = $666	$7,321

Calculating Present Value

Definition and Formula

Present value is the value today of a future payment, discounted to reflect the time value of money. The formula is:

$PV = \frac{FV}{(1 + r)^n}$

Where:

• PV = Present Value

• FV = Future Value

• r = Interest rate per period

• n = Number of periods

Example: To receive $5,000 one year from now at a 10% interest rate, you would invest $4,545 today.

Present Value Example (Two Years)

If $5,000 is to be received two years from now at a 10% interest rate:

$PV = \frac{5,000}{(1 + 0.10)^2} = 4,132$

Breakdown:

• Amount invested (present value): $4,132

• Expected earnings for first year: $4,132 \times 0.10 = $413

• Value after one year: $4,545

• Expected earnings for second year: $4,545 \times 0.10 = $455

• Amount to be received after two years: $5,000

Using Microsoft Excel to Calculate Present Value

Excel Functions for Time Value of Money

• Excel provides financial functions such as PV (Present Value) and FV (Future Value) to calculate time value of money problems.

• To calculate the present value of an annuity in Excel:

  1. Open a blank spreadsheet.

  2. Click the insert function button (fx).

  3. Select the "financial" category.

  4. Choose the PV function.

  5. Enter the interest rate, number of periods, and payment (as a negative number).

• Example: An investment returns $20,000 per year for 20 years at 8% interest. Use the PV function to calculate its present value.

Applications: Car Payments and Retirement Savings

Calculating Car Payments

• To determine monthly payments for a car loan, use a financial calculator or Excel.

• Inputs required:

  • N: Number of periods (months)

  • I/Y: Interest rate per year

  • PV: Present value (loan amount)

  • FV: Future value (usually 0 for loans)

• Example: $18,000 car, 5-year loan, 6% annual interest rate compounded monthly.

Calculating Future Values for Retirement

• Consistent investing over time, with compound interest, can lead to significant growth.

• Example: Investing $75 per week for 45 years at a 7% annual return, compounded weekly, can result in a substantial retirement fund.

Present Value of an Investment in Bonds

Bond Valuation Example

• Calculate the present value of 9% five-year bonds (Southwest Airlines) from the investor's perspective:

• Face value: $100,000

• Face interest rate: 9% annually (4.5% semiannually)

• Market interest rate: 10% annually (5% semiannually)

• Present value (market price): $96,149

TI BAII+ Key Inputs for Bond Valuation

Function	Input	Explanation
N	10	5 years × 2 semiannual periods/year
I/Y	5	Market interest rate per period (10% ÷ 2)
PMT	4,500	Semiannual interest payment
FV	100,000	Face value of the bond
CPT → PV	?	Compute present value (should be ≈ $96,149)

Summary

• The time value of money is essential for understanding investments, loans, and financial decision-making.

• Key formulas for present and future value allow for the calculation of investment growth and the valuation of future cash flows.

• Financial tools such as Excel and financial calculators simplify these calculations for practical applications.