Quantitative Methods: Time Value of Money

Introduction to the Time Value of Money (TVM)

• Money accumulates value over time (compounding).
• Concept learned early: saving today + positive return z more money in the future.

Key Verbs in TVM Application

• Interpret: Understand components of interest rates.
• Explain: Clarify different types of interest rates.
• Demonstrate: Use timelines and financial calculators.

Financial Calculators

• Familiarize with the functions for calculating TVM.
• Solve for future value, present value, interest rates with varying compounding frequencies.

Basic TVM Concepts

• Thousand dollars today vs. in five years: Better to have money now (investment = more money later).
• Compounding: Earning interest on both principal and accumulated interest.
• Example: $100 now = $110 in one year at 10% rate; not just $110 over two years due to compounding.

Math Methods

• Old Long Way Math: Multiplying and adding in steps.
• New Short Way Math: Using the time value factor $FV = PV \times (1 + r)^n$.

Financial Calculator Steps

• Setting up the calculator: Adjusting settings for compounding periods.
• Calculating Future Value: Input values to compute results.

TVM Problem Scenarios

1. Bob: $500 today, 7% interest, 5 years = $701.
2. Betty: $1000 in 11 years, 9% return = $387 today.
3. Bill: $49 today to $92 in 6 years = 11.07% interest.
4. Bonnie: $200/year for 8 years, 11% return = FV $2371.
5. Bruce: $9000 today, annual payments for 7 years at 3% = PMT $1444.55.

Components of Interest Rates

• Required Rate of Return: Minimum rate to accept investment.
• Discount Rate: Used to discount future cash flows back to present.
• Opportunity Cost: Value of the best foregone alternative.

Interest Rate Components

• Real Risk-Free Rate: Single period rate for risk-free security without inflation.
• Inflation Premium: Compensation for expected inflation.
• Default Risk Premium: Compensation for potential default.
• Liquidity Premium: Compensation for lack of liquidity.
• Maturity Premium: Compensation for risk over a longer period.

Effective Annual Interest Rate (EAR)

• Calculated based on stated annual interest rates and compounding frequency.
• Formula: $EAR = (1 + \frac{r}{m})^m - 1$
• Example: 10% compounded quarterly = EAR 10.38%

Present Value and Future Value Formulas

• Future Value: $FV = PV \times (1 + r)^n$
• Present Value: $PV = \frac{FV}{(1 + r)^n}$

Multiple Compounding Periods

• Adjust for different compounding periods manually or using functions on financial calculators.
• Example: $10,000 at end of 3 years compounded quarterly = $7435 present value.

Annuities

• Ordinary Annuity: Payments begin one year from today.
• Annuity Due: Payments begin immediately.
• Adjustments for present value and future value formulas.
• Use financial calculators to toggle between end and begin modes for accurate calculations.

Timelines in TVM

• Visualize cash flows and discount them to present value.
• Example: Unequal cash flows discounted back to present value = $889.

Final Summary

• Key learning outcomes include explaining components of interest rates and using financial calculators.
• Understanding the TVM concepts through practical examples and varied scenarios.