The Time Value of Money

1
The Time Value of Money

• Chapter Three

2
Time Value of Money

• A dollar received today is worth more than a
  dollar received in the future.
• The sooner your money can earn interest, the
  faster the interest can earn interest.

3
Interest and Compound Interest

• Interest -- is the return you receive for
  investing your money.
• Compound interest -- is the interest that your
  investment earns on the interest that your
  investment previously earned.

4
Future Value Equation

• FVn PV(1 i)n
• FV the future value of the investment at the
  end of n year
• i the annual interest (or discount) rate
• PV the present value, in todays dollars, of a
  sum of money
• This equation is used to determine the value of
  an investment at some point in the future.

5
Compounding Period

• Definition -- is the frequency that interest is
  applied to the investment
• Examples -- daily, monthly, or annually

6
Reinvesting -- How to Earn Interest on Interest

• Future-value interest factor (FVIFi,n) is a value
  used as a multiplier to calculate an amounts
  future value, and substitutes for the (1 i)n
  part of the equation.

7
The Future Value of a Wedding

• In 1998 the average wedding cost 19,104.
  Assuming 4 inflation, what will it cost in 2028?
• FVn PV (FVIFi,n)
• FVn PV (1 i)n
• FV30 PV (1 0.04)30
• FV30 19,104 (3.243)
• FV30 61,954.27

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The Rule of 72

• Estimates how many years an investment will take
  to double in value
• Number of years to double
• 72 / annual compound growth rate
• Example -- 72 / 8 9 therefore, it will take
  9 years for an investment to double in value if
  it earns 8 annually

9
Compound Interest With Nonannual Periods

• The length of the compounding period and the
  effective annual interest rate are inversely
  related therefore, the shorter the compounding
  period, the quicker the investment grows.

10
Compound Interest With Nonannual Periods (contd)

• Effective annual interest rate
• amount of annual interest earned
• amount of money invested
• Examples -- daily, weekly, monthly, and
  semi-annually

11
The Time Value of a Financial Calculator

• The TI BAII Plus financial calculator keys
• N stores the total number of payments
• I/Y stores the interest or discount rate
• PV stores the present value
• FV stores the future value
• PMT stores the dollar amount of each annuity
  payment
• CPT is the compute key

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The Time Value of a Financial Calculator (contd)

• Step 1 -- input the values of the known
  variables.
• Step 2 -- calculate the value of the remaining
  unknown variable.
• Note be sure to set your calculator to end of
  year and one payment per year modes unless
  otherwise directed.

13
Tables Versus Calculator

• REMEMBER -- The tables have a discrepancy due to
  rounding error therefore, the calculator is more
  accurate.

14
Compounding and the Power of Time

• In the long run, money saved now is much more
  valuable than money saved later.
• Dont ignore the bottom line, but also consider
  the average annual return.

15
The Power of Time in Compounding Over 35 Years

• Selma contributed 2,000 per year in years 1
  10, or 10 years.
• Patty contributed 2,000 per year in years 11
  35, or 25 years.
• Both earned 8 average annual return.

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The Importance of the Interest Rate in Compounding

• From 1926-1998 the compound growth rate of stocks
  was approximately 11.2, whereas long-term
  corporate bonds only returned 5.8.
• The Daily Double -- states that you are earning
  a 100 return compounded on a daily basis.

17
Present Value

• Is also know as the discount rate, or the
  interest rate used to bring future dollars back
  to the present.
• Present-value interest factor (PVIFi,n) is a
  value used to calculate the present value of a
  given amount.

18
Present Value Equation

• PV FVn (PVIFi,n)
• PV the present value, in todays dollars, of a
  sum of money
• FVn the future value of the investment at the
  end of n years
• PVIFi,n the present value interest factor
• This equation is used to determine todays value
  of some future sum of money.

19
Calculating Present Value for the Prodigal Son

• If promised 500,000 in 40 years, assuming 6
  interest, what is the value today?
• PV FVn (PVIFi,n)
• PV 500,000 (PVIF6, 40 yr)
• PV 500,000 (.097)
• PV 48,500

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Annuities

• Definition -- a series of equal dollar payments
  coming at the end of a certain time period for a
  specified number of time periods.
• Examples -- life insurance benefits, lottery
  payments, retirement payments.

21
Compound Annuities

• Definition -- depositing an equal sum of money at
  the end of each time period for a certain number
  of periods and allowing the money to grow
• Example -- saving 50 a month to buy a new stereo
  two years in the future
• By allowing the money to gain interest and
  compound interest, the first 50, at the end of
  two years is worth 50 (1 0.08)2 58.32

22
Future Value of an Annuity Equation

• FVn PMT (FVIFAi,n)
• FVn the future value, in todays dollars, of a
  sum of money
• PMT the payment made at the end of each time
  period
• FVIFAi,n the future-value interest factor for
  an annuity

23
Future Value of an Annuity Equation (contd)

• This equation is used to determine the future
  value of a stream of payments invested in the
  present, such as the value of your 401(k)
  contributions.

24
Calculating the Future Value of an Annuity An IRA

• Assuming 2000 annual contributions with 9
  return, how much will an IRA be worth in 30
  years?
• FVn PMT (FVIFA i, n)
• FV30 2000 (FVIFA 9,30 yr)
• FV30 2000 (136.305)
• FV30 272,610

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Present Value of an Annuity Equation

• PVn PMT (PVIFAi,n)
• PVn the present value, in todays dollars, of a
  sum of money
• PMT the payment to be made at the end of each
  time period
• PVIFAi,n the present-value interest factor for
  an annuity

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Present Value of an Annuity Equation (contd)

• This equation is used to determine the present
  value of a future stream of payments, such as
  your pension fund or insurance benefits.

27
Calculating Present Value of an Annuity Now or
Wait?

• What is the present value of the 25 annual
  payments of 50,000 offered to the soon-to-be
  ex-wife, assuming a 5 discount rate?
• PV PMT (PVIFA i,n)
• PV 50,000 (PVIFA 5, 25)
• PV 50,000 (14.094)
• PV 704,700

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Amortized Loans

• Definition -- loans that are repaid in equal
  periodic installments
• With an amortized loan the interest payment
  declines as your outstanding principal declines
  therefore, with each payment you will be paying
  an increasing amount towards the principal of the
  loan.
• Examples -- car loans or home mortgages

29
Buying a Car With Four Easy Annual Installments

• What are the annual payments to repay 6,000 at
  15 interest?
• PV PMT(PVIFA i,n yr)
• 6,000 PMT (PVIFA 15, 4 yr)
• 6,000 PMT (2.855)
• 2,101.58 PMT

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Perpetuities

• Definition an annuity that lasts forever
• PV PP / i
• PV the present value of the perpetuity
• PP the annual dollar amount provided by the
  perpetuity
• i the annual interest (or discount) rate

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Summary

• Future value the value, in the future, of a
  current investment
• Rule of 72 estimates how long your investment
  will take to double at a given rate of return
• Present value todays value of an investment
  received in the future

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Summary (contd)

• Annuity a periodic series of equal payments for
  a specific length of time
• Future value of an annuity the value, in the
  future, of a current stream of investments
• Present value of an annuity todays value of a
  stream of investments received in the future

33
Summary (contd)

• Amortized loans loans paid in equal periodic
  installments for a specific length of time
• Perpetuities annuities that continue forever