Ch. 4

Chapter 4: Introduction to Valuation: The Time Value of Money

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Key Concepts and Skills

• Future Value (FV): Determine the future value of an investment made today.

• Present Value (PV): Determine the present value of future cash flows.

• Return on Investment (ROI): Calculate the return on an investment.

• Time to Reach Desired Value: Predict the duration for an investment to reach a specified value.

Chapter Outline

4.1: Future Value and Compounding

4.2: Present Value and Discounting

4.3: More on Present and Future Values

Basic Definitions

Present Value (PV)

• Definition: The current value of future cash flows discounted at the appropriate discount rate.

• Representation: Value at t = 0 on a time line.

Future Value (FV)

• Definition: The amount an investment is worth after one or more periods.

• Representation: “Later” money on a time line.

Interest Rate (r)

• Alternative Names:

  • Discount rate

  • Cost of capital

  • Opportunity cost of capital

  • Required return

• Note: Terminology varies based on context.

The Time Value of Money (TVM)

Choice Example

You have two options:

1. Option A: Receive $10,000 now.

2. Option B: Receive $10,000 in three years.

• Analysis: Choosing now is preferable due to the time value of money.

Future Values

General Formula

• Formula: $FV = PV (1 + r)^{t}$

  • Where:

  • FV = Future Value

  • PV = Present Value

  • r = Period interest rate (in decimal)

  • t = Number of periods

• Future value interest factor should be noted.

Example 1: Investment at 10%

• Investment: $100 for one year at 10% per year.

• Calculation:

  • Interest:
    $Interest = 100 imes .10 = 10$

  • Total Value in One Year:
    $100 + 10 = 110$

  • Future Value (FV):
    $FV = 100(1 + .10) = 110$

Effects of Compounding

Simple Interest vs. Compound Interest

• Simple Interest: Interest earned only on the principal.

• Compound Interest:

  • Interest earned on both the principal and on interest already received.

  • Known as “interest on interest.”

Example Calculation

1. Future Value with Simple Interest:

   • $FV = 100 + 10 + 10 = 120$

2. Future Value with Compound Interest:

   • $FV = 100 + 10 + 1 = 111$

   • Extra comes from interest on the earlier interest of $10, which is $1.

Compound Interest Formula

• Amount of compound interest: $A = P (1 + i)^{n} - P$

  • Where:

  • P = Principal

  • i = Annual interest rate (as a decimal)

  • n = Number of compounding periods

Example: 3-Year loan at 5%

1. Given: Principal = $10,000, Interest Rate = 5%

• Calculation:
  $10,000 imes [(1 + 0.05)^{3} - 1] = 10,000 imes (1.157625 - 1) = 1,576.25$

Texas Instruments BA-II Plus Calculator

1. FV = Future Value

2. PV = Present Value (one must be negative)

3. N = Number of periods

4. r = Period interest rate (expressed as a whole number)

Future Values: Continued Examples

Example 2: Investments Over Time

• 
  

• Investment: $100 for 5 years at 10%

• 
  

• Calculation Table:
  

  Year	Beginning Amount	Interest Earned	Ending Amount
  1	$100.00	$10.00	$110.00
  2	$110.00	$11.00	$121.00
  …	…	…	…
  5	$146.41	$14.64	$161.05
  Total Interest: $61.05
  Compounding Impact: Historical

  If $10 deposited at 5.5% compounded for 200 years is now worth $447,189.84 with compounding, but only $120 with simple interest ($10 original + $200 x 10 x 0.055).

  Excel Spreadsheet Functions for Time Value of Money

  1. Future Value: =FV(rate, nper, pmt, pv)

  2. Present Value: =PV(rate, nper, pmt, fv)

  3. Rate: =RATE(nper, pmt, pv, fv)

  4. Number of Periods: =NPER(rate, pmt, pv, fv)

  Important Relationships

  Relationship 1

  For a given interest rate:

  • The longer the time period, the higher the future value.

  Relationship 2

  For a given time period:

  • The higher the interest rate, the larger the future value.

  Quick Quiz

  What is the difference between simple and compound interest?

  1. Calculating Compound Interest: Using $500 at 8% over 15 years, expect a substantial future value.

  2. Calculating Simple Interest: Would yield $500 more as compared to compound interest calculations.

  Present Values

  • Definition: The current value of future cash flows discounted at an appropriate discount rate.

  • Purpose: To determine how much to invest today for a future amount.

  Example: Present Value Calculation

  • What is the PV of $100 due in 3 Years at r = 10%?

  • PV formula:
    $PV = rac{FV}{(1 + r)^{t}}$

  • Calculation:
    If $FV = 100$, $r = 0.10$, $t = 3$,
    $PV = rac{100}{(1 + 0.10)^{3}} = 75.13$

  Discount Rate

  Rearranging for Implied Rate Calculation

  • Use the formula to find r if investing amounts are known.

  Example Scenarios

  1. Investment of $1,000 yielding $1,200 in 5 years can be evaluated using both calculator and Excel for the implied rate.

  2. Investment of $10,000, aiming to double in 6 years also uses the same concept.

  Finding Number of Periods

  • Rearrange the basic equation to calculate time required for an investment to reach a specified amount.

  Example Calculation

  For $20,000 vehicle purchase required in 3.02 years with a starting amount of $15,000, when a 10% interest rate is achieved you can use a representative calculator

  Summary of Time Value of Money Calculations

  Key Symbols

  • PV: Present value

  • FV: Future value

  • r: Interest rate per period

  • t: Number of periods

  • C: Cash amount

  General Calculation Factors

  1. Future Values:
     $FV = C(1 + r)^{t}$

  2. Present Values:
     $PV = rac{C}{(1 + r)^{t}}$

  Comprehensive Problems

  1. Calculate additional interest from different rates on a $10,000 investment over five years.

  2. Finding the time it would take to double an investment of $10,000 earning 5% annually.

  3. Determine the annual rate if $1,000 grows to $4,000 over 20 years.