Time Value of Money

Time Value of Money

• The time value of money is an important concept relating the value of money across time
• Money received today is worth more than the same amount received in the future
• Financial strategies and projects should move all dollar flows to a common future date for logical comparison

Compound Interest, Future Value, and Present Value

• Compound interest occurs when interest earned during the first period is added to the principal
• In the second period, you can then earn interest on the original principal, alongside the interest earned in the first period

Timelines to Visualize Cash flows

• Use a timeline to visualize cash flows as a linear representation of the timing of cash flows
• For example, a $100 that pays $30 at the end of year 1, $20 at the end of year 2, costs $10 at the end of year 3, and pays $50 at the end of year 4 is a cash flow.

Future Value

• Future value is the amount to which an investment will grow over time
• Future value is impacted by the number of years and the compounding rate

Future Value Equation

• The future value of an investment, if compounded annually can be calculated
• Use the equation FVn = PV (1 + r)^n
• n represents the number of years during which the compounding occurs
• r means the annual interest (or discount) rate
• PV means the present value or original amount invested at the beginning of the first period
• FVn is the abbreviation for the future value of the investment at the end of n years
• The future value factor can be represented as (1 + r)^n
• The future value factor is used as a multiplier to calculate an amount's future value

Simple Interest

• Simple interest = interest on the starting investment
• With simple interest, you cant earn interest on previous interest

Strategies for Increasing Future Value

• Increase the number of years for investment compounding
• Compound the investment at a higher interest rate

Techniques for Moving Money Through Time

• Time value of money problems can be solved through three approaches
• Use mathematical calculations with the correct formula
• Use financial calculators with specific inputs
• Cash outflows (money disbursed) are usually negative numbers
• Cash inflows (money received) are usually positive numbers
• All variables require a value, even zero.
• Enter the interest rate as a percent, not a decimal
• Use spreadsheets, as many financial calculations are preprogrammed
• Cash outflows are entered as a negative number
• Cash inflows are entered as a positive number
• Enter the interest rate as a decimal rather than as a percent

Additional Uses for Present and Future Value Problems

• Present and future value formulas can be used to determine the interest rate, r, or the number of periods, n, needed to accumulate a certain amount
• To solve, have at least three of the four necessary variables

Applying Compounding

• Apply the future value/present value formula to any problem involving compound growth
• An example could be predicting the total number of turtles in a preserve based on a certain rate per year

Present Value

• Present value refers to how much a sum of money to be received in the future is worth in today’s dollars
• Present value involves inverse compounding

Present Value Equation

• Present value can be mathematically determined as: PV= FVn / (1+r)^n
  • n = the number of years until payment will be received
  • r = the annual interest rate or discount rate
• PV = the present value of the future sum of money
• FVn = the future value of the investment at the end of nyyeears
• The present value factor is denoted as 1/(1 + r)^n
• It Multiplies to calculate an amount's present value
• A future sum of money's present value is inversely related to the number of years until payment and the opportunity rate

Annuities

• An annuity refers to a series of equal dollar payments over a specified number of years
• Annuities are common in finance, such as bond interest payments
• Compound annuities are when an equal sum of money is deposited or invested at the end of each year for a certain number of years, and allowed to grow

Compound Annuities Equation

• Compound annuities FVn = PMT [(1+r)^n -1] /r
  • PMT = the annuity value deposited at the end of each year
  • r = the annual interest or discount rate
  • n = the number of years for which the annuity will last
  • FVn = the future value of the annuity at the end of the nth year
• (1+r)^n -1 /r is referred to as the annuity value factor
• And it is used as a multiplier to determine the future value of an annuity

Present Value of an Annuity

• Pension funds, insurance obligations, and interest received from bonds all involve annuities
• The present value of each of these annuities can be calculated by the equation: PV = PMT [1 - 1/(1+r)^n] / r
  • PMT equals annuity withdrawn at the end of each year
  • r is the annual interest or discount rate
  • PV equals the present value of the future annuity
  • n is the number of years for which the annuity will last
• [1 - 1/(1+r)^n] / r is referred to as the annuity present value factor
• It is used as a multiplier to calculate the present value of an annuity

Annuities Due

• An annuity due is one in which payments occur at the beginning of each period
• Annuities due are ordinary annuities where all payments have been shifted forward by one year
• Future value formula: FVn (annuity due) = PMT [(1+r)^n -1] (1+r) where:
  • PMT = the annuity deposited at the beginning of each year
  • r = the annual interest or discount rate
  • FVn = future value of the annuity
  • n = the number of years for which the annuity will last
• Present value formula for an annuity due: PV (annuity due) = PMT [1 - 1/(1+r)^n] (1 + r) where:
  • PMT = the annuity withdrawn at the beginning of each year
  • r = the annual interest/ discount rate
  • PV = the present value of the annuity
  • n = the number of years for which the annuity will last

Amortized Loans

• Solving for PMT uses the annuity value when r, n, and PV are known
• This procedure determines the payments associated with paying off a loan in equal installments, called amortized loans

Making Interest Rates Comparable

• Interest rates on investments or loans may be compounded at different intervals
• The U.S. Truth-in-Lending Act dictates the calculation of the annual percentage rate (APR) when comparing interest rates
• The annual percentage rate (APR) is the interest rate reflecting the amount of interest earned in one year, excluding compounding
• To Calculate APR: APR = (interest rate per period) x (compounding periods per year, m)
• The APR is the interest rate typically quoted/ stated

Effective Annual Rate

• The APR can be unhelpful if the interest rates being compared are not compounded for the same number of periods each wear
• Comparison between interest rates with different compounding periods uses the effective annual rate (EAR)
• EAR, the annual compound rate, produces the same return as the nominal (or quoted) rate when interest is compounded on a nonannual basis
• The EAR presents the true rate of return
• To calculate EAR: EAR= (1+ quoted rate/ m)^m -1

Periodic Rate

• An annual percentage rate (APR) can be converted to a periodic rate: Periodic rate = APR/Compounding periods per year

Nonannual Periods

• The logic that applies to calculating the EAR also applies to calculating present and future values with nonannual periods
• FVn= PV(1+ r/m)^mn where
• FVn = the future value of the investment at the end of n years
• n = the number of years during which the compounding occurs
• r = APR
• PV = the present value or original amount invested at the beginning of the first year
• m = the number of times compounding occurs during the year

Uneven Streams and Perpetuities

• The present value of projects with uneven cash flows can be calculated by discounting or accumulating individual cash flows
• Sum the cash flows as a function of the appropriate interest rate
• If some sequential payments are the same, use the formula for the present value/future value of an annuity
• Individually discount any payments that are not the same before summing them

Perpetuity

• A perpetuity refers to an annuity that continues forever
• That is, every year from now on, the investment pays the same dollar amount
• An example of a perpetuity is preferred stock that yields a constant dollar dividend infinitely
• The present value of a perpetuity can be calculated as: PV= PP/ r where
  • PV = the present value of the perpetuity
  • PP = the constant dollar amount provided by the perpetuity
  • r = the annual interest or discount rate

Learn about the time value of money and why it is a critical concept in finance. Explore compound interest, future value, and present value calculations. Visualize cash flows using timelines for effective financial planning and decision-making.