Time Value of Money – Present Value via Simple & Compound Discounting

Problem Setup

  • Task: Determine how much money (Present Value, PV) must be invested today in a bank account yielding a 10 % annual interest rate for 2 years to receive €100 at the end of year 2 (Future Value, FV).
  • Given values
    • Future Value (cash received in year 2): €100
    • Annual nominal interest rate: 10 % (0.10)
    • Time horizon: 2 years
  • Core question: “What is the present value of €100 due in 2 years?”

Key Time-Value-of-Money Concepts

  • Time line terminology
    • Moment 0 → “today” (present)
    • Moment 1 → end of year 1
    • Moment 2 → end of year 2 (future)
  • Directions on the timeline
    • Moving right (today → future) = capitalizing/compounding
    • Moving left (future → present) = discounting
  • Interest interpretation for this problem
    • Each year you “earn” 10 % on the amount that is invested at the start of that year.
    • Two alternative assumptions are compared
    • Simple interest (10 % is not reinvested after year 1, i.e., it is “taken home”)
    • Compounded interest (the 10 % earned in year 1 is reinvested for year 2)

Simple Discounting Approach

  • Formula for simple interest capitalization
    • FV=PV×(1+i t)FV = PV \times \bigl(1 + i\,t\bigr)
    • ii = annual rate (0.10)
    • tt = number of years (2)
  • Rearranged for present value (discounting)
    • PV=FV1+i tPV = \frac{FV}{1 + i\,t}
    • Alternative view: PV=FV×11+i tPV = FV \times \frac{1}{1 + i\,t}
  • Numerical substitution (simple)
    • PV=100×11+0.10×2=100×11.20PV = 100 \times \frac{1}{1 + 0.10\times2} = 100 \times \frac{1}{1.20}
    • Discount factor (simple) =0.8333 (rounded)= 0.8333\,\text{(rounded)}
    • PV≈100×0.8333=83.33PV \approx 100 \times 0.8333 = 83.33
  • Interpretation
    • Under simple interest, each euro receivable in year 2 is worth about €0.8333 today.
    • You would need to invest €83.33 now (take home the 10 % each year) to end with €100 after 2 years.

Discount Factor (Simple Interest)

  • Definition: The multiplier that converts a future cash flow to its present value under simple interest.
  • Expressed generically: DFsimple(t)=11+i tDF_{simple}(t) = \frac{1}{1 + i\,t}
  • For this case: DFsimple(2)≈0.8333DF_{simple}(2) \approx 0.8333
  • Practical significance: Provides a quick way to translate any euro amount due in 2 years to its value today under simple interest.

Compounded Discounting Approach

  • Compounded (annual) capitalization formula
    • FV=PV×(1+i)tFV = PV \times \bigl(1 + i\bigr)^{t}
  • Rearranged for PV (discounting with compounding)
    • PV=FV(1+i)t=FV×1(1+i)tPV = \frac{FV}{(1 + i)^{t}} = FV \times \frac{1}{(1 + i)^{t}}
  • Numerical substitution (compound)
    • PV=100×1(1+0.10)2=100×11.21PV = 100 \times \frac{1}{(1 + 0.10)^{2}} = 100 \times \frac{1}{1.21}
    • Discount factor (compound) =0.8264 (rounded)= 0.8264\,\text{(rounded)}
    • PV≈100×0.8264=82.64PV \approx 100 \times 0.8264 = 82.64
  • Interpretation
    • Assuming reinvestment, €1 receivable in year 2 is worth about €0.8264 today.
    • You must invest €82.64 now (reinvesting interest) to accumulate €100 after 2 years.

Discount Factor (Compound Interest)

  • Definition: Multiplier for converting a future cash flow to its present value with compounding.
  • Generic form: DFcompound(t)=1(1+i)tDF_{compound}(t) = \frac{1}{(1 + i)^{t}}
  • For this case: DFcompound(2)≈0.8264DF_{compound}(2) \approx 0.8264
  • Notice the compounded discount factor is smaller than the simple one (0.8264 < 0.8333), reflecting the stronger effect of reinvested interest.

Comparative Insights & Practical Takeaways

  • Simple vs. Compounded
    • Simple: interest is not reinvested → higher PV (less discounting).
    • Compounded: interest is reinvested → lower PV (more discounting).
  • Capitalizing vs. Discounting
    • Capitalizing: multiplying today’s money by growth factors to reach the future.
    • Discounting: dividing future money by growth factors (or multiplying by discount factors) to find today’s worth.
  • Discount factors
    • Function as “exchange rates” between €1 in the future and its present-day value.
    • Essential for valuing bonds, project cash flows, pensions, etc.
  • Ethical / philosophical angle (implicit)
    • Illustrates “time preference”: money today is preferred to the same amount tomorrow because of earning potential.
  • Connection to earlier lectures (mentioned by speaker)
    • Same formulas apply in opposite directions (capitalization vs. discounting).
    • Distinction between “moment-0, moment-1, moment-2” remains a consistent analytical framework.

Numerical Summary

  • Simple-interest PV of €100 in 2 yrs @10 %: €83.33
  • Compound-interest PV of €100 in 2 yrs @10 %: €82.64
  • Discount factors
    • Simple (2 yrs): 0.8333
    • Compound (2 yrs): 0.8264

Memory Triggers & Study Hints

  • Remember the mnemonic “DF = 1 / growth factor.”
  • Always specify whether interest is simple or compounded; results differ.
  • Draw a timeline; label cash flows and interest arrows to keep direction clear.
  • Check units: rate in decimals (0.10) and time in years must align with interest compounding convention.