Time Value of Money

What is Time Value of Money?

  • Money received today is worth more than money received in the future

    • Saving (or investing) today means more money tomorrow as a result of interest earnings

    • Spending today results in a loss of potential interest earnings

  • Every personal finance decision involves time value of money

Interest Calculations

  • The amount of the savings (commonly called the principal).

  • The annual interest rate

  • The length of time the money is on deposited

Three methods for calculating time value or money:

  1. Online calculator and apps

  2. Spreadsheet software (excel)

  3. Financial calculator

Definitions

  • Future value - money paid or received later in time

  • Present value - money paid or received earlier in time

  • Interest rate - “exchange rate” between present value and future value

  • Lump sum - a single, one-time payment

Future Value and Compounding

  • What amount will an investment today grow to in the future after earning interest? FV=PV(1+i)nFV=PV(1+i)^{n}

  • Where FV = future value, PV = presen value, i = interest rate for each period, n = number of periods

  • Compounding - earning interest on interest

Future Value and Compounding

  • Simple interest - interest on principal only

    • Interest = principal x periodic interest rate x number of periods

  • Compounding allows for the future value of a deposit to grow faster than it would if interest were paid only the original deposit

Future Value: Example

Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year?

  • FV = $1,000(1 + 0.05) = $1,050

Suppose you leave the money in for another year. How much will you have two years from now?

  • FV = $1,000(1.05)(1.05) = $1,000(1.05)² = $1,102.50

How much would you have after two years if you were only earning simple interest?

  • $1,000 + ($1,000×0.05×2) = $1,000 + $50×2 = $1,000 + $100 = $1,100

Power of Compounding: Example

Suppose you invest $1000 at an 8% annual interest rate. How much would investment be worth in 50 years?

  • FV = $1,000(1.08)^50 = $46,901.61

What is the effect of compounding?

  • How much would you have under simple interest?

    • $1000 + $1000×0.08×50= $1000 + 4000 = $5000

  • Compounding added $41,901.61 to the value of the investment

  • Time value of money table (Exhibit 1-A) solution:

    • FV=PV(1+i)n=PVxFVIi,nFV=PV(1+i)^{n}=PVxFVIi,n

    • Where FVIF I,n (future value interest factor = future value of $1 after n periods given interest rate I

    • FVIF (I=8%, n=50 years) = 46.902

    • FV = $1,000 × 46.902 = $46,902 (difference due to rounding)

Power of Compounding at Different Interest Rates

  • The higher the interest rate, the faster your money will grow

Rule of 72

  • How long will it take for your money to double?

    • # of years to double = 72/annual interest rate

  • Example: If a $1000 investment grows at an annual rate of 8% per years, then it should take 72/8 = 9 years to double to $2000

Present Value and Discounting

  • How much do I have to invest today to have some amount in the future?

    • PV=FV/(1+i)nPV=FV/(1+i)^{n}

  • Discounting - finding the present value of some future amount

  • Time Value of money table (Exhibit 1-C) solution:

    • PV = FV/(1+I)^n= FV x PVIF I,n

    • Where PVIF I,n (present value interest factor = present value of $1 to be received at the end of n periods given interest rate

Annuity

  • Annuity - a series of equal payments made at equal time intervals (e.g. monthly, annual)

  • Types of annuities:

    • Ordinary annuity - payments occur at the end of each period

    • Annuity due - payments occur at the beginning of each period

  • Examples - payments on installment loans (car loans, student loans, mortgages), rent payments

Assignment

E.g. Saved 5 years from now toward down payment, depositing $1,852 a year, earning 2% interest

After 5 years you will have…

Solution: FV = Savings × TVM (using the chart)

= $1,852 × 5.204 = $9,638