Chapter 7 Notes: Time Value of Money (TVM)

TIME VALUE OF MONEY (TVM) — CHAPTER 7

  • TVM is a mathematical concept that determines the value of money at a given rate of interest over time. It reflects that a dollar today is worth more than a dollar tomorrow.
  • Key values: Present Value (PV) and Future Value (FV).
    • PV: the value today of one or more future cash payments discounted at an appropriate interest rate.
    • FV: the value at some point in the future of a present amount after earning a return over time.

TVM CALCULATION APPROACH

  • Four-step method to avoid data-entry errors:
    1) Start with a timeline.
    2) Write down the TVM variables.
    3) Clear all registers in the financial calculator.
    4) Populate the TVM variables in the calculator.
  • This approach helps manage keystroke errors and ensures consistent data entry.

TIMELINE AND VARIABLES

  • Visual timeline uses: 0, 1, 2, 3, 4, …
  • Variables shown on the timeline:
    • Interest Rate (i)
    • Period (N)
    • Present Value in $ (PV)
    • Payment in $ (PMT)
    • Future Value in $ (FV)
  • Definitions:
    • The value of the cash flow today (present value).
    • Any recurring payments (income stream or debt repayments).
    • The dollar value at a future time of a present amount earning a rate of return.

STEP TWO: TIMELINE INTERPRETATION

  • Cash inflows (money received) are entered as positive.
  • Cash outflows (money paid or invested) are entered as negative.
  • Enter TVM values in the same order as they appear on the timeline: N, i, PV, PMT, FV.

STEP THREE: CLEAR ALL REGISTERS IN THE CALCULATOR

  • Clear the calculator to avoid legacy data influencing results.
  • Keystrokes (example):
    • [SHIFT]
      ightarrow [C ext{ ALL}]
  • Always clear before inputting new TVM problems.

STEP FOUR: POPULATE THE TVM VARIABLES IN THE CALCULATOR

  • Populate input values in the same order as Step 2.
  • To solve for a TVM register, press the register you want to solve (e.g., i, N, PV, PMT, FV).

PRACTICAL EXAMPLES AND RESULTS

How much is $100 deposited today worth in one year at 5%?

  • TVM inputs: N=1, i=5ext(percentperyear), PV=−100, PMT=0, FV=?N=1,\, i=5 ext{ (percent per year)},\, PV = -100,\, PMT = 0,\, FV = ?
  • Answer: FV=105.00FV = 105.00
  • Explanation: FV after 1 year with no additional payments and a 5% return is $100 × (1+0.05) = $105.

Future value of a present amount

  • Example: Holly deposits PV=−30,000PV = -30{,}000 at an interest rate of i=4 ext{%} compounded annually for N=3N=3 years.
  • Formula: FV=PV(1+i)NFV = PV (1+i)^N (Payment PMT = 0)
  • FV: FV=−30,000×(1+0.04)3=−30,000×1.124864=−33,745.92FV = -30{,}000 × (1+0.04)^3 = -30{,}000 × 1.124864 = -33{,}745.92
  • Reported FV: 33,745.9233{,}745.92 (sign convention shows a positive future value in this context).

Periods of compounding other than annual

  • Calculators can handle compounding periods beyond one year without manual adjustment of the rate.
  • Steps to adjust: set periods per year (P/YR) accordingly.
    • Examples:
    • Semiannual: P/YR=2P/YR = 2 → adjust N and i per period.
    • Quarterly: P/YR=4P/YR = 4
    • Monthly: P/YR=12P/YR = 12
  • Tip: Always switch back to annual when needed.

Annual vs non-annual compounding: effect on FV

  • Greater frequency of compounding increases FV for the same nominal rate.
  • Example comparisons with the same nominal rate:
    • Annual: lower FV than semiannual/quarterly/monthly.
    • More frequent compounding yields higher FV due to more frequent application of interest.

Example: Semiannual, quarterly, and monthly compounding for $30{,}000 at 4%

  • Semiannual: N=6, i per period = 0.02 (2%), FV ≈ 33,784.8733{,}784.87
  • Quarterly: N=12, i per period = 0.01 (1%), FV ≈ 33,804.7533{,}804.75
  • Monthly: N=36, i per period ≈ 0.003333… (1/12%), FV ≈ 33,818.1633{,}818.16
  • Note: These values illustrate how higher compounding frequency increases the FV relative to simple annual compounding.

Present value of a future amount

  • If you need FV=25,000FV = 25{,}000 in N=5N=5 years and the annual return is i=6 ext{%}, what is today’s deposit?
  • Formula: PV=FV(1+i)NPV = \frac{FV}{(1+i)^N}
  • Calculation: PV=25,000(1+0.06)5≈−18,681.45PV = \frac{25{,}000}{(1+0.06)^5} ≈ -18{,}681.45
  • Sign convention: Depositing today is an outflow; hence PV is negative.

The Present Value problem (Arnold and Karen)

  • Goal: Have FV=45,000FV = 45{,}000 in 5 years, return 8% compounded annually.
  • Solve for PV: PV=FV(1+i)N=45,000(1+0.08)5≈−30,626PV = \frac{FV}{(1+i)^N} = \frac{45{,}000}{(1+0.08)^5} ≈ -30{,}626
  • Answer choice: A. 30,62630{,}626 (negative sign indicates today’s deposit). The computed PV is approximately −30,626.24-30{,}626.24.

Learning Objective 7.2: Ordinary vs Annuity Due; calculations for annuities

  • Annuity: recurring cash flow of equal amount at regular intervals.
  • Ordinary annuity: first payment at end of the period (time 1).
  • Annuity due: first payment at the beginning (time 0).
  • Examples and common uses:
    • Ordinary annuity: most loan payments (end of period).
    • Annuity due: rent, tuition payments, retirement income at period start, etc.

Annuities: practical examples

  • Ordinary annuity example: $100 deposited at end of each year for 3 years at 5%:
    • FV = PMTimes(1+i)N−1i=100imes(1.05)3−10.05=100imes3.1525=315.25PMT imes \frac{(1+i)^N - 1}{i} = 100 imes \frac{(1.05)^3 - 1}{0.05} = 100 imes 3.1525 = 315.25
  • Annuity due example: $100 at beginning for 3 years at 5%:
    • FV{due} = FV{ordinary} imes (1+i) = 315.25 × 1.05 = 331.01

Annuities: Dollar-cost averaging example (529 plan)

  • Sonya invests $500 at the beginning of each month for 10 years (120 months) at an 8% annual return (monthly compounding).
  • Beginnings mode (ANNUITY DUE) used.
  • Result: Approximately 92,08392{,}083 in 18 years (when her daughter turns 18).
  • Calculator setup (BEGIN mode):
    • P/YR = 12, N = 12 × 10 = 120, I/YR = 8, PV = 0, PMT = -500, FV ≈ 92{,}083.

Ordinary annuity payments from a lump-sum deposit (loan amortization)

  • Example: Borrower deposits to repay a loan with equal payments at end of each period.
  • Example: $200{,}000 loan; 12 years; 7% annual interest (compounded annually).
  • PMT (payment) required to amortize the loan: approximately 25,180.4025{,}180.40 per year.
  • TVM inputs: N=12, i=7, PV = -200{,}000, PMT = ?, FV = 0.

Annuity due payments from a lump-sum deposit

  • Example: Peter won $3,000,000; wants income at the beginning of each year for 12 years; investment return 3% per year.
  • Using BEGIN mode: PV = -3{,}000{,}000, N = 12, i = 3%, FV = 0, PMT ≈ 292,608.02292{,}608.02 per year.

Solving for term (N)

  • When there is no payment (PMT = 0) and FV is known, you can solve for N:
    • Example: Double an investment at 9%: PV = -25{,}000, FV = -50{,}000, N ≈ 8.0432 years.
  • Quick rule of 72 (approximation to N):
    • N ≈ 72 ÷ i(%)
    • Example: If inflation is 3.5%, N ≈ 72/3.5 ≈ 20.57 years (actual N ≈ 20.15 years for exact values shown in the material).

Patricia example: savings toward a down payment

  • Goal: $75,000 in 10 years, starting now, contributing $1,300 at the beginning of each month, 5% annual return, compounded monthly.
  • Result: About 51.6 months needed (≈ 52 months) to reach the goal (BEGIN mode required).
  • Calculator setup (BEGIN mode): P/YR = 12, I/YR = 5, PV = 0, PMT = -1300, FV = 75{,}000, N ≈ 51.62 months.

Solving for interest rate (I)

  • Example: Car loan: PV = $25{,}000; monthly payment $600 at month end; 4 years; monthly compounding.
  • Result: Interest rate ≈ 7.1158 ext{%} per year (i.e., per annum nominal rate with monthly compounding).
  • Calculator setup: P/YR = 12, N = 48, PV = -25{,}000, PMT = -600, FV = 0, I/YR ≈ 7.12

Solving for rate of return (I) in a growth goal

  • Example: Sebastian invests $5{,}000; aims to reach $15{,}000 by age 18 (15 years away).
  • Required rate of return: ≈ 7.60 ext{%} per year (I ≈ 7.60%).
  • Calculator setup: N = 15, PV = -5{,}000, PMT = 0, FV = 15{,}000, I ≈ 7.60

Practical tips and weblike notes

  • Sign conventions are important: inflows vs. outflows determine the sign of PV/PMT in your inputs.
  • Always confirm whether the problem uses BEGIN (annuity due) or END (ordinary annuity) mode; this choice changes the solution by a factor of (1+i).
  • For annuity due problems, use BEGIN mode or multiply the ordinary annuity result by (1+i).
  • When solving for N or I, ensure you choose the right input pattern (PMT = 0 when solving for N or I, etc.).

Quick reference formulas (LaTeX)

  • Future value of a present lump sum:
    FV=PV(1+i)NFV = PV (1+i)^N

  • Present value of a future lump sum:
    PV=FV(1+i)NPV = \frac{FV}{(1+i)^N}

  • Future value of an ordinary annuity:
    FVann=PMT(1+i)N−1iFV_{ann} = PMT \frac{(1+i)^N - 1}{i}

  • Future value of an annuity due:
    FVann,due=PMT(1+i)N−1i(1+i)FV_{ann,due} = PMT \frac{(1+i)^N - 1}{i} (1+i)

  • Present value of an ordinary annuity:
    PVann=PMT1−(1+i)−NiPV_{ann} = PMT \frac{1 - (1+i)^{-N}}{i}

  • Present value of an annuity due:
    PVann,due=PMT1−(1+i)−Ni(1+i)PV_{ann,due} = PMT \frac{1 - (1+i)^{-N}}{i} (1+i)

  • Time to reach a target with no PMT (solve for N):
    N=extln(FV/PV)extln(1+i)N = \frac{ ext{ln}(FV/PV)}{ ext{ln}(1+i)}

  • Rule of 72 (approximate doubling time at rate i%):
    N \n

    \approx \frac{72}{i}\,\text{(percent per period)}

Note: Cash inflows are treated as positive and cash outflows (including deposits) as negative in the TVM calculator models. Signs in PV/PMT/FV reflect the cash-flow perspective (borrower vs. lender). Always align the problem statement with the origin of cash flows to select the correct sign convention.