Time Value of Money - Present Value vs Future Value Analysis

Time Value of Money

Concept of Present Value (PV)

  • Present Value refers to the current worth of a sum of money that is to be received in the future, discounted at a specific interest rate.

Given Problem

  • Question: Which amount is worth more at 12%, compounded annually: $4,000 in hand today or $3,500 due in 6 years?
Given Data
  1. Amount Today: $4,000
  2. Amount in Future: $3,500
  3. Interest Rate: 12% per annum
  4. Time: 6 years

Calculation of Future Value (FV)

  • To evaluate the amount due in the future in present value terms, we utilize the formula for future value: FV=PVimes(1+r)nFV = PV imes (1 + r)^n Where:
    • FVFV = future value
    • PVPV = present value
    • rr = annual interest rate (expressed as a decimal)
    • nn = number of years
Calculation of Present Value of $3,500

  1. Convert the interest rate from percentage to decimal:

    • r=12100=0.12r = \frac{12}{100} = 0.12

  2. Calculate Present Value (PV) of $3,500

    • Rearranging the FV formula:
      PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}
    • Plug in the values:
      PV=3500(1+0.12)6PV = \frac{3500}{(1 + 0.12)^6}
    • Calculate:
      PV=3500(1.12)6PV = \frac{3500}{(1.12)^6}
    • Now calculate $(1.12)^6$
    • Approximately: (1.12)^6
      ightarrow 1.9738
    • Thus:
      PV
      ightarrow rac{3500}{1.9738}
      ightarrow 1775.77 (approximately)
Comparison of PV and Cash Amount Today
  • Amount in Hand Today: $4,000
  • Present Value of Future Amount ($3,500 in 6 years): $1775.77 (approximately)

Conclusion

  • Comparing the two:
    • $4,000 (today) is greater than $1,775.77 (the present value of $3,500 due in 6 years).
  • Therefore, $4,000 in hand today is worth more than $3,500 due in 6 years at an interest rate of 12%, compounded annually.