Chapter 4: Discount Rate and Number of Periods

Exam Information

  • Your first exam is scheduled between September 26th and 28th, covering Chapters 1, 2, and 3.

  • Chapter 3 quiz information will be released today.

  • The exam consists of 25 questions and you will have 1 hour to complete it.

  • The format is similar to the quizzes: approximately 60%60\% calculation questions and 40%40\% definition or true/false questions.

  • It's crucial to take the exam as scheduled. In case of an emergency or technical issue, you must email the instructor first to explain the situation.

Finding the Implied Interest Rate (Discount Rate)

  • Concept: To find the implied interest rate (or discount rate), you need to rearrange the basic present value equation to solve for 'r', which is the unknown in this case.

  • Formula: The future value formula, FV=PV(1+r)tFV = PV(1+r)^t, can be rearranged to solve for 'r':
    r=(FVPV)(1t)−1r = \left(\frac{FV}{PV}\right)^{\left(\frac{1}{t}\right)} - 1

  • Methods of Calculation:

    • Formula: While possible, it can be more cumbersome due to the power operation.

    • Financial Calculator: This is a convenient method, but requires careful attention to the sign convention.

    • Excel Spreadsheet: Offers a dedicated function for this calculation.

    • Time Value of Money (TVM) Tables: These tables are generally difficult to use for finding the discount rate because they provide multipliers based on known interest rates and periods, not for solving for the rate itself.

  • Sign Convention for Financial Calculators:

    • When calculating an interest rate or number of periods using a financial calculator, one of Present Value (PV) or Future Value (FV) must have a negative sign, and the other must be positive.

    • The negative sign represents a cash outflow (e.g., an initial investment you make), and the positive sign represents a cash inflow (e.g., money you receive in the future).

    • If both PV and FV have the same sign, the calculator will typically produce an error message.

    • Example: If you invest −$1,000-\$1,000 (cash outflow) and receive +$1,200+\$1,200 (cash inflow), this is correctly represented. If you're borrowing, the initial loan is a cash inflow (PV=+amountPV = +\text{amount}), and future payments (or repayment of principal) are cash outflows (FV=−amountFV = -\text{amount}).

  • Example 1: Implied Rate of an Investment

    • Scenario: Investing PV=$1,000PV = \$1,000 today to receive FV=$1,200FV = \$1,200 in t=5t = 5 years.

    • Using Formula: Plugging in the values yields r=(12001000)(15)−1=(1.2)0.2−1≈0.03714r = \left(\frac{1200}{1000}\right)^{\left(\frac{1}{5}\right)} - 1 = (1.2)^{0.2} - 1 \approx 0.03714 or 3.714%3.714\%.

    • Using Financial Calculator:

      • Set N=5N = 5.

      • Set PV=−1000PV = -1000 (cash outflow).

      • Set FV=1200FV = 1200 (cash inflow).

      • Compute I/YI/Y (interest per year) to get 3.714%3.714\%.

    • Using Excel: Use the RATE function.

      • =RATE(nper, pmt, pv, fv)

      • =RATE(5, 0, -1000, 1200) will yield 3.714%3.714\%.

  • Example 2: Doubling Your Money

    • Scenario: An investment allows you to double PV=$10,000PV = \$10,000 in t=6t = 6 years, meaning FV=$20,000FV = \$20,000.

    • Using Excel: =RATE(6, 0, -10000, 20000) yields 12.246%12.246\% (or 12.25%12.25\% rounded).

    • Using Financial Calculator: N=6N = 6, PV=−10000PV = -10000, FV=20000FV = 20000. Compute I/YI/Y to get 12.246%12.246\%.

    • Rule of 72: While primarily used for the number of periods, it can approximate the interest rate when an investment doubles. Divide 7272 by the number of periods:
      r≈72t=726=12%r \approx \frac{72}{t} = \frac{72}{6} = 12\% (This is an approximation).

  • Example 3: College Education Goal

    • Scenario: You have PV=$5,000PV = \$5,000 to invest and need FV=$75,000FV = \$75,000 in t=17t = 17 years for your son's college education. You need to find the required interest rate 'r'. (The calculation was not fully presented in the transcript, but the setup follows the previous examples).

Finding the Number of Periods

  • Concept: To find out how long it will take to reach a financial goal, you rearrange the future value equation to solve for 't' (number of periods).

  • Formula: Rearranging FV=PV(1+r)tFV = PV(1+r)^t for 't' involves logarithms:
    t=ln⁡(FVPV)ln⁡(1+r)t = \frac{\ln(\frac{FV}{PV})}{\ln(1+r)}
    (where ln⁡\ln denotes the natural logarithm).

  • Methods of Calculation:

    • Formula: Can be cumbersome, requiring a calculator with a natural log function.

    • Financial Calculator: A straightforward method, but requires the sign convention for PV/FV.

    • Excel Spreadsheet: Uses a dedicated function.

  • Excel Function: Use the NPER function.

    • =NPER(rate, pmt, pv, fv)

  • Example: Time to Buy a New Car

    • Scenario: You want to buy a car for FV=$20,000FV = \$20,000. You currently have PV=$15,000PV = \$15,000 and can invest it at r=10%r = 10\% per year. How long until you have enough money?

    • Using Excel: =NPER(0.10, 0, -15000, 20000) yields 3.023.02 years.

    • Using Financial Calculator:

      • Set I/Y=10I/Y = 10.

      • Set PV=−15000PV = -15000 (cash outflow).

      • Set FV=20000FV = 20000 (cash inflow).

      • Compute NN (number of periods) to get 3.023.02 years.

  • This calculation is valuable for both individuals setting financial goals and corporations evaluating project profitability timelines.