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Fnce 09/09 Time Value of Money - Key Concepts and Formulas

Present Value and Future Value basics

  • Time value of money: money today is worth more than the same amount in the future due to earning potential.

  • Present Value (PV): value today of a future cash flow.

  • Future Value (FV): value in the future of a present cash flow.

  • Core formulas (lump-sum):

    • PV = \dfrac{FV}{(1+i)^n}

    • FV = PV \cdot (1+i)^n

  • Higher discount rate (i) or longer period (n) lowers PV for the same FV.

  • Present value answers: "What is this future cash flow worth today at a given discount rate?"

Compounding and periods

  • Compounding frequency changes n and the per-period rate:

    • Number of periods: n = t \cdot m (t = years, m = periods per year)

    • Per-period rate: i = \dfrac{r}{m} (r = annual rate)

  • For monthly compounding (m = 12):

    • FV = PV \cdot \left(1 + \dfrac{r}{12}\right)^{12t}

    • PV with monthly compounding: PV = \dfrac{FV}{\left(1 + \dfrac{r}{12}\right)^{12t}}

  • Practical note: keep the same time unit and compounding assumption throughout a problem to avoid errors.

Present value of a lump-sum example (formulas only)

  • Example form: "You will receive $FV$ in $n$ years with discount rate $i$; what is PV?"

    • PV = \dfrac{FV}{(1+i)^n}

  • If you know the future cash flow and the discount rate, PV will be less than FV (assuming i>0).

  • Example structure you might see: PV of $7000 in 4 years at 6% annual, monthly compounding:

    • PV = \dfrac{7000}{\left(1 + \dfrac{0.06}{12}\right)^{4 \cdot 12}}

Sign convention and calculator tips

  • Sign convention matters for problem setup:

    • Cash outflow (investing today) often treated as negative; inflows positive.

    • If you enter today’s investment as a negative cash flow, future inflows should be positive.

  • If a calculator shows an error, check the sign conventions first, not the concept.

  • Calculator settings:

    • Consistency of periods per year matters if you switch between annual, monthly, etc.

    • You can adjust periods per year, but you must re-enter for each example if you change the compounding basis.

Implied rate of return (return on investment)

  • If you know PV, FV, and time, solve for the rate r in:

    • FV = PV \cdot (1+r)^n \quad \Rightarrow \quad r = \left(\dfrac{FV}{PV}\right)^{1/n} - 1

  • Annualization: if n is years, the result is the annualized return.

  • Example idea: if you spend $PV$ today and in $n$ years you have $FV$, the implied return is given by the formula above.

Risk, discount rate, and cost of capital

  • The discount rate reflects required return given risk; higher risk implies higher required return and higher discount rate.

  • For firms evaluating projects, the discount rate is tied to cost of capital (e.g., WACC).

  • Market terms: the discount rate can be the market-required return for a project or investment.

Lump sum vs. series of cash flows (preview)

  • So far: we’ve focused on lump-sum cash flows (single future cash flow).

  • Next steps in this course: discounting multiple cash flows (annuities, streams), and capital budgeting decisions.

  • Tools: formula, financial calculator, Excel, or websites; AI tools can assist but you should know the underlying method.

Quick recap and last-minute tips

  • PV decreases as i increases (for the same FV and n).

  • PV decreases with more frequent compounding if not adjusted (i and n must reflect periods).

  • Always confirm sign conventions before solving; this is a common exam pitfall.

  • Distinguish between discount rate (used to find PV) and interest rate (earned on funds); in many contexts they are synonyms.

  • When you see a future value problem, identify: FV, n, i, and solve for PV; or if solving for r, use the implied-rate formula above.