Time Value of Money Notes

Understanding and Appreciating the Time Value of Money

Learning Objectives

  • Explain the mechanics of compounding.
  • Understand the power of time and the importance of the interest rate in compounding.
  • Calculate the present value of money to be received in the future.
  • Define an annuity and calculate its compound or future value.

Content

  • Introduction
  • Future Value
  • Present Value
  • Annuity
    • Compound annuity
    • Present value of annuity

Calculations and Formulas

Future Value
  • Formula:
    1. Single amount: FV=PV×Future Value Interest Factor (table)FV = PV \times \text{Future Value Interest Factor (table)} OR FV=PV×(1+i)nFV = PV \times (1+i)^n
Present Value
  • Formula:
    1. Single amount: PV=FV×Present Value Interest Factor (table)PV = FV \times \text{Present Value Interest Factor (table)} OR PV=FV×[1(1+i)n]PV = FV \times [\frac{1}{(1+i)^n}] OR PV=FV×1(1+i)nPV = FV \times \frac{1}{(1+i)^n}

Introduction

  • Always comparing money from different time periods
  • A dollar received today is worth more than a dollar received in the future
  • Everything in personal finance involves time value of money

Compound Interest and Future Values

  • Interest paid on interest.
  • Reinvestment of interest paid on an investment’s principal
  • Principal is the face value of the deposit or debt instrument.

How Compound Interest Works

  • Future value (FV)
    • PV×Amount it has increased by the end of 1 year(1+i)PV \times \text{Amount it has increased by the end of 1 year} (1+i)
    • The value of an investment at some point in the future
  • Present value
    • The current value in today’s dollars of a future sum of money

How Compound Interest Works

  • Annual compounding—reinvesting interest at end of each year for more than 1 year
  • FV=PV×Amount Present Value has increased by the end of n years (1+i)nFV = PV \times \text{Amount Present Value has increased by the end of n years } (1+i)^n
  • n is equal to the number of years during which compounding occurs

The Future-Value Interest Factor

  • The value of (1+i)n(1+i)^n used as a multiplier to calculate an amount’s future value.
  • Found in certain tables
  • FV=PV×Future-Value Interest FactorFV = PV \times \text{Future-Value Interest Factor}

The Rule of 72

  • How long will it take to double your money?
  • Numbers of years for a given sum to double by dividing the investment’s annual growth or interest rate into 72.
  • Example: If an investment grows at an annual rate of 9% per year, then it should take 72/9=872/9 = 8 years to double.

Compound Interest with Nonannual Periods

  • Compounding may be quarterly, monthly, daily, or even a continuous basis.
  • Money grows faster as the compounding period becomes shorter.
  • Interest earned on interest more frequently grows money faster.

The Importance of the Interest Rate

  • The interest rate plays a critical role in how much an investment grows.
  • Higher interest rate—“Daily double”
  • “Compound interest is the eighth wonder of the world.”

Present Value

  • What’s it worth in today’s dollars?
  • Strip away inflation to see what future cash flows are worth today.
  • Inverse of compounding.
  • Discount rate is the interest rate used to bring future money back to present.

Present Value

  • The present value of a future sum of money is inversely related to both the number of years until payment will be received and the discount rate.
  • PV=FV(1+i)n=FV×[1(1+i)n]PV = \frac{FV}{(1+i)^n} = FV \times [\frac{1}{(1+i)^n}]

Present Value

  • Tables can be used to calculate the [1(1+i)n][\frac{1}{(1+i)^n}] part of the equation.
  • This is the Present-Value Interest Factor
  • PV=FV×Present-Value Interest FactorPV = FV \times \text{Present-Value Interest Factor}

Present Value Example

  • What is the PV of the $100 to be received 10 years from today if your discount rate is 6%?
    • You discover that the $100 is in the form of a savings bond that will not pay you the $100 for 10 years.
    • You’re on vacation in Florida and you see an advertisement stating that you’ll receive $100 simply for taking a tour of a model condominium.

Annuities

  • An annuity is a series of equal dollar payments coming at the end of each time period for a specific number of time period.

Compound Annuities

  • A compound annuity involves depositing an equal sum of money at the end of each year for a certain number of years, allowing it to grow.
  • You want to know how much your savings will have grown by some point in the future.
  • Sum up a number of future values.

Compound Annuities

  • Future value of an annuity = Annual payment (PMT) x Future-Value Interest Factor of an annuity (from table).

Compound Annuities Example

  • You’ll need $10,000 for education in 8 years. How much must you put away at the end of each year at 6% interest to have the college money ready?

Present Value of an Annuity

  • To compare the relative value of annuities, you need to know the present value of each.
  • Need to know what $500 received at the end of the next 5 years is worth given discount rate of 6%.
  • Sum up the present values.

Present Value of an Annuity

  • PV of an annuity = Annuity Payment or (PMT) x Present-Value Interest Factor of Annuity (from table)

Perpetuities

  • A perpetuity is an annuity that continues to pay forever.
  • Present value of a perpetuity = annual dollar amount provided by the perpetuity divided by the annual interest (or discount) rate.

Summary

  • The cornerstone of time value of money is compound interest.
  • Using future-value interest factors from tables, you can determine how much investments will grow over time.
  • The interest rate or the number of years that your money is compounded for increase future values.

Summary

  • Use the present-value interest factor to find present value of future value.
  • An annuity is a equal dollar periodic payment of investment earnings or paying off installment loans.