Time Value of Money

Time Value of Money

Introduction

  • This module introduces the concepts of future value, present value, compounding, and discounting.
  • It explains how the value of money changes over time due to its potential to grow.
  • The module uses real-life examples and requires a spreadsheet program like Microsoft Excel and potentially a calculator.

Core Elements

  • N: Number of periods.
  • I: Interest rate.
  • PV: Present value (value of future money today).
  • PMT: Payment (not used in this module).
  • FV: Future value (worth of money today in the future).

Future Value

  • Future value is the value of an asset at a specified date in the future, based on an assumed rate of growth.
  • The fundamental concept is that money available today is worth more than the same amount in the future due to its potential earning capacity.
  • Example: 100 invested in a bank at 5\%$ interest will be worth 105 in one year.
Basic Formula
  • The basic time value of money formula is: FV=PV∗(1+i)FV = PV * (1 + i), where:
    • FVFV = Future Value
    • PVPV = Present Value
    • ii = Interest rate
Multiple Periods
  • For multiple periods, the formula extends to: FV=PV∗(1+i)nFV = PV * (1 + i)^n, where nn is the number of periods.
  • Example: 100 invested for two years at 5\%$:
    • Year 1: 100 * (1 + 0.05) = $105
    • Year 2: 105 * (1 + 0.05) = $110.25

Present Value

  • Present value is the current value of a future sum of money or stream of cash flows, given a specified rate of return.
  • It's used to determine how much a future amount of money is worth today.
  • Consumers are indifferent to 100 today or 105 in one year with a 5\%$ interest rate.
Formula
  • The formula to calculate present value is: PV = \frac{FV}{(1 + i)}
Multiple Periods
  • The present value formula over multiple periods is: PV = \frac{FV}{(1 + i)^n}
  • Example, for 1,000 in two years with a 5\%$ interest rate:
    • PV = \frac{1000}{(1 + 0.05)^2} = $907.03

Applications

  • Time value of money has many real-life applications, such as saving up for a vacation.
  • Example: Calculating how much to deposit today to have 5,000 for a vacation in three years, earning an 8\%$ return.
Example Calculation
  • The calculation is as follows:
    • N = 3
    • I = 8\%
    • FV = $5,000
    • PV = $3,969

Historical Context

  • The purchase of Manhattan in 1626 for 24 provides a historical example to illustrate time value of money.
Calculation
  • If 24 had been invested at 10\%$ annually from 1626 to 2026 (400 years):
    • FV = PV * (1 + i)^n
    • FV = -24 * (1 + 0.10)^{400} = $865 quadrillion

Calculating Investment Return

  • To solve for investment return (I), you need the other four inputs: N, PV, PMT, and FV.
  • Example: Investment of 1,000 today returns 1,500 in five years. The rate of return can be calculated using Excel's RATE function.
Input
  • n = 5 (number of periods)
  • PV = -1000 (present value is negative because it's an investment or cash outflow)
  • PMT = 0 (no periodic payment)
  • FV = 1500 (future value)
Formula
  • Rate = rate(nper, pmt, pv, fv)
    Rate = rate(5, 0, -1000, 1500)
  • This gives an annual yield or rate of return of approximately 8.45\%.

Determining the Number of Periods

  • The time value of money concepts can be used to determine how long it will take to reach a savings goal.
  • Example : How long would it take to save \$4,000 for a home stereo system if you have \$3,000 and can earn 6\% interest?
Calculation
  • N is what we are solving for.
  • I = 6\%
  • PV = -\$3,000
  • FV = \$4,000

The calculation shows it would take approximately 4.94 years to save enough money.

Compounding

  • Albert Einstein referred to compounding as the most powerful force in the universe.
  • Compounding is the process by which an asset's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time.
Example
  • Starting with \$10,000 and growing it at different interest rates (5\%, 10\%, 15\%, 20\%, 25\%) over a hundred years illustrates the exponential growth potential of compounding.
Visualizing Compounding
  • The growth starts slow, but as time increases, the growth becomes faster and faster, which is the essence of exponential growth.

Discounting

  • The opposite of compounding is discounting, where we take money expected in the future and discount it back to its present value today.
  • It is the procedure used to determine the present value of a payment or a stream of payments that is to be received in the future.
Factors
  • The risk associated with an investment impacts the discount rate used. Higher risk translates to a higher discount rate.
  • The more certain we are that we will earn this cash flow, the lower the discount rate.
Example
  • If $$1,000,000 expected in different timeframes (5, 10, 15, 20 years) is discounted at rates of 5\%, 10\%, 15\%, and 20\%, its present value varies significantly: the further out in time and the greater the discount rate use, the smaller the present value today is going to be.
Assessing Risk
  • Choosing an appropriate discount rate that reflects the risk and the time value of money for the expectation of the future cash flow is important.