Fnce 09/02 Time Value of Money: Comprehensive

Money's value changes over time due to investment potential and risks (e.g., inflation).
Purpose: manipulate money through time to understand its value at different points on a timeline.
Cash flows at different times are not directly comparable without adjusting for time and risk.

Visualize cash flows using a timeline (inflows positive, outflows negative).
Cash flows must be converted to the same point in time (Future Value or Present Value) for comparison.

Present value (PV): Earlier money on a timeline (often today's value).
Future value (FV): Later money on a timeline (value after time has passed).
Core relationships:
- Future value of a lump sum: FV = PV \times (1 + r)^n
- Present value of a future amount: PV = \frac{FV}{(1 + r)^n}
Example: 100 today invested at 5\% for 1 year yields FV = 100 \times (1 + 0.05)^1 = 105.
Example: 100 in one year at 5\% has PV \approx 95.24 today: PV = \frac{100}{(1 + 0.05)^1} \approx 95.24.

Simple Interest: Interest earned only on the original principal.
- Formula: FV_{\text{simple}} = PV \times (1 + r \times t)
- Example: 100 at 5\% simple interest for 2 years = 100 \times (1 + 0.05 \times 2) = 110.
Compound Interest: Interest earned on the accumulated amount (principal + prior interest).
- Formula: FV = PV \times (1 + r)^n
- Example: 100 at 5\% compound interest for 2 years = 100 \times (1 + 0.05)^2 = 110.25.
In finance, compound interest is typically assumed.

PV: Present Value (earlier money)
FV: Future Value (later money)
N: Number of periods
I (or I/Y): Interest rate per period
r: Per-period interest rate in formulas
PMT: Payment per period (for annuities; 0 for lump sums)
These concepts map to financial calculator keys (N, I/Y, PV, PMT, FV) and Excel functions.
Per-period rate: annual rate / periods per year (e.g., 12% annually, monthly periods -> 0.12/12 = 0.01 per month).

Formula: Directly use FV = PV \times (1 + r)^n. Example: 100 \times (1 + 0.05)^5 \approx 127.63.
Tables (Time Value of Money Tables): Use a precomputed factor (1 + r)^n based on r and n, then multiply by PV.
Financial calculator: Input four known values (N, I/Y, PV, PMT) and compute the fifth (FV).
- Sign convention: Cash inflows are positive (+), outflows are negative (-). Input PV as negative for an investment outflow.

Lump-sum investment: PV = 100 today, r = 5\% per year, n = 5 years, PMT = 0. FV after 5 years: FV = 100 \times (1 + 0.05)^5 \approx 127.63.
Calculator workflow: N=5, I/Y=5, PV=-100, PMT=0, CPT FV \rightarrow \approx 127.63.

Tables: Historical, precomputed factors.
Calculators: Fastest for lump-sum and multi-period tasks; requires sign handling.
Excel: Uses built-in functions like =FV() and =PV().

Future value of a lump sum: FV = PV \times (1 + r)^n
Present value of a future amount: PV = \frac{FV}{(1 + r)^n}
Simple interest (fixed-time horizon): FV_{\text{simple}} = PV \times (1 + r \times t)
Compound interest (per-period compounding): FV = PV \times (1 + r)^n
Per-period vs per-year rate: If annual rate is r{\text{annual}} and compounding is monthly (m periods per year): per-period rate r = \frac{r{\text{annual}}}{m}..

Underpins corporate finance, investing, and personal finance.
Used for investment appraisal (discount rate) and comparing investments with inflation.