Accounting and The Time Value of Money

Basic Time Value Concepts

  • Time Value of Money: A relationship between time and money, where a dollar today is worth more than a dollar in the future.
  • Importance: Essential for comparing investment or borrowing alternatives by putting today’s and tomorrow’s dollars on the same footing.

Understanding Interest

  • Definition: Payment for the use of money. It's the excess cash received or repaid over the principal amount.
  • Variables in Interest Computation:
    • Principal: The amount borrowed or invested.
    • Interest Rate: Percentage of the outstanding principal.
    • Time: Number of years or fraction of a year that the principal is outstanding.

Simple Interest

  • Definition: Interest computed only on the principal.
  • Formula: Interest=p×i×nInterest = p \times i \times n
    • p = Principal
    • i = Interest rate
    • n = Time (in years or fraction of a year)

Simple Interest (1 year)

  • Example: Hillfarm borrows $10,000 for one year at 6% simple interest.
  • Calculation: Interest = $10,000 \times 0.06 \times 1 = $600

Simple Interest (3 months)

  • Example: Hillfarm borrows $10,000 for three months at 6% per year.
  • Calculation: Interest = $10,000 \times 0.06 \times (3/12) = $150

Compound Interest

  • Definition: Interest computed on the principal and accumulated interest.
  • Application: Commonly used in business situations.

Simple vs. Compound Interest

  • Illustration: Vasquez deposits $10,000 in Last National Bank at 9% simple interest and $10,000 in First State Bank at 9% compound interest annually.
  • Observation: Compound interest yields more due to interest being computed on the accumulated balance each year.

Compound Interest Tables

  • Tables:
    • Table 5.1: Future Value of 1
    • Table 5.2: Present Value of 1
    • Table 5.3: Future Value of an Ordinary Annuity of 1
    • Table 5.4: Present Value of an Ordinary Annuity of 1
    • Table 5.5: Present Value of an Annuity Due of 1
  • Calculations:
    • Number of Periods = Number of years × Number of compounding periods per year
    • Interest Rate = Annual rate ÷ Number of compounding periods per year

Excerpt from Table 5.1 (Future Value of 1)

  • Shows how much a dollar accumulates to at different compound interest rates over five periods.

Formula for Future Value Factor (FVF) for 1

  • FVFn,iFVF_{n,i}
    • n = Number of periods
    • i = Interest rate for a single period

Frequency of Compounding

  • Demonstrates how different compounding frequencies affect the number of periods and interest rate per period.
  • Example: 12% annual interest rate over 5 years:
    • Annually: 12% interest, 5 periods
    • Semiannually: 6% interest, 10 periods
    • Quarterly: 3% interest, 20 periods
    • Monthly: 1% interest, 60 periods

Comparison of Different Compounding Periods

  • A 9% annual interest compounded daily yields 9.42%.
  • The more frequent the compounding, the higher the yield.

Fundamental Variables

  • Rate of Interest
  • Number of Time Periods
  • Future Value
  • Present Value

Single-Sum Problems

  • Two Categories:
    • Unknown Present Value
    • Unknown Future Value

Future Value of a Single Sum

  • Definition: Value at a future date of a given amount invested with compound interest.
  • Formula: FV=PV×FVFn,iFV = PV \times FVF_{n, i}
    • FV = Future Value
    • PV = Present Value (principal or single sum)
    • FVFn,iFVF_{n, i} = Future value factor for n periods at i interest

Example

  • Bruegger Co. invests $50,000 for 5 years at 6% compounded annually.

Alternate Calculation with Table

  • Using Table 5.1, the future value factor for 5 periods at 6% is 1.33823.
  • FV = $50,000 \times 1.33823 = $66,912

Another Example

  • Amazon deposited $250 million in 2025 for a construction project to be completed by the end of 2028.
  • Interest is 10%, compounded semi-annually.

Alternate Calculation with Table

  • Using Table 5.1, with 8 compounding periods (4 years x 2) and 5% interest rate (10% / 2), future value factor is 1.47746.
  • FV = $250,000,000 \times 1.47746 = $369,365,000

Present Value of a Single Sum

  • Definition: Value now of a given amount to be received in the future, assuming compound interest.
  • Formula: PV=FV×PVFn,iPV = FV \times PVF_{n, i}
    • PV = Present Value (principal or single sum)
    • FV = Future Value
    • PVFn,iPVF_{n, i} = Present value factor for n periods at i interest

Example

  • You will receive $73,466 in 5 years.
  • The appropriate rate is 8% with annual compounding.

Alternate Calculation with Table

  • Using Table 5.2, the present value factor for 5 periods at 8% is 0.68058.
  • PV = $73,466 \times 0.68058 = $50,000

Another Example

  • Rich uncle gives you $2,000 in 3 years for a Europe trip.
  • He invests a sum now at 8% compound annual interest.

Solution with Present Value Table of 1

  • Using Table 5.2, the present value factor for 3 periods at 8% is 0.79383.
  • PV = $2,000 \times 0.79383 = $1,587.66

Annuities

  • Requirements:
    • Periodic payments or receipts (rents) of the same amount.
    • Same-length interval between rents.
    • Compounding of interest once each interval.
  • Two Types:
    • Ordinary Annuity: Rents occur at the end of each period.
    • Annuity Due: Rents occur at the beginning of each period.

Present Value of an Ordinary Annuity Formula

  • PresentValue=R×(PVF−OAn,i)Present Value = R \times (PVF-OA_{n,i})
    • R = Periodic rent (ordinary annuity)
    • PVF−OAn,iPVF-OA_{n,i} = Present value of an ordinary annuity of 1 for n periods at i interest

Example

  • Receiving rental receipts of $6,000 each at the end of each of the next 5 years, discounted at 6%.
  • Using Table 5.4, the present value of an ordinary annuity factor is 4.21236.
  • PV = $6,000 \times 4.21236 = $25,274.16

Another Example

  • Lucky Louie wins a lottery prize of $4,000,000, receiving $200,000 at the end of each of the next 20 years.
  • Assume an interest rate of 10%.
  • PV = $200,000 \times (PVF-OA_{20, 10%}) = $200,000 \times 8.51356 = $1,702,712

Present Value of an Annuity Due

Ordinary Annuity versus Annuity Due

  • Illustrates the difference in present value calculation.

Example

  • Space Odyssey, Inc., leases a communications satellite for 4 years with annual rental payments of $4.8 million at the beginning of each year.
  • Relevant annual interest rate is 5%.

Solution

  • The present value of an ordinary annuity of 1 for 4 periods at 5% (Table 5.4) = 3.54595
  • Factor (1 + 0.05) = 1.05
  • Present value of an annuity due of 1 for 4 periods at 5% = 3.54595 x 1.05 = 3.72325
  • Periodic deposit (rent) = $4,800,000
  • Present value of payments = $4,800,000 x 3.72325 = $17,871,600

Other Time Value of Money Issues

Valuation of Long-Term Bonds

  • Two Cash Flows:
    • Periodic interest payments (annuity).
    • Principal paid at maturity (single-sum).

Example

  • Alltech Corporation issues $100,000 of 5% bonds due in 5 years with annual interest payable at year-end ($100,000 × 0.05 = $5,000).
  • Market rate of interest is 6%.

Present Value of Principal

  • Using Table 5.2, the present value factor for 5 periods at 6% is 0.74726.
  • PV = $100,000 \times 0.74726 = $74,726

Present Value of Interest

  • Using Table 5.4, the present value factor for 5 periods at 6% is 4.21236.
  • PV = $5,000 \times 4.21236 = $21,061.80

Principal plus Interest

  • Bond current market value = Present value of interest + Present value of principal
  • Bond current market value = $21,061.80 + $74,726.00 = $95,787.80