Comprehensive Study Notes on Time Value of Money, Annuities, and Statistical Analysis

Course Logistics, Schedule, and Performance Data

• Seating Policy:

  • Assigned seating is strictly enforced based on the official seating chart.
  • If a student finds someone sitting in their assigned seat and feels uncomfortable addressing it, course administration will handle the seating enforcement directly.

• Exam 2 Policies and Schedule:

  • Exam Date: Wednesday, October 14.
  • Syllabus Navigation: Located on D2L under the "Syllabus" link at the top of the page, navigate to "Content" and then "Schedule".
  • Cumulative Scope: Exam 2 is non-cumulative regarding conceptual material. No conceptual questions from Chapter 1 or Chapter 3 will appear on Exam 2 or subsequent exams.

• Exam 1 Performance Statistics:

  • Class Average: The Exam 1 score clustered at an average of $84.8\%$ to $84.9\%$ (approximately $84\%$).
  • Statistical Evaluation: Statistically, comparing performance between different class sections requires a paired samples t-test to control for sample variances. Without conducting a paired samples t-test, definitive comparative conclusions cannot be drawn.

Applied Statistical Analysis: Utility Bill Dispute Case Study

• Context and Background:

  • Location: Graduate school at Florida State University (FSU).
  • Instructor Demographics: Age 25 to 26, residing in a standard college apartment.
  • Teaching Curriculum Difference: At FSU, business statistics was taught within the business school because there was no independent economics department inside the business school. At other institutions (e.g., KSU), the economics department teaches business statistics courses.

• The Utility Problem:

  • Bill Escalation: The monthly electric/air conditioning bill suddenly spiked from approximately \\text{\120}$per month to$\ ext{\250} per month.
  • Mechanical Failure: The air conditioner ran non-stop without cooling the apartment to the set thermostat temperature.
  • Management Non-Response: Initial polite inquiries to apartment management yielded no repairs over a multi-month period, creating severe financial strain.

• Statistical Resolution ("Statistical Can of Bullpast"):

  • Data Collection: A public records request was submitted to the city to obtain historical utility bills for surrounding neighboring apartment units in the same area.
  • Hypothesis Testing: A paired samples t-test was performed comparing the target apartment's power usage against neighboring units.
  • Econometric Controls: The empirical analysis explicitly controlled for heteroskedasticity in the error terms.
  • Statistical Proof: The test demonstrated that the targeted unit's electric bill was statistically significantly higher than neighboring units despite identical environmental conditions ("give or take 10 feet from the sun").

• Outcome:

  • Presenting the empirical findings and regression diagnostics caused apartment management to concede.
  • Full financial reimbursement for the excess electric bills was delivered within two weeks.

Macroeconomic Mandates: The Federal Reserve and Interest Rates

• Federal Reserve System:

  • Definition: The Federal Reserve Bank (the Fed) is the central banking authority of the United States.
  • Policy Benchmark: Implemented a benchmark interest rate increase—the first rate hike in six years.

• Dual Mandate of the Federal Reserve:

  • Congress legally mandates the Federal Reserve to maintain control over two primary economic indicators:
    1. Inflation Rate: The rate at which the purchasing power of currency declines and price levels for goods (e.g., a gallon of milk over 30 years) increase.
    2. Unemployment Rate: The percentage of the labor force that is unemployed and actively seeking employment.

• Mechanics of Interest Rate Adjustment:

  • Rationale: When inflation is running high ("running hot"), the Fed raises interest rates.
  • Behavioral Impact: Higher interest rates raise borrowing costs across the economy, making consumer loans, credit cards, and mortgages more expensive.
  • Economic Outcome: Increased borrowing costs disincentivize borrowing and spending. Reduced aggregate spending cools down market demand and lowers inflation.
  • Net Impact on Consumers vs. Savers:
    • Borrowers/Consumers: Disadvantaged due to elevated costs of debt financing.
    • Net Savers: Advantaged due to higher annual interest yields earned on cash savings balances.

Valuation of Unequal Cash Flow Streams

• Transition from Chapter 3 to Chapter 4 Setup:

  • Chapter 3 Assumptions: Defaulted to $PMT = 0$ and payments per year $P/Y = 1$ (annual cash flows implied unless explicitly stated otherwise).
  • Chapter 4 Assumptions: Active use of the $PMT$ key for recurring periodic cash flows. The $P/Y$ key must be adjusted whenever non-annual frequencies occur (e.g., monthly, daily, weekly, quarterly, semiannually).

• Unequal Cash Flows Definition & Rules:

  • Definition: A sequence of periodic cash flows where the dollar amounts differ across periods.
  • Key Operational Rule: The payment key ($PMT$) cannot be utilized when cash flows are unequal. Each cash flow must be calculated independently.
  • Exam Expectation: Exactly one question (at least and at most one) on Exam 2 will cover unequal cash flow streams (valued at approximately 4 points).

• Compounding Procedure for Future Value ($FV$) of Unequal Streams:

  1. Treat every individual cash flow as a distinct Chapter 3 present value ($PV$) problem.
  2. Calculate the future value for each cash flow based on the number of remaining compounding periods ($n$) until the target target future period.
  3. Special Condition ($n = 0$): Any cash flow occurring precisely at the final evaluation period has $n = 0$ compounding periods remaining. It earns no interest, so its future value equals its nominal dollar amount.
  4. Sum the individual calculated future values to obtain the total future value of the payment stream.

• Worked Example 1: Nest Egg Unequal Stream at $r = 5\%$ ($I/Y = 5$):

  • Cash Flow Schedule:
    • $t = 0$ (Today): \2,000\n * t = 1$:$\$3,000$
    • $t = 2$: \4,000\n * t = 3$:$\$5,000$
  • Compounding Calculations to $t = 3$:
    • For \4,000$cash flow at$t = 2$: Compounds for$n = 1$period ($t = 2$to$t = 3).\n * For \$5,000$ cash flow at $t = 3$: Compounds for $n = 0$ periods; future value is exactly \5,000.\n\n* **Worked Example 2: Jim's Escalating Deposit Plan at r = 10\%$($I/Y = 10):**\n * **Deposit Structure:**\n * Initial Deposit (t = 0$):$\$3,000$ (cash outflow: $-3,000$).
    • Three subsequent annual deposits, each \2,000 greater than the preceding deposit:\n * t = 1$Deposit:$\3,000 + \2,000 = \$5,000$
      • $t = 2$ Deposit: \5,000 + \2,000 = \7,000\n * t = 3$Deposit:$\7,000 + \2,000 = \$9,000$
  • Financial Calculator Inputs ($P/Y = 1, I/Y = 10$):
    • Deposit 1 ($t = 0$): $n = 3$, $PV = -3000$, $PMT = 0$ $\rightarrow$ Compute $FV_0$
    • Deposit 2 ($t = 1$): $n = 2$, $PV = -5000$, $PMT = 0$ $\rightarrow$ Compute $FV_1$
    • Deposit 3 ($t = 2$): $n = 1$, $PV = -7000$, $PMT = 0$ $\rightarrow$ Compute $FV_2$
    • Deposit 4 ($t = 3$): $n = 0$, $PV = -9000$, $PMT = 0$ $\rightarrow$ FV_3 = \9,000\n * **Total Balance at t = 3$:**$FV_{\text{total}} = FV_0 + FV_1 + FV_2 + FV_3\n\n# Annuities: Classification and Structures\n\n* **Definition of an Annuity:**\n * An annuity is a finite series of equal, periodic cash flows (inflows or outflows) occurring over a fixed time horizon.\n * *Finite Property:* Mathematically, an annuity must have a defined end point (n is finite).\n\n* **Common Real-World Examples:**\n * Residential mortgages\n * Car loans and auto leases\n * Apartment lease rentals\n * Health and auto insurance policy payments\n * Retirement payment streams\n\n* **Structured Legal Settlements and Discounting Exploitation:**\n * Court settlements (e.g., injury cases) are frequently structured as annual annuity payouts over fixed durations (e.g., 10 equal annual payments).\n * Specialized financial acquisition companies (e.g., J.G. Wentworth) offer single upfront lump-sum payouts in exchange for acquiring the victim's future legal annuity stream.\n * These firms apply steep discount rates to determine the present value of the stream, yielding an upfront lump sum that is substantially lower than the cumulative face value of the periodic annuity payments.\n\n* **Classification of Annuity Types:**\n 1. **Ordinary Annuity:**\n * Cash flows occur at the **end** of each period.\n * *Example:* Residential Mortgages. Mortgage payments made at the beginning of a calendar month cover the preceding 30 days of home occupancy (paid in arrears).\n 2. **Annuity Due:**\n * Cash flows occur at the **beginning** of each period ($t = 0$ / today).\n * *Example:* Apartment Rent and Insurance Premiums. Landlords require rental payments upfront before issuing keys to cover the upcoming occupancy interval; insurance providers require premium payments upfront to initiate coverage.\n\n# Valuation Formulas and Applications for Ordinary Annuities\n\n* **Future Value of an Ordinary Annuity Formula:**\n         FV = PMT \times \left[ \frac{(1 + r)^n - 1}{r} \right]     \n    Where:\n * FV = Future value of the annuity\n * PMT = Equal periodic payment amount\n * r = Periodic interest rate in decimal form\n * n = Total number of compounding periods\n\n* **Worked Example 1: Jill's Savings Plan:**\n * **Problem Statement:** Jill deposits \$2,000$ at the end of each year for $10$ years into an account earning $8\%$ annual interest. Calculate the accumulated balance at the end of 10 years.
  • Classification: Ordinary Annuity ($PMT$ occurs at the end of each year).
  • Calculator Execution:
    • $P/Y = 1$
    • $N = 10$
    • $I/Y = 8$
    • $PV = 0$ (No initial starting balance reported)
    • $PMT = -2000$
    • Compute $FV$
  • Result: FV = \28,009.73\n\n* **Worked Example 2: John's College Tuition Funding Plan:**\n * **Problem Statement:** John needs to make four annual tuition payments of \$40,000$ (\40\text{k}$) for his daughter's college education. The discount rate is$7\%$. Calculate how much money John must have saved in the account one year prior ($t = 0) to the first tuition payment.\n * **Calculator Execution:**\n * P/Y = 1\n * N = 4\n * I/Y = 7\n * PMT = 40000 (Positive value representing withdrawals)\n * FV = 0 (Account is fully depleted after the 4th tuition withdrawal)\n * Compute PV\n * **Result:** PV = -\$135,488$ (Negative value indicates the required initial funding deposit at $t = 0$).

Annuity Due Valuation and Mathematical Adjustments

• Comparative Value Property:

  • For any identical set of inputs ($PMT$, $n$, and non-zero positive interest rate $r$), the value of an Annuity Due is strictly greater than the value of an Ordinary Annuity:         \n        PV_{\text{due}} > PV_{\text{ordinary}}\n                 \n        FV_{\text{due}} > FV_{\text{ordinary}}\n        
  • Reasoning: Each cash flow in an annuity due occurs one period earlier, allowing it to earn one additional period of compound interest (or discount for one fewer period).

• Mathematical Conversion Formulas (Bypassing Calculator Mode Changes):

  • To avoid leaving financial calculators in BEGIN mode during exams, calculate the standard Ordinary Annuity value first, then multiply by $(1 + r)$, where $r$ is the interest rate in decimal form:         \n        PV_{\text{due}} = PV_{\text{ordinary}} \times (1 + r)\n                 \n        FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + r)\n        

• Worked Example: 20-Year Savings Comparison (Ordinary Annuity vs. Annuity Due):

  • Problem Setup: Compare saving \3,000$per year ($PMT = 3000$) for$20$years ($N = 20$) at an annual rate of$8\%$($I/Y = 8) under an Ordinary Annuity structure versus an Annuity Due structure.\n * **Calculations:**\n * *Ordinary Annuity Future Value:* Calculated using standard TVM inputs (P/Y = 1, N = 20, I/Y = 8, PV = 0, PMT = 3000$$\rightarrow$Compute$FV_{\text{ordinary}}).\n * *Annuity Due Future Value:* Multiply Ordinary Annuity result by (1 + 0.08).\n                         FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + 0.08)             \n * **Financial Impact:** Depositing money at the beginning of each period (Annuity Due) yields almost \$11,000$ (\approx \10,900$$+) more after 20 years than waiting until the end of each period (Ordinary Annuity) due to the compounding effect on each payment over time.